[
  {
    "path": ".github/workflows/build-book.yml",
    "content": "name: build book\n\non:\n  push:\n    branches:\n      - main\n  pull_request:\n    branches:\n      - main\n\njobs:\n  build-book:\n    runs-on: ubuntu-latest\n    steps:\n      - uses: actions/checkout@v3\n        with:\n          fetch-depth: 0\n\n      - name: Install dependencies\n        run: |\n          sudo apt-get update -y -qq\n          sudo apt-get -qq -y install texlive texlive-latex-extra texlive-fonts-extra texlive-font-utils build-essential\n\n      - name: Build\n        run: |\n          make\n          mkdir out\n          mv CompHydroTutorial.pdf out\n\n      - name: Archive\n        uses: actions/upload-artifact@v3\n        with:\n          name: book-pdf\n          path: out/CompHydroTutorial.pdf\n\n      - name: Deploy\n        uses: peaceiris/actions-gh-pages@v3\n        with:\n          github_token: ${{ secrets.GITHUB_TOKEN }}\n          publish_dir: ./out\n          keep_files: true\n\n"
  },
  {
    "path": ".gitignore",
    "content": "*.pyc\n*~\n\\#*\n.\\#*\n\n\n*.log\n*.aux\n*.dvi\n*.bbl\n*.blg\n*-converted-to.pdf\n*.lof\n*.exc\n*.toc\n*.png\n*.bcf\n*.brf\nCompHydroTutorial.out\nCompHydroTutorial.pdf\n\nfigures/*/*.eps\nfigures/*/*.pdf\n"
  },
  {
    "path": "ADDITIONS",
    "content": "* convergence testing via Richardson extrapolation\n\n  -- always test the convergence of the initial conditions\n\n* PPM vis\n\n* well-balancing\n"
  },
  {
    "path": "CompHydroTutorial.tex",
    "content": "\\documentclass[11pt]{book}\n\n%\\usepackage{microtype}\n\n\\input conditional.tex\n%\\def\\debugmode{}\n\n% include figures\n\\usepackage{epsfig,wrapfig}\n\n% rotating table in the appendix\n\\usepackage{rotating}\n\n% prefer PDF to PNG\n\\DeclareGraphicsExtensions{%\n  .pdf, .png}\n\n% Bibliographic Package\n\\usepackage[numbers,sort&compress]{natbib}\n\n% code listings\n\\usepackage[procnames]{listings}\n\n\\lstset{basicstyle=\\ttfamily\\small, \n        keywordstyle=\\color{keywords},\n        commentstyle=\\color{comments},\n        stringstyle=\\color{red},\n        showstringspaces=false,\n        identifierstyle=\\color{blue},\n        procnamekeys={def,class}}\n\n\\graphicspath{{preface/}{intro/}{finite-volume/}{advection/}{burgers/}{diffusion/}{Euler/}{incompressible/}{multigrid/}{multiphysics/}{low_mach/}{radiation/}{pyro/}{higher-order/}}\n\n\\usepackage{longtable}\n\n% AMS symbols\n\\usepackage{amsmath,amssymb}\n\n% for forcing a figure to appear on a single page\n% see http://tex.stackexchange.com/questions/109219/force-position-of-full-page-sized-figure\n\\usepackage{afterpage}\n\n% cancel\n\\usepackage{cancel}\n\n% Palatino font (and math symbols) -- looks nicer than the standard\n% LaTeX font\n\\usepackage[osf]{mathpazo}\n\\linespread{1.05}        % Palatino needs more leading\n%\\usepackage[scaled]{helvet} % ss\n\\usepackage{raleway}\n\\normalfont\n\\usepackage[T1]{fontenc}\n\n% URLs (special font for monospace)\n\\usepackage{inconsolata}\n%\\usepackage[T1]{fontenc}\n\n\n% footnotes use symbols\n\\renewcommand*{\\thefootnote}{\\fnsymbol{footnote}}\n\n\n% margins and paper size -- this needs to come before fncychap\n% see http://stackoverflow.com/questions/3729099/latex-paper-size\n\\usepackage[top=0.75in,\n            bottom=0.75in,\n            inner=0.65in,\n            outer=0.65in]{geometry}\n\\geometry{papersize={7.0in,10in}}\n\n% crop marks (for printing)\n%\\usepackage[\n%  noinfo,\n%  cam,\n%  cross,                % crosses as marks\n%  width=7.25in,         % the width of the galley\n%  height=10.25in,        % the height of the galley\n%  center                % actual page is centered on the galley\n%]{crop}\n\n\n% this needs to be done before the fncychap style, since that will\n% redo the chapter stuff\n\\usepackage{sectsty}\n\\allsectionsfont{\\sffamily}\n\n\n% chapter title styles\n%\\usepackage[Bjornstrup]{fncychap}\n%\\ChNumVar{\\fontsize{76}{80}\\usefont{OT1}{pzc}{m}{n}\\selectfont}\n%\\ChTitleVar{\\raggedleft\\Large\\sffamily\\bfseries}\n\n\\usepackage[PetersLenny]{fncychap}\n\\ChNameVar{\\fontsize{16}{16}\\sffamily\\selectfont}%usefont{OT1}{raleway}{m}{n}\\selectfont}\n\\ChNumVar{\\fontsize{60}{62}\\sffamily\\selectfont}%\\fontsize{60}{62}}\\usefont{OT1}{raleway}{m}{n}\\selectfont}\n\\ChTitleVar{\\Huge\\bfseries\\sffamily}\n\\ChRuleWidth{1pt}\n\n% part page style\n% see http://tex.stackexchange.com/questions/6609/problems-with-part-labels-using-titlesec\n\\usepackage{titlesec}\n\n\\titleformat{\\part}[display]\n   {\\bfseries\\sffamily\\Huge\\filcenter}\n   {{\\partname{}} \\thepart}\n   {0em}\n   {\\hrule}\n\n\n\n% hyperlinks -- load after fncychap\n\\usepackage{hyperref}\n\\usepackage[hyperpageref]{backref}\n\\renewcommand*{\\backref}[1]{}% for backref < 1.33 necessary\n\\renewcommand*{\\backrefalt}[4]{%\n\\ifcase #1 %\n  No citations.%\n\\or\n  (Cited on page #2)%\n\\else\n  (Cited on pages #2)%\n\\fi\n}\n\n% back references\n\n\n% color package\n\\usepackage{xcolor}\n\\definecolor{mygray}{gray}{0.5}\n\n\n% custom hrule for title page\n\\newcommand{\\HRule}{\\rule{\\linewidth}{0.125mm}}\n\n\\newcommand{\\pyro}{{\\sf pyro}}\n\\newcommand{\\hydrooned}{{\\sf hydro1d}}\n\\newcommand{\\git}{{\\tt git}}\n\n\n\\usepackage{xfrac}\n%% \\newcommand{\\sfrac}[2]{\\mathchoice\n%%   {\\kern0em\\raise.5ex\\hbox{\\the\\scriptfont0 #1}\\kern-.15em/\n%%    \\kern-.15em\\lower.25ex\\hbox{\\the\\scriptfont0 #2}}\n%%   {\\kern0em\\raise.5ex\\hbox{\\the\\scriptfont0 #1}\\kern-.15em/\n%%    \\kern-.15em\\lower.25ex\\hbox{\\the\\scriptfont0 #2}}\n%%   {\\kern0em\\raise.5ex\\hbox{\\the\\scriptscriptfont0 #1}\\kern-.2em/\n%%    \\kern-.15em\\lower.25ex\\hbox{\\the\\scriptscriptfont0 #2}}\n%%   {#1\\!/#2}}\n\n\n\\newcommand{\\myhalf}{{\\sfrac{1}{2}}}\n\\newcommand{\\mythreehalf}{\\sfrac{3}{2}}\n\\newcommand{\\mydxh}{\\sfrac{\\Delta x}{2}}\n\\newcommand{\\mydth}{\\sfrac{\\Delta t}{2}}\n\n\n% footer\n%% \\usepackage{fancyhdr}\n%% \\pagestyle{fancy}\n%% \\fancyfoot[LO,LE]{\\footnotesize \\sffamily \\color{mygray} M.\\ Zingale---Notes on the Euler equations}\n%% \\fancyfoot[RO,RE]{\\footnotesize \\sffamily \\color{mygray} (\\today)}\n%% \\fancyfoot[CO,CE]{\\thepage}\n%% \\fancyhead{}\n%% \\renewcommand{\\headrulewidth}{0.0pt}\n%% \\renewcommand{\\footrulewidth}{0.0pt}\n\n\n\n% don't make the chapter/section headings uppercase.  See the fancyhdr\n% documentation (section 9)\n\\usepackage{fancyhdr}\n\\renewcommand{\\chaptermark}[1]{%\n\\markboth{\\chaptername\n\\ \\thechapter.\\ #1}{}}\n\n\\renewcommand{\\sectionmark}[1]{\\markright{\\thesection---#1}}\n\n\n% don't put a header on blank pages, see\n% http://www.latex-community.org/forum/viewtopic.php?f=4&p=51559\n% change ``plain'' to ``empty'' to eliminate the page number\n\\makeatletter\n\\renewcommand*\\cleardoublepage{\\clearpage\\if@twoside\n\\ifodd\\c@page\\else\n\\hbox{}\n\\thispagestyle{empty}\n\\newpage\n\\if@twocolumn\\hbox{}\\newpage\\fi\\fi\\fi}\n\\makeatother\n\n\n% no page number of the part page\n% http://latex.org/forum/viewtopic.php?f=47&t=4293&p=16738\n\\makeatletter\n\\renewcommand\\part{%\n   \\if@openright\n     \\cleardoublepage\n   \\else\n     \\clearpage\n   \\fi\n   \\thispagestyle{empty}%\n   \\if@twocolumn\n     \\onecolumn\n     \\@tempswatrue\n   \\else\n     \\@tempswafalse\n   \\fi\n   \\null\\vfil\n   \\secdef\\@part\\@spart}\n\\makeatother\n\n\n\n\n% skip a bit of space between paragraphs, to enhance readability\n\\usepackage{parskip}\n\n\n% captions\n\\usepackage{caption}\n\\renewcommand{\\captionfont}{\\footnotesize}\n\\renewcommand{\\captionlabelfont}{\\footnotesize}\n\\setlength{\\captionmargin}{3em}\n\n\n%-----------------------------------------------------------------------------\n% define a new environment for exercises\n%\\newcounter{exercise}    % simple way -- no TOC list\n% toclot allows us to make a list of exercises\n% the new environment stuff came from http://www.dickimaw-books.com/latex/novices/html/newenv.html\n\\usepackage{tocloft}\n\\newcommand{\\listexercisename}{List of Exercises}\n\n% the [chapter] argumenet here means that we reset the numbers each chapter\n% exc is the short name we give to this new list\n\\newlistof[chapter]{exercise}{exc}{\\listexercisename}\n\\usepackage{changepage}   % used to adjust the margins within the environment\n\n% environment name, with optional argument for display in the list of\n% exercises\n\\newenvironment{exercise}[1][]\n{% begin code\n  %\n  % this indents our text from the left and right\n  \\begin{adjustwidth}{1.0cm}{1.0cm}%\n  %\n  % this skips a line before\n  %\\par\\vspace{\\baselineskip}\\noindent\n  %\n  % this increments our counter\n  \\refstepcounter{exercise}%\n  \\begin{tcolorbox}[enhanced, breakable, title={\\sffamily Exercise \\theexercise}]\n  %\n  % this displays the exercise number and starts italics\n  \\begin{itshape}%\n  %\n  % this adds an entry to our custom List of Exercises\n  \\addcontentsline{exc}{exercise}{\\protect\\numberline{\\theexercise}~~ #1}\n}%\n{% end code -- this is done at the end of the environment\n  %\n  % stop the italics\n  \\end{itshape}%\n  %\n  \\end{tcolorbox}\n  % this stops our custom margins\n  \\end{adjustwidth}%\n  %\n  % this skips a line at the end\n  %\\vspace{\\baselineskip}\\ignorespacesafterend\n}\n\n% this makes the List of Exercises have a line break before each chapter,\n% just like in the main ToC.\n%\n% this has a bug -- it does not omit the space if the chapter has 0 exercises\n\\usepackage{etoolbox}\n\\preto\\section{%\n  \\ifnum\\value{chapter}=1{}\\else   % don't skip before Ch 1\n    \\ifnum\\value{section}=0\\addtocontents{exc}{\\vskip10pt}\\fi\n  \\fi\n}\n\n% indent these the standard amount for a chaptered section\n\\setlength{\\cftexerciseindent}{1.5em}\n\n%-----------------------------------------------------------------------------\n\n% license\n\\usepackage{ccicons}\n\n% computer keyboard symbol\n\\usepackage{marvosym}\n\n% for dotted lines in the matrics/arrays\n\\usepackage{arydshln}\n\n\n\n% fonts for TOC, list of figures, etc\n\\renewcommand*\\listfigurename{\\bf\\textsf{List of Figures}}\n\\renewcommand*\\listexercisename{\\bf\\textsf{List of Exercises}}\n\\renewcommand*\\contentsname{\\bf\\textsf{Table of Contents}}\n\n\n\n% short table of contents\n\\usepackage{shorttoc}\n\n\n% pack more figures on a page\n\\usepackage{float}\n\\renewcommand\\floatpagefraction{.9}\n\\renewcommand\\topfraction{.9}\n\\renewcommand\\bottomfraction{.9}\n\\renewcommand\\textfraction{.1}\n\n\n\\newcommand{\\hydroex}{{\\sf hydro\\_examples}}\n\\newcommand{\\hydroexdoit}[1]{{\\color{red} \\LARGE \\Keyboard}\\/ \\hydroex: {\\tt #1}}\n\\newcommand{\\hydroexsymb}{{\\color{red} \\LARGE \\Keyboard}}\n\n\\usepackage{multirow}\n\n\n% MarginPars\n\\setlength{\\marginparwidth}{0.75in}\n\\newcommand{\\aMarginPar}[1]{\\marginpar{\\vskip-\\baselineskip\\raggedright\\tiny\\sffamily\\hrule\\smallskip{\\color{red}#1}\\par\\smallskip\\hrule}}\n\\newcommand{\\MarginPar}[1]{\\ifdefined\\debugmode\\aMarginPar{#1}\\fi}\n\n\\usepackage{eso-pic}\n\\newcommand\\BackgroundPic{%\n\\put(-0,-0){%\n\\parbox[b][\\paperheight]{\\paperwidth}{%\n\\vfill\n\\centering\n\\includegraphics[width=\\paperwidth,%height=\\paperheight,%\nkeepaspectratio]{images/crop_cover.png}%\n\\vfill\n}}}\n\n\n\\usepackage[many]{tcolorbox}\n\n\\newtcolorbox{mybox}[1][]{\n    width=\\textwidth,\n    arc=0mm,\n    auto outer arc,\n    boxsep=0.05cm,\n    toprule=2pt,\n    leftrule=2pt,\n    bottomrule=2pt,\n    rightrule=2pt,\n    colframe=black,\n    colback=white,\n    fontupper=\\centering\\fontsize{28pt}{28pt}\\sffamily,\n    breakable,\n    nobeforeafter,\n    enhanced jigsaw,\n    opacityframe=0.7,\n    opacityback=0.75,\n    title={\\hfill\\sffamily part of the open astrophysics bookshelf},\n}\n\n\\pdfpageattr{/Group <</S /Transparency /I true /CS /DeviceRGB>>}\n\n\n\\input mysymbols\n\n\\begin{document}\n\n\\AddToShipoutPicture*{\\BackgroundPic}\n\n\\frontmatter\n\n\\begin{titlepage}\n\n\\ \\\\[1.25in]\n\n%% \\HRule\\\\[0.5em]\n%% {\\Huge \\textsf{{\n%% Lecture Notes on\\\\[0.1em]\n%% Computational Hydrodynamics\\\\[0.3em]\n%% for Astrophysics}}\n%% }\n%% \\HRule\n%% \\\\[2em]\n%\n%% {\\Large \\sf  Michael Zingale} \\\\ {\\sf Stony Brook University}\n%\n%% \\begin{mybox}[]\n%% \\vskip 3mm\n%% {Introduction to} \\\\%[-0.75em]\n%% {Computational Hydrodynamics \\\\%[-0.75em]\n%% for Astrophysics}\n%% \\vspace{0.5em}\n%% \\end{mybox}\n\n{\\sffamily \\color{white}%\n\\begin{center}\n\\fontsize{22}{18}\\selectfont\nIntroduction to \\\\\n\\fontsize{32}{26}\\selectfont\nComputational Astrophysical Hydrodynamics\n\\end{center}\n}\n\\vspace{0.5em}\n\n\\vfill\n\n\n{\\color{red}\\sffamily the Open Astrophysics Bookshelf \\hfill\\color{white} Michael Zingale}\n\n\\end{titlepage}\n\n\\pagestyle{plain}\n\n\\null \\vfill\n\n\\noindent \\ccCopy\\ 2013, 2014, 2015, 2016, 2017 Michael Zingale \\\\\n\\noindent document git version: \\input git_info.tex $\\ldots$ \\\\\n\\noindent \\today\n\n\\noindent the source for these notes are available online (via \\git):\\\\\n\\url{https://github.com/Open-Astrophysics-Bookshelf/numerical_exercises}\n\n\\noindent \\ccbyncsa \\\\\n\\noindent This work is licensed under the Creative Commons\nAttribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA\n4.0) license.\n\n\n\\clearpage\n\n\n\\shorttoc{Chapter Listing}{0}\n\n\\clearpage\n\n\n\\setcounter{tocdepth}{2}\n\\tableofcontents\n\n\\clearpage\n\n\\listoffigures\n\\addcontentsline{toc}{chapter}{list of figures}\n\n\\clearpage\n\n\\listofexercise\n\\addcontentsline{toc}{chapter}{list of exercises}\n\n\\clearpage\n\n\\chapter*{preface}\n\\chaptermark{preface}\n\\addcontentsline{toc}{chapter}{preface}\n\n\\input preface/preface.tex\n\n\\clearpage\n\n\\pagestyle{headings}\n\n\\renewcommand{\\chaptermark}[1]{%\n\\markboth{\\chaptername\n\\ \\thechapter.\\ #1}{}}\n\n\\renewcommand{\\sectionmark}[1]{\\markright{\\thesection---#1}}\n\n% put the git hash on the first page of each chapter -- see section\n% 7 of the fancyhdr docs to see how to override the plain\n% style\n\\fancypagestyle{plain}{%\n\\fancyhf{} % clear all header and footer fields\n\\fancyfoot[C]{\\thepage} % except the center\n\\fancyfoot[L]{\\scriptsize git version: \\input git_info.tex $\\ldots$}\n\\renewcommand{\\headrulewidth}{0pt}\n\\renewcommand{\\footrulewidth}{0pt}}\n\n\n\\mainmatter\n\n\\part{Basics}\n\n\\chapter{Simulation Overview}\n\n\\input intro/intro.tex\n\n\n\\chapter{Classification of PDEs}\n\n\\input pde-classes/pde-classes.tex\n\n\n%\\addtocontents{exc}{\\protect\\addvspace{10pt}}%\n\\chapter{Finite-Volume Grids}\n\n\\input finite-volume/finite-volume.tex\n\n\\part{Advection and Hydrodynamics}\n\n%\\addtocontents{exc}{\\protect\\addvspace{10pt}}%\n\\chapter{Advection Basics}\n\n\\input advection/advection-basics.tex\n\n\\chapter{Second- (and Higher-) Order Advection}\n\n\\input advection/advection-higherorder.tex\n\n%\\addtocontents{exc}{\\protect\\addvspace{10pt}}%\n\\chapter{Burgers' Equation}\n\n\\input burgers/burgers.tex\n\n\\input higher-order/higher-order-burgers.tex\n\n%\\addtocontents{exc}{\\protect\\addvspace{10pt}}%\n\\chapter{Euler Equations: Theory}\n\n\\input Euler/Euler-theory.tex\n\n\n\\chapter{Euler Equations: Numerical Methods}\n\n\\input Euler/Euler-methods.tex\n\n\\input higher-order/higher-order-euler.tex\n\n\\ifdefined\\debugmode\n\\chapter{Hydrodynamics Test Problems}\n\n\\input hydro-test-problems/hydro-test-problems.tex\n\n\n\\chapter{Instabilities and Turbulence}\n\n\\input instabilities/instabilities.tex\n\n\\fi\n\n\n\n\n\\part{Elliptic and Parabolic Problems}\n\n\n%\\addtocontents{exc}{\\protect\\addvspace{10pt}}%\n\\chapter{Elliptic Equations and Multigrid}\n\n\\input multigrid/multigrid.tex\n\n%\\addtocontents{exc}{\\protect\\addvspace{10pt}}%\n\\chapter{Diffusion}\n\n\\input diffusion/diffusion.tex\n\n\n\n\n\\part{Multiphysics applications}\n\n%\\addtocontents{exc}{\\protect\\addvspace{10pt}}%\n\\chapter{Model Multiphysics Problems}\n\n\\input multiphysics/multiphysics.tex\n\n\\chapter{Reactive Flow}\n\n\\input reactive_flow/reactive_flow.tex\n\n\\ifdefined\\debugmode\n\\chapter{Self-gravity}\n\n\\input gravity/gravity.tex\n\\fi\n\n\\chapter{Planning a Simulation}\n\n\\input simulations/simulations.tex\n\n\n\n\n\\part{Low Speed Hydrodynamics}\n\n%\\addtocontents{exc}{\\protect\\addvspace{10pt}}%\n\\chapter{Incompressible Flow and Projection Methods}\n\n\\input incompressible/incompressible.tex\n\n\\chapter{Low Mach Number Methods}\n\n\\input low_mach/low_mach.tex\n\n\\ifdefined\\debugmode\n\\chapter{Radiation Hydrodynamics}\n\n\\input radiation/radiation.tex\n\\fi\n\n\\part{Code Descriptions}\n\n\\appendix\n\n\\chapter{Using \\hydroex}\n\n\\input hydro_examples/hydro_examples.tex\n\n\\chapter{Using \\pyro}\n\n\\input pyro/pyro.tex\n\n\\chapter{Using \\hydrooned}\n\n\\input hydro1d/hydro1d.tex\n\n%% \\chapter{Software Engineering Practices}\n\n%% \\input software-engineering/software-engineering.tex\n\n\\backmatter\n\n\\addcontentsline{toc}{chapter}{References}\n\\bibliographystyle{plain}\n\\bibliography{refs}\n\n\\end{document}\n"
  },
  {
    "path": "Euler/Euler-methods.tex",
    "content": "\\label{ch:compressible}\n\n\\section{Introduction}\n\n\\label{sec:eulermeth:intro}\n\nThe numerical methods we use for the Euler equations mirror those used\nwith advection in Ch.~\\ref{ch:advection}.  The basic process is:\n\\begin{itemize}\n\\item Reconstruct the state variables to the interfaces.\n\\item Solve the Riemann problem (as described in\n  \\S~\\ref{euler:sec:riemann}) and construct the fluxes through\n  the interfaces.\n\\item Perform the conservative update on the state variables.\n\\end{itemize}\nThe Euler equations are much more complicated than linear\nadvection---we already saw this with the Riemann problem.  But we will\nalso see that in different parts of the algorithm we will use\nconservative or primitive variables.  Additional physics can enter as\nsource terms, which we'll see how to add to both the interface states\nand conservative update here.\n\n%-----------------------------------------------------------------------------\n\\section{Reconstruction of interface states}\n\n\\label{sec:onedrecon}\n\nWe will solve the Euler equations using a high-order {\\em Godunov\n  method}---a finite volume method whereby the fluxes through the\ninterfaces are computed by solving the Riemann problem for our system.\nThe finite-volume update for our system appears as:\n\\begin{equation}\n\\Uc^{n+1}_i = \\Uc^n_i + \\frac{\\Delta t}{\\Delta x} \\left ( \\Fb_{i-\\myhalf}^{n+\\myhalf} - \\Fb_{i+\\myhalf}^{n+\\myhalf} \\right )\n\\end{equation}\nThis says that each of the conserved quantities in $\\Uc$ change only due\nto the flux of that quantity through the boundary of the cell.\n\nInstead of approximating the flux itself on the interface, we find an\napproximation to the state on the interface, $\\Uc_{i-\\myhalf}^{n+\\myhalf}$ and\n$\\Uc_{i+\\myhalf}^{n+\\myhalf}$ and use this with the flux function to define the\nflux through the interface:\n\\begin{align}\n\\Fb_{i-\\myhalf}^{n+\\myhalf} &= \\Fb(\\Uc_{i-\\myhalf}^{n+\\myhalf}) \\\\\n\\Fb_{i+\\myhalf}^{n+\\myhalf} &= \\Fb(\\Uc_{i+\\myhalf}^{n+\\myhalf})\n\\end{align}\nTo find this interface state, we predict left and right states at each\ninterface (centered in time), which are the input to the Riemann\nsolver.  The Riemann solver will then look at the characteristic wave\nstructure and determine the fluid state on the interface, which is\nthen used to compute the flux.  This is illustrated in\nFigure~\\ref{fig:riemann}.\n\nFinally, although we use the conserved variables for the final update,\nin constructing the interface states it is often easier to work with\nthe primitive variables.  These have a simpler characteristic\nstructure.  The interface states in terms of the primitive variables\ncan be converted into the interface states of the conserved variables\nthrough a simple algebraic transformation,\n\\begin{equation}\n\\Uc_{i+\\myhalf,L}^{n+\\myhalf} = \\Uc(\\qb_{i+\\myhalf,L}^{n+\\myhalf})\n\\end{equation}\n\n% figure made with figures/Euler/riemann-states.py\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{riemann_comp}\n\\caption[The left and right states for the Riemann\n  problem]{\\label{fig:riemann} The left and right states at interface\n  $i+\\myhalf$.  The arrow indicates the flux through the interface, as\n  computed by the Riemann solver using these states as input.}\n\\end{figure}\n\nAs we saw with linear advection in \\S~\\ref{ch:adv:sndorder},\nat the start of a timestep, each zones contains cell-averages.\nConsider a cell average quantity, $\\qb_i$.\nConstructing the interface states requires reconstructing the\ncell-average data as a continuous function, $\\qb(x)$, defined for\neach cell.  This polynomial is only {\\em piecewise} continuous, since\neach cell has its own $\\qb(x)$.  Standard choices are piecewise constant,\n\\begin{equation}\n\\qb(x) = \\qb_i\n\\end{equation}\npiecewise linear,\n\\begin{equation}\n\\qb(x) = \\qb_i + \\frac{\\Delta \\qb_i}{\\Delta x}(x - x_i)\n\\end{equation}\nor piecewise parabolic,\n\\begin{equation}\n\\qb(x) = \\boldsymbol{\\alpha} (x - x_i)^2 + \\boldsymbol{\\beta} (x - x_i) + \\boldsymbol{\\gamma}\n\\end{equation}\nwhere in each case, the average of $\\qb(x)$ over the cell gives $\\qb_i$.\nMost of the complexity of the method is then figuring out the coefficients\nof the polynomial.\n\nCharacteristic tracing is then done under this polynomial to see how\nmuch of each characteristic quantity comes to the interface over the\ntimestep.  The jump in the primitive variables is projected into the\ncharacteristic variables, and only jumps moving toward the interface\nare included in our reconstruction.  We look at several methods below\nthat build off of these ideas below.\n\n\\subsection{Piecewise constant}\n\nThe simplest possible reconstruction of the data is piecewise constant.\nThis is what was done in the original Godunov method~\\cite{godunov:1959}.  For the interface\nmarked by $i+\\myhalf$, the left and right states on the interface are simply:\n\\begin{align}\n\\Uc_{i+\\myhalf,L} &= \\Uc_i \\\\\n\\Uc_{i+\\myhalf,R} &= \\Uc_{i+1}\n\\end{align}\nThis does not take into account in any way how the state $\\Uc$ may be changing\nthrough the cell.  As a result, it is first-order accurate in space, and since\nno attempt was made to center it in time, it is first-order accurate in time.\n\n\\subsection{Piecewise linear}\n\nFor higher-order reconstruction, we first convert from the conserved\nvariables, $\\Uc$, to the primitive variables, $\\qb$.  These have a simpler\ncharacteristic structure, making them easier to work with.  Here we\nconsider piecewise linear reconstruction---the cell average data is\napproximated by a line with non-zero slope within each cell.\nFigure~\\ref{fig:plm} shows the piecewise linear reconstruction of some\ndata.\n\n\n% figure made with figures/Euler/ppm.py\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{piecewise-linear}\n\\caption[Piecewise linear reconstruction of cell average\n  data]{\\label{fig:plm} Piecewise linear reconstruction of the cell\n  averages.  The dotted line shows the unlimited center-difference\n  slopes and the solid line shows the limited slopes.}\n\\end{figure}\n\nConsider constructing the left state at the interface $i+\\myhalf$ (see\nFigure~\\ref{fig:riemann}).  Just like for the advection equation, we\ndo a Taylor expansion through $\\Delta x/2$ to bring us to the\ninterface, and $\\Delta t/2$ to bring us to the midpoint in time.\nStarting with $\\qb_i$, the cell-centered primitive variable, expanding\nto the right interface (to create the left state there) gives:\n\\begin{align}\n\\qb_{i+\\myhalf,L}^{n+\\myhalf} &= \\qb_i^n +\n    \\left . \\frac{\\Delta x}{2} \\frac{\\partial \\qb}{\\partial x} \\right |_i +\n    \\frac{\\Delta t}{2} \\underbrace{\\left .\\frac{\\partial \\qb}{\\partial t} \\right |_i}_{= -\\Ab \\partial \\qb / \\partial x} + \\ldots \\\\\n&= \\qb_i^n + \\frac{\\Delta x}{2} \\left . \\frac{\\partial \\qb}{\\partial x} \\right |_i\n          - \\frac{\\Delta t}{2} \\left ( \\Ab \\frac{\\partial \\qb}{\\partial x} \\right )_i\\\\\n&= \\qb_i^n + \\frac{1}{2} \\left [ 1 - \\frac{\\Delta t}{\\Delta x} \\Ab_i \\right ] \\Delta \\qb_i \\label{eq:taylorstate}\n\\end{align}\nwhere $\\Delta \\qb_i$ is the reconstructed slope of the primitive\nvariable in that cell (similar to how we compute it for the advection\nequation).  We note that the terms truncated in the first line are\n$O(\\Delta x^2)$ and $O(\\Delta t^2)$, so our method will be second-order\naccurate in space and time.\n\nAs with the advection equation, we limit the slope such that no new minima\nor maxima are introduced.  Any of the slope limiters used for linear advection\napply here as well.  We represent the limited slope as $\\overline{\\Delta \\qb}_i$.\n\nWe can decompose $\\Ab \\Delta \\qb$ in terms of the left and right\neigenvectors and sum over all the waves that move {\\em toward} the\ninterface.  First, we recognize that $\\Ab = R \\Lambda L$ and recognizing\nthat the `$1$'\nin Eq.~\\ref{eq:taylorstate} is the identity, $I = LR$, we rewrite this\nexpression as:\n\\begin{equation}\n\\qb_{i+\\myhalf,L}^{n+\\myhalf} = \\qb_i^n +\n     \\frac{1}{2} \\left [ \\Rb\\Lb - \\frac{\\Delta t}{\\Delta x} \\Rb\\Lambdab \\Lb \\right ]_i \\overline{\\Delta \\qb}_i\n\\end{equation}\nWe see the common factor of $\\Lb \\Delta \\qb$.  We now write this back in\ncomponent form.  Consider:\n\\begin{equation}\n\\renewcommand{\\arraystretch}{1.5}\n\\Rb\\Lambdab \\Lb \\overline{\\Delta \\qb} =\n   \\left ( \\begin{array}{c:c:c}\n             r_1^\\evm & r_1^\\evz & r_1^\\evp \\\\\n             r_2^\\evm & r_2^\\evz & r_2^\\evp \\\\\n             r_3^\\evm & r_3^\\evz & r_3^\\evp \\end{array} \\right )\n   \\left ( \\begin{array}{ccc}\n             \\lambda^\\evm &              & \\\\\n                          & \\lambda^\\evz & \\\\\n                          &              & \\lambda^\\evp \\end{array} \\right )\n   \\left ( \\begin{array}{ccc}\n             l_1^\\evm & l_2^\\evm & l_3^\\evm \\\\\n             \\hdashline\n             l_1^\\evz & l_2^\\evz & l_3^\\evz \\\\\n             \\hdashline\n             l_1^\\evp & l_2^\\evp & l_3^\\evp \\end{array} \\right )\n   \\left ( \\begin{array}{c}\n            \\overline{\\Delta \\rho} \\\\\n            \\overline{\\Delta u} \\\\\n            \\overline{\\Delta p} \\end{array} \\right )\n\\renewcommand{\\arraystretch}{1.0}\n\\end{equation}\nStarting with $\\Lb \\overline{\\Delta \\qb}$, which is a vector with each component\nthe dot-product of a left eigenvalue with $\\overline{\\Delta \\qb}$, we have\n\\begin{equation}\n\\renewcommand{\\arraystretch}{1.5}\n\\Rb\\Lambdab \\Lb \\overline{\\Delta \\qb} =\n   \\left ( \\begin{array}{c:c:c}\n             r_1^\\evm & r_1^\\evz & r_1^\\evp \\\\\n             r_2^\\evm & r_2^\\evz & r_2^\\evp \\\\\n             r_3^\\evm & r_3^\\evz & r_3^\\evp \\end{array} \\right )\n   \\left ( \\begin{array}{ccc}\n             \\lambda^\\evm &              & \\\\\n                          & \\lambda^\\evz & \\\\\n                          &              & \\lambda^\\evp \\end{array} \\right )\n   \\left ( \\begin{array}{c}\n            \\lb^\\evm \\cdot \\overline{\\Delta \\qb} \\\\\n            \\lb^\\evz \\cdot \\overline{\\Delta \\qb} \\\\\n            \\lb^\\evp \\cdot \\overline{\\Delta \\qb} \\end{array} \\right )\n\\renewcommand{\\arraystretch}{1.0}\n\\end{equation}\nNext we see that multiplying this vector by $\\Lambdab$ simply puts the\neigenvalue with its respective eigenvector in the resulting column vector:\n\\begin{equation}\n\\renewcommand{\\arraystretch}{1.5}\n\\Rb\\Lambdab \\Lb \\overline{\\Delta \\qb} =\n   \\left ( \\begin{array}{c:c:c}\n             r_1^\\evm & r_1^\\evz & r_1^\\evp \\\\\n             r_2^\\evm & r_2^\\evz & r_2^\\evp \\\\\n             r_3^\\evm & r_3^\\evz & r_3^\\evp \\end{array} \\right )\n   \\left ( \\begin{array}{c}\n            \\lambda^\\evm \\, \\lb^\\evm \\cdot \\overline{\\Delta \\qb} \\\\\n            \\lambda^\\evz \\, \\lb^\\evz \\cdot \\overline{\\Delta \\qb} \\\\\n            \\lambda^\\evp \\, \\lb^\\evp \\cdot \\overline{\\Delta \\qb} \\end{array} \\right )\n\\renewcommand{\\arraystretch}{1.0}\n\\end{equation}\nFinally, the last multiply results in a column vector:\n\\begin{equation}\n\\renewcommand{\\arraystretch}{1.5}\nR\\Lambda L \\overline{\\Delta \\qb} =\n   \\left ( \\begin{array}{c}\n            r_1^\\evm \\, \\lambda^\\evm \\, \\lb^\\evm \\cdot \\overline{\\Delta \\qb} +\n            r_1^\\evz \\, \\lambda^\\evz \\, \\lb^\\evz \\cdot \\overline{\\Delta \\qb} +\n            r_1^\\evp \\, \\lambda^\\evp \\, \\lb^\\evp \\cdot \\overline{\\Delta \\qb} \\\\\n%\n            r_2^\\evm \\, \\lambda^\\evm \\, \\lb^\\evm \\cdot \\overline{\\Delta \\qb} +\n            r_2^\\evz \\, \\lambda^\\evz \\, \\lb^\\evz \\cdot \\overline{\\Delta \\qb} +\n            r_2^\\evp \\, \\lambda^\\evp \\, \\lb^\\evp \\cdot \\overline{\\Delta \\qb} \\\\\n%\n            r_3^\\evm \\, \\lambda^\\evm \\, \\lb^\\evm \\cdot \\overline{\\Delta \\qb} +\n            r_3^\\evz \\, \\lambda^\\evz \\, \\lb^\\evz \\cdot \\overline{\\Delta \\qb} +\n            r_3^\\evp \\, \\lambda^\\evp \\, \\lb^\\evp \\cdot \\overline{\\Delta \\qb} \\\\\n   \\end{array} \\right )\n\\renewcommand{\\arraystretch}{1.0}\n\\end{equation}\nWe can rewrite this compactly as:\n\\begin{equation}\n\\sum_\\nu \\lambda^{(\\nu)} (\\lb^{(\\nu)} \\cdot \\overline{\\Delta \\qb}) \\rb^{(\\nu)}\n\\end{equation}\nwhere we use $\\nu$ to indicate which wave we are summing over.  A similar\nexpansion is used for $\\Rb\\Lb \\overline{\\Delta \\qb}$.  In fact, any vector\ncan be decomposed in this fashion:\n\\begin{equation}\n{\\boldsymbol{\\chi}} = {\\bf I} {\\boldsymbol{\\chi}} = \\Rb \\Lb {\\boldsymbol{\\chi}} = \n   \\sum_\\nu (\\lb^\\enu \\cdot {\\boldsymbol{\\chi}}) \\rb^\\enu\n\\end{equation}\nAnd then it is easy to see that the above manipulations for $\\Ab \\Delta \\qb$\ncan be expressed as:\n\\begin{equation}\n\\Ab \\Delta \\qb =  \\Ab \\sum_\\nu (\\lb^\\enu \\cdot \\overline{\\Delta \\qb}) \\rb^\\enu = \n  \\sum_\\nu (\\lb^\\enu \\cdot \\overline{\\Delta \\qb}) \\Ab \\rb^\\enu = \n  \\sum_\\nu (\\lb^\\enu \\cdot \\overline{\\Delta \\qb}) \\lambda^\\enu \\rb^\\enu\n\\end{equation}\nwhere we used $\\Ab \\rb^\\enu = \\lambda^\\enu \\rb^\\enu$.  The quantity $(\\lb^\\enu\n\\cdot \\overline{\\Delta \\qb})$ that shows up here is the projection of\nthe vector $\\overline{\\Delta \\qb}$ into the characteristic variable\ncarried by wave $\\nu$.  This sum shows, as discussed earlier, that each wave\ncarries a jump in the characteristic variable, with the jump in the primitive\nvariables proportion to the right eigenvector, $\\rb^\\enu$.\n\nThe resulting vector\nfor the left state is:\n\\begin{equation}\n\\qb_{i+\\myhalf,L}^{n+\\myhalf} = \\qb_i^n + \\frac{1}{2} \\sum_{\\nu; \\lambda^{(\\nu)} \\ge 0}\n  \\left [ 1 - \\frac{\\Delta t}{\\Delta x} \\lambda_i^{(\\nu)} \\right ] (\\lb_i^{(\\nu)} \\cdot \\overline{\\Delta \\qb}_i) \\rb_i^{(\\nu)}\n\\end{equation}\nNote that we make a slight change here, and only include a term in the sum if\nits wave is moving toward the interface ($\\lambda^\\enu \\ge 0$).  The quantity\n$\\Delta t \\lambda^\\enu/\\Delta x$ inside the brackets is simply the CFL\nnumber for the wave $\\nu$.\n\nStarting with the data in the $i+1$ zone and expanding to the left, we can\nfind the right state on the $i+\\myhalf$ interface:\n\\begin{equation}\n\\qb_{i+\\myhalf,R}^{n+\\myhalf} = \\qb_{i+1}^n - \\frac{1}{2} \\sum_{\\nu; \\lambda^{(\\nu)} \\le 0}\n  \\left [ 1 + \\frac{\\Delta t}{\\Delta x} \\lambda_{i+1}^{(\\nu)} \\right ] (\\lb_{i+1}^{(\\nu)} \\cdot \\overline{\\Delta \\qb}_{i+1}) \\rb_{i+1}^{(\\nu)}\n\\end{equation}\n\n\nA good discussion of this is in Miller \\& Colella\n\\cite{millercolella:2002} (Eq.\\ 85).  This expression is saying that\neach wave carries a jump in $\\rb^{(\\nu)}$ and only those jumps moving\ntoward the interface contribute to our interface state.  This\nrestriction of only summing up the waves moving toward the interface\nis sometimes called {\\em characteristic tracing}.  This decomposition\nin terms of the eigenvectors and eigenvalues is commonly called a {\\em\n  characteristic projection}.  In terms of an operator, $P$, it can be\nexpressed as:\n\\begin{equation}\nP {\\boldsymbol{\\chi}} = \\sum_\\nu (\\lb^{(\\nu)} . {\\boldsymbol{\\chi}}) \\rb^{(\\nu)}\n\\end{equation}\n\\begin{exercise}[Characteristic projection]\n{\nShow that $P \\qb = \\qb$, using the eigenvectors corresponding to the primitive\n variable form of the Euler equations.}\n\\end{exercise}\nIn the literature, sometimes a `$>$' or `$<$' subscript on $P$ is used\nto indicate the characteristic tracing.\n\nWe could stop here, but Colella \\& Glaz~\\cite{colellaglaz:1985}\n(p.\\ 278) argue that the act of decomposing $\\Ab$ in terms of the left\nand right eigenvectors is a linearization of the quasi-linear system,\nand we should minimize the size of the quantities that are subjected\nto this characteristic projection.  To accomplish this, they suggest\nsubtracting off a {\\em reference state}.  Saltzman (Eq.\\ 8) further\nargues that since only jumps in the solution are used in constructing\nthe interface state, and that the characteristic decomposition simply adds\nup all these jumps, we can subtract off the reference state and\nproject the result.  In other words, we can write:\n\\begin{equation}\n\\qb_{i+\\myhalf,L}^{n+\\myhalf} - \\qb_\\mathrm{ref} = \\qb_i^n - \\qb_\\mathrm{ref} +\n  \\frac{1}{2} \\left [ 1 - \\frac{\\Delta t}{\\Delta x} \\Ab_i \\right ] \\overline{\\Delta \\qb}_i\n\\label{eq:lin_decomp}\n\\end{equation}\nThen we subject the RHS to the characteristic projection---this tells\nus how much of the quantity $\\qb_{i+\\myhalf,L}^{n+\\myhalf} - \\qb_\\mathrm{ref}$\nreaches the interface.  Colella \\& Glaz (p.\\ 278) and Colella\n(Eq.\\ 2.11) suggest\n\\begin{equation}\n\\qb_\\mathrm{ref} = \\tilde{\\qb}_{i,L} \\equiv \\qb_i^n +\n   \\frac{1}{2} \\left [ 1 - \\frac{\\Delta t}{\\Delta x}\n \\max(\\lambda_i^\\evp, 0) \\right ] \\overline{\\Delta \\qb}_i \\label{eq:qref}\n\\end{equation}\nwhere $\\lambda^\\evp$ is the fastest eigenvalue, and thus will see\nthe largest portion of the linear profiles.  Physically, this\nreference state represents the jump carried by the fastest wave\nmoving toward the interface.  Then,\n\\begin{equation}\n\\qb_{i+\\myhalf,L}^{n+\\myhalf} - \\tilde{\\qb}_{i,L} = \\frac{1}{2} \\frac{\\Delta t}{\\Delta x}\n  \\left [ \\max(\\lambda_i^\\evp,0) - \\Ab_i \\right ] \\overline{\\Delta \\qb}_i\n\\end{equation}\nand projecting this RHS (see Colella \\& Glaz Eq. 43; Miller \\& Colella Eq. 87),\nand isolating the interface state, we have\n\\begin{align}\n\\qb_{i+\\myhalf,L}^{n+\\myhalf} &= \\tilde{\\qb}_{i,L} + \\frac{1}{2} \\frac{\\Delta t}{\\Delta x}\n       \\sum_{\\nu; \\lambda^{(\\nu)} \\ge 0} \\lb_i^{(\\nu)} \\cdot \\left [ \\max(\\lambda_i^\\evp,0) - \\Ab_i \\right ]\n                                           \\overline{\\Delta \\qb}_i \\, \\rb_i^{(\\nu)} \\\\\n                    &= \\tilde{\\qb}_{i,L} + \\frac{1}{2} \\frac{\\Delta t}{\\Delta x}\n       \\sum_{\\nu; \\lambda^{(\\nu)} \\ge 0} \\left [ \\max(\\lambda_i^\\evp,0) - \\lambda_i^{(\\nu)} \\right ]\n                                          (\\lb_i^{(\\nu)} \\cdot \\overline{\\Delta \\qb}_i) \\, \\rb_i^{(\\nu)}\n\\end{align}\nThis is equivalent to the expression in Saltzman~\\cite{saltzman:1994}\n(p.\\ 161, first column, second-to-last equation) and\nColella~\\cite{colella:1990} (p.\\ 191, the group of expressions at the\nend)\\footnote{Since the combination $\\lb^\\enu \\cdot \\Delta \\qb$\n  appears so frequently, some sources represent this as $\\beta^\\enu$\n  and then write the interface states as a linear combination of the\n  $\\beta$'s.  Some variation in the literation, depending on whether $\\rho$\n  or $\\tau = 1/\\rho$ is used as a primitive variable}.  The corresponding\n  state to the right of this interface is:\n\\begin{equation}\n\\qb_{i+\\myhalf,R}^{n+\\myhalf} = \\tilde{\\qb}_{i+1,R} + \\frac{1}{2} \\frac{\\Delta t}{\\Delta x}\n       \\sum_{\\nu; \\lambda^\\enu \\le 0}\n       \\left [ \\min(\\lambda_{i+1}^\\evm,0) - \\lambda_{i+1}^\\enu \\right ]\n       (\\lb_{i+1}^\\enu \\cdot \\overline{\\Delta \\qb}_{i+1}) \\, \\rb_{i+1}^\\enu\n\\end{equation}\nwhere now the reference state captures the flow from the $i+1$ zone\nmoving to the {\\em left} to this interface (hence the appearance of\n$\\lambda^\\evm$, the leftmost eigenvalue):\n\\begin{equation}\n\\tilde{\\qb}_{i+1,R} = \\qb_{i+1} - \\frac{1}{2} \\left [ 1 + \\frac{\\Delta t}{\\Delta x} \\min(\\lambda_{i+1}^\\evm,0) \\right ] \\overline{\\Delta \\qb}_{i+1}\n\\end{equation}\n\nSide note: the data in zone $i$ will be used to construct the right\nstate at $i-\\myhalf$ (the left interface) and the left state at $i+\\myhalf$\n(the right interface) (see Figure~\\ref{fig:states}).  For this reason,\ncodes usually compute the eigenvectors/eigenvalues for that zone and\nthen compute $\\qb_{i-\\myhalf,R}^{n+\\myhalf}$ together with $\\qb_{i+\\myhalf,L}^{n+\\myhalf}$\nin a loop over the zone centers.\n\n% figure made by figures/Euler/states.py\n\\begin{figure}\n\\centering\n\\includegraphics[width=3.5in]{states}\n\\caption[The two interface states derived from a cell-center quantity]{\\label{fig:states} The two interface states that are constructed\nusing $\\qb_i$ as the starting point.}\n\\end{figure}\n\n\\subsection{Piecewise parabolic}\n\n\\label{sec:hydro:ppm}\n\nThe piecewise parabolic method uses a parabolic reconstruction in each\ncell.  This is more accurate than the linear reconstruction.\nFigure~\\ref{fig:ppm} shows the reconstructed parabolic profiles within\na few cells.  Since the original PPM\npaper~\\cite{colellawoodward:1984}, there have been many discussions of\nthe method, with many variations.  Here we focus on the presentation\nby Miller \\& Colella~\\cite{millercolella:2002}, since that is the most\nstraightforward.  Note: even though a parabolic profile could be\nthird-order accurate, the temporal discretization and prediction in\nthis method is still only second-order.\n%\n% figure made by figures/Euler/ppm.py\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{piecewise-parabolic}\n\\caption[Piecewise parabolic reconstruction of the cell\n  averages]{\\label{fig:ppm} Piecewise parabolic reconstruction of the\n  cell averages.  The dotted line shows the unlimited parabolas---note\n  how they touch at each interface, since the interface values come\n  from the same interpolant initially.  The solid line shows the\n  limited parabolas.}\n\\end{figure}\n\n\nMiller \\& Colella give an excellent description of how to take the\nresults for piecewise linear reconstruction and generalize it to the case of\nPPM~\\cite{colellawoodward:1984} (see Eqs.\\ 88-90).  Starting with\nEq.~\\ref{eq:lin_decomp}, we can write this (after the characteristic\nprojection) as\n\\begin{equation}\n\\qb_{i+\\myhalf,L}^{n+\\myhalf} = \\tilde{\\qb}_+ -\n   \\sum_{\\nu;\\lambda^{(\\nu)}\\ge 0} \\lb_i^{(\\nu)} \\cdot \\left \\{\n        \\tilde{\\qb}_+ - \\left [ \\qb_i^n +\n            \\frac{1}{2} \\left ( 1 - \\frac{\\Delta t}{\\Delta x} \\lambda_i^{(\\nu)} \\right ) \\overline{\\Delta \\qb}_i \\right ]\n       \\right \\} \\rb_i^{(\\nu)}\n\\end{equation}\nMiller \\& Colella rewrite the portion inside the $[\\ldots]$\nrecognizing that (similar to M\\&C Eq.\\ 88, but for the $i+\\myhalf,L$ interface):\n\\begin{equation}\n  \\qb_i^n + \\frac{1}{2} \\left (1 - \\frac{\\Delta t}{\\Delta x} \\lambda_i^{(\\nu)} \\right ) \\overline{\\Delta \\qb}_i\n  \\approx \\frac{1}{\\lambda \\Delta t} \\int_{x_{i+\\myhalf} - \\lambda \\Delta t}^{x_{i+\\myhalf}}\n           \\qb(x) dx\n\\end{equation}\nwhere $\\qb(x)$ is the reconstructed functional form of $\\qb$ in the zone.\n\n\\begin{exercise}[The average state reacting the interface]\n{Show that this is exactly true for a linear reconstruction of $\\qb(x)$, i.e.,\n$\\qb(x) = \\qb_i + (\\partial \\qb/\\partial x) (x - x_i)$.}\n\\end{exercise}\n\n\\noindent The integral on the right represents the average of $\\qb$ that\ncan reach the right interface of the cell $i$ over timestep $\\Delta\nt$, moving at the wavespeed $\\lambda$.  This suggests that we can\nreplace the linear reconstruction of $\\qb$ with a parabolic one, and\nkeep our expressions for the interface states.\n\n% figure made by figures/Euler/ppm-trace.py\n\\begin{figure}\n\\centering\n\\includegraphics[width=3.5in]{ppm-trace}\n\\caption[Integration under the parabola profile for to an\n  interface]{\\label{fig:ppm_trace} Integration under the parabolic\n  profile.  For each of the waves, $\\sigma^\\enu$ is the fraction of\n  the cell that they cross in a timestep, and $\\sigma^\\enu \\Delta x =\n  \\lambda^\\enu \\Delta t$ is the distance they can travel.  Here we are\n  integrating under the parabola to the right interface of cell $i$ to\n  define $\\Ic_+^\\enu$ (this is indicated by the shaded\n  region).  The $\\Ic_+^\\enu$ carried by this wave will be\n  added to those carried by the other waves to form the left state at\n  interface $i+\\myhalf$.}\n\\end{figure}\n\nIn particular, we define\n\\begin{equation}\n\\Ic_+^{(\\nu)}(\\qb_i) = \\frac{1}{\\sigma^{(\\nu)} \\Delta x} \\int _{x_{i+\\myhalf} - \\sigma^{(\\nu)} \\Delta x}^{x_{i+\\myhalf}} \\qb(x) dx\n\\end{equation}\nwith $\\sigma^{(\\nu)} = |\\lambda^{(\\nu)}|\\Delta t / \\Delta x$ (see Almgren et\nal. Eq. 31)  (see Figure~\\ref{fig:ppm_trace}).  Then\n\\begin{equation}\n\\qb_{i+\\myhalf,L}^{n+\\myhalf} = \\tilde{\\qb}_+ -\n   \\sum_{\\nu;\\lambda^{(\\nu)}\\ge 0} \\lb_i^{(\\nu)} \\cdot \\left (\n        \\tilde{\\qb}_+ - \\Ic_+^{(\\nu)}(\\qb_i)\n       \\right ) \\rb_i^{(\\nu)}  \\label{eq:ppmtrace}\n\\end{equation}\nMiller \\& Colella choose the reference state as\n\\begin{equation}\n\\tilde{\\qb}_+ = \\left \\{ \\begin{array}{cc}\n       \\Ic_+^{(+)}(\\qb_i) & \\mathrm{if~} u + c > 0 \\\\\n       \\qb_i                    & \\mathrm{otherwise}\n\\end{array}\n\\right .\n\\end{equation}\nwhere the superscript $(+)$ on $\\Ic$ indicates that the\nfastest eigenvalue ($\\lambda^\\evp = u + c$) is used.  This is similar in\nspirit to Eq.~\\ref{eq:qref}.  Note: in the original PPM paper, if the\nwave is not approaching the interface, instead of using the\ncell-average, $\\qb_i$, they use the limit of the quadratic interpolant.\nIn contrast to the above, the Castro paper~\\cite{almgren:2010} just\nuses $\\qb_i$ for the reference state regardless of whether the wave is\nmoving toward or away from the interface.  Note that if the system\nwere linear, then the choice of reference state would not matter.\n\nTo finish the reconstruction, we need to know the parabolic form\nof $\\qb(x)$.  Here, we do the reconstruction from the original PPM\npaper:\n\\begin{equation}\n\\qb(x) = \\qb_{-} + \\xi(x) \\left ( \\Delta \\qb + \\qb_6 (1 - \\xi(x) ) \\right )\n\\end{equation}\nwith $\\Delta \\qb = \\qb_+ - \\qb_-$, and $\\qb_-$, $\\qb_+$ the values of the polynomial\non the left and right edges, respectively, of the current cell, and\n\\begin{equation}\n\\qb_6 \\equiv 6 \\left [ \\qb_i - \\frac{1}{2} (\\qb_- + \\qb_+) \\right ]\n\\end{equation}\nand\n\\begin{equation}\n\\xi(x) = \\frac{x - x_{i-\\myhalf}}{\\Delta x}\n\\end{equation}\n\nTo complete the description, we need to determine the parameters of\nthe parabola.  The values of $\\qb_-$ and $\\qb_+$ are computed and limited\nas described in the original PPM paper.  With this definition, we can\ndo the integral $\\Ic_+$:\n\\begin{equation}\n\\Ic_+^{(\\nu)}(\\qb_i) = \\qb_{+,i} - \\frac{\\sigma_i^{(\\nu)}}{2}\n   \\left [ \\Delta \\qb_i - \\qb_{6,i} \\left ( 1 - \\frac{2}{3} \\sigma_i^{(\\nu)} \\right ) \\right ]\n\\end{equation}\nFigure~\\ref{fig:ppm_trace} illustrates the process of integrating under\nthe parabolic profile.\n\n\n\\begin{exercise}[Conservative interpolation]\n{Show that $\\qb(x)$ is a conservative interpolant.  That is\n\\begin{equation}\n\\frac{1}{\\Delta x} \\int_{x_{i-\\myhalf}}^{x_{i+\\myhalf}} \\qb(x) dx = \\qb_i\n\\end{equation}\nYou can also see that the average over the left half of the zone is\n$\\qb_i -\\frac{1}{4}\\Delta \\qb$ and the average over the right half of the\nzone is $\\qb_i + \\frac{1}{4}\\Delta \\qb$.  This means that there are equal\nareas between the integral and zone average on the left and right\nsides of the zone.  This can be seen by looking at\nFigure~\\ref{fig:ppm}.  }\n\\end{exercise}\n\n\\begin{quote}\n\\noindent\\makebox[\\linewidth]{\\rule{0.9\\textwidth}{1pt}}\n{\\em Aside}:\nNote that this characteristic projection of $\\tilde{\\qb}_+ -\n\\Ic_+^{(\\nu)}$ is discussed in the original PPM paper in the\nparagraph following their Eq.~3.5.  They do not keep things in this form\nhowever, and instead explicitly multiply out the $l\\cdot [\\ldots] r$\nterms to arrive at their Eq.~3.6.  For example, starting with\nEq.~\\ref{eq:ppmtrace},\nwe can write the left velocity state as (leaving off the $i$\nsubscripts on the vectors):\n\\begin{equation}\nu_{i+\\myhalf,L}^{n+\\myhalf} =\n  \\tilde{u}_+ - \\sum_\\nu \\lb^{(\\nu)} \\cdot\n      ( \\tilde{\\qb}_+ -  \\Ic_+^{(\\nu)}(\\qb) )\n    \\underbrace{\\rb^{(\\nu)}}_{\\begin{smallmatrix}\\mathrm{only~the}\\\\ u~\\mathrm{`slot'} \\end{smallmatrix}}\n\\end{equation}\n(where, as above, the $\\sim$ indicates the reference state).\nHere the $r$ eigenvector on the end is representative---we only pick\nthe row corresponding to $u$ in the $\\qb$ vector (in our case, the\nsecond row).\n\nPutting in the eigenvectors and writing out the sum, we have:\n\\begin{align}\n u_{i+\\myhalf,L}^{n+\\myhalf} =\n     \\tilde{u}_+ &-\n       \\left ( \\begin{array}{ccc}\n                  0 & -\\frac{\\rho}{2c} & \\frac{1}{2c^2} \\end{array}\n       \\right )\n    \\left ( \\begin{array}{c}\n           \\tilde{\\rho}_+ - \\Ic_+^\\evm(\\rho) \\\\\n           \\tilde{u}_+ - \\Ic_+^\\evm(u) \\\\\n           \\tilde{p}_+ - \\Ic_+^\\evm(p)\n            \\end{array} \\right )\n    {\\color{mygray} \\left ( \\begin{array}{c}\n           1  \\\\\n           {\\color{black} -c/\\rho} \\\\\n           c^2\n    \\end{array} \\right ) } \\nonumber \\\\\n%\n     &-\\left ( \\begin{array}{ccc}\n                  1 & 0 & -\\frac{1}{c^2} \\end{array}\n       \\right )\n    \\left ( \\begin{array}{c}\n           \\tilde{\\rho}_+ - \\Ic_+^\\evz(\\rho) \\\\\n           \\tilde{u}_+ - \\Ic_+^\\evz(u) \\\\\n           \\tilde{p}_+ - \\Ic_+^\\evz(p)\n            \\end{array} \\right )\n    {\\color{mygray} \\left ( \\begin{array}{c}\n           1  \\\\\n           {\\color{black} 0} \\\\\n           0\n    \\end{array} \\right ) } \\nonumber \\\\\n%\n    &-\\left ( \\begin{array}{ccc}\n                  0 & \\frac{\\rho}{2c} & \\frac{1}{2c^2} \\end{array}\n       \\right )\n    \\left ( \\begin{array}{c}\n           \\tilde{\\rho}_+ - \\Ic_+^\\evp(\\rho) \\\\\n           \\tilde{u}_+ - \\Ic_+^\\evp(u) \\\\\n           \\tilde{p}_+ - \\Ic_+^\\evp(p)\n            \\end{array} \\right )\n    {\\color{mygray} \\left ( \\begin{array}{c}\n           1  \\\\\n           {\\color{black} c/\\rho} \\\\\n           c^2\n    \\end{array} \\right ) }\n%\n\\end{align}\nHere again we show the entire right eigenvector for illustration, but\nonly the element that comes into play is drawn in black.  This shows\nthat the second term is $0$---the contact wave does not carry a jump\nin velocity.  Multiplying out $\\lb^{(\\nu)} \\cdot (\\tilde{\\qb}_+ -\n\\Ic_+^{(\\nu)})$ we have:\n\\begin{align}\nu_{i+\\myhalf,L}^{n+\\myhalf} =\n   \\tilde{u}_+\n  &- \\frac{1}{2} \\left [\n      (\\tilde{u}_+ - \\Ic_+^\\evm(u) ) -\n       \\frac{\\tilde{p}_+ - \\Ic_+^\\evm(p)}{C} \\right ] \\nonumber \\\\\n  &- \\frac{1}{2} \\left [\n      (\\tilde{u}_+ - \\Ic_+^\\evp(u) ) +\n       \\frac{\\tilde{p}_+ - \\Ic_+^\\evp(p)}{C} \\right ]\n\\label{eq:ufull}\n\\end{align}\nwhere $C$ is the Lagrangian sound speed ($C = \\sqrt{\\gamma p \\rho}$).\nDefining\n\\begin{align}\n\\beta^\\evp &= - \\frac{1}{2C}\n  \\left [\n      (\\tilde{u}_+ - \\Ic_+^\\evp(u) ) +\n       \\frac{\\tilde{p}_+ - \\Ic_+^\\evp(p)}{C} \\right ] \\\\\n%\n\\beta^\\evm &= + \\frac{1}{2C}\n  \\left [\n      (\\tilde{u}_+ - \\Ic_+^\\evm(u) ) -\n       \\frac{\\tilde{p}_+ - \\Ic_+^\\evm(p)}{C} \\right ]\n\\end{align}\nwe can write our left state as:\n\\begin{equation}\nu_{i+\\myhalf,L}^{n+\\myhalf} =\n   \\tilde{u}_+ + C ( \\beta^\\evp - \\beta^\\evm)\n\\end{equation}\nThis is Eqs.~3.6 and 3.7 in the PPM paper.  Note that in their\nconstruction appears to use the reference state in defining the\nLagrangian sound speed (in their $\\beta$ expressions is written as\n$\\tilde{C}$).  This may follow from the comment before Eq.~3.6,\n``modified slightly for the present application''.  Similarly,\nthe expressions for $\\rho_L$ and $p_L$ can be written out. \\\\\n\\noindent\\makebox[\\linewidth]{\\rule{0.9\\textwidth}{1pt}}\n\\end{quote}\n\nSimilar expressions can be derived for the right state at the left interface\nof the zone ($\\qb_{i-\\myhalf,R}^{n+\\myhalf}$).  Here, the integral under the parabolic\nreconstruction is done over the region of each wave that can reach the left\ninterface over our timestep:\n\\begin{equation}\n\\Ic_-^{(\\nu)}(\\qb) = \\frac{1}{\\sigma^{(\\nu)} \\Delta x}\n  \\int_{x_{i-\\myhalf}}^{x_{i-\\myhalf} + \\sigma^{(\\nu)} \\Delta x} \\qb(x) dx\n\\end{equation}\nThe right state at $i-\\myhalf$ using zone $i$ data is:\n\\begin{equation}\n\\qb_{i-\\myhalf,R}^{n+\\myhalf} = \\tilde{\\qb}_- - \\sum_{\\nu; \\lambda_\\nu \\le 0}\n   \\lb_i^{(\\nu)} \\cdot \\left ( \\tilde{\\qb}_- - \\Ic_-^{(\\nu)}(\\qb_i) \\right ) \\rb_i^{(\\nu)}\n\\end{equation}\nwhere the reference state is now:\n\\begin{equation}\n\\tilde{\\qb}_- = \\left \\{ \\begin{array}{cc}\n   \\Ic_-^{(-)}(\\qb_i) & \\mathrm{if~} u - c < 0 \\\\\n    \\qb_i                   & \\mathrm{otherwise}\n\\end{array} \\right .\n\\end{equation}\nwhere the $(-)$ superscript on $\\Ic$ indicates that the most\nnegative eigenvalue $(\\lambda^- = u - c)$ is used.  The integral\n$\\Ic_-^{(\\nu)}(\\qb)$ can be computed analytically by\nsubstituting in the parabolic interpolant, giving:\n\\begin{equation}\n\\Ic_-^{(\\nu)}(\\qb_i) = \\qb_{-,i} + \\frac{\\sigma_i^{(\\nu)}}{2}\n   \\left [ \\Delta \\qb_i + \\qb_{6,i} \\left ( 1 - \\frac{2}{3} \\sigma_i^{(\\nu)} \\right ) \\right ]\n\\end{equation}\nThis is equivalent to Eq.~31b in the Castro paper.\n\n\\subsubsection{New PPM limiters}\n\nRecent work \\cite{colellasekora} has formulated improved limiters for\nPPM that do not clip the profiles at extrema.  This only changes the\nlimiting process in defining $\\qb_+$ and $\\qb_-$, and does not affect the\nsubsequent parts of the algorithm.\n\n\n\\subsection{Flattening and contact steepening}\n\n  Shocks are self-steepening (this is how we\n  detect them in the Riemann solver---we look for converging characteristics).\n  This can cause trouble with the methods here, because the shocks may become\n  too steep.\n\n  Flattening is a procedure to add additional dissipation at shocks,\n  to ensure that they are smeared out over $\\sim 2$ zones.  The\n  flattening procedure is a multi-dimensional operation that looks at\n  the pressure and velocity profiles and returns a coefficient, $\\chi\n  \\in [0,1]$ that multiplies the limited slopes.  The convention most\n  sources use is that $\\chi = 1$ means no flattening (the slopes are\n  unaltered), while $\\chi = 0$ means complete flattening---the slopes\n  are zeroed, dropping us to a first-order method.\n  See for example in Saltzman~\\cite{saltzman:1994}.  Once the flattening\n  coefficient is determined, the interface state is blended with the\n  cell-centered value via:\n  \\begin{equation}\n  \\qb_{i+\\myhalf,\\{L,R\\}}^{n+\\myhalf} \\leftarrow (1 - \\chi) \\qb_i + \\chi \\qb_{i+\\myhalf,\\{L,R\\}}^{n+\\myhalf}\n  \\end{equation}\n\n  Note that the flattening algorithm increases the stencil size of\n  piecewise-linear and piecewise-parabolic reconstruction to 4 ghost cells\n  on each side.  This is because the flattening procedure itself looks\n  at the pressure 2 zones away, and we need to construct the flattening\n  coefficient in both the first ghost cell (since we need the interface\n  values there) and the second ghost cell (since the flattening procedure\n  looks at the coefficients in its immediate upwinded neighbor).\n\n  In contrast to shocks, contact waves\n  do not steepen (they are associated with the middle characteristic\n  wave, and the velocity does not change across that, meaning there\n  cannot be any convergence).  The original PPM\n  paper advocates a contact steepening method to artificially steepen\n  contact waves.  While it shows good results in 1-d, it can be\n  problematic in multi-dimensions.\n\n  Overall, the community seems split over whether this term should be\n  used.  Many people advocate that if you reach a situation where you\n  think contact steepening may be necessary, it is more likely that\n  the issue is that you do not have enough resolution.\n\n\n\\subsection{Limiting on characteristic variables}\n\nSome authors (see for example, \\cite{athena} Eqs.~37, 38) advocate\nlimiting on the characteristic variables rather than the primitive\nvariables.  The characteristic slopes for the quantity carried by the\nwave $\\nu$ can be found from the primitive variables\nas: \n%\n\\begin{equation} \n\\Delta w^{(\\nu)} = \\lb^{(\\nu)} \\cdot \\Delta \\qb \n\\end{equation} \n%\nany limiting would then be done to $\\Delta w^{(\\nu)}$ and the limited\nprimitive variables would be recovered as:\n\\begin{equation}\n  \\overline{\\Delta \\qb} = \\sum_\\nu \\overline{\\Delta w}^{(\\nu)}\n  \\rb^{(\\nu)} \n\\end{equation} \n(here we use an overline to indicate limiting).\n\nThis is attractive because it is more in the spirit of the linear\nadvection equation and the formalism that was developed there.  A\npotential downside is that when you limit on the characteristic\nvariables and convert back to the primitive, the primitive variables\nmay now fall outside of valid physical ranges (for example, negative\ndensity).\n\n\n\n%-----------------------------------------------------------------------------\n\\section{Riemann solvers}\n\n\nOnce the interface states are created, the Riemann solver is called.  This\nreturns the solution at the interface:\n\\begin{equation}\n\\qb_{i+\\myhalf}^{n+\\myhalf} = \\mathcal{R}(\\qb_{i+\\myhalf,L}^{n+\\myhalf}, \\qb_{i+\\myhalf,R}^{n+\\myhalf})\n\\end{equation}\nThis is done for every interface, as illustrated in Figure~\\ref{fig:euler:multiple_interfaces}.\n\n% figure drawn with figures/Euler/multiple_interfaces.py\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{multiple_interfaces}\n\\caption[Riemann wave structure at each interface]{\\label{fig:euler:multiple_interfaces}\n  The Riemann wave structure resulting from the separate Riemann problems at each interface.  For each\n  interface, we find the state on the interface and use this to evaluate the flux through the interface.} \n\\end{figure}\n\nAs discussed in \\S~\\ref{Euler:riemann:solution}, we need to determine\nwhich state is on our interface to compute the fluxes through the\ninterface.  The full solution of the Riemann problem can be quite expensive,\nso in practice, approximate Riemann solvers are used to speed\nthe computation.  Different Riemann solvers\nwill have different approximations for finding the speeds of the left,\ncenter, and right wave, and evaluating the star state.  In the `star'\nregion, only $\\rho$ jumps across the middle (contact) wave, the\npressure and velocity are constant across that wave (see $\\rb^\\evz$).\nWe determine the state in the star region ($\\rho_l^*, \\rho_r^*, u^*,\np^*$) by using the jump conditions for the Euler equations or the\nRiemann invariants, as shown in \\S~\\ref{Euler:riemann:starstate}.\nSome approximate Riemann solvers assume that both waves are shocks or\nboth are rarefactions, aimplying the form of the equation that\nmust be solved to find $p_\\star$.\n\nTwo-shock solvers, like the one in \\cite{colellaglaz:1985} are quite\ncommon in astrophyiscs.  Figure~\\ref{fig:euler:riemann-2shock-curve}\nshows the Hugoniot curves for the Sod problem under the 2-shock\nassumption---they are quite close to those for the true solution.  To\nfurther save on cost, approximate Riemann solvers for general\nequations of state often include additional thermodynamic information\nat the interfaces that overconstrains the system, but can remove the\nneed to call an expensive equation of state routine in solving for the\nstar state.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{riemann-2shock-sod-phase}\n\\caption[The approximate (2-shock) Hugoniot curves corresponding to\n  the Sod problem]{\\label{fig:euler:riemann-2shock-curve} The\n  approximate Hugoniot curves under the 2-shock approximation\n  corresponding to the Sod problem.  Also shown as the gray dotted\n  line is the exact Hugoniot curves (compare to\n  Figure~\\ref{fig:euler:riemann-curve}).  We see that the 2-shock\n  approximation does a reasonable job near the intersection and only\n  diverges significantly for small $p$ (which is where the solution\n  should really be a\n  rarefaction).\\\\ \n  \\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/compressible/riemann-2shock.py}{riemann-2shock.py}}}\n\\end{figure}\n\nAn additional approximation concerns rarefactions.  Recall that a\nrarefaction involves diverging flow---it spreads out with time.\nSpecial consideration needs to be taken if the rarefaction wave spans\nthe interface (a {\\em transonic rarefaction},\n\\S~\\ref{Euler:riemann:solution}).  In this case, most approximate\nRiemann solvers simply linearly interpolate between the left or right\nstate and the appropriate star state instead of solving for the\nstructure inside the rarefaction.\n\nWith this approximate state, the fluxes are computed as:\n\\begin{equation}\n\\renewcommand{\\arraystretch}{1.5}\n\\Fb_{i+\\myhalf}^{n+\\myhalf} = \\left ( \\begin{array}{c}\n                             \\rho_{i+\\myhalf}^{n+\\myhalf} u_{i+\\myhalf}^{n+\\myhalf} \\\\\n                             \\rho_{i+\\myhalf}^{n+\\myhalf} (u_{i+\\myhalf}^{n+\\myhalf})^2 + p_{i+\\myhalf}^{n+\\myhalf} \\\\\n                             u_{i+\\myhalf}^{n+\\myhalf} p_{i+\\myhalf}^{n+\\myhalf} / (\\gamma - 1)  +\n                             \\frac{1}{2} \\rho_{i+\\myhalf}^{n+\\myhalf} (u_{i+\\myhalf}^{n+\\myhalf})^3 +\n                             u_{i+\\myhalf}^{n+\\myhalf} p_{i+\\myhalf}^{n+\\myhalf}\n                            \\end{array} \\right )\n\\renewcommand{\\arraystretch}{1.0}\n\\end{equation}\n\nA different class of approximate Riemann solvers (the HLL family)\napproximate the fluxes directly instead of approximating the state\nfirst.  These require estimates of the wave speeds, and care must be\ntaken to ensure those estimates are valid for a general equation of\nstate.  These wave speeds are then used together with the\nRankine-Hugoniot jump conditions to give the fluxes.\n\n\n%-----------------------------------------------------------------------------\n\\section{Conservative update}\n\nOnce we have the fluxes, the conservative update is done as\n\\begin{equation}\n\\Uc^{n+1}_i = \\Uc^n_i + \\frac{\\Delta t}{\\Delta x} \n   \\left ( \\Fb_{i-\\myhalf}^{n+\\myhalf} - \\Fb_{i+\\myhalf}^{n+\\myhalf} \\right )\n\\end{equation}\nThe timestep, $\\Delta t$ is determined by the time it takes for the\nfastest wave to cross a single zone:\n\\begin{equation}\n\\Delta t < \\min_d \\left \\{ \\frac{\\Delta x}{|\\Ub \\cdot {\\bf e}_d| + c}\\right \\}\n\\end{equation}\n(This is the limit for the CTU unsplit scheme here, but see the discussion\nin \\S~\\ref{sec:adv:timestep}).\n\nOften simulation codes will further restrict the timestep.  Commonly\nused restrictions for pure hydrodynamics include a limit on the factor\nby which a timestep can grow from one step to the next (a typical\nvalue is $1.2$), and an initial scaling, say of 1/10, for the first\ntimestep.  Together, these will force the code to take a few initial\nsteps before working up to the CFL limit.\n\n\\subsection{Artificial viscosity}\n\nColella and Woodward argue that behind slow-moving shocks these\nmethods can have oscillations.  The fix they propose is to use some\nartificial viscosity---this is additional dissipation that kicks in at\nshocks.  (They argue that flattening alone is not enough).\n\nWe use a multidimensional analog of their artificial viscosity\n(\\cite{colellawoodward:1984}, Eq.\\ 4.5) which modifies the fluxes.  By\ndesign, it only kicks in for converging flows, such that you would\nfind around a shock.\n\n\n%-----------------------------------------------------------------------------\n\\section{Boundary conditions}\n\nBoundary conditions are implemented through ghost cells.  The following\nare the most commonly used boundary conditions.  For the expressions\nbelow, we use the subscript $\\mathrm{lo}$ to denote the spatial index\nof the first valid zone in the domain (just inside the left boundary).\n\n\\begin{itemize}\n\n\\item {\\em Outflow}: the idea here is that the flow should gracefully\n  leave the domain.  The simplest form is to simply give all variables\n  a zero-gradient:\n  \\begin{equation}\n  \\left ( \\begin{array}{c} \\rho_{\\mathrm{lo-1},j} \\\\\n                           (\\rho u)_{\\mathrm{lo-1},j} \\\\\n                           (\\rho v)_{\\mathrm{lo-1},j} \\\\\n                           (\\rho E)_{\\mathrm{lo-1},j} \\end{array} \\right ) =\n  \\left ( \\begin{array}{c} \\rho_{\\mathrm{lo},j} \\\\\n                           (\\rho u)_{\\mathrm{lo},j} \\\\\n                           (\\rho v)_{\\mathrm{lo},j} \\\\\n                           (\\rho E)_{\\mathrm{lo},j} \\end{array} \\right )\n  \\end{equation}\n\n  Note that these boundaries are not perfect.  At the boundary, one\n  (or more) of the waves from the Riemann problem can still enter\n  the domain.  Only for supersonic flow, do all waves point outward.\n\n\n\\item {\\em Reflect}: this is appropriate at a solid wall or symmetry\n  plane.  All variables are reflected across the boundary, with the\n  normal velocity given the opposite sign.  At the $x$-boundary, the\n  first ghost cell is:\n  \\begin{equation}\n  \\left ( \\begin{array}{c} \\rho_{\\mathrm{lo-1},j} \\\\\n                           (\\rho u)_{\\mathrm{lo-1},j} \\\\\n                           (\\rho v)_{\\mathrm{lo-1},j} \\\\\n                           (\\rho E)_{\\mathrm{lo-1},j} \\end{array} \\right ) =\n  \\left ( \\begin{array}{c} \\rho_{\\mathrm{lo},j} \\\\\n                           -(\\rho u)_{\\mathrm{lo},j} \\\\\n                           (\\rho v)_{\\mathrm{lo},j} \\\\\n                           (\\rho E)_{\\mathrm{lo},j} \\end{array} \\right )\n  \\end{equation}\n  The next is:\n  \\begin{equation}\n  \\left ( \\begin{array}{c} \\rho_{\\mathrm{lo-2},j} \\\\\n                           (\\rho u)_{\\mathrm{lo-2},j} \\\\\n                           (\\rho v)_{\\mathrm{lo-2},j} \\\\\n                           (\\rho E)_{\\mathrm{lo-2},j} \\end{array} \\right ) =\n  \\left ( \\begin{array}{c} \\rho_{\\mathrm{lo+1},j} \\\\\n                           -(\\rho u)_{\\mathrm{lo+1},j} \\\\\n                           (\\rho v)_{\\mathrm{lo+1},j} \\\\\n                           (\\rho E)_{\\mathrm{lo+1},j} \\end{array} \\right )\n  \\end{equation}\n  and so on $\\ldots$\n\n\n  \\item {\\em Inflow}: inflow boundary conditions specify the state\n    directly on the boundary.  Technically, this state is on the\n    boundary itself, not the cell-center.  This can be accounted for\n    by modifying the stencils used in the reconstruction near inflow\n    boundaries.\n\n  \\item {\\em Hydrostatic}: a hydrostatic boundary can be used at the\n    base of an atmosphere to provide the pressure support necessary to\n    hold up the atmosphere against gravity while still letting\n    acoustic waves pass through.  An example of this is described\n    in \\cite{hse}.\n\n\\end{itemize}\n\n\n%-----------------------------------------------------------------------------\n\\section{Multidimensional problems}\n\nThe multidimensional case is very similar to the multidimensional\nadvection problem.  Our system of equations is now:\n\\begin{equation}\n\\Uc_t + [\\Fb^{(x)}(\\Uc)]_x + [\\Fb^{(y)}(\\Uc)]_y = 0\n\\end{equation}\nwith\n\\begin{equation}\n\\Uc = \\left ( \\begin{array}{c} \\rho \\\\ \\rho u \\\\ \\rho v \\\\ \\rho E \\end{array} \\right )\n%\n\\qquad\n%\n\\Fb^{(x)}(\\Uc) = \\left ( \\begin{array}{c} \\rho u \\\\ \\rho uu + p \\\\ \\rho v u \\\\ \\rho u E + up \\end{array} \\right )\n%\n\\qquad\n\\Fb^{(y)}(\\Uc) = \\left ( \\begin{array}{c} \\rho v \\\\ \\rho vu     \\\\ \\rho v v + p \\\\ \\rho v E + vp \\end{array} \\right )\n\\end{equation}\n\nWe note that there is no transformation that can convert the multidimensional\nsystem into characteristic variables, since we cannot simultaneously\ndiagonalize the Jacobians corresponding to $\\Fb^{(x)}$ and $\\Fb^{(y)}$.  Related\nto this is that when limiting, we limit one-dimensional slopes instead of\ndoing a full multidimensional reconstruction and limiting (see \\cite{BDS}\nfor a multidimensional limiting procedure for linear advection.  For\nthe Euler equations, since we cannot write the multidimensional system\nin a characteristic form, we cannot use this type of method).\n\nFor a directionally-unsplit discretization, we predict the\ncell-centered quantities to the edges by Taylor expanding the\nconservative state, $\\Uc$, in space and time.  Now, when replacing the\ntime derivative ($\\partial \\Uc/\\partial t$) with the divergence of the\nfluxes, we gain a transverse flux derivative term.  For example,\npredicting to the upper $x$ edge of zone $i,j$, we have:\n\\begin{align}\n\\Uc_{i+\\myhalf,j,L}^{n+\\myhalf} &= \\Uc_{i,j}^n + \\frac{\\Delta x}{2} \\frac{\\partial \\Uc}{\\partial x}\n                            + \\frac{\\Delta t}{2} \\frac{\\partial \\Uc}{\\partial t} + \\ldots \\\\\n&= \\Uc_{i,j}^n + \\frac{\\Delta x}{2} \\frac{\\partial \\Uc}{\\partial x}\n                            - \\frac{\\Delta t}{2} \\frac{\\partial \\Fb^{(x)}}{\\partial x}\n                            - \\frac{\\Delta t}{2} \\frac{\\partial \\Fb^{(y)}}{\\partial y} \\\\\n&= \\Uc_{i,j}^n + \\frac{1}{2} \\left [ 1 - \\frac{\\Delta t}{\\Delta x} \\Ab^{(x)}(\\Uc) \\right ] \\Delta \\Uc\n                            - \\frac{\\Delta t}{2} \\frac{\\partial \\Fb^{(y)}}{\\partial y} \\label{eq:Utaylorstate}\n\\end{align}\nwhere $\\Ab^{(x)}(\\Uc) \\equiv \\partial \\Fb^{(x)} / \\partial \\Uc$.  We decompose\nthis into a {\\em normal state} and a {\\em transverse flux difference}.\nAdopting the notation from Colella (1990), we use $\\hat{\\Uc}$ to denote\nthe normal state:\n\\begin{align}\n\\hat{\\Uc}_{i+\\myhalf,j,L}^{n+\\myhalf} &\\equiv \\Uc_{i,j}^n\n      + \\frac{1}{2} \\left [ 1 - \\frac{\\Delta t}{\\Delta x} \\Ab^{(x)}(\\Uc) \\right ] \\Delta \\Uc \\\\\n\\Uc_{i+\\myhalf,j,L}^{n+\\myhalf} &= \\hat{\\Uc}_{i+\\myhalf,j,L}^{n+\\myhalf}\n                            - \\frac{\\Delta t}{2} \\frac{\\partial \\Fb^{(y)}}{\\partial y}  \\label{eq:fullleftstate}\n\\end{align}\n\nThe primitive variable form for this system is\n\\begin{equation}\n\\qb_t + \\Ab^{(x)}(\\qb) \\qb_x + \\Ab^{(y)}(\\qb) \\qb_y = 0\n\\end{equation}\nwhere\n\\begin{equation}\n\\qb = \\left ( \\begin{array}{c} \\rho \\\\ u \\\\ v \\\\ p \\end{array} \\right )\n%\n\\qquad\n\\Ab^{(x)}(\\qb) = \\left ( \\begin{array}{cccc} u  & \\rho     & 0 &  0 \\\\\n                                         0  &  u       & 0 &  1/\\rho \\\\\n                                         0  &  0       & u &  0 \\\\\n                                         0  & \\gamma p & 0 &  u \\end{array} \\right )\n\\qquad\n\\Ab^{(y)}(\\qb) = \\left ( \\begin{array}{cccc} v  & 0 & \\rho &  0 \\\\\n                                         0  & v & 0    &  0 \\\\\n                                         0  & 0 & v    &  1/\\rho \\\\\n                                         0  & 0 & \\gamma p & v \\end{array} \\right )\n\\end{equation}\nThere are now 4 eigenvalues.  For $\\Ab^{(x)}(\\qb)$, they are $u-c$, $u$,\n$u$, $u+c$.  If we just look at the system for the $x$ evolution, we\nsee that the transverse velocity (in this case, $v$) just advects with\nvelocity $u$, corresponding to the additional eigenvalue.\n\\begin{exercise}[Eigenvectors for the 2-d Euler equations]\n{\nDerive the form of $\\Ab^{(x)}(\\qb)$ and $\\Ab^{(y)}(\\qb)$ and find their left and right eigenvectors.}\n\\end{exercise}\n\nWe note here that $\\hat{\\Uc}_{i+\\myhalf,j,L}^{n+\\myhalf}$ is essentially\none-dimensional, since only the $x$-fluxes are involved (through\n$\\Ab^{(x)}(\\Uc)$).  This means that we can compute this term using the\none-dimensional techniques developed in \\S~\\ref{sec:onedrecon}.  In\nparticular, Colella (1990) suggest that we switch to primitive variables\nand compute this as:\n\\begin{equation}\n\\hat{\\Uc}_{i+\\myhalf,j,L}^{n+\\myhalf} = \\Uc(\\hat{\\qb}_{i+\\myhalf,j,L}^{n+\\myhalf})\n\\end{equation}\nSimilarly, we consider the system projected along the $y$-direction to\ndefine the normal states on the $y$-edges, again using the one-dimensional\nreconstruction on the primitive variables from \\S~\\ref{sec:onedrecon}:\n\\begin{equation}\n\\hat{\\Uc}_{i,j+\\myhalf,L}^{n+\\myhalf} = \\Uc(\\hat{\\qb}_{i,j+\\myhalf,L}^{n+\\myhalf})\n\\end{equation}\n\nTo compute the full interface state (Eq.~\\ref{eq:fullleftstate}), we\nneed to include the transverse term.\nColella (1990) gives two different procedures for evaluating the\ntransverse fluxes.  The first is to simply use the cell-centered\n$\\Uc_{i,j}$ (Colella 1990,\nEq. 2.13); the second is to use the reconstructed normal states (the\n$\\hat{\\Uc}$'s) (Eq. 2.15).  In both cases, we need to solve a {\\em\n  transverse Riemann problem} to find the true state on the transverse\ninterface.  This latter approach is what we prefer.  In particular,\nfor computing the full $x$-interface left state, $\\Uc_{i+\\myhalf,j,L}^{n+\\myhalf}$, we need the\ntransverse ($y$) states, which we define as\n\\begin{align}\n\\Uc^T_{i,j+\\myhalf} &= \\mathcal{R}(\\hat{\\Uc}^{n+\\myhalf}_{i,j+\\myhalf,L},\n                            \\hat{\\Uc}^{n+\\myhalf}_{i,j+\\myhalf,R}) \\\\\n\\Uc^T_{i,j-\\myhalf} &= \\mathcal{R}(\\hat{\\Uc}^{n+\\myhalf}_{i,j-\\myhalf,L},\n                            \\hat{\\Uc}^{n+\\myhalf}_{i,j-\\myhalf,R})\n\\end{align}\n\nTaken together, the full interface state is now:\n\\begin{equation}\n\\Uc_{i+\\myhalf,j,L}^{n+\\myhalf} = \\Uc(\\hat{\\qb}_{i+\\myhalf,j,L}^{n+\\myhalf})\n   - \\frac{\\Delta t}{2} \\frac{\\Fb^{(y)}(\\Uc^T_{i,j+\\myhalf}) - \\Fb^{(y)}(\\Uc^T_{i,j-\\myhalf})}{\\Delta y}\n\\end{equation}\n\nThe right state at the $i+\\myhalf$ interface can be similarly computed (starting with the\ndata in zone $i+1,j$ and expanding to the left) as:\n\\begin{equation}\n\\Uc_{i+\\myhalf,j,R}^{n+\\myhalf} = \\Uc(\\hat{\\qb}_{i+\\myhalf,j,R}^{n+\\myhalf})\n   - \\frac{\\Delta t}{2} \\frac{\\Fb^{(y)}(\\Uc^T_{i+1,j+\\myhalf}) - \\Fb^{(y)}(\\Uc^T_{i+1,j-\\myhalf})}{\\Delta y}\n\\end{equation}\nNote the indices on the transverse states---they are now to the right of the interface (since\nwe are dealing with the right state).\n\nWe then find the $x$-interface state by solving the Riemann problem\nnormal to our interface:\n\\begin{equation}\n\\Uc_{i+\\myhalf,j}^{n+\\myhalf} = \\mathcal{R}(\\Uc_{i+\\myhalf,j,L}^{n+\\myhalf}, \\Uc_{i+\\myhalf,j,R}^{n+\\myhalf})\n\\end{equation}\nTherefore, construction of the interface states now requires two\nRiemann solves: a transverse and normal one.  The fluxes are then evaluated as:\n\\begin{equation}\n\\Fb^{(x),n+\\myhalf}_{i+\\myhalf,j} = \\Fb^{(x)}(\\Uc_{i+\\myhalf,j}^{n+\\myhalf})\n\\end{equation}\nNote, for multi-dimensional problems, in the Riemann solver, the transverse\nvelocities are simply selected based on the speed of the contact, giving\neither the left or right state.\n\nThe final conservative update is done as:\n\\begin{equation}\n\\Uc^{n+1}_{i,j} = \\Uc^n_{i,j}\n   + \\frac{\\Delta t}{\\Delta x} \\left ( \\Fb^{(x),n+\\myhalf}_{i-\\myhalf,j} - \\Fb^{(x),n+\\myhalf}_{i+\\myhalf,j} \\right )\n   + \\frac{\\Delta t}{\\Delta y} \\left ( \\Fb^{(y),n+\\myhalf}_{i,j-\\myhalf} - \\Fb^{(y),n+\\myhalf}_{i,j+\\myhalf} \\right )\n\\end{equation}\n\n\n\n\\subsection{3-d unsplit}\n\nThe extension of the unsplit methodology to 3-d is described by\nSaltzman~\\cite{saltzman:1994}.  The basic idea is the same as in 2-d,\nexcept now additional transverse Riemann solve are needed to fully\ncouple in the corners.\n\n\n\\section{Source terms}\n\\label{euler:sec:sourceterms}\n\nAdding source terms is straightforward.  For\na system described by\n\\begin{equation}\n\\Uc_t + [\\Fb^{(x)}(\\Uc)]_x + [\\Fb^{(y)}(\\Uc)]_y = \\Hb\n\\end{equation}\nwe predict to the edges in the same fashion as described above, but now\nwhen we replace $\\partial \\Uc/\\partial t$ with the divergence of the\nfluxes, we also pick up the source term.  This appears as:\n\\begin{align}\n\\Uc_{i+\\myhalf,j,L}^{n+\\myhalf} &= \\Uc_{i,j}^n\n            + \\frac{\\Delta x}{2} \\frac{\\partial \\Uc}{\\partial x}\n            + \\frac{\\Delta t}{2} \\frac{\\partial \\Uc}{\\partial t} + \\ldots \\\\\n&= \\Uc_{i,j}^n + \\frac{\\Delta x}{2} \\frac{\\partial \\Uc}{\\partial x}\n              - \\frac{\\Delta t}{2} \\frac{\\partial \\Fb^{(x)}}{\\partial x}\n              - \\frac{\\Delta t}{2} \\frac{\\partial \\Fb^{(y)}}{\\partial y}\n              + \\frac{\\Delta t}{2} \\Hb_{i,j} \\\\\n&= \\Uc_{i,j}^n\n + \\frac{1}{2} \\left [1 -\\frac{\\Delta t}{\\Delta x} \\Ab^{(x)}(\\Uc)\\right ] \\Delta \\Uc\n - \\frac{\\Delta t}{2} \\frac{\\partial \\Fb^{(y)}}{\\partial y}\n + \\frac{\\Delta t}{2} \\Hb_{i,j}\n  \\label{eq:Utaylorstatesource}\n\\end{align}\nWe can compute things as above, but simply add the source term to the\n$\\hat{\\Uc}$'s and carry it through.\n\nNote that the source here is cell-centered.  This expansion is\nsecond-order accurate.  This is the approach outlined in Miller\n\\& Colella \\cite{millercolella:2002}.  Also notice the \nsimilarity of this source term to a second-order Euler method\nfor integrating ODEs.\n\nAlternately, we can include the source terms in the characteristic\ntracing of the interface states.  This is the approach taken in, e.g.,\nthe original PPM paper.  To make things more concrete, let's consider just\ngravity as the source.  Our primitive variable equations in this case are:\n\\begin{equation}\n\\qb_t + \\Ab(\\qb) \\qb_x = \\Gb\n\\end{equation}\nwhere $\\Gb = (0, g, 0)^T$---i.e. the gravitational source only affects\n$u$, not $\\rho$ or $p$\\footnote{Note that in the PPM paper, they put $\\Gb$ on\nthe lefthand side of the primitive variable equation, so our signs are\nopposite.}.\n\nFirst we construct a parabolic profile of $g$ in each zone and\nintegrate under that profile to determine the average $g$ carried by\neach wave to the interface, we'll denote these as $\\Ic^\\enu_\\pm(g)$.  Then we include the gravitational source\nterm in the characteristic projection itself.  Our projections are\nnow:\n\\begin{equation}\n\\sum_{\\nu; \\lambda^\\enu \\ge 0}\\lb^\\enu \\cdot (\\tilde{\\qb} - \\Ic^\\enu_+(\\qb) - \\tfrac{\\Delta t}{2} \\Gb) \\rb^\\enu\n\\end{equation}\nfor the left state, and\n\\begin{equation}\n\\sum_{\\nu; \\lambda^\\enu \\le 0} \\lb^\\enu \\cdot (\\tilde{\\qb} - \\Ic^\\enu_-(\\qb) - \\tfrac{\\Delta t}{2} \\Gb) \\rb^\\enu\n\\end{equation}\nfor the right state.  Since $\\Gb$ is only non-zero for velocity, only\nthe velocity changes.  Writing out the sum (and performing the vector products), we\nget:\n\\begin{align}\nu_{i+\\myhalf,L}^{n+\\myhalf} =\n   \\tilde{u}_+\n  &- \\frac{1}{2} \\left [\n      \\left (\\tilde{u}_+ - \\Ic_+^\\evm(u) - \\frac{\\Delta t}{2} \\Ic^\\evm_+(g) \\right ) -\n       \\frac{\\tilde{p}_+ - \\Ic_+^\\evm(p)}{C} \\right ] \\nonumber \\\\\n  &- \\frac{1}{2} \\left [\n      \\left (\\tilde{u}_+ - \\Ic_+^\\evp(u) - \\frac{\\Delta t}{2} \\Ic^\\evp_+(g) \\right ) +\n       \\frac{\\tilde{p}_+ - \\Ic_+^\\evp(p)}{C} \\right ]\n\\end{align}\nwhere the only change from Eq.~\\ref{eq:ufull} are the\n$\\Ic^\\evm_+(g)$ and $\\Ic^\\evp_+(g)$ terms.\n\\footnote{These differ from the expression in the PPM paper, where $\\Delta t \\Gb$,\nnot $(\\Delta t/2) \\Gb$ is used in the projection, however this appears to\nbe a typo.  To see this, notice that if both waves are moving toward\nthe interface, then the source term that is added to the interface\nstate is $(\\Delta t/4) (\\Ic_+^\\evm(g) +\n\\Ic_+^\\evp(g))$ for the left state, which reduces to $(\\Delta\nt/2) g$ for constant g---this matches the result from the Taylor\nexpansion above (Eq.~\\ref{eq:Utaylorstatesource}).}\n\nRegardless of how the source term information is included in the\ninterface states, we also need to include it in the conservative\nupdate.  To second-order, we need it to be time-centered, which\nusually means averaging the time-level $n$ and $n+1$ sources:\n\\begin{align}\n\\Uc^{n+1}_{i,j} = \\Uc^n_{i,j}\n   &+ \\frac{\\Delta t}{\\Delta x} \\left ( \\Fb^{(x),n+\\myhalf}_{i-\\myhalf,j} - \\Fb^{(x),n+\\myhalf}_{i+\\myhalf,j} \\right ) \\nonumber \\\\\n   &+ \\frac{\\Delta t}{\\Delta y} \\left ( \\Fb^{(y),n+\\myhalf}_{i,j-\\myhalf} - \\Fb^{(y),n+\\myhalf}_{i,j+\\myhalf} \\right ) + \\frac{\\Delta t}{2} \\left (\\Hb(\\Uc^n_{i,j}) + \\Hb(\\Uc^{n+1}_{i,j})\\right )\n\\end{align}\nAs written, this appears to be an implicit update (since $\\Uc^{n+1}$\ndepends on $\\Hb^{n+1}$), but often, the form of the source terms allows\nyou to update the equations in sequence explicitly.  \n\nAgain, using a constant gravitational acceleration as an example, and\nlooking in 1-d for simplicity, $\\Uc = (\\rho, \\rho u, \\rho E)^\\intercal$\nand $\\Hb = (0, \\rho g, \\rho u g)^\\intercal$, so our update sequence is:\n\\begin{align}\n\\rho^{n+1}_i = \\rho^n_i &+ \\frac{\\Delta t}{\\Delta x} \n   \\left [\\rho_{i-\\myhalf}^{n+\\myhalf} u_{i-\\myhalf}^{n+\\myhalf} - \n          \\rho_{i+\\myhalf}^{n+\\myhalf} u_{i+\\myhalf}^{n+\\myhalf} \\right ] \\\\\n(\\rho u)^{n+1}_i = (\\rho u)^n_i &+ \\frac{\\Delta t}{\\Delta x} \n   \\left [ \\rho_{i-\\myhalf}^{n+\\myhalf} (u_{i-\\myhalf}^{n+\\myhalf})^2  - \n           \\rho_{i+\\myhalf}^{n+\\myhalf} (u_{i+\\myhalf}^{n+\\myhalf})^2  \\right ]\n         + \\frac{\\Delta t}{\\Delta x} \\left ( p_{i-\\myhalf}^{n+\\myhalf} - \n                                              p_{i+\\myhalf}^{n+\\myhalf} \\right ) \\nonumber \\\\\n       &+ \\frac{\\Delta t}{2}(\\rho^n_i + \\rho^{n+1}_i) g \\\\\n(\\rho E)^{n+1}_i = (\\rho E)^n_i &+ \\frac{\\Delta t}{\\Delta x} \n   \\left [ \\left (\\rho_{i-\\myhalf}^{n+\\myhalf} E_{i-\\myhalf}^{n+\\myhalf}  + p_{i-\\myhalf}^{n+\\myhalf} \\right ) u_{i-\\myhalf}^{n+\\myhalf} - \\right . \\nonumber \\\\\n   &\\phantom{+ \\frac{\\Delta t}{\\Delta x} \\left[ \\right.}   \\left .   \\left (\\rho_{i+\\myhalf}^{n+\\myhalf} E_{i+\\myhalf}^{n+\\myhalf}  + p_{i+\\myhalf}^{n+\\myhalf} \\right ) u_{i+\\myhalf}^{n+\\myhalf} \\right ]\n    + \\frac{\\Delta t}{2} \\left [ (\\rho u)^n + (\\rho u)^{n+1} \\right] g\n\\end{align}\nThese updates can be done one after another without any implicit\ncoupling.  Other sources, like the Coriolis force, will involve an\nimplicit update, but even then, it is local to a single zone and can\nbe solved analytically.\n\n%-----------------------------------------------------------------------------\n\\section{Simple geometries}\n\\label{ch:euler:geom}\n\nSo far we have been working only with Cartesian geometries, but it is\neasy to extend these methods to simple non-Cartesian geometies, like\nspherical and cylindrical.  These geometies allow us to capture 3-d\nvolume expansion effects in lower dimensions.\n\nFor a 1-d solver, a spherical geometry means that our coordinate is\nthe radius in the sphere, and as we move outward from the origin, the\nvolume of a zone (which is actually now a spherical shell) grows.\nThere are two places where we need to take the geometry into account:\nthe reconstruction of the interface states and the conservative update.\n\nOur 1-d system in spherical coordinates appears as:\n\\begin{equation}\n\\ddt{\\Uc} + \\frac{1}{r^2} \\ddr{r^2 \\Fb} = 0\n\\end{equation}\nThe geometry factors that appear are from the spherical form of the\ndivergence.  Expanding out the radial derivative, we have:\n\\begin{equation}\n\\ddt{\\Uc} + \\ddr{\\Fb} = -\\frac{2\\Fb}{r}\n\\end{equation}\nLikewise, the primitive variable equations now have source terms:\n\\begin{align}\n\\ddt{\\rho} + u \\ddx{\\rho} + \\rho \\ddx{u} &= -\\frac{2\\rho u}{r} \\\\\n\\ddt{u} + u \\ddx{u} + \\frac{1}{\\rho} \\ddx{p} &= 0 \\\\\n\\ddt{p} + u \\ddx{p} + \\Gamma_1 p \\ddx{u} &= -\\frac{2\\Gamma_1 p u}{r}\n\\end{align}\n\n\\begin{exercise}[Spherical form of primitive variable equations]\n{Derive the above system of 1-d spherical primitive variable equations\n  starting from the conservative equations.  Be sure to include the\n  geometry factors everywhere there is a divergence.}\n\\end{exercise}\n\nFor the prediction of the interface states, we now include these\ngeometric source terms as sources to the interface state, following\nthe same ideas as in \\S~\\ref{euler:sec:sourceterms}.\n\nThe conservative update now needs to include the geometry factors.  However,\nit is complicated by the fact that in the momentum equation, the pressure \nterm is a gradient, not a divergence, and therefore has different \ngeometic factors.  The update of the system appears as:\n\\begin{align}\n\\rho^{n+1}_i = \\rho^n_i &+ \\frac{\\Delta t}{V_i} \n   \\left [A_{i-\\myhalf} \\rho_{i-\\myhalf}^{n+\\myhalf} u_{i-\\myhalf}^{n+\\myhalf} - \n          A_{i+\\myhalf} \\rho_{i+\\myhalf}^{n+\\myhalf} u_{i+\\myhalf}^{n+\\myhalf} \\right ] \\\\\n(\\rho u)^{n+1}_i = (\\rho u)^n_i &+ \\frac{\\Delta t}{V_i} \n   \\left [ A_{i-\\myhalf} \\rho_{i-\\myhalf}^{n+\\myhalf} (u_{i-\\myhalf}^{n+\\myhalf})^2  - \n           A_{i+\\myhalf} \\rho_{i+\\myhalf}^{n+\\myhalf} (u_{i+\\myhalf}^{n+\\myhalf})^2  \\right ]\\nonumber \\\\\n        & + \\frac{\\Delta t}{\\Delta r} \\left ( p_{i-\\myhalf}^{n+\\myhalf} - \n                                              p_{i+\\myhalf}^{n+\\myhalf} \\right ) \\\\\n(\\rho E)^{n+1}_i = (\\rho E)^n_i &+ \\frac{\\Delta t}{V_i} \n   \\left [ A_{i-\\myhalf} \\left (\\rho_{i-\\myhalf}^{n+\\myhalf} E_{i-\\myhalf}^{n+\\myhalf}  + p_{i-\\myhalf}^{n+\\myhalf} \\right ) u_{i-\\myhalf}^{n+\\myhalf} - \\right . \\nonumber \\\\\n   &\\phantom{+ \\frac{\\Delta t}{V_i} \\left[ \\right.}   \\left .   A_{i+\\myhalf} \\left (\\rho_{i+\\myhalf}^{n+\\myhalf} E_{i+\\myhalf}^{n+\\myhalf}  + p_{i+\\myhalf}^{n+\\myhalf} \\right ) u_{i+\\myhalf}^{n+\\myhalf} \\right ]\n\\end{align}\nHere, the geometry factors are:\n\\begin{align}\nA_{i-\\myhalf} &= (r_{i-\\myhalf})^2\\\\\nV_i &= (r_i)^2 \\Delta r\n\\end{align}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.8\\linewidth]{axisymmetry}\n\\caption[The axisymmetric computational domain]\n{\\label{fig:axisymmetri} The axisymmetric computational domain.  Here we\n  imagine that the 2-d plane rotates through the cylindrical $\\theta$ angle to\n  create a volume.}\n\\end{figure}\n\nIt is also common to do 2-d axisymmetric models---here the $r$ and $z$\ncoordinates from a cylindrical geometry are modeled.  Again, this\nappears Cartesian, except there is a volume factor implicit in the\ndivergence that must be accounted for.  Our system in axisymmetric\ncoordinates\\footnote{Some sources will call this cylindrical\n  coordinates, but note that the grid is not a polar grid---it is\n  still Cartesian} is:\n\\begin{equation}\n\\frac{\\partial \\Uc}{\\partial t}\n  + \\frac{1}{r}\\frac{\\partial r \\Fb^{(r)}}{\\partial r}\n  + \\frac{\\partial \\Fb^{(z)}}{\\partial z} = 0\n\\end{equation}\nExpanding out the $r$ derivative, we can write this as:\n\\begin{equation}\n\\frac{\\partial \\Uc}{\\partial t}\n  + \\frac{\\partial \\Fb^{(r)}}{\\partial r}\n  + \\frac{\\partial \\Fb^{(z)}}{\\partial z} = -\\frac{\\Fb^{(r)}}{r}\n\\end{equation}\nAgain, the primitive variable version of this expanded form is used\nfor the interface state prediction.  The conservative update\nfollows the same idea as the 1-d spherical version.  The area\nand volume factors only differ from their Cartesian counterparts\nin the radial direction, and take the form:\n\\begin{align}\nA_{i-\\myhalf,j} &= r_{i-\\myhalf} \\Delta z \\\\\nA_{i,j-\\myhalf} &= r_i \\Delta z\\\\\nV_{i,j} &= r_i \\Delta r \\Delta z\n\\end{align}\nThese choices of geometric factors reproduce a discretized\nform of the cylindrical divergence:\n\\begin{equation}\n\\nabla \\cdot {\\boldsymbol{\\phi}} = \\frac{1}{r} \\ddr{(r\\phi^{(r)})} + \\ddz{\\phi^{(z)}}\n\\end{equation}\nJust as with the 1-d spherical case, the pressure term in the momentum\nequation needs to be treated separately from the flux, since it enters\nas a gradient and not a divergence.\\footnote{It is common to see the divergence\nterm expressed as \n\\begin{equation}\n\\nabla \\cdot {\\boldsymbol{\\phi}} = \\frac{1}{r^\\alpha} \\ddr{(r^\\alpha \\phi^{(r)})} + \\ldots\n\\end{equation}\nwhere $\\alpha = 1$ for axisymmetric and $\\alpha = 2$ for 1-d\nspherical.  This allows for the geometry correction to be expressed\nmore concisely, as the source term takes the form $-\\alpha F/r$.  }\n\n\n%-----------------------------------------------------------------------------\n\\section{Some Test problems}\n\nThere are a large number of standard test problems that are used to\ntest out our methods.  Like we say with advection, it is best to have\na problem with an analytic solution.  Here we show just a few of the most\npopular test problems.\n\n\\subsection{Shock tubes}\n\\label{sec:euler-methods:shocktubes}\n\nShock tubes are Riemann problems---consider the evolution of an\ninitial discontinuity in the domain.  The evolution with time will\njust be the solution to the Riemann problem that we described\n\\S~\\ref{euler:sec:riemann}.  The initial conditions can be varied to\nproduce any combination of shocks and rarefactions as the left and\nright waves.  A popular initial condition is the Sod\nproblem~\\cite{sod:1978} which results is a right moving shock and\ncontact and a left moving rarefaction.\n\nNote: it is a good test of your code's ability to preserve symmetry to\nflip the initial conditions left/right and rerun.  The results should\nbe the same to machine precision, but atleast to roundoff error.  A\ncommon reason for breaking symmetry is using inequalities in your code\nthat are biases in a direction, e.g.,\n\\begin{lstlisting}\nif u > 0:\n   # positive velocity test case\nelse:\n   # negative or zero velocity test case\n\\end{lstlisting}\nThis has a left-right bias, since we don't handle the case where $u =\n0$ separately.  A better construction would test on $u < 0$ alone, and\nthen have a final {\\em else} clause to catch $u = 0$.\n\nSince these tests start out with a discontinuity, they are not the\nbest tests to use for convergence testing.  Wherever there is an\ninitiali discontinuity, the limiters will kick in and drop your\nmethod to first-order accurate.\n\nFor a general equation of state, you can still solve the Riemann\nproblem exactly and define analogous test problems to those commonly\nused for a gamma-law gas.  Some shock tube test problems for a stellar\nequation of state are shown in \\cite{zingalekatz}.\n\nFor the tests shown below, we use the \\hydrooned\\ code described in\nAppendix~\\ref{app:hydro1d}.\n\n\\subsubsection{Sod problem}\n\nThe initial conditions for the Sod problem \\cite{sod:1978} are:\n\\begin{align}\n\\rho_l &= 1      &  \\rho_r &= 1/8 \\nonumber \\\\\nu_l   &= 0       &  u_r    &= 0   \\\\\np_l    &= 1      &  p_r    &= 1/10 \\nonumber\n\\end{align}\nusually with $\\gamma = 1.4$\\MarginPar{double check}\n\nThese result in a left wave moving contact and rightward moving\ncontact and shock.  The shock is not particularly strong, but this\nproblem is a nice demonstration of the types of hydrodynamic waves.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.7\\linewidth]{hydro1d_god_sod}\n\\caption[Piecewise constant reconstruction Sod problem]{\\label{fig:Euler:sod:god} Piecewise constant reconstruction\n  with the Sod problem, using 128 zones, $\\cfl = 0.8$, and the CGF\n  Riemann solver.  This was run with \\hydrooned\\ using the {\\tt sod}\n  problem setup, setting {\\tt godunov\\_type=0} and visualized with the\n  {\\tt sod\\_compare.py} script there.}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.7\\linewidth]{hydro1d_ppm_sod}\n\\caption[Piecewise parabolic reconstruction Sod problem]{\\label{fig:Euler:sod:ppm} Piecewise parabolic reconstruction\n  with the Sod problem, using 128 zones, $\\cfl = 0.8$, and the CGF\n  Riemann solver.  This was run with \\hydrooned\\ using the {\\tt sod}\n  problem setup, setting {\\tt godunov\\_type=2} and visualized with the\n  {\\tt sod\\_compare.py} script there.}\n\\end{figure}\n\n\n\\subsubsection{Double rarefaction}\n\nThe double rarefaction problem starts with initially diverging flow\nthat creates a vacuum state in the middle.  The initial conditions (as\ngiven in \\cite{toro:1997} are\n\\begin{align}\n\\rho_l &= 1      &  \\rho_r &= 1 \\nonumber \\\\\nu_l    &= -2.0   &  u_r    &= 2.0   \\\\\np_l    &= 0.4    &  p_r    &= 0.4 \\nonumber\n\\end{align}\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.7\\linewidth]{hydro1d_god_double_rare}\n\\caption[Piecewise constant reconstruction double rarefaction problem]{\\label{fig:Euler:doublerare:god} Piecewise constant\n  reconstruction with the double rarefaction problem, using 128 zones,\n  $\\cfl = 0.8$, and the CGF Riemann solver.  This was run with\n  \\hydrooned\\ using the {\\tt sod} problem setup, setting {\\tt\n    godunov\\_type=0} and visualized with the {\\tt sod\\_compare.py}\n  script there.}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.7\\linewidth]{hydro1d_ppm_double_rare}\n\\caption[Piecewise constant reconstruction double rarefaction problem]{\\label{fig:Euler:doublerare:ppm} Piecewise parabolic\n  reconstruction with the double rarefaction problem, using 128 zones,\n  $\\cfl = 0.8$, and the CGF Riemann solver.  This was run with\n  \\hydrooned\\ using the {\\tt sod} problem setup, setting {\\tt\n    godunov\\_type=2} and visualized with the {\\tt sod\\_compare.py}\n  script there.}\n\\end{figure}\n\n\n\\subsection{Sedov blast wave}\n\nThe Sedov (or Sedov-Taylor) blast wave~\\cite{sedov:1959} is a point\nexplosion---energy, $\\mathcal{E}_\\mathrm{expl}$, is placed at a point\nin a uniform domain.  A spherical shockwave propagates outward,\nevacuating the region in the center.  The Sedov problem also has an\nanalytic solution\\footnote{See \\cite{timmes_sedov_code} for a nice\n  code to general the solutions}, and this problem is a good way of\ntesting out the geometric factors for 1-d spherical and 2-d\naxisymmetric geometries.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.7\\linewidth]{hydro1d_sedov_1dsph}\n\\caption[1-d spherical Sedov problem]{\\label{fig:Euler:sedov1d} 1-d Sedov explosion with\n  piecewise parabolic reconstruction, using 128 zones (on $r \\in\n  [0,1]$), $\\cfl = 0.8$, and the CGF Riemann solver.  This was run\n  with \\hydrooned\\ using the {\\tt sedov} problem setup and visualized\n  with the {\\tt sedov\\_compare.py} script there.  The 1-d spherical\n  geometry used makes this act as if it were a sphere.}\n\\end{figure}\n\nThe major difficulty with initializing the Sedov problem is\nrepresenting a point, where all the energy desposited, on the grid.\nIf you just initialize a singe zone, then the size of the point\nchanges as you change resolution.  Additionally, in 2- or 3-d\nCartesian coordinates, the point will be squared off.  A standard way\nof initializing this (see, e.g., \\cite{omang:2006}) is to imagine the\nenergy deposition inflating a region of radius $r_\\mathrm{init}$ like\na balloon, resulting in an energy density, $E =\n\\mathcal{E}_\\mathrm{expl}/V_\\mathrm{init}$, where $V_\\mathrm{init} =\n4\\pi r_\\mathrm{init}^3/3$ for a spherical blast wave and\n$V_\\mathrm{init} = \\pi r_\\mathrm{init}^2$ for a cylindrical blast\nwave. Then the pressure is\n\\begin{equation}\np = \\frac{\\mathcal{E}_\\mathrm{expl}}{V_\\mathrm{init}} (\\gamma - 1)\n\\end{equation}\n\n\nFigure~\\ref{fig:Euler:sedov1d} shows the solution to the Sedov problem\nin 1-d in a spherical geometry.  This is compared to the spherical\nSedov solution (solid line). There is good agreement in the position of the\nshock and density and velocity profiles.  The pressure behind the shock\nis a little low---this is likely an artifact of the initialization process.\n\nIn 2-d, we run the problem in Cartesian coordinates---this produces a cylindrical\nblast wave.  Figure~\\ref{fig:Euler:sedov2d} shows the solution in 2-d.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.85\\linewidth]{sedov_pyro}\n\\caption[2-d cylindrical Sedov problem]{\\label{fig:Euler:sedov2d} 2-d Sedov explosion with\n  piecewise linear reconstruction, using 128$^2$ zones (on $r \\in\n  [0,1]\\times[0,1]$), $\\cfl = 0.8$, and the HLLC Riemann solver.  This was run\n  with \\pyro\\ as {\\tt ./pyro.py compressible sedov inputs.sedov}.  An initial\n  perturbation size of $r_\\mathrm{init} = 0.01$ was used.}\n\\end{figure}\nThe density and pressure look very symmetric.  In the velocity magnitude\nplot, we see an imprint of the grid along coordinate axes, and likewise\nin the internal energy.  The bump in internal energy near the origin arises\nbecause of the error in defining $e$ from $E$ and $U$.  We can produce an angle-averaged\nprofile of this and compare to the analytic solution, shown in Figure~\\ref{fig:Euler:sedov2d_compare}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.75\\linewidth]{sedov_compare}\n\\caption[2-d cylindrical Sedov problem]{\\label{fig:Euler:sedov2d_compare} Angle-average \n  profile for the 2-d Sedov explosion from Figure~\\ref{fig:Euler:sedov2d} shown\n  with the analytic solution.  This was constructed using the {\\tt sedov\\_compare.py}\n  script in \\pyro.}\n\\end{figure}\n\n\n\\subsection{Advection}\n\nWe can run a simple advection test analogous to the tests we used in\nCh.~\\ref{ch:advection}.  However, because we are now doing hydrodynamics,\nwe need to suppress the dynamics.  This is accomplished by putting the \nprofile we want to advect in the density field and then put it in\npressure equilibrium by adjusting the internal energy.  For example, consider\na Gaussian profile.  We initialize the density as:\n\\begin{equation}\n\\rho = (\\rho_1 - \\rho_0) e^{-(x - x_c)^2/\\sigma^2} + \\rho_0\n\\end{equation}\nTo advect the profile to the right, we choose a constant velocity,\n\\begin{equation}\nu = u_0\n\\end{equation}\nand to suppress dynamics, we make the pressure constant,\n\\begin{equation}\np = p_0\n\\end{equation}\nFinally, we compute the total energy as\n\\begin{equation}\n\\rho E = \\frac{p}{\\gamma - 1} + \\frac{1}{2}\\rho u^2\n\\end{equation}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.7\\linewidth]{advect_hydro1d}\n\\caption[Simple advection test]{\\label{fig:Euler:advect:1d} Piecewise\n  parabolic reconstruction with an advection test, using 128 zones,\n  $\\cfl = 0.8$, and the CGF Riemann solver.  A Gaussian profile was\n  advected for 10 periods.  This was run with \\hydrooned\\ using the\n  {\\tt advect} problem setup and\n  visualized with the {\\tt advect\\_compare.py} script there.}\n\\end{figure}\n\nFigure~\\ref{fig:Euler:advect:1d} shows an example of a Gaussian\nprofile advected for 10 periods.  The initial conditions used $\\rho_0\n= 10^{-3}$, $\\rho_1 = 1$, $p_0 = 10^{-6}$, $u_0 = 1$, and $\\sigma =\n0.1$ (these coincide with the choices used in \\cite{flash}).  The\nnon-zero value for the ambient density, $\\rho_0$, ensures that any\nquantities that are derived by dividing by density remain\nwell-defined.  The result is similar to what we saw when we considered\npure advection---the shape is mostly preserved, but the peak of the\nGaussian is clipped.\n\n\\subsection{Slow moving shock}\n\nSlow moving (or stationary) shocks can be difficult to model, as\noscillations can be setup behind the shock (this is discussed a little\nin \\cite{colellawoodward:1984,leveque:2002}).  We can produce a slow\nmoving shock as a shock tube, and we can use the jump conditions \nacross a shock that were derived for the Riemann problem to find the\nconditions to setup a stationary (or slow-moving) shock.  \n\nThe speed of a right-moving shock was found (see\nEq.~\\ref{eq:euler:shockspeedjump}) as:\n\\begin{equation}\nS = u_r + c_r \\left [ \\left ( \\frac{p_\\star}{p_r} \\right ) \\frac{\\gamma+1}{2\\gamma} + \\frac{\\gamma-1}{2\\gamma} \\right ]^{\\myhalf}\n\\end{equation}\nWe want $S = 0$, which allows us to express the pre-shock velocity, $u_r$, as:\n\\begin{equation}\nu_r = -c_r \\left [ \\left ( \\frac{p_\\star}{p_r} \\right ) \\frac{\\gamma+1}{2\\gamma} + \\frac{\\gamma-1}{2\\gamma} \\right ]^{\\myhalf}\n\\end{equation}\nWe have the freedom to choose the density and pressure ahead of the\nshock, $\\rho_r$ and $p_r$, which in turn gives us $c_r$.  Next, we can\npick the strength of the shock by choosing the jump in pressure,\n$p_\\star/p_r$.  Together, this allows us to compute $u_r$, and thus we\nknow the entire pre-shock state.  We can compute the post-shock state\n(which was the star state when we discussed the Riemann problem) using\nthe jump conditions, Eqs.~\\ref{eq:euler:shockrhojump} and\n\\ref{eq:euler:shockujump}.\\footnote{The script\n  \\href{https://github.com/zingale/hydro_examples/blob/master/compressible/slow_shock.py}{\\tt\n    slow\\_shock.py} will find the initial conditions to generate a\n  stationary shock.}\n\nFor a pressure jump of 100 across a shock, the following conditions will\ngenerate a stationary right-facing shock (with $\\gamma = 1.4$):\n\\begin{align}\n\\rho_l &= 5.6698      &  \\rho_r &= 1 \\nonumber \\\\\nu_l   &= -1.9336      &  u_r    &= -10.9636   \\\\\np_l    &= 100         &  p_r    &= 1 \\nonumber\n\\end{align}\n\nBy adjusting the velocity of both the left and right state, we can\nproduce a strong shock that moves slowly across the grid.  For\ninstance, a shock with $S = 0.4$ results from\n\\begin{align}\n\\rho_l &= 5.6698      &  \\rho_r &= 1 \\nonumber \\\\\nu_l   &= -1.5336      &  u_r    &= -10.5636   \\\\\np_l    &= 100         &  p_r    &= 1 \\nonumber\n\\end{align}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.7\\linewidth]{slowshock}\n\\caption[1-d spherical Sedov problem]{\\label{fig:Euler:slowshock:ppm} 1-d slow moving shock\n  problem with\n  piecewise parabolic reconstruction, using 128 zones (on $r \\in\n  [0,1]$), $\\cfl = 0.8$, and the CGF Riemann solver.  This was run\n  with \\hydrooned\\ using the {\\tt sod} problem setup and visualized\n  with the {\\tt sod\\_compare.py} script there.}\n\\end{figure}\n\n\n\\subsection{Two-dimensional Riemann problems}\n\nSeveral different 2-d Riemann problems were introduced in\n\\cite{hydro_test_quad}, to explore the multi-dimensional\ninteractive of the different hydrodynamic waves.  These\nproblems initialize the 4 quadrants of the domain with different\nstates, and watch the ensuing evolution.  There are some\nanalytic estimates that can be compared to, but also these\ntests can provide a means of assessing the symmetry of \na code in the presence of complex flows.  We use the setup\ncorresponding to {\\em configuration 3} in that paper (this\nsame setup is used in \\cite{leveque:1997}).\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.85\\linewidth]{quad}\n\\caption{\\label{fig:Euler:quad} Two-dimensional Riemann problem\nfrom \\cite{hydro_test_quad}.}\n\\end{figure}\n\n\\ifdefined \\debugmode\n\\subsection{Odd-even decoupling}\n\n\\subsection{Gresho vortex}\n\nThe Gresho vortex is a vortex in with a stabilizing pressure gradient\ndesigned such that the overall structure is unchanging in time.  It\nhas a nice feature in that it allows you to set the Mach number of\nthe flow as one of the parameters.\n\n\n\\subsection{Hydrostatic equilibrium}\n\nHydrodynamics codes can have a difficult time keeping an atmosphere in\nhydrostatic equilibrium.  This is because any departure from $\\nabla p\n= \\rho {\\bf g}$ due to truncation error in the method will be seen as an\nacceleration by the momentum equation:\n\\begin{equation}\n\\rho \\DDt{\\Ub} = \\rho {\\bf g} - \\nabla p\n\\end{equation}\nThe result will be growing velocities in the atmosphere, which can\noverwhelm any signal in the flow that you are interested in studying.\nSince this acceleration is driven by truncation error, higher\nresolution will result in lower velocities.\n\n% top of the star\n\n% scale height resolved -- do uniform T atmosphere\n\n\n%\\subsection{Rayleigh-Taylor instability}\n\n\\fi\n\n\n%-----------------------------------------------------------------------------\n\\section{Method of lines integration and higher order}\n\\label{sec:comp:mol}\nJust like we explored with linear advection (\\S~\\ref{adv:sec:mol_2d}),\ninstead of doing the characteristic tracing, we could rely on the\nintegrator to do the work for us.\n\n\nDiscretizing our system in space leads to the following system:\n\\begin{equation}\n\\frac{d\\Uc_{i,j}}{dt} = -\\frac{\\Fb^{(x)}(\\Uc_{i+\\myhalf,j}) - \\Fb^{(x)}(\\Uc_{i-\\myhalf,j})}{\\Delta x}\n                      -\\frac{\\Fb^{(y)}(\\Uc_{i,j+\\myhalf}) - \\Fb^{(y)}(\\Uc_{i,j-\\myhalf})}{\\Delta y}\n\\end{equation}\nNote that there is no time superscript in the $\\Uc$ used to evaluate the\nfluxes on the righthand side---we have not done any time\ndiscretization yet.  Now we can use an ODE integrator to solve this system.\n\nConsider\nsecond-order Runge-Kutta.  We evaluate two slopes,\n\\begin{alignat}{3}\n{\\bf k}_1 &= \\Delta t \\bigg [ &-&\\frac{\\Fb^{(x)}(\\Uc^n_{i+\\myhalf,j})\n                               - \\Fb^{(x)}(\\Uc^n_{i-\\myhalf,j})}{\\Delta x} \\nonumber \\\\\n    &\\,                 &-&\\frac{\\Fb^{(y)}(\\Uc^n_{i,j+\\myhalf})\n                               - \\Fb^{(y)}(\\Uc^n_{i,j-\\myhalf})}{\\Delta y} \\bigg ] \\\\\n{\\bf k}_2 &= \\Delta t \\bigg [ &-&\\frac{\\Fb^{(x)}([\\Uc^n + {\\bf k}_\\myhalf]_{i+\\myhalf,j})\n                               - \\Fb^{(x)}([\\Uc^n + {\\bf k}_\\myhalf]_{i-\\myhalf,j})}{\\Delta x} \\nonumber \\\\\n    &\\,                 &-&\\frac{\\Fb^{(y)}([\\Uc^n + {\\bf k}_\\myhalf]_{i,j+\\myhalf})\n                               - \\Fb^{(y)}([\\Uc^n + {\\bf k}_\\myhalf]_{i,j-\\myhalf})}{\\Delta y} \\bigg ]\n\\end{alignat}\nand then\n\\begin{equation}\n\\Uc^{n+1}_{i,j} = \\Uc^n_{i,j} + {\\bf k}_2\n\\end{equation}\nIn the construction of the interface states, $\\Uc^n_{i+\\myhalf,j}$ or $[\\Uc^n\n+ {\\bf k}_\\myhalf]_{i+\\myhalf,j}$, there is no explicit transverse term, since that\narose from Taylor expanding $\\Uc^n_{i,j}$ in time through $\\Delta t/2$.\nInstead, we simply construct these interface states using a\none-dimensional reconstruction and solve a Riemann problem at each\ninterface.  The evaluation of the second slope, ${\\bf k}_2$, implicitly\nincludes the transverse information since we add ${\\bf k}_\\myhalf$ to\n$\\Uc^n_{i,j}$ before doing the prediction to the interfaces.  Also note,\nhowever, that in this construction of the interface states, there is\nno characteristic projection, since that arises from predicting the\ninterface states forward in time.  Again, these interface states are\nat a constant time, not predicted into the future.\n\nGenerally speaking we want the order of accuracy in time to match that\nin space.  The fourth-order Runge-Kutta method is a popular method for\nintegrating ODEs, so it makes sense to couple this with a\nfourth-order-in-space method.  However, going higher-order than\nsecond-order is more challenging.  The key issue is that we can no\nlonger simply approximate the cell average as the cell-center value,\ni.e., $\\langle \\phi\\rangle_i \\ne \\phi_i$.  This comes into play, for\ninstance, in translating between the conserved and primitive variables.\nA fully fourth-order method is presented in ~\\cite{mccorquodalecolella}\n\nNote that when using a Runge-Kutta method-of-lines integrator for the\ntime-discretization of a multidimensional system, the timestep is\nactually more restrictive than the cases presented above that\npredicted the interface states to the half-time and performed\ncharacteristic tracing.  Titarev \\& Toro \\cite{titarevtoro} claim that\nyou need $0 < C < \\myhalf$ for 2-d flows and $0 < C < 1/3$ for 3-d flows.\n\nAn additional complexity arises when doing multiphysics.  Often we\nsplit the different physical processes up and treat them in turn.\nThere are standard methods to do this with second-order accuracy in time,\nbut higher-order is more tricky.\n\n\n\n\n\n\\section{Thermodynamic issues}\n\n\\subsection{Defining temperature}\n\n  Although not needed for the pure\n  Euler equations, it is sometimes desirable to define the temperature\n  for source terms (like reactions) or complex equations of state.\n  The temperature can typically be found from the equation of state\n  given the internal energy:\n  \\begin{eqnarray}\n  e &=& E - \\frac{1}{2} u^2 \\\\\n  T &=& T(e, \\rho)\n  \\end{eqnarray}\n  Trouble can arise when you are in a region of flow where the kinetic\n  energy dominates (high Mach number flow).  In this case, the $e$ defined\n  via subtraction can become negative due to truncation error in the\n  evolution of $u$ compared to $E$.  In this instance, one must either\n  impose a floor value for $e$ or find an alternate method of deriving\n  it.\n\n  In~\\cite{bryan:1995}, an alternate formulation of the Euler equations\n  is proposed.  Both the total energy equation {\\em and} the internal\n  energy equation are evolved in each zone.  When the flow is dominated\n  by kinetic energy, then the internal energy from the internal energy\n  evolution equation is used.  The cost of this is conservation---the\n  internal energy is not a conserved quantity, and switching to it\n  introduces conservation of energy errors.\n\n\\subsection{General equation of state}\n\nThe above methods were formulated with a constant gamma equation of\nstate.  A general equation of state (such as degenerate electrons)\nrequires a more complex method.  Most methods are designed to \nreduce the need to call a complex equation of state frequently,\nand work by augmenting the vector of primitive variables with \nadditional thermodynamic information.  There are two parts of the \nadaption to a general equation of state: the interface states and \nthe Riemann problem. \n\n\\subsubsection{Carrying $\\gamma_e$}\n\n\nThe classic prescription for\nextending this methodology is presented by Colella and\nGlaz~\\cite{colellaglaz:1985}.  They construct a thermodynamic index,\n\\begin{equation}\n\\gamma_e = \\frac{p}{\\rho e} + 1\n\\end{equation}\nand derive an evolution equation for $\\gamma_e$ (C\\&G, Eq.\\ 26).\nWe can derive a similar expression as\n\\begin{align}\n\\frac{D\\gamma_e}{Dt} &= \\frac{D}{Dt} \\left ( \\frac{p}{\\rho e} + 1 \\right )\n   = - \\frac{p}{(\\rho e)^2} \\frac{D(\\rho e)}{Dt} + \\frac{1}{\\rho e} \\frac{Dp}{Dt} \\nonumber \\\\\n   &= (\\gamma_e - 1) (\\gamma_e  - \\Gamma_1) \\nabla \\cdot \\Ub \\label{eq:gammae}\n\\end{align}\nwhere we used Eqs.~\\ref{eq:euler:pgeneral} and \\ref{eq:euler:econs}, and the definition\nof the sound speed.\n\nThis evolution equation is used to predict $\\gamma_e$ to interfaces,\nand these interface values of $\\gamma_e$ are used in the Riemann\nsolver presented there to find the fluxes through the interface.  A\ndifferent adiabatic index (they call $\\Gamma$, we call $\\Gamma_1$)\nappears in the definition of the sound speed.  They argue that this\ncan be brought to interfaces in a piecewise constant fashion while\nstill making the overall method second order, since $\\Gamma_1$ does\nnot explicitly appear in the fluxes (see the discussion at the top of\npage 277).\n\nWe can derive the characteristic structure of this system for use\nin the tracing in the construction of interface states%\n\\footnote{A {\\sf Jupyter} notebook using {\\sf SymPy} that derives these\neigenvectors is available here:\n\\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/compressible/euler-generaleos.ipynb}{euler-generaleos.ipynb}}}.\nIf we write \nour system as\n\\begin{equation}\n\\qb = \\left ( \\begin{array}{c} \\tau \\\\ u \\\\ p \\\\ \\gamma_e \\end{array} \\right )\n\\end{equation}\n(we use $\\tau = 1/\\rho$ here for consistency with CG), we have\n\\begin{equation}\n\\Ab = \\left ( \\begin{array}{cccc} u & -\\tau    & 0      & 0 \\\\\n                                0 & u        & \\tau   & 0 \\\\\n                                0 & c^2/\\tau & u      & 0 \\\\\n                                0 & -\\alpha  & 0      & u\n            \\end{array} \\right )\n\\end{equation}\nwhere we define $\\alpha = (\\gamma_e - 1)(\\gamma_e - \\Gamma_1)$ for\nconvenience.  The right eigenvectors are:\n\\begin{equation}\n\\rb^\\evm = \\left( \\begin{array}{c} 1\\\\ {c}/{\\tau}\\\\ -{c^{2}}/{\\tau^{2}}\\\\ {\\alpha}/{\\tau}\\end{array}\\right)\n%\n\\qquad\n\\rb^\\evz = \\left( \\begin{array}{c} 1\\\\0\\\\0\\\\0 \\end{array}\\right)\n%\n\\qquad\n\\rb^\\evzs{\\gamma_e} = \\left( \\begin{array}{c} 0\\\\0\\\\0\\\\1\\end{array}\\right)\n%\n\\qquad\n\\rb^\\evp = \\left( \\begin{array}{c} 1\\\\ -{c}/{\\tau}\\\\ -{c^{2}}/{\\tau^{2}}\\\\ {\\alpha}/{\\tau} \\end{array}\\right )\n\\end{equation}\nand corresponding left eigenvectors are:\n\\begin{align}\n\\lb^\\evm &= \\left( \\begin{array}{cccc} 0 & \\frac{\\tau}{2 c} & - \\frac{\\tau^{2}}{2 c^{2}} & 0\\end{array}\\right) \\\\\n%\n\\lb^\\evz &= \\left( \\begin{array}{cccc} 0 & \\frac{\\tau}{2 c} & - \\frac{\\tau^{2}}{2 c^{2}} & 0\\end{array}\\right) \\\\\n%\n\\lb^\\evzs{\\gamma_e} &= \\left( \\begin{array}{cccc} 0 & 0 & \\frac{\\alpha \\tau}{c^{2}} & 1\\end{array}\\right) \\\\\n%\n\\lb^\\evp &= \\left ( \\begin{array}{cccc} 0 & - \\frac{\\tau}{2 c} & - \\frac{\\tau^{2}}{2 c^{2}} & 0\\end{array}\\right)\n\\end{align}\n\n\\subsubsection{Carrying $(\\rho e)$}\n\nAlternately, the Castro paper~\\cite{almgren:2010} relies on an idea\nfrom an unpublished manuscript by Colella, Glaz, and Ferguson that\npredicts $\\rho e$ to edges in addition to $\\rho$, $u$, and $p$.  Since\n$\\rho e$ comes from a conservation-like equation\n(Eq.~\\ref{eq:euler:econs}), predicting it to the interface in the\nunsplit formulation is straightforward.  This over-specifies the\nthermodynamics, but eliminates the need for $\\gamma_e$.\n\n\nWith the addition of $\\rho e$, our system becomes:\n\\begin{equation}\n\\qb = \\left ( \\begin{array}{c} \\rho \\\\ u \\\\ p \\\\ \\rho e \\end{array} \\right )\n%\n\\qquad\n%\n\\Ab = \\left ( \\begin{array}{cccc} u & \\rho     & 0      & 0 \\\\\n                                0 & u        & 1/\\rho & 0 \\\\\n                                0 & \\rho c^2 & u      & 0 \\\\\n                                0 & \\rho h   & 0      & u\n            \\end{array} \\right )\n\\end{equation}\nwhere $h = e + p/\\rho$ is the specific enthalpy.  The eigenvalues of this\nsystem are:\n\\begin{equation}\n\\lambda^\\evm = u - c \\qquad\n\\lambda^\\evz = u  \\qquad\n\\lambda^\\evzs{\\rho e} = u  \\qquad\n\\lambda^\\evp = u + c\n\\end{equation}\nand the eigenvectors are:\n\\begin{equation}\n\\rb^\\evm = \\left ( \\begin{array}{c} 1 \\\\ -c/\\rho \\\\ c^2 \\\\ h \\end{array} \\right )\n\\qquad\n\\rb^\\evz = \\left ( \\begin{array}{c} 1 \\\\ 0 \\\\ 0 \\\\ 0 \\end{array} \\right )\n\\qquad\n\\rb^\\evzs{\\rho e} = \\left ( \\begin{array}{c} 0 \\\\ 0 \\\\ 0 \\\\ 1 \\end{array} \\right )\n\\qquad\n\\rb^\\evp = \\left ( \\begin{array}{c} 1 \\\\ c/\\rho \\\\ c^2 \\\\ h \\end{array} \\right )\n\\end{equation}\nand\n\\begin{eqnarray}\n&&\n\\lb^\\evm = ( \\begin{array}{cccc} 0 & -\\frac{\\rho}{2c} & \\frac{1}{2c^2} & 0\n            \\end{array} ) \\nonumber \\\\\n&&\n\\lb^\\evz = ( \\begin{array}{cccc} 1 & 0 & -\\frac{1}{c^2} & 0\n            \\end{array} ) \\nonumber \\\\\n&&\n\\lb^\\evzs{\\rho e} = ( \\begin{array}{cccc} 0 & 0 & -\\frac{h}{c^2} & 1\n            \\end{array} ) \\nonumber \\\\\n&&\n\\lb^\\evp = ( \\begin{array}{cccc} 0 & \\frac{\\rho}{2c} & \\frac{1}{2c^2} & 0\n            \\end{array} )\n\\end{eqnarray}\nRemember that the state variables in the $\\qb$ vector are mixed into the\nother states by $\\lb \\cdot \\qb$.  Since all $\\lb^{(\\nu)}$'s have $0$ in the\n$\\rho e$ `slot' (the last position) except for $\\lb^\\evzs{\\rho e}$, and the\ncorresponding $\\rb^\\evzs{\\rho e}$ is only non-zero in the $\\rho e$ slot, this\nmeans that $\\rho e$ is not mixed into the other state variables.  This\nis as expected, since $\\rho e$ is not needed in the system.\n\nAlso recall that the jump carried by the wave $\\nu$ is proportional\nto $\\rb^{(\\nu)}$---since $\\rb^\\evm$, $\\rb^\\evzs{\\rho e}$, and $\\rb^\\evp$ have\nnon-zero $\\rho e$ elements, this means that $\\rho e$ jumps across\nthese three waves.\n\nWorking through the sum for the $(\\rho e)$ state, and using a $\\sim$ to\ndenote the reference states, we arrive at:\n\\begin{align}\n(\\rho e)_{i+\\myhalf,L}^{n+\\myhalf} = \\widetilde{(\\rho e)} &-\n   \\frac{1}{2} \\left [ -\\frac{\\rho}{c} \\left (\\tilde{u} - \\Ic^{(1)}_+(u) \\right )\n                       +\\frac{1}{c^2} \\left (\\tilde{p} - \\Ic^{(1)}_+(p) \\right )\n               \\right ] h \\nonumber \\\\\n  &-\n    \\left [ -\\frac{h}{c^2} \\left (\\tilde{p} - \\Ic^{(3)}_+(p) \\right )\n                       + \\left (\\widetilde{(\\rho e)} - \\Ic^{(3)}_+(\\rho e) \\right )\n               \\right ] \\nonumber \\\\\n  &-\n   \\frac{1}{2} \\left [ \\frac{\\rho}{c} \\left (\\tilde{u} - \\Ic^{(4)}_+(u) \\right )\n                       +\\frac{1}{c^2} \\left (\\tilde{p} - \\Ic^{(4)}_+(p) \\right )\n               \\right ] h\n\\end{align}\nThis is the expression that is found in the Castro code.  If you are dealing with\na simple geometry, then the divergences have metric terms.  In 1-d, we then have:\n\\begin{equation}\n \\ddt{(\\rho e)} + \\frac{1}{r^\\alpha} \\ddr{(r^\\alpha \\rho u e)} + p \\frac{1}{r^\\alpha} \\ddr{(r^\\alpha u)}\n\\end{equation}\nExpanding out the derivatives gives use the equation in Cartesian form with a geometric source term:\n\\begin{equation}\n \\ddt{(\\rho e)} + \\ddr{(\\rho u e)} + p \\ddr{u} = -\\frac{\\alpha \\rho hu}{r}\n\\end{equation}\nwhere $h$ is the specific enthalpy.  This can be accommodated using\nthe procedure described in \\S~\\ref{ch:euler:geom}.\n\nAll of these methods are designed to avoid EOS calls where possible,\nsince general equations of state can be expensive.\n\nExtending these to an unsplit formulation requires carrying an additional\nauxiliary variable from the primitive state back to the conserved state\nand adding the transverse gradients to its interface state.  Castro\ndeals with a conserved state of $\\Uc = (\\rho, \\rho \\Ub, \\rho E, p)$, and\nexplicitly adds the transverse terms found in the multi-dimensional form\nof Eq.~\\ref{eq:euler:pgeneral}\nto the normal states of $p$.\n\n\n\n"
  },
  {
    "path": "Euler/Euler-theory.tex",
    "content": "\\label{ch:compressible-theory}\n\n\n%-----------------------------------------------------------------------------\n\\section{Euler equation properties}\n\nThe Euler equations\\footnote{ We focus on the Euler equations, which\nare the most commonly modeled set of fluid equations in astrophysics.\nThe more general equation set, the Navier-Stokes equations, includes\ndissipative terms.  However, for astrophysical flows, the scales on\nwhich these dissipative terms operate are usually much smaller than\nthe system of interest (equivalently, Reynolds numbers of\nastrophysical flows are very large).} describe conservation of \nmass, momentum, and energy in the fluid approximation.  Their general\nform, without any source terms, is:  \\MarginPar{need to add a discussion of the N-S equations and dimensionless numbers}\n\\begin{align}\n\\ddt{\\rho} + \\nabla \\cdot (\\rho \\Ub) &= 0 \\\\\n\\ddt{(\\rho \\Ub)} + \\nabla \\cdot (\\rho \\Ub \\Ub) + \\nabla p &= 0 \\\\\n\\ddt{(\\rho E)} + \\nabla \\cdot (\\rho E \\Ub + p \\Ub ) &= 0 \n\\end{align}\nHere $\\rho$ is the density, $\\Ub$ is the velocity vector, $\\Ub =\nu\\hat{x} + v\\hat{y}$, $p$ is the pressure, and $E$ is the total energy\n/ mass, and can be expressed in terms of the specific internal energy,\n$e$, and kinetic energy as:\n\\begin{equation}\nE = e + \\frac{1}{2} |\\Ub|^2\n\\end{equation}\nThe equations are closed with the addition of an equation of state.  A common\nchoice is the gamma-law EOS:\n\\begin{equation}\np = \\rho e(\\gamma - 1)\n\\end{equation}\nwhere $\\gamma$ is the ratio of specific heats for the gas/fluid (for\nan ideal, monatomic gas, $\\gamma = 5/3$), but any relation of the form:\n\\begin{equation}\np = p(\\rho, e)\n\\end{equation}\nwill work.  For many astrophysical environments, we may not be able to\nexpress this relation analytically, but instead will solve it via\nnumerical integration or by interpolating from tabulated results.\n\nFor a derivation of the equations of hydrodynamics using moments of\nthe Boltzmann equation see \\cite{shu,choudhuri}.  For a\nphysically-motivated derivation from conservation, see\n\\cite{choudhuri,leveque:2002}.\n\nIn one dimension, they appear as\\footnote{assuming Cartesian coordinates}:\n\\begin{align}\n\\frac{\\partial \\rho}{\\partial t} +\n    \\frac{\\partial (\\rho u)}{\\partial x} &= 0 \\\\\n%\n\\frac{\\partial(\\rho u)}{\\partial t} +\n    \\frac{\\partial (\\rho uu + p)}{\\partial x} &= 0 \\\\\n%\n\\frac{\\partial(\\rho E)}{\\partial t} +\n    \\frac{\\partial(\\rho u E + u p)}{\\partial x} &= 0\n\\end{align}\n\nOne thing that we can notice immediately is that there is no need for\ntemperature in this equation set, although often, when source terms\nare present, we will need to obtain temperature from the equation of\nstate.\n\nIn this form, the equations are said to be in {\\em conservative form},\ni.e.\\ they can be written as:\n\\begin{equation}\n\\Uc_t + \\left [\\Fb(\\Uc) \\right ]_x = 0\n\\end{equation}\nwith\n\\begin{equation}\n\\Uc = \\left ( \\begin{array}{c} \\rho \\\\ \\rho u \\\\ \\rho E \\end{array} \\right )\n%\n\\qquad\n%\n\\Fb(\\Uc) = \\left ( \\begin{array}{c} \\rho u \\\\ \\rho uu + p \\\\ \\rho u E + up \\end{array} \\right )\n\\end{equation}\n%\nWe can write this in {\\em quasi-linear} form by first expressing the\nflux vector in terms of the conserved variables directly.  Taking $m\n\\equiv \\rho u$, $\\mathcal{E} \\equiv \\rho E$, and assuming a gamma-law\nEOS%\n\\footnote{we can relax this assumption by writing $p = p(\\rho, e)$, and then\ntaking the derivatives of this as needed:  $\\partial p/\\partial \\rho$, \n$\\partial p/\\partial m = \\partial p /\\partial e|_\\rho \\partial e/\\partial m$,\nand $\\partial p/\\partial \\mathcal{E} = \\partial p/\\partial e|_\\rho \\partial e/\\partial \\mathcal{E}$,\nwith $e = (\\mathcal{E} - \\myhalf m^2/\\rho)/\\rho$.  But as we'll see, there are\nsimpler systems to work with.},\n\\begin{equation}\np = \\rho e (\\gamma-1) =  \\left (\\mathcal{E} - \\frac{1}{2} \\frac{m^2}{\\rho}\\right )(\\gamma - 1)\n\\end{equation}\nwe have\n\\begin{equation}\n\\Fb(\\Uc) = \\left ( \\begin{array}{c}\n      m \\\\\n      \\frac{1}{2}\\frac{m^2}{\\rho} \\left (3 - \\gamma \\right ) +\n          \\mathcal{E} (\\gamma - 1) \\\\\n      \\frac{m\\mathcal{E}}{\\rho} \\gamma -\\frac{1}{2} \\frac{m^3}{\\rho^2} (\\gamma -1) \\end{array} \\right )\n\\end{equation}\nThe Jacobian%\n\\footnote{\nThe Jacobian, ${\\bf J}$ of a vector $\\Fb(\\Uc)$ with $\\Fb = (f_1, f_2, \\ldots, f_n)^\\intercal$ and\n$\\Uc = (u_1, u_2, \\ldots, u_n)^\\intercal$ is \n\\begin{equation*}\n{\\bf J} \\equiv \\frac{\\partial \\Fb}{\\partial \\Uc} = \\left (\n  \\begin{array}{cccc}\n     \\partial f_1/\\partial u_1 & \\partial f_1/\\partial u_2 & \\ldots & \\partial f_1/\\partial u_n \\\\\n     \\partial f_2/\\partial u_1 & \\partial f_2/\\partial u_2 & \\ldots & \\partial f_2/\\partial u_n \\\\\n     \\vdots                    & \\vdots                    & \\ddots & \\vdots \\\\\n     \\partial f_n/\\partial u_1 & \\partial f_n/\\partial u_2 & \\ldots & \\partial f_n/\\partial u_n \n  \\end{array} \\right )\n\\end{equation*}\n}\nof this flux vector can now be computed as $\\Ab = \\partial\n\\Fb/\\partial \\Uc$:\n\\begin{equation}\n\\Ab(\\Uc) = \\left ( \\begin{array}{ccc}\n   0  & 1 & 0 \\\\\n   -\\frac{1}{2}u^2(3 -\\gamma) & u (3 -\\gamma) & \\gamma - 1 \\\\\n   \\frac{1}{2}(\\gamma -2)u^3 - \\frac{uc^2}{\\gamma -1} &\n       \\frac{3-2\\gamma}{2} u^2 + \\frac{c^2}{\\gamma -1} & u \\gamma\n  \\end{array} \\right )\n\\end{equation}\\MarginPar{SymPy notebook?}\nwhere the speed of sound is $c = \\sqrt{\\gamma p/\\rho}$.   With this, our\nsystem can be written as:\n\\begin{equation}\n\\Uc_t + \\Ab(\\Uc) \\Uc_x = 0\n\\end{equation}\nThis matrix is quite complex and difficult to work with.  The\neigenvectors of this matrix can be found in a variety of sources\n(e.g. \\cite{toro:1997,athena}).\n\n\nAn alternate way to express these equations is using the {\\em\n  primitive variables}: $\\rho, u, p$.\n\\begin{exercise}[Primitive variable form of the Euler equations]\n{Show that the Euler equations in primitive form can\n  be written as\n\\begin{align}\n\\ddt{\\rho} + u \\ddx{\\rho} + \\rho \\ddx{u} &= 0 \\\\\n\\ddt{u} + u \\ddx{u} + \\frac{1}{\\rho} \\ddx{p} &= 0 \\\\\n\\ddt{p} + u \\ddx{p} + \\gamma p \\ddx{u} &= 0 \\label{eq:prim:p}\n\\end{align}\n}\n\\end{exercise}\nNotice that the velocity equation looks like Burgers' equation, and \nis nonlinear.  This nonlinearity will admit shock and rarefaction\nsolutions like we saw with Burgers' equation.\n\n%\nThe primitive variable system can be written compactly as:\n\\begin{equation}\n\\qb_t + \\Ab(\\qb) \\qb_x = 0\n\\end{equation}\nwhere\n\\begin{equation}\n\\qb = \\left ( \\begin{array}{c} \\rho \\\\ u \\\\ p \\end{array} \\right )\n%\n\\qquad\n\\Ab(\\qb) = \\left ( \\begin{array}{ccc} u  & \\rho     & 0 \\\\\n                                  0  &  u       & 1/\\rho \\\\\n                                  0  & \\gamma p & u \\end{array} \\right )\n\\label{eq:primitivesystem}\n\\end{equation}\n\n\n\nThe eigenvalues of $\\Ab$ can be found via $| \\Ab - \\lambda {\\bf I} | = 0$,\nwhere $|\\ldots|$ indicates the determinant and $\\lambda$ are the eigenvalues.\n\\begin{exercise}[The eigenvalues of the Euler system]\n{\nShow that the eigenvalues of $\\Ab$ are $\\lambda^\\evm = u -c$, $\\lambda^\\evz = u$, $\\lambda^\\evp = u+c$.\n}\n\\end{exercise}\nNote that both the conserved Jacobian matrix, $\\Ab(\\Uc)$, and the\nprimitive variable matrix, $\\Ab(\\qb)$, have the same eigenvalues,\n since they represent the same physics.\n\nIn Eq.~\\ref{eq:prim:p}, we used the algebraic gamma-law\nequation of state to replace $e$ with $p$, however, for a general\nequation of state, we can get the appropriate expression by writing $p\n= p(\\rho, s)$:\n\\begin{equation}\n\\label{eq:euler:presnosource}\n\\frac{Dp}{Dt} = \\left . \\frac{\\partial p}{\\partial \\rho} \\right |_s\n     \\frac{D\\rho}{Dt} +\n     \\left . \\frac{\\partial p}{\\partial s} \\right |_\\rho\n     \\cancelto{0}{\\frac{Ds}{Dt}}\n\\end{equation}\nwhere $Ds/Dt = 0$ when no entropy sources are present (we will relax this\nassumption in \\S~\\ref{ch:lm:constraints}).  Recognizing\nthat $\\Gamma_1 \\equiv \\partial \\log p/\\partial \\log \\rho |_s$, we have:\n\\begin{equation}\n\\frac{\\partial p}{\\partial t} + u \\frac{\\partial p}{\\partial x}\n  + \\Gamma_1 p \\frac{\\partial u}{\\partial x} = 0  \\label{eq:euler:pgeneral}\n\\end{equation}\nas the generalization of the pressure equation\\footnote{$\\Gamma_1$ is one of the\nmany adiabatic indices that describe the coupling between different\nthermodynamic quantities under compression/expansion.  Any stellar structure\nbook should give a good background, e.g., \\cite{HKT}.  Note that for an ideal gas, $\\Gamma_1 = \\gamma$}.\nThe sound speed is then $c^2 = \\Gamma_1 p /\\rho$.\n\nThese eigenvalues are the speeds at which information propagates\nthrough the fluid.  Since the eigenvalues are real, this system (the\nEuler equations) is said to be {\\em hyperbolic}.  Additionally, since\n$\\Ab = \\Ab(\\qb)$, the system is said to be {\\em quasi-linear}.  Note,\nphysically, these eigenvalues are telling us that information is\ncommunicated at the speed of the fluid ($u$), as well as the speed of\nsound moving with respect to the fluid ($u \\pm c$).\n\nWe'll use the symbols $\\{-,\\circ,+\\}$ to denote the eigenvalues and their\ncorresponding eigenvectors throughout these notes.\n%\nThe right and left eigenvectors can be found via:\n\\begin{equation}\n\\Ab \\, \\rb^\\enu = \\lambda^\\enu \\rb^\\enu \\; ;\n\\qquad\n\\lb^\\enu \\, \\Ab  = \\lambda^\\enu \\lb^\\enu\n\\end{equation}\nwhere $\\nu = \\{-,\\circ,+\\}$ corresponding to the three waves, and\nthere is one right and one left eigenvector for each of the eigenvalues.\n%\n\\begin{exercise}[Eigenvectors of the Euler system]\n{\nShow that the right eigenvectors are:\n\\begin{equation}\n\\label{eq:euler:primRevs}\n\\rb^\\evm = \\left ( \\begin{array}{c} 1 \\\\ -c/\\rho \\\\ c^2 \\end{array} \\right )\n%\n\\qquad\n\\rb^\\evz = \\left ( \\begin{array}{c} 1 \\\\ 0 \\\\ 0  \\end{array} \\right )\n%\n\\qquad\n\\rb^\\evp = \\left ( \\begin{array}{c} 1 \\\\ c/\\rho \\\\ c^2 \\end{array} \\right )\n\\end{equation}\nand the left eigenvectors are:\n\\begin{align}\n\\lb^\\evm &= \\left ( \\begin{array}{ccc} 0 & -\\frac{\\rho}{2c} & \\frac{1}{2c^2}\n                  \\end{array} \\right ) \\\\\n%\n\\qquad\n\\lb^\\evz &= \\left ( \\begin{array}{ccc} 1 & 0 & -\\frac{1}{c^2}  \\end{array} \\right ) \\\\\n%\n\\qquad\n\\lb^\\evp &= \\left ( \\begin{array}{ccc} 0 & \\phantom{+}\\frac{\\rho}{2c} & \\frac{1}{2c^2} \\end{array} \\right )\n\\end{align}\nNote that in general, there can be an arbitrary constant in front of each\neigenvector.  Here they are normalized such that\n$\\lb^{(i)} \\cdot \\rb^{(j)} = \\delta_{ij}$.\n}\n\\end{exercise}\nA {\\sf Jupyter} notebook using {\\sf SymPy} that derives these\neigenvectors is available:\n\\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/compressible/euler.ipynb}{euler.ipynb}}\n\\footnote{if you don't have {\\sf Jupyter}, you can view this rendered online via the github link.}\n\nA final form of the equations is called the {\\em characteristic form}.  Here,\nwe wish to diagonalize the matrix $\\Ab$.  We take the matrix $\\Rb$ to be the\nmatrix of right eigenvectors,\n\\begin{equation}\n\\Rb = (\\rb^\\evm | \\rb^\\evz | \\rb^\\evp )\n\\end{equation}\n(i.e., $\\Rb$ is a square matrix whose columns are the eigenvectors),\nand $\\Lb$\nis the corresponding matrix of left eigenvectors:\n\\begin{equation}\n\\Lb = \\left ( \\begin{array}{c} \\lb^\\evm \\\\\n                             \\hdashline\n                             \\lb^\\evz \\\\\n                             \\hdashline\n                             \\lb^\\evp \\end{array} \\right )\n\\end{equation}\nNote that $\\Lb\\, \\Rb = {\\bf I} = \\Rb\\, \\Lb$, and $\\Lb = \\Rb^{-1}$.\n\\begin{exercise}[Characteristic form of the Euler equations]\n{\nShow that $\\Lambdab = \\Lb \\Ab \\Rb$ is a diagonal matrix with the diagonal elements\nsimply the 3 eigenvalues we found above:\n\\begin{equation}\n\\Lambdab =\n   \\left ( \\begin{array}{ccc}\n             \\lambda^\\evm &              & \\\\\n                          & \\lambda^\\evz & \\\\\n                          &              & \\lambda^\\evp \\end{array} \\right )\n\\end{equation}\n}\n\\end{exercise}\nTo transform our primitive variable system into the characteristic form, we \nstart by multiplying by $\\Lb$:\n\\begin{equation}\n\\Lb\\qb_t + \\Lb \\Ab \\qb_x = 0\n\\end{equation}\nNext, recalling that $\\Rb\\Lb = {\\bf I}$ (we can insert\nthe identity matrix where we please), we can rewrite our system as:\n\\begin{equation}\n\\Lb\\qb_t + \\Lb \\Ab \\Rb \\Lb \\qb_x = 0\n\\end{equation}\nFinally, defining $d\\wb = \\Lb d\\qb$,\n\\begin{equation}\n\\wb_t + \\Lambda \\wb_x = 0\n\\end{equation}\nHere, the $\\wb$ are the {\\em characteristic variables}.  Note that we cannot\nin general integrate $d\\wb = \\Lb d\\qb$ to write down the characteristic\nquantities.  Since $\\Lambdab$ is diagonal, this system is a set of\ndecoupled advection-like equations:\n\\begin{align}\nw_t^\\evm + \\lambda^\\evm w_x^\\evm &= 0 \\\\\nw_t^\\evz + \\lambda^\\evz w_x^\\evz &= 0 \\\\\nw_t^\\evp + \\lambda^\\evp w_x^\\evp &= 0 \n\\end{align}\n\nIf the system were linear, then the solution to each would simply be\nto advect the quantity $w^\\enu$ at the wave speed $\\lambda^\\enu$.  We\ncould then get back the primitive variables as $d\\qb = \\Lb^{-1} d\\wb = \\Rb d\\wb$.\n\nThe basic idea of hyperbolic systems of PDEs is that there is one wave\nfor each eigenvalue, and these define the characteristic curves,\n$dx/dt = \\lambda^\\enu$.  As we discussed with linear advection\nand Burgers' equation, along the characteristics the solution is\nconstant.  Across them, the solution jumps.  For a linear system, the\ncharacteristics are straight lines.\n\nImagine initial conditions consisting of a jump in the primitive\nvariables, $\\Delta \\qb$.  The corresponding jumps in characteristic\nvariables are $\\Delta \\wb \\sim \\Lb \\Delta \\qb$ (where the $\\sim$ accounts\nfor the fact that in a nonlinear system, $\\Lb = \\Lb(\\qb)$).  The\ncharacteristic form of the equations says that each of the waves will\ncarry its respective jump, $\\Delta \\wb$.  Since $d\\qb = \\Lb^{-1}d\\wb = \\Rb d\\wb$,\nthe jump in the primitive variable across each wave is proportional to\nthe right-eigenvector associated with that wave.  So, for example,\nsince $\\rb^\\evz$ is only non-zero for the density element (see\nEq.~\\ref{eq:euler:primRevs}), this then means that only density jumps\nacross the $\\lambda^\\evz = u$ wave---pressure and velocity are\nconstant across this wave (see for example, Toro~\\cite{toro:1997},\nCh.\\ 2, 3 or LeVeque~\\cite{leveque:2002} for a thorough discussion).\n\nFigure~\\ref{fig:sod} shows the three waves emanating from an initial\ndiscontinuity.  This is a classic problem called the Sod\nproblem~\\cite{sod:1978}.  The initial conditions are\n\\begin{align}\n\\rho_l &= 1      &  \\rho_r &= 1/8 \\nonumber \\\\\nu_l   &= 0       &  u_r    &= 0   \\\\\np_l    &= 1      &  p_r    &= 1/10 \\nonumber\n\\end{align}\nHere we see the three waves propagating away from the initial\ndiscontinuity.  The left ($u-c$) wave is a rarefaction, the middle\n($u$) is the contact discontinuity, and the right ($u+c$) is a\nshock. Note that all 3 primitive variables jump across the left and\nright waves, but only the density jumps across the middle wave.  This\nreflects the right eigenvectors.  Also note that no waves have reached\nthe far left and far right yet, the conditions there are the same as\nthe initial conditions.  \\MarginPar{we need a python version of the Riemann exact script}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.8\\linewidth]{riemann-sod}\n\\caption[The Sod problem]{\\label{fig:sod} Evolution following from an initial\n  discontinuity at $x = 0.5$.  These particular conditions are called\n  the {\\em Sod problem}, and in general, a setup with two states\n  separated by a discontinuity is called a shock-tube problem.  We see three\n  waves carrying changes in the solution. \\\\\n  \\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/compressible/riemann-sod.py}{riemann-sod.py}}}\n\\end{figure}\n\n\n\n\n\n\n%-----------------------------------------------------------------------------\n\\section{The Riemann problem}\n\n\\label{euler:sec:riemann}\n\nJust like with advection and Burgers' equation, we will need to solve\na Riemann problem.  However, as our system is nonlinear, and has 3\nwaves, the solution is considerably more complex, but the general\nideas are straightforward.  Here we review the basic outline of\noperations, and refer to Toro~\\cite{toro:1997} for full details on a\nvariety of methods for solving the Riemann problem.\n\nThe Riemann problem consists of a left and right state separated by an\ninterface.  For the Euler equations, there are three eigenvalues,\nwhich are the speeds at which information propagates.  Each of these\ncorrespond to a wave that will move out from the interface with time,\nand each wave will carry with it a jump in the characteristic\nvariables.  Figure~\\ref{fig:riemann:waves} shows the\nthree waves moving out from the interface, separating space into 4\nregions, marked: $L$, $L_*$, $R_*$, and $R$.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=\\linewidth]{riemann-waves}\n\\caption[The Riemann problem wave structure for the Euler\n  equations]{\\label{fig:riemann:waves} The wave structure and 4\n  distinct regions for the Riemann problem.  Time can be thought of as\n  the vertical axis here, so we see the waves moving outward from the\n  interface.}\n\\end{figure}\nWe typically work in terms of primitive variables.  The states in the\n$L$ and $R$ regions are simply the left and right input states---the\nwaves have not had time to reach here, so they are unmodified.\n\nLooking at the middle right eigenvector, $\\rb^\\evz$, we identify the\nmiddle wave as a {\\em contact discontinuity}.  There is neither\ncompression nor expansion across this wave (since the velocity does\nnot jump across it) and it simply separates two states.  We also know\nfrom the eigenvectors that the pressure is constant across the\ncontact.\n\nThe left and right waves can be either a rarefaction or a shock.  It will\nbe a shock if the flow is compressing and a rarefaction otherwise.\n\nWe already see that the only unknowns are $u_\\star$, $p_\\star$, $\\rho_{\\star,L}$,\nand $\\rho_{\\star,R}$.\n\n\\subsection{Rarefactions}\n\nThe conditions across a rarefaction can be understood by considering\na system where entropy replaces pressure, $\\qb_s = (\\rho, u, s)^\\intercal$.\nThe entropy equation is simply $Ds/Dt = 0$.  We need to express the\npressure gradient in the velocity equation in terms of $\\qb_s$.\n\\begin{equation}\n\\frac{\\partial p(\\rho, s)}{\\partial x} =\n  \\left . \\frac{\\partial p}{\\partial s} \\right |_\\rho \\frac{\\partial s}{\\partial x} +\n  \\left . \\frac{\\partial p}{\\partial \\rho} \\right |_s \\frac{\\partial \\rho}{\\partial x}\n=\n  \\left . \\frac{\\partial p}{\\partial s} \\right |_\\rho \\frac{\\partial s}{\\partial x} +\n  \\frac{p\\Gamma_1}{\\rho} \\frac{\\partial \\rho}{\\partial x}\n\\end{equation}\ngiving\n\\begin{equation}\n\\frac{\\partial u}{\\partial t} + u \\frac{\\partial u}{\\partial x} + \\frac{1}{\\rho} \\left [\n     \\left . \\frac{\\partial p}{\\partial s} \\right |_\\rho \\frac{\\partial s}{\\partial x} +\n          \\frac{p\\Gamma_1}{\\rho} \\frac{\\partial \\rho}{\\partial x} \\right ] = 0\n\\end{equation}\nThis gives a system:\n\\begin{equation}\n{\\qb_s}_t + \\Ab_s(\\qb_s) {\\qb_s}_x = 0\n\\end{equation}\nwhere\n\\begin{equation}\n\\Ab_s =\n \\left ( \\begin{array}{ccc} u & \\rho & 0 \\\\\n        c^2/\\rho & u & \\frac{1}{\\rho} \\left . \\frac{\\partial p}{\\partial s}\\right |_\\rho \\\\\n        0 & 0 & u \\end{array} \\right )\n\\end{equation}\nThe eigenvalues of $\\Ab_s$ are again $u$, $u-c$, and $u+c$.  The right eigenvectors\nare\\footnote{Again, see the {\\sf SymPy} {\\sf Jupyter} notebook:\n\\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/compressible/euler.ipynb}{euler.ipynb}}}\n\\begin{align}\n\\rb_s^\\evm = \\left ( \\begin{array}{c} 1 \\\\ -c/\\rho \\\\ 0 \\end{array} \\right )\n%\n\\qquad\n\\rb_s^\\evz = \\left ( \\begin{array}{c} 1 \\\\ 0 \\\\ -c^2/p_s  \\end{array} \\right )\n%\n\\qquad\n\\rb_s^\\evp = \\left ( \\begin{array}{c} 1 \\\\ c/\\rho \\\\ 0 \\end{array} \\right )\n\\end{align}\nHere, we define $p_s \\equiv \\partial p/\\partial s |_\\rho$.\nSince the jump in $\\qb_s$ is proportional to these eigenvectors, we see that\nentropy does not change across the left, $(-)$, and right, $(+)$, waves.\n\nConsider the $\\evp$ wave, which moves at a speed $\\lambda^\\evp = u\n+c$.  Across this wave, the characteristic variable $\\wb^\\evp$ will\njump, but the others will not.  Similarly, $\\wb^\\evp$ will be constant\nacross the $\\evm$ wave.  So we can use this constancy to tell us how\nthe evolution behaves across that wave.  The constancy of the middle and right characteristic variables across the left\nwave gives us the conditions (using our original primitive variable system):\n\\begin{align}\n\\lb^\\evp \\cdot d\\qb &= 0 \\\\\n\\lb^\\evz \\cdot d\\qb &= 0 \n\\end{align}\nor \n\\begin{equation}\n\\left ( \\begin{array}{ccc} 0 & \\frac{\\rho}{2c} & \\frac{1}{2c^2} \\end{array} \\right)\n   \\left ( \\begin{array}{c} d\\rho \\\\ du \\\\ dp \\end{array} \\right ) = 0\n\\qquad\n%\n\\left ( \\begin{array}{ccc} 1 & 0 & -\\frac{1}{c^2} \\end{array} \\right)\n   \\left ( \\begin{array}{c} d\\rho \\\\ du \\\\ dp \\end{array} \\right ) = 0\n\\end{equation}\nExpanding these, we have the system:\n\\begin{align}\ndu + \\frac{1}{\\rho c} dp &= 0 \\\\\nd\\rho - \\frac{1}{c^2} dp &= 0 \n\\end{align}\nDefining the {\\em Lagrangian sound speed}, $C \\equiv \\rho c$, and the \nspecific volume, $\\tau = 1/\\rho$, we can rewrite this system as:\n\\begin{equation}\ndu = -\\frac{dp}{C} , \\,\\,\\, d\\tau = -\\frac{dp}{C^2} \\quad \\mbox{across the left wave}\n\\end{equation}\n\nSimilar arguments would give the condition across the right wave as:\n\\begin{equation}\ndu = \\frac{dp}{C} , \\,\\,\\, d\\tau = -\\frac{dp}{C^2} \\quad \\mbox{across the right wave}\n\\end{equation}\n\nThese are completely general at this point.  Note that the condition\nof the second wave, $d\\tau/dp$ is essentially the definition of the\nadiabatic index $\\Gamma_1$, $dp/p - \\Gamma_1 d\\rho/\\rho = 0$ for\nconstant entropy.  Finding the solution across the rarefaction then\ninvolves integrating the system from $p_{l,r}$ to $p_\\star$, where $l,\nr$ is the original left or right state, depending on which wave you\nare considering.  Coupled with this, we need our equation of state to\nprovide $C = C(\\tau, p)$.  These solutions are called the {\\em Riemann\n  invariants}.\n\nConsiderable simplification can be made if we assume a gamma-law gas.\nThe eigensystem with entropy showed us that entropy is constant across\nthe solution.  This allows us to write our equation of state as:\n\\begin{equation}\np = K \\rho^\\gamma\n\\end{equation}\nwhere $K$ is a constant that depends on the entropy, and do the \nintegrals analytically.\n\n\\begin{exercise}[Riemann invariants for gamma-law gas]\nAssume a gamma-law gas, then you only need to integrate a single\nequation to find the solution across the left rarefaction:\n\\begin{equation}\nu = - \\int \\frac{dp}{\\rho c}\n\\end{equation}\nwhere $c = \\sqrt{\\gamma p/\\rho}$.  Using $p = K\\rho^\\gamma$, show that\n\\begin{equation}\nu + \\frac{2c}{\\gamma - 1} = \\mbox{constant} \\quad \\mbox{across the left rarefaction}\n\\end{equation}\n\nThis allows you to link the state to the left of the rarefaction to the star state:\n\\begin{equation}\nu_L + \\frac{2c_L}{\\gamma - 1} = u_\\star + \\frac{2c_\\star}{\\gamma -1}\n\\end{equation}\nand\n\\begin{equation}\n\\frac{p_L}{\\rho_L^\\gamma} = \\frac{p_\\star}{\\rho_\\star^\\gamma}\n\\end{equation}\nfrom our constant-entropy equation of state.  Together, show that this\ngives:\n\\begin{equation}\nu_{\\star,L} = u_L + \\frac{2c_L}{\\gamma - 1} \\left [ 1 - \\left ( \\frac{p_\\star}{p_L}\\right )^{(\\gamma - 1)/2\\gamma} \\right ]\n\\end{equation}\nA similar construction can be done for the right rarefaction, yielding:\n\\begin{equation}\nu_{\\star,R} = u_R - \\frac{2c_R}{\\gamma - 1} \\left [ 1 - \\left ( \\frac{p_\\star}{p_R}\\right )^{(\\gamma - 1)/2\\gamma} \\right ]\n\\end{equation}\n\n\\end{exercise}\n\nFor a general equation of state, there is not a closed form, but we would need\nto integrate our system to get $u_\\star = u(p_\\star)$ and $\\rho_{\\star,s} = 1/\\tau(p_\\star)$.\n\nWe'll denote the solution across the left rarefaction as $u_\\star =\nu_L^{rare}(p_\\star)$ and across the right rarefaction as $u_\\star =\nu_R^{rare}(p_\\star)$.\n\n\\subsection{Shocks}\n\nEntropy is not constant across a shock---there is dissipation, so we\nneed a different way to connect the states across the shock.  As with\nBurgers' equation, we can understand the shock by looking at the\nRankine-Hugoniot jump conditions.  There will be one condition for\neach of our conservation laws, and together they tell us the speed of\nthe shock and how density and pressure jump across\nit.  \n\nThe Rankine-Hugoniot conditions are:\n\\begin{equation}\n\\frac{\\Fb(\\Uc_\\star) - \\Fb(\\Uc_s)}{\\Uc_\\star - \\Uc_s} = S\n\\end{equation}\nwhere $S$ is the shock speed.  It is easiest to work in the frame of the shock.  If the shock speed\nis $S$, then we define the velocity in the shock frame as:\n\\begin{align}\n\\hat{u}_s &= u_s - S  \\\\\n\\hat{u}_\\star &= u_\\star - S\n\\end{align}\nand the Rankine-Hugoniot conditions become:\n\\begin{equation}\n\\frac{\\Fb(\\hat{\\Uc}_\\star) - \\Fb(\\hat{\\Uc}_s)}{\\hat{\\Uc}_\\star - \\hat{\\Uc}_s} = 0\n\\end{equation}\n\nFor the one-dimensional system of Euler equations, we have the conditions:\n\\begin{align}\n\\rho_\\star \\hat{u}_\\star &= \\rho_s \\hat{u}_s \\\\\n\\rho_\\star \\hat{u}_\\star^2 + p_\\star &= \\rho_s \\hat{u}_s^2 + p_s \\\\\n\\rho_\\star \\hat{u}_\\star e_\\star + \\frac{1}{2} \\rho_\\star \\hat{u}_\\star^3 + \\hat{u}_\\star p_\\star &= \n  \\rho_s \\hat{u}_s e_s + \\frac{1}{2} \\rho_s \\hat{u}_s^3 + \\hat{u}_s p_s\n\\end{align}\nOur goal is to find how each variable jumps across the shock.  We'll \nwork this out for the general EOS case, following the ideas in \\cite{colellaglaz:1985}.\n\nStarting with the mass flux, we can express:\n\\begin{equation}\n\\hat{u}_\\star = \\frac{\\rho_s}{\\rho_\\star} \\hat{u}_s\n\\end{equation}\nand then insert this into the momentum equation to get:\n\\begin{equation}\n\\rho_s^2 \\left ( \\frac{1}{\\rho_s} - \\frac{1}{\\rho_\\star} \\right ) \\hat{u}_s^2 = p_\\star - p_s\n\\end{equation}\nIntroducing compact notation for the jump across the shock:\n\\begin{equation}\n[q] \\equiv q_\\star - q\n\\end{equation}\nwe have:\n\\begin{equation}\n-[\\tau] = \\frac{[p]}{\\rho_s^2 \\hat{u}_s^2}\n\\end{equation}\n\nWe now introduce the mass flux, $W_s$.  For a shock separating\n$L$/$L\\star$ (the 1-shock), mass will be moving through the shock to the\nright, so, in the frame of the shock, the mass flux is\n\\begin{equation}\nW_L \\equiv \\rho_L \\hat{u}_L = \\rho_\\star \\hat{u}_\\star\n\\end{equation}\nLikewise, for a shock separating $R$/$R\\star$ (the 3-shock), mass will\nmove to the left passing through the shock, and in the frame of the\nshock, the mass flux is\n\\begin{equation}\nW_R \\equiv -\\rho_R \\hat{u}_R = -\\rho_\\star \\hat{u}_\\star\n\\end{equation}\n\nThus, our first jump condition becomes\n\\begin{equation}\n-[\\tau] = \\frac{[p]}{W_s^2}\n\\end{equation}\n\nNext, going back to the momentum equation, we can substitute in the\nmass flux.  For the left (1-shock), we have:\n\\begin{equation}\nW_L \\hat{u}_\\star + p_\\star = W_L \\hat{u}_L + p_L\n\\end{equation}\ngiving\n\\begin{equation}\n[u] = -\\frac{[p]}{W_L}\n\\end{equation}\nwhere we recognized that $\\hat{u}_\\star - \\hat{u}_L = u_\\star - u_L =\n[u]$, since the shock speed is the same for $u_\\star$ and $u$.\n\nLikewise, for the right (3-shock), we have:\n\\begin{equation}\n-W_R \\hat{u}_\\star + p_\\star = -W_R \\hat{u}_R + p_R\n\\end{equation}\ngiving\n\\begin{equation}\n[u] = \\frac{[p]}{W_R}\n\\end{equation}\n\nThe last jump condition is for energy.  Since all the terms in the \nenergy equation are proportional to velocity, there will be no sign\ndifference between the left and right shock jump conditions.  \nWe start by introducing the mass flux:\n\\begin{equation}\nW_s e_\\star + \\frac{1}{2} W_s \\hat{u}_\\star^2 + \\frac{W_s}{\\rho_\\star} p_\\star =\n  W_s e_s + \\frac{1}{2} W_s \\hat{u}_s^2 + \\frac{W_s}{\\rho_s} p_s\n\\end{equation}\nThe mass flux cancels, leaving\n\\begin{equation}\n[e] + \\frac{p_\\star}{\\rho_\\star} - \\frac{p_s}{\\rho_s} + \\frac{1}{2} \\left ( \\hat{u}_\\star^2 - \\hat{u}_s^2 \\right ) = 0\n\\end{equation}\ngetting rid of the velocities using $\\hat{u}_\\star^2 = W_s^2/\\rho_\\star^2$ and \n$\\hat{u}_s^2 = W_s^2/\\rho_s^2$, we have:\n\\begin{equation}\n[e] + \\frac{p_\\star}{\\rho_\\star} - \\frac{p_s}{\\rho_s} + \\frac{1}{2} W_s^2 \\left ( \\frac{1}{\\rho_\\star^2} - \\frac{1}{\\rho_s^2} \\right ) = 0\n\\end{equation}\nThen introducing $W_s^2 = -[p]/[\\tau]$, and after a lot of algebra, we arrive at:\n\\begin{equation}\n[e] = -\\frac{p_\\star + p_s}{2} [\\tau]\n\\end{equation}\nSome sources write $\\bar{p} \\equiv (p_\\star + p_s)/2$.\n\nTo summarize, our jump conditions across the shock are:\n\\begin{align}\n[\\tau] &= -\\frac{[p]}{W_s^2} \\\\\n[u] &= \\mp \\frac{[p]}{W_s} \\quad\\mbox{`$-$' for left, `+' for right}\\\\\n[e] &= - \\frac{p_\\star + p_s}{2} [\\tau] \\label{eq:euler:shock:ejump}\n\\end{align}\n\nAs with the rarefaction, the goal is to express this as a function, $u_\\star\n= u_s^\\mathrm{shock}(p_\\star)$.  The general solution procedure is starts with\na proposed value for $p_\\star$, then:\n\\begin{enumerate}\n\\item root find to find $\\rho_\\star$ corresponding to the $p_\\star$:\n  \\begin{enumerate}\n  \\item guess a value for $\\rho_\\star$\n  \\item using the equation of state, express $e_\\star = e(p_\\star, \\rho_\\star)$\n  \\item use Newton's method (or another technique) with $[e] = -\\bar{p} [\\tau]$\n     to find a correction to $\\rho_\\star$\n  \\end{enumerate}\n\\item compute \n  \\begin{equation}\n    \\frac{1}{W_s^2} = - \\frac{[\\tau]}{[p]}\n  \\end{equation}\n\\item find the star velocity:\n  \\begin{equation}\n    u_\\star = u_s \\mp \\frac{[p]}{W_s}\n  \\end{equation}\n  where we use `$-$' for the left shock, and `+' for the right shock\n\\end{enumerate}\n\nThe one other piece of information we need is the shock speed.  We can\nget this from the mass flux, $W_s$, definition, e.g., $W_L = \\rho_L\n\\hat{u}_L = \\rho_L (u_L - S)$,:\n\\begin{equation}\n\\label{eq:euler-theory:shockgeneral}\nS = u_s \\mp \\frac{W_s}{\\rho_s} \\quad\\mbox{`$-$' for left, `+' for right}\\\\\n\\end{equation}\nThis procedure gives us $\\rho_{\\star,s}$, $p_\\star$, and $S$, the\nshock speed.\n\nJust as with the rarefaction, considerable simplification can be made\nif we assume a gamma-law gas.\n\n\\begin{exercise}[Shock jump conditions for $\\gamma$-law EOS]\n{\nIntroducing \n\\begin{equation}\ne = \\frac{p}{\\rho} \\frac{1}{\\gamma -1}\n\\end{equation}\ninto the jump condition for energy, Eq.~\\ref{eq:euler:shock:ejump},\nshow that we can express the jump in density in terms of the\nratio of pressure, $p_\\star/p_s$, as:\n\\begin{equation}\n\\label{eq:euler:shockrhojump}\n\\rho_{\\star,s} = \\rho_s \\left [ \\frac{ \\frac{p_\\star}{p_s} (\\gamma + 1) + (\\gamma - 1)}\n   {(\\gamma + 1) + \\frac{p_\\star}{p_s} (\\gamma -1)} \\right ]\n\\end{equation}\n}\n\nNow, compute the mass flux, $W_s$ starting with\n\\begin{equation}\nW_s^2 = -\\frac{[p]}{[\\tau]}\n\\end{equation}\nuse the $\\gamma$-law equation of state and show that\n\\begin{equation}\nW_s^2 = \\frac{1}{2} p_s \\rho_s \\left [ \\left(\\frac{p_\\star}{p_s}\\right) (\\gamma + 1) + (\\gamma -1) \\right ]\n\\end{equation}\n\nWith this, show that the star velocity is:\n\\begin{equation}\n\\label{eq:euler:shockujump}\nu_\\star = u_s \\pm c \\left [\\frac{2}{\\gamma(\\gamma - 1)}\\right]^{1/2} \\frac{1 - \\frac{p_\\star}{p_s}}{\\left ( \\frac{p_\\star}{p_s} \\frac{\\gamma + 1}{\\gamma - 1} + 1\\right)^{1/2}}\n\\end{equation}\nwith `+` for the left shock and `$-$` for the right shock.\nAlso show that the shock speed is:\n\\begin{equation}\n\\label{eq:euler:shockspeedjump}\nS = u_s \\mp c \\left [ \\left ( \\frac{p_\\star}{p_s} \\right ) \\frac{\\gamma+1}{2\\gamma} + \\frac{\\gamma-1}{2\\gamma} \\right ]^{1/2}\n\\end{equation}\nwith the `$-$' or the left shock and the `+' for the right shock.\n\\end{exercise}\n  \n\\subsection{Finding the Star State}\n\n\\label{Euler:riemann:starstate}\n\nThe left and right states are connected to the state in the star\nregion by a Hugoniot curve---this is a curve in the $u$-$p$ plane that\nshows all of the possible states one can reach from the current state\nthrough either a shock or rarefaction.  There are two such curves, one\ncorresponding to the left and one to the right state:\n\\begin{equation}\nu_{\\star,L}(p) = \\begin{cases}\n   u_{\\star,L}^\\mathrm{shock}(p) & p > p_L \\\\\n   u_{\\star,L}^\\mathrm{rare}(p) & p \\le p_L \n   \\end{cases}\n\\qquad\nu_{\\star,r}(p) = \\begin{cases}\n   u_{\\star,R}^\\mathrm{shock}(p) & p > p_R \\\\\n   u_{\\star,R}^\\mathrm{rare}(p) & p \\le p_R \n   \\end{cases}\n\\end{equation}\nThe solution\nto the Riemann problem is the point in the $u$-$p$ plane where these\ntwo curves intersect, e.g., we solve for $p_\\star$ in:\n\\begin{equation}\nu_{\\star,l}(p_\\star) - u_{\\star,r}(p_\\star) = 0\n\\end{equation}\nThis is equivalent to saying that pressure and velocity do not jump\nacross the contact wave.\n\nFor a gamma-law equation of state, using the results from the\nexercises, we have:\n\\begin{equation}\n\\renewcommand{\\arraystretch}{1.5}\nu_{\\star,s}(p_\\star) = \\begin{cases}\n  u_s \\pm \\frac{2c}{\\gamma - 1} \\left [ 1 - \\left ( \\frac{p_\\star}{p_s}\\right )^{(\\gamma - 1)/2\\gamma} \\right ] & p_\\star \\le p_s \\\\\n\\\\\n%\n  u_s \\pm c \\left [\\frac{2}{\\gamma(\\gamma - 1)}\\right]^{1/2} \\frac{1 - \\frac{p_\\star}{p_s}}{\\left ( \\frac{p_\\star}{p_s} \\frac{\\gamma + 1}{\\gamma - 1} + 1\\right)^{1/2}} & p_\\star > p_s\n\\end{cases}\n\\renewcommand{\\arraystretch}{1.0}\n\\end{equation}\n\n\n\nFigure~\\ref{fig:euler:riemann-curve} shows the Hugoniot curves for the\nSod problem.  Comparing to Figure~\\ref{fig:sod}, we see that the right\nstate is linked to the star state with a shock while the left state is\nlinked to the star state with a rarefaction.\n\n\\begin{exercise}[Hugoniot curves]\nDownload the code demonstrated in Figure~\\ref{fig:euler:riemann-curve}\nand experiment with different initial conditions to see how the\nsolution changes based on the states.  Try to create initial states\nthat give rise to two rarefactions and a separate set of states that\ngive rise to two shocks.\n\\end{exercise}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{riemann-phase}\n\\caption[The Hugoniot curves corresponding\nto the Sod problem]{\\label{fig:euler:riemann-curve} The Hugoniot\ncurves corresponding to the Sod problem.  The shock and rarefaction\ncurves are shown.  The solution to the Riemann problem is the point\nwhere the curves intersect.  The line style of the curve indicates\nwhere we are a shock or rarefaction.  Where $p > p_s$, where $s \\in {L,R}$, we have a shock.\\\\\n\\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/compressible/riemann-phase.py}{riemann-phase.py}}}\n\\end{figure}\n\n\n\\subsection{Complete Solution}\n\n\\label{Euler:riemann:solution}\n\nTo complete the solution, we need to find which of the 4 regions, $L,\nL\\star, R\\star, R$ we fall in.  For hydrodynamics, we are usually\ninterested only in the solution on the interface, but we can look\nalong any ray in the $x$-$t$ plane by defining $\\xi = (x -\nx_\\mathrm{interface})/t$.  If the initial discontinuity is on the\ninterface, then $\\xi = 0$.\n\n% figure from figures/Euler/riemann_cartoon.py\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\linewidth]{riemann_waves_ifc_R}\n\\includegraphics[width=0.49\\linewidth]{riemann_waves_ifc_Rstar} \\\\\n\\includegraphics[width=0.49\\linewidth]{riemann_waves_ifc_Lstar}\n\\includegraphics[width=0.49\\linewidth]{riemann_waves_ifc_L} \n\\caption[Wave configuration for the Riemann problem]\n        {\\label{fig:euler:riemann_sample} An illustration of the 3\n          waves emanating from an initial discontinuity (the origin of\n          the axes).  The 4 cases show all possible cases of waves\n          being to the left or right of the interface (corresponding\n          to the choice $\\xi = 0$).  In all cases, the middle wave is\n          a contact and the left (1) and right (3) waves are either a\n          shock or a rarefaction.  The state on the interface is: $R$\n          (case a); $R\\star$ (case b); $L\\star$ (case c); $L$ (case\n          d).}\n\\end{figure}\n\nFigure~\\ref{fig:euler:riemann_sample} shows the possible\nconfigurations of the waves.  Note that the middle wave is always a\ncontact but the left (1) and right (3) waves can be either a shock or\nrarefaction.  To find out which region the interface falls in, we simply\nlook at the speeds.  \n\nThe first speed to consider is the contact wave, that has a speed of\nsimply $S_c = u_\\star$.  If $S_c < \\xi$, then we are choosing between\n$R$ and $R_\\star$ states (cases a and b).  If $S_c > \\xi$, then we are\nchoosing between the $L$ and $L_\\star$ states (cases c and d in\nFigure~\\ref{fig:euler:riemann_sample})\\footnote{for the special case\n  where $S_c = 0$, we usually take the solution to be the average of\n  the $L_\\star$ and $R_\\star$ states.  Testing for this explicitly\n  rather than using $\\ge$ in a check is important for maintaining\n  symmetry (see \\S~\\ref{sec:euler-methods:shocktubes}).}.\n\nThe direction of the contact reduces the problem down to choosing\nbetween just two regions, either $L$-$L_\\star$ or $R$-$R_\\star$.  This\nleaves just a single wave to consider: the right wave for cases a and\nb; and the left wave for cases c and d.  We need the wave speed for\nthis---it will depend on whether it is a shock or a rarefaction.  \n\nFor a shock, the wave speed is given by\nEq.~\\ref{eq:euler-theory:shockgeneral}.  We do the same procedure as\nbefore.  For example, for cases a and b, we look at $S_R$, the\nright-moving shock speed for the 3-wave.  If $S_R > 0$, then the\ninterface is in the $R_\\star$ state (case b), while if $S_R < 0$, then\nthe interface is in the $R$ state (case a).  A similar process\nholds for the left-moving shock and cases c and d.\n\nFor a rarefaction, the\nrelevant speeds are the characteristic speeds on either side of the\nrarefaction.  We usually talk about the leading part of the\nrarefaction (the {\\em head}) and trailing part (the {\\em tail}).  The\ncorresponding wave speeds are:\n\\begin{itemize}\n\\item left (1) rarefaction:\n  \\begin{itemize} \n  \\item $\\lambda_\\mathrm{head} = u_L - c_L$\n  \\item $\\lambda_\\mathrm{tail} = u_\\star - c_\\star$\n  \\end{itemize}\n  \n\\item right (3) rarefaction:\n  \\begin{itemize} \n  \\item $\\lambda_\\mathrm{head} = u_R + c_R$\n  \\item $\\lambda_\\mathrm{tail} = u_\\star + c_\\star$\n  \\end{itemize}\n\\end{itemize}\nThe nature of the rarefaction is such that it spreads out,\n$|\\lambda_\\mathrm{head}| > |\\lambda_\\mathrm{tail}|$.  To determine\nwhich state we are in, we look to see where both the head and tail are\nwith respect to the interface.\nFigure~\\ref{fig:euler:rarefaction_sample} shows the possibilities for\na left (1) rarefaction.  In case (a), $\\lambda_\\mathrm{head},\n\\lambda_\\mathrm{tail} < 0$, so the $L_\\star$ state is on the\ninterface.  In case (c), $\\lambda_\\mathrm{head}, \\lambda_\\mathrm{tail}\n> 0$, so the $L$ state is on the interface.  Case (b) is more\ncomplicated---we have $\\lambda_\\mathrm{head} <0, \\lambda_\\mathrm{tail}\n> 0$, so the rarefaction spans the interface.  We need to therefore\nsolve for the state in the rarefaction itself.\n\nTo find the structure inside of a rarefaction, consider some point,\n$(x, t)$, in the $x$-$t$ plane.  For an initial discontinuity at the\norigin, we can imagine a line connecting this point and the origin,\nwhich has the form:\n\\begin{equation}\n\\frac{x}{t} = u - c\n\\end{equation}\n\n% figure from figures/Euler/rarefaction_cartoon.py\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.6\\linewidth]{rarefaction_left} \\\\\n\\includegraphics[width=0.6\\linewidth]{rarefaction_center} \\\\\n\\includegraphics[width=0.6\\linewidth]{rarefaction_right}\n\\caption[Rarefaction configuration for the Riemann problem]\n        {\\label{fig:euler:rarefaction_sample} An illustration of the \n          structure of the left rarefaction wave, for the case\n          where the contact and right wave are to the right of the\n          interface.  Here, we are choosing between states $L$ and $L_\\star$.\n          In (a), both the head and tail of the rarefaction are to the \n          left of the interface, so state $L_\\star$ is on the interface.\n          In (c), both the head and tail of the rarefaction are to the \n          right of the interface, so state $L$ is on the interface.\n          Inbetween, case (b), shows a rarefaction that spans the \n          interface.  For this case, we need to integrate the Riemann\n          invariants to find the state on the interface.}\n\\end{figure}\n\n\n%-----------------------------------------------------------------------------\n\\section{Other thermodynamic equations}\n\n\\label{sec:euler:otherthermo}\n\nAt times we will want to use alternate forms of the energy equation.  The\ninternal energy is governed by the first law of thermodynamics.  In the\nabsence of any heat sources, we have:\n\\begin{equation}\ndq = 0 = de + pd(1/\\rho)\n\\end{equation}\nwhere $e$ is the specific internal energy.\nApplying this to a Lagrangian fluid element, we have:\n\\begin{align}\n\\DDt{e} + p \\DDt{(1/\\rho)} &= 0 \\\\\n\\DDt{e} - \\frac{p}{\\rho^2} \\frac{D\\rho}{Dt} &= 0 \\\\\n\\rho \\frac{De}{Dt} + p \\nabla \\cdot \\Ub &= 0\n\\end{align}\nwhere we used the continuity equation in the last step to eliminate\n$D\\rho/Dt$.  This can be rewritten by adding $e \\times$ the continuity\nequation to give:\n\\begin{equation}\n\\ddt{(\\rho e)} + \\nabla \\cdot (\\rho \\Ub e) + p \\nabla \\cdot \\Ub = 0 \\label{eq:euler:econs}\n\\end{equation}\n\nNotice that internal energy, $(\\rho e)$ is not a conserved quantity\n(in particular, the $p\\nabla \\cdot \\Ub$ term is not in conservative\nform).  \n\nAnother energy-like quantity that we can consider is specific enthalpy,\n\\begin{equation}\nh = e + \\frac{p}{\\rho}\n\\end{equation}\nDifferentiating this, and using the internal energy equation,\n\\begin{align}\n\\DDt{h} &= \\DDt{e} - \\frac{p}{\\rho^2} \\DDt{\\rho} + \\frac{1}{\\rho} \\DDt{p} \\\\\n        &= \\frac{p}{\\rho^2}\\DDt{\\rho} - \\frac{p}{\\rho^2} \\DDt{\\rho} + \\frac{1}{\\rho} \\DDt{p}\n\\end{align}\nso\n\\begin{equation}\n\\rho \\DDt{h} = \\DDt{p}\n\\end{equation}\n\nThis form is useful for very subsonic flows (see,\ne.g.~\\cite{SNpaper}), where $Dp/Dt = 0$, which then shows that\nenthalpy is conserved:\n\\begin{equation}\n\\ddt{(\\rho h)} + \\nabla \\cdot (\\rho h \\Ub) = 0\n\\end{equation}\n(here we added $h \\times$ the continuity equation to transform from\nthe Lagrangian derivative to a conservation equation).\n\n\nWe can also look at the temperature evolution.  It is interesting to\napproach this from both the internal energy and enthalpy equations.\nStarting with internal energy, writing it as $e(\\rho, T)$, we have:\n\\begin{equation}\n\\DDt{e} = \\left . \\frac{\\partial e}{\\partial \\rho} \\right |_T \\DDt{\\rho} +\n          \\left . \\frac{\\partial e}{\\partial T} \\right |_\\rho \\DDt{T} = -\\frac{p}{\\rho} \\nabla \\cdot \\Ub\n\\end{equation}\nThe specific heat at constant volume is defined as $c_v = \\partial\ne/\\partial T|_\\rho$, allowing us to write:\n\\begin{equation}\nc_v \\DDt{T} = \\left ( \\left . \\frac{\\partial e}{\\partial \\rho} \\right |_T - \\frac{p}{\\rho} \\right ) \\nabla \\cdot \\Ub\n\\end{equation}\n\nIf we alternately start with enthalpy, expressing it as $h = h(p, T)$,\nwe have:\n\\begin{equation}\n\\DDt{h} = \\left . \\frac{\\partial h}{\\partial p} \\right |_T \\DDt{p} +\n          \\left . \\frac{\\partial h}{\\partial T} \\right |_p \\DDt{T} = \\frac{1}{\\rho} \\DDt{p}\n\\end{equation}\nThe specific heat at constant pressure is defined as $c_p = \\partial\nh/\\partial T|_p$, letting us write:\n\\begin{equation}\nc_p \\DDt{T} = \\left ( \\frac{1}{\\rho} - \\left . \\frac{\\partial h}{\\partial p} \\right |_T \\right ) \\DDt{p}\n\\end{equation}\n\nThese two temperature evolution equations are equivalent, but one\ndescribes the evolution in terms of density and the other in terms of\npressure.  Later, we'll see how these can be useful when we\napproximate the evolution of reactive flow under constant density or\nconstant pressure behaviors.\n\n\\if debug\n\\subsection{Eigensystem with temperature}\n\nHere we illustrate how the eigensystem changes when we replace pressure with\ntemperature in our primitive variable system.\n\nWe write this set of variables as $\\hat{\\qb} = (\\tau, u,\nT)^\\intercal$---note that we keep $\\tau$ instead of $\\rho$ here.  The\nmotivation for this comes from the fact that with temperature in the\nmix, the temperature will jump across the contact wave.  Since\npressure should be constant across the contact, and, even with the\ngeneral EOS, a temperature jump needs a density drop for pressure to\nremain constant, $\\tau$ should counteract the behavior of $T$ across\nthe contact.\n\nThe temperature evolution equation appears as:\n\\begin{equation}\n\\frac{\\partial T}{\\partial t} = -u\\ddx{T} +\n  \\frac{1}{\\rho c_p} \\left [ (1 - \\rho h_p) \\frac{Dp}{Dt} \\right ]\n\\end{equation}\n(see, e.g.\\ \\cite{ABRZ:I}) where $c_p$ is the specific heat at\nconstant pressure, $c_p = \\partial h/\\partial T|_p$ and $h_p \\equiv\n\\partial h / \\partial p |_T$, with $h = e + p/\\rho$ the specific\nenthalpy.  We can use the standard pressure evolution equation\n(Eq.~\\ref{eq:euler:pgeneral}) to eliminate the Lagrangian pressure term,\nresulting in:\n\\begin{equation}\n\\frac{\\partial T}{\\partial t} = -u\\ddx{T} - \\eta \\ddx{u}\n\\end{equation}\nwhere we defined\n\\begin{equation}\n\\eta \\equiv \\frac{1 - \\rho h_p}{c_p} c^2\n\\end{equation}\n\nThe density equation remains unchanged from the traditional primitive\nvariable formulation, but for the velocity, we need to write the\n$\\partial p/\\partial x$ term in terms of $\\hat{\\qb}$.  We do this via\nthe chain rule.  We define $p_\\rho \\equiv {\\partial p}/{\\partial \\rho}\n|_T$ and $p_T \\equiv {\\partial p}/{\\partial T} |_\\rho$, then our\nvelocity equation is:\n\\begin{equation}\n\\frac{\\partial u}{\\partial t} + u \\frac{\\partial u}{\\partial x} - \\frac{p_\\rho}{\\tau} \\frac{\\partial \\tau}{\\partial x} + {\\tau p_T} \\frac{\\partial T}{\\partial x} = 0\n\\end{equation}\nNote that here we neglected any composition dependence in the EOS.\n\nOur primitive variable system in matrix form is then:\n\\begin{equation}\n\\hat{\\qb}_t + \\hat{\\Ab} \\hat{\\qb}_x = 0\n\\end{equation}\nwith\n\\begin{equation}\n\\renewcommand{\\arraystretch}{1.5}\n\\hat{\\Ab}(\\hat{\\qb}) =\n\\left (\n\\begin{array}{ccc}\nu                   & -\\tau & 0 \\\\\n-\\dfrac{p_\\rho}{\\tau} & u    & \\tau p_T \\\\\n0                   & \\eta & u\n\\end{array}\n\\right )\n\\end{equation}\nThe eigenvalues can be found through the characteristic polynomial,\n$|\\hat{\\Ab} - \\lambda I| = 0$:\n\\begin{equation}\n(u - \\lambda)^3 - (u -\\lambda) \\left ( {p_T \\eta\\tau} + p_\\rho \\right ) = 0\n\\end{equation}\nWe can simplify the last term in parenthesis:\n\\begin{equation}\n{p_T \\eta\\tau} + p_\\rho = \\frac{p_T (1 - \\rho h_p) c^2}{\\rho c_p} + p_\\rho\n                               = \\frac{c^2}{c_p}\\left ( \\frac{p p_T}{\\rho^2 p_\\rho} - \\frac{p_T e_\\rho}{p_\\rho} \\right ) + p_\\rho \\label{eq:step1}\n\\end{equation}\nwhere we substituted in\n\\begin{equation}\nh_p = \\frac{1}{\\rho} \\left ( 1 - \\frac{p}{\\rho p_\\rho} \\right ) + \\frac{e_\\rho}{p_\\rho}\n\\end{equation}\n(see \\cite{ABRZ:I}, appendix A) where $e_\\rho = \\partial e/\\partial\n\\rho |_T$.  The term in parenthesis in Eq.~\\ref{eq:step1} is simply\n$c_p - c_v$ (\\cite{cg}, Eq.~9.81) where $c_v$ is the specific heat\nat constant volume, and Eq.~\\ref{eq:step1} further reduces to:\n\\begin{equation}\n{p_T \\eta\\tau} + p_\\rho = \\frac{c^2}{c_p} (c_p - c_v) + p_\\rho\n                               = c^2 - c^2 \\frac{\\chi_p}{\\Gamma_1} + p_\\rho = c^2\n\\end{equation}\nwhere we used $c_v/c_p = \\chi_p/\\Gamma_1$ and $\\chi_p = \\rho p_\\rho /\np$ (\\cite{cg}, Eqs.~9.87, 9.82).  Putting this into our\ncharacteristic polynomial, we see that the eigenvalues are, as expected,\n$\\lambda = u, u \\pm c$.  It is also useful to note that\n\\begin{equation}\n\\eta = \\frac{c^2 - p_\\rho}{\\tau p_T}\n\\end{equation}\n\n\nWe construct the left and right eigenvectors such that they are orthonormal,\nand find:\n\\begin{equation}\n\\renewcommand{\\arraystretch}{1.3}\n\\hat{R} = \\left ( \\begin{array}{ccc}\n     1                       & 1                 & 1 \\\\\n   c/\\tau                    & 0                 & -c/\\tau \\\\\n  -(c^2 - p_\\rho)/\\tau^2 p_T & p_\\rho/\\tau^2 p_T & -(c^2 - p_\\rho)/\\tau^2 p_T\n  \\end{array} \\right )\n\\end{equation}\nand\n\\begin{equation}\n\\hat{L} = \\left ( \\begin{array}{ccc}\n      {p_\\rho}/{2 c^2}   & {\\tau}/{2c}  & -{\\tau^2 p_T}/{2 c^2} \\\\\n      1 - {p_\\rho}/{c^2} &0             & {\\tau^2 p_T}/{c^2} \\\\\n      {p_\\rho}/{2 c^2}   & -{\\tau}/{2c} & -{\\tau^2 p_T}/{2 c^2} \\end{array} \\right )\n\\end{equation}\nWe note that all\nthermodynamic derivatives are expressed in terms of $\\rho$ or $T$ with\nthe other quantity held constant.  This is in the form we expect a\n$(T, \\rho)$-based EOS to return derivatives.  Notice also that the\ntemperature jumps across the $\\evz$ wave (the contact discontinuity;\nthis is seen from the non-zero value in $\\hat{r}^\\evz$ for the\ntemperature).\n\nWe write:\n\\begin{eqnarray}\n\\hat{\\beta}^\\evm_s &\\equiv& (\\hat{l}^\\evm \\cdot \\Delta \\hat{\\qb}^\\evm ) =\n   \\frac{1}{2C} \\left (\\frac{\\rho^2 p_\\rho}{C} \\Delta \\tau^\\evm\n                        + \\Delta u^\\evm - \\frac{p_T}{C}\\Delta T^\\evm \\right ) \\\\\n%\n\\hat{\\beta}^\\evz_s &\\equiv& (\\hat{l}^\\evz \\cdot \\Delta \\hat{\\qb}^\\evz ) =\n    \\Delta \\tau^\\evz +\n        \\frac{1}{C^2} \\left (-\\rho^2 p_\\rho \\Delta \\tau^\\evz\n                        + {p_T}\\Delta T^\\evz \\right ) \\\\\n%\n\\hat{\\beta}^\\evp_s &\\equiv& (\\hat{l}^\\evp \\cdot \\Delta \\hat{\\qb}^\\evp ) =\n   \\frac{1}{2C} \\left (\\frac{\\rho^2 p_\\rho}{C} \\Delta \\tau^\\evp\n                        - \\Delta u^\\evp - \\frac{p_T}{C}\\Delta T^\\evp \\right )\n\\end{eqnarray}\nNote that since $p_\\tau = -\\rho^2 p_\\rho$, we can form a $\\Delta p^\\enu$ as\n\\begin{equation}\n\\Delta p^\\enu = p_\\tau \\Delta \\tau^\\enu + p_T \\Delta T^\\enu\n\\end{equation}\nand then we see that the $\\hat{\\beta}^\\enu$'s above have the same\nfunctional form as the $\\mathring{\\beta}^\\enu$ from the $\\mathring{\\qb}\n= (\\tau, u, p, e)^\\intercal$ eigensystem.  This is no surprise, since\nthe $l\\cdot \\Delta \\qb$ are the characteristic variables of the Euler\nequations.  The numerical values will differ though, because the\n$\\hat{\\beta}^\\enu$ and $\\mathring{\\beta}^\\enu$ use different reference\nstates and reconstructed variables.\n\nFinally, we can write out the interface states:\n\\begin{eqnarray}\n\\tau_s &=& \\tilde{\\tau} - (\\hat{\\beta}^\\evm + \\hat{\\beta}^\\evz + \\hat{\\beta}^\\evp) \\\\\n%\nu_s &=& \\tilde{u} - (C \\hat{\\beta}^\\evm - C \\hat{\\beta}^\\evp) \\\\\n%\nT_s &=& \\tilde{T} - \\frac{1}{p_T} \\left \\{\n   \\left [ -C^2 \\hat{\\beta}^\\evm - C^2 \\hat{\\beta}^\\evp \\right ] +\n   \\rho^2 p_\\rho \\left [ \\hat{\\beta}^\\evm + \\hat{\\beta}^\\evz + \\hat{\\beta}^\\evp \\right ] \\right \\}\n\\end{eqnarray}\nFor $T_s$, we recognize the first quantity in the square brackets as\nbeing the same as $-(p_s - \\tilde{p})$ in the $\\mathring{\\qb}$ system, and the\nsecond term in the square brackets being the same as $-(\\tau_s - \\tilde{\\tau})$\nin the $\\mathring{\\qb}$ system, then we see\n\\begin{equation}\np_T (T_s - \\tilde{T} ) \\approx (p_s - \\tilde{p} ) - p_\\tau (\\tau_s - \\tilde{\\tau})\n\\end{equation}\nwhich is what we would expect for jumps in $p$ when applying the chain\nrule.  Note that this is not a strict equality, since the reference\nstates and interpolation between the two eigensystems are different.\nNevertheless, this demonstrates the connection between the two\nmethods.\n\nThe above did not consider variations in the composition of the fluid.\nMultiple species complicate things---now the replacement of the\npressure gradient picks up a composition gradient term.\nThe eigensystem will change with this addition.  We don't explore this here at this\ntime.\n\\fi\n\n"
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    "content": "sod.eps: made by hydro1d in the exact/ subdirector\n"
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    "content": "\\documentclass[11pt]{article}\n\n% margins\n\\usepackage[margin=0.75in]{geometry}\n\n%\n\\usepackage{amsmath}\n\n% figures\n\\usepackage{graphicx}\n\n% font\n\\usepackage{mathpazo}\n\n\\usepackage{helvet}\n\n% coloring\n\\usepackage{color}\n\\definecolor{mygray}{gray}{0.5}\n\n% footer\n\\usepackage{fancyhdr}\n\\pagestyle{fancy}\n\\fancyfoot[LO,LE]{\\footnotesize \\sffamily \\color{mygray} M.\\ Zingale---Notes on the Euler equations}\n\\fancyfoot[RO,RE]{\\footnotesize \\sffamily \\color{mygray} (\\today)}\n\\fancyfoot[CO,CE]{\\thepage}\n\\fancyhead{}\n\\renewcommand{\\headrulewidth}{0.0pt}\n\\renewcommand{\\footrulewidth}{0.0pt}\n\n% captions\n\\usepackage{caption}\n\\renewcommand{\\captionfont}{\\footnotesize}\n\\renewcommand{\\captionlabelfont}{\\footnotesize}\n\\setlength{\\captionmargin}{5em}\n\n\\newcommand{\\evm}{{(-)}}\n\\newcommand{\\evz}{{(\\circ)}}\n\\newcommand{\\evp}{{(+)}}\n\\newcommand{\\enu}{{(\\nu)}}\n\n% for dotted lines in the matrics/arrays\n\\usepackage{arydshln}\n\n\\usepackage{sectsty}\n\\allsectionsfont{\\sffamily}\n\n\\begin{document}\n\n\\begin{center}\n{\\LARGE \\textsf{\\textbf{\nNotes on the Euler Equations}}\n}\n\\end{center}\n\n\\input Euler\n\n\\bibliographystyle{plain}\n\\bibliography{../refs}\n\n\\end{document}\n"
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    "content": "# by default, typing make will build the `production' version of the \n# Computational Hydrodynamics notes.\n#\n# if you instead for 'make DEBUG=t', then it will make the draft version.\n# this will enable margin comments and build in any chapters that are\n# not yet complete.\n\nALL: CompHydroTutorial.pdf\n\nDEBUG := \n\nDIRS := preface \\\n        intro \\\n\tpde-classes \\\n        finite-volume \\\n        advection \\\n        burgers \\\n        Euler \\\n        hydro-test-problems \\\n        multigrid \\\n        diffusion \\\n        multiphysics \\\n        incompressible \\\n        low_mach \\\n        pyro \\\n        hydro_examples \\\n        software-engineering \\\n        symbols \\\n        radiation\n\n# dependencies\nTEXS := $(foreach dir, $(DIRS), $(wildcard $(dir)/*.tex))\nEPSS := $(foreach dir, $(DIRS), $(wildcard $(dir)/*.eps))\n\n# for PDFs, pdflatex will automagically convert file.eps to\n# file-converted-to.pdf at build time.  These lines create\n# a variable, PDFS, that just contains the original PDF files\n# as dependencies\ntPDFS := $(foreach dir, $(DIRS), $(wildcard $(dir)/*.pdf))\ncPDFS := $(foreach dir, $(DIRS), $(wildcard $(dir)/*converted-to.pdf))\nPDFS := $(filter-out $(cPDFS), $(tPDFS)) \n\n\nconditional.tex:\nifdef DEBUG\n\techo \"\\def\\debugmode{}\" > conditional.tex\nelse\n\techo \"\" > conditional.tex\nendif\n\nCompHydroTutorial.pdf: CompHydroTutorial.tex conditional.tex $(TEXS) $(EPSS) $(PDFS) refs.bib\n\tgit rev-parse --short=12 HEAD > git_info.tex\n\tpdflatex CompHydroTutorial  < /dev/null\n\tbibtex CompHydroTutorial.aux\n\tpdflatex CompHydroTutorial  < /dev/null\n\tpdflatex CompHydroTutorial  < /dev/null\n\t#pdflatex CompHydroTutorial  < /dev/null\n\t$(RM) git_info.tex\n\t$(RM) conditional.tex\n\n\nclean:\n\t$(RM) *.aux *.log *.dvi *.bbl *.blg *.lof *.toc *.exc *.out *.brf\n\t$(RM) *~ conditional.tex\n\nrealclean: clean\n\t$(RM) CompHydroTutorial.pdf\n\tfind . -name \"*-converted-to.pdf\" -exec $(RM) {} \\;\n\n.PHONY: clean\n\n\nprint-%: ; @echo $* is $($*)\n"
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  },
  {
    "path": "NOTES",
    "content": "-- add to Ch 1 a review of numerical methods:\n\n   . truncation vs. roundoff error\n   . numerical differentiation and integration\n   . ODEs\n   . linear algebra\n\n-- add section to each chapter: \"Numerical Explorations\"\n\n   . create a separate, non-pyro git repo that has basic examples for all \n     of the methods in 1-d also add discussions from pyro.\n   . start with simple grid class and BC discussion \n   . 1-d advection example\n\n\n-- preface:\n\n  \"Each of the main chapters has 2 sections at the end: \"Numerical\n  Explorations\" and \"Going further\".  The first guides the reader\n  through some exercises using the codes in the git repo associated\n  with these notes.  The latter briefly discusses some of the popular\n  extensions done to the core algorithms discussed here, with pointers\n  to the relevant papers in the literature.\n\n-- burgers:\n\nhttp://onlinelibrary.wiley.com/doi/10.1002/fld.1650030302/abstract\n\nGenerating exact solutions of the two-dimensional Burgers' equations\nClive A. J. Fletcher\n\n\n"
  },
  {
    "path": "README.md",
    "content": "# Computational Hydrodynamics for Astrophysics\n\n*part of the Open Astrophysics Bookshelf*\n\nNotes on numerical methods for computational astrophysical hydrodynamics.\n\nThese notes describe the way I think about the numerical methods commonly\nused with grid-based codes in astrophysical hydrodynamics.  The notes\nare written in LaTeX, and should build by typing 'make' in the main\ndirectory.\n\nWorking implementations for all of the solvers are contained either in\nthe main pyro code or in the set of hydro examples, both referenced\nbelow.\n\n## Chapters\n\nThe following chapters are mostly written:\n\n- Simulation Overview\n- Finite-Volume Grids\n- Advection\n- Burgers' Equation\n- Euler Equations: Theory\n- Euler Equations: Numerical Methods\n- Elliptic Equations and Multigrid\n- Diffusion\n- Model Multiphysics Problems\n- Reactive Flow\n- Planning a Simulation\n- Incompressible Flow and Projection Methods\n- Low Mach Number Methods\n\nThe following are things I'd like to add in the next 1-2 years:\n\n- Fluid Instabilities\n- Rotation and Self-gravity\n- Radiation Hydrodynamics\n- MHD\n- AMR\n- Mapped Grids\n\nThe following are things hopefully will eventually get written:\n\n- Relativisitc Flows\n- Higher-Order Methods\n- Implicit Hydrodynamics\n\n\n## Getting PDFs and Source\n\nA PDF version of these notes is available at:\n\nhttp://open-astrophysics-bookshelf.github.io/numerical_exercises/\n\n(this PDF is automatically generated by a Github action each time changes are pushed to `main`.)\n\nThere are two sets of companion codes that go along with these notes:\n\n - *hydro_examples*: https://github.com/zingale/hydro_examples\n\n   simple, standalone, 1-d solvers that illustrate the basic ideas.\n\n - *pyro*: https://github.com/python-hydro/pyro2\n\n   pyro is a 2-d full simulation code designed with simplicity in\n   mind that implements the core solvers described in these notes\n   along with various test problems.\n\n\n\n\n"
  },
  {
    "path": "TODO",
    "content": "Add implementation details throughout, explaining how to translate stuff into code\n   -- maybe in the margins with the Tufte class?\n\nhttps://en.wikipedia.org/wiki/Shallow_water_equations#/media/File:Characteristics_saint-venant.svg\n\nnew chapters:\n\n  -- self-gravity\n\n  -- radiation hydro\n\n  -- AMR\n\n  -- mapped grids\n\n     * general quadralateral grids\n\n     * moving grids\n\n       - 1-d moving grid showing \n     \n  -- MHD\n\n  -- relativisitc\n\n  -- WENO\n\n  -- higher-order\n\n     o method of lines integration with RK3/4 and high-order reconstruction\n\n  -- implicit hydro\n\n  -- shallow water\n\n  -- cosmological flows\n\n  -- understanding simulations\n\n     * convergence tests\n\n     * sanity checks\n\n     * parameter studies vs. hero calculations\n\n  -- volume of fluid\n\n\n\n\nsample of public codes\n\n  -- Athena\n  -- Castro\n  -- Enzo\n  -- Flash\n  -- Maestro\n  -- Pluto\n  -- Zeus\n\n\n=======================\n\nChapter 2:\n\n -- in \"what is a simulation\", add a discussion of DNS, LES, and ILES\n\n -- add BC and difference on a grid examples perhaps through a ipython\n    notebook.\n\n\n\n=======================\n\nChapter 4:\n\n -- add a notebook showing FTCS and upwind on a simple FV grid\n \n pyro exercises:\n\n -- try out different limiters\n\n\n=======================\n\nChapter 5:\n\n\n -- exercise: notebook showing burgers' solution\n\n\n=======================\n\nChapter 6/7:\n\n -- add a figure for section 5.2.1, showing piecewise constant\n    reconstruction\n\n -- add a discussion (interlude) motivating how limiters are derived\n\n -- bulletted list on different Riemann solvers\n\n -- derive the jump conditions and the entropy conditions for the \n    Riemann problem and show the function we zero.\n\n -- implementation detail: for Riemann solvers, usually the godunov\n    velocity is set to 0 at symmetry boundaries\n\n -- it is important to odd-reflect the gravitational acceleration \n    at reflecting boundaries\n\n examples:\n\n   -- 1-d Riemann solver\n   -- 1-d Godunov\n\n pyro...\n\n\n=======================\n\nMultigrid chapter:\n\n examples:\n\n   -- 1-d relaxation\n   -- 1-d MG\n\n\n=======================\n\nDiffusion chapter:\n\n\n  examples:\n    \n    -- 1-d diffusion solver\n    \n  pyro\n\n\n----\n\nreference for the choice of epsilon in numerical derivatives as\n\\sqrt{machine epsilon} -- this gives some suggestions:\nhttp://mathoverflow.net/questions/28463/optimum-small-number-for-numerical-differentiation\n\n\n----\n\ntcolorbox for exercises:\n\nhttp://tex.stackexchange.com/questions/172475/how-can-i-define-a-custom-tcolorbox-environment-with-color-as-a-parameter\n\n\nhttp://www.scriptscoop2.com/t/e6de0140d668/counters-for-chapter-and-section-environments-when-using-tcolorbox-for.html\n\n\nhttp://tex.stackexchange.com/questions/254401/use-the-exercise-environment-from-legrand-book-template\n\nhttp://www.latextemplates.com/template/the-legrand-orange-book\n\n"
  },
  {
    "path": "advection/advection-basics.tex",
    "content": "\\label{ch:advection}\n\n\n\\section{The linear advection equation}\n\nThe linear advection equation is simply:\n\\begin{equation}\n\\label{eq:advect}\na_t + u a_x = 0\n\\end{equation}\nwhere $a(x,t)$ is some scalar quantity and $u$ is the velocity at\nwhich it is advected ($u > 0$ advects to the right).  The solution to\nEq.~\\ref{eq:advect} is to simply take the initial data, $a(x,t=0)$,\nand displace it to the right at a speed $u$.  The shape of the initial\ndata is preserved in the advection.  Many hyperbolic systems of PDEs,\ne.g.\\ the equations of hydrodynamics, can be written in a form that\nlooks like a system of (nonlinear) advection equations, so the\nadvection equation provides important insight into the methods used\nfor these systems.\n%\n\\begin{exercise}[Linear advection analytic solution]\n{Show via substitution that $a(x - ut)$ is a solution to\n  Eq.~\\ref{eq:advect} for any choice of a.  This means that\nthe solution is constant along the lines $x = u t$\n(the curves along which the solution is constant are called the\ncharacteristics).}\n\\end{exercise}\n\nFigure~\\ref{fig:advection_char} shows an initial profile, $a(x)$, and\nthe corresponding characteristics in the $t$-$x$ plane.  With time,\nsince the solution is constant along these characteristics, it simply\neach point simply follows the characteristic curve, resulting in a\nshift of the profile to the right.\n\nAn important concept that we will discuss shortly is {\\em stability}.\nNot every discretization that we write down will be well behaved.  For\nsome, our initial state will begin to ``blow-up'', and take on\nobscenely large and unphysical values after just a few steps.  This is\nthe hallmark of a method that is unstable.  Some methods have\nrestrictions on the size of the timestep that result in a stable\nmethods as well.\n\n% this figure is produced by figures/advection/characteristics.py\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.6\\linewidth]{advection-characteristics}\n\\caption[Characteristics for linear advection]\n{\\label{fig:advection_char} (top) Initially sinusoidal\ndistribution (bottom) Characteristic\nstructure for the linear advection equation $u = 1$.  Note\nthat for this linear equation the characteristics are all parallel\n(they have the same slope in the $t$-$x$ plane.}\n\\end{figure}\n\n\\section{First-order advection in 1-d and finite-differences}\n\nTo get a flavor of the methods for advection, we will use a simple\nfinite-difference discretization---here, the domain is divided into\na sequence of points where we store the solution.\nWe will solve\nEq.~\\ref{eq:advect} numerically by discretizing the solution at\nthese points.  The index $i$ denotes the point's location, and $a_i$\ndenotes the discrete value of $a(x)$ in zone $i$.  The data in each\nzone can be initialized as $a_i = a(x_i)$.  Figure~\\ref{fig:fdgrid}\nshows the grid.\n\nWe also need to discretize in time.  We denote the time-level of the\nsolution with a superscript, so $a_i^n = a(x_i,t^n)$.  For a fixed\n$\\Delta t$, time level $n$ corresponds to a time of $t = n\\Delta t$.\n\n\nA simple first-order accurate discretization is:\n\\begin{equation}\n\\frac{a_i^{n+1} - a_i^n}{\\Delta t} = - u \\frac{a_i^n - a_{i-1}^n}{\\Delta x}\n\\label{eq:fo}\n\\end{equation}\nThis is an {\\em explicit} method, since the new solution, $a_i^{n+1}$,\ndepends only on information at the old time level, $n$.  \n\nFinally, we also need to specify a boundary condition for this.  Our\nchoice of spatial derivative is one-sided---it uses information from\nthe zone to the left of the zone we are updating.  This is because\ninformation is flowing from left to right in this problem ($u > 0$).\nThis choice of the derivative is called {\\em upwinding}---this choice\nof derivative results in a stable method.\nNotice that if we use Eq.~\\ref{eq:fo} to update the data in the first\nzone inside the boundary, we need data to the left of this\nzone---outside of the domain.  This means that we need a single\n{\\em ghost point} to implement the boundary conditions for our method.  The\npresence of the ghost points allow us to use the same update equation\n(e.g.\\ Eq.~\\ref{eq:fo}) for all zones in the domain.\n\n\n% figure created by figures/advection/fd-ghost.py\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{fd_ghost}\n\\caption[A simple finite-difference grid]{\\label{fig:fdgrid} A simple\n  finite-difference grid.  The solution is stored at each of the\n  labeled points.  The dotted lines show the ghost points used to\n  extend our grid past the physical boundaries to accommodate boundary\n  conditions.  Note that if we are periodic, then points $0$ and $N-1$\n  are at the same physical point in space, so we would only need to\n  update one of them.}\n\\end{figure}\n\nThe last piece of information needed to update the solution is the\ntimestep, $\\Delta t$.  It can be shown that for the solution to be\n{\\em stable}, the timestep must be less than the time it takes information\nto propagate across a single zone.  That is:\n\\begin{equation}\n\\Delta t \\le \\frac{\\Delta x}{u} \\enskip .\n\\end{equation}\nThis is called the {\\em Courant-Friedrichs-Lewy} or {\\em CFL}\ncondition.  A dimensionless quantity called the {\\em CFL number} is \ndefined as \n\\begin{equation}\n\\cfl = \\frac{\\Delta t u }{\\Delta x} \n\\end{equation}\nStability requires $\\cfl \\le 1$.\n%\nWe traditionally write the timestep as\n\\begin{equation}\n\\label{eq:timestep}\n\\Delta t = \\cfl \\frac{\\Delta x}{u}\n\\end{equation}\nand specify $\\cfl$ as part of the problem (a typical value may be $\\cfl = 0.7$).\n\n\\begin{exercise}[Perfect advection with a Courant number of 1]\n{Show analytically that when you use $\\cfl=1$ in the\n  first-order differenced advection equation (Eq.~\\ref{eq:fo}) that\n  you advect the profile exactly, without any numerical error.}\n\\end{exercise}\n\nKeep in mind that, in general, we will be solving a non-linear\nsystem of equations, so it is not possible to run with $\\cfl=1$, \nsince $u$ (and therefore $\\cfl$) will change from zone to zone.\nInstead, one looks at the most restrictive timestep over all the\nzones and uses that for the entire system.\n\n\n\\begin{exercise}[A 1-d finite-difference solver for linear advection]\n{Write a code to solve the 1-d linear advection equation\n  using the discretization of Eq.~\\ref{eq:fo} on the domain $[0,1]$ with\n  $u=1$ and periodic boundary conditions.  For initial conditions,\n  try both a Gaussian profile and a top-hat:}\n  \\begin{equation}\n  a = \\left \\{\n      \\begin{array}{lllll}0 & \\mathit{~if~~} &         &x& < 1/3 \\\\\n                          1 & \\mathit{~if~~} & 1/3 \\le &x& < 2/3 \\\\\n                          0 & \\mathit{~if~~} & 2/3 \\le &x&\n      \\end{array}\n      \\right .\n  \\end{equation}\n\n  \\noindent Note: For a general treatment of boundary conditions, you would\n    initialize the ghost points to their corresponding periodic data\n    and apply the difference equations to zones $0, \\ldots, N-1$.\n    However, for periodic BCs on this grid, points $0$ and $N-1$ are\n    identical, so you could do the update in this special case on\n    points $1, \\ldots, N-1$ without the need for ghost points and then\n    set $a_0 = a_{N-1}$ after the update. \\\\\n\n  \\noindent Run you program for one or more periods (one period\n    is $T=1/u$) with several different CFL numbers and notice that\n    there is substantial numerical dissipation (see Figure~\\ref{fig:fdadvect}).\n\\end{exercise}\n\n\\begin{figure}[t]\n\\centering\n% figure generated by hydro_examples/advection/fdadvect.py\n\\includegraphics[width=0.8\\linewidth]{fdadvect-upwind}\n\\caption[First-order finite-difference solution to linear advection]\n{\\label{fig:fdadvect} Finite-difference solution to the first-order\nfinite-difference upwind method for advection, using 65 points and\na variety of CFL numbers. \\\\\n\\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/advection/fdadvect.py}{fdadvect.py}}}\n\\end{figure}\n%\nThis method is first-order accurate.  \n\n\nUltimately we will want higher-order accurate methods.  The most obvious\nchange from our initial discretization is to try a higher-order spatial\nderivative.\n\n\\begin{exercise}[FTCS and stability]\n{You may think that using a centered-difference for\n  the spatial derivative, $a_x \\sim (a_{i+1} - a_{i-1})/(2 \\Delta x)$\n  would be more accurate.  This method is called FTCS (forward-time,\n  centered-space).  Try this on the same test problems.}\n\\end{exercise}\n\n\\begin{figure}[t]\n\\centering\n% figure generated by hydro_examples/advection/fdadvect.py\n\\includegraphics[width=0.48\\linewidth]{fdadvect-FTCS-C0_1}\n\\includegraphics[width=0.48\\linewidth]{fdadvect-FTCS-C0_5}\n\\caption[FTCS finite-difference solution to linear advection]\n{\\label{fig:fdadvect-ftcs} Finite-difference solution using the FTCS\nfinite-difference method for advection using 65 points, modeling for\nonly 1/10$^\\mathrm{th}$ of a period.  The panel of the left is\nwith $\\cfl = 0.1$ and the panel on the right is $\\cfl = 0.5$.\\\\\n\\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/advection/fdadvect.py}{fdadvect.py}}}\n\\end{figure}\n\nYou will find that no matter what value of $\\cfl$ you choose with the\nFTCS method, the solution is unconditionally {\\em unstable} (see\nFigure~\\ref{fig:fdadvect-ftcs}).  If you continue to evolve the\nequation with this method, you would find that the amplitude grows\nwithout bound.  There is something about that discretization that\nsimply gets the physics wrong.\n\n\n\\section{Stability}\n\n\\MarginPar{add a discussion of domain of dependence}\n\nThe classic method for understanding stability is to consider the growth \nof a single Fourier mode in our discretization.  That is, substitute in\n$a_i^n = A^n e^{Ii\\theta}$, where $I = \\sqrt{-1}$\\footnote{our use of $i$ and $j$ as spatial indices presents an unfortunate clash of notation here, hence the use of $I$ for the imaginary unit}, and $\\theta$ represents a\nphase.  A method is stable if $|A^{n+1}/A^n| \\le 1$.  FTCS appears as:\n\\begin{equation}\n  a_i^{n +1} = a_i^n - \\frac{\\cfl}{2} (a_{i+1}^n - a_{i-1}^n)\n\\end{equation}\nExamining a Fourier mode shows that:\n\\begin{align}\n  A^{n+1}e^{Ii\\theta} &= A^n e^{Ii\\theta} - \\frac{\\cfl}{2} \\left (\n      A^n e^{I(i+1)\\theta} - A^n e^{I(i-1)\\theta} \\right ) \\\\\n  A^{n+1} &= A^n  - \\frac{\\cfl}{2} A^n \\left ( e^{I\\theta} - e^{-I\\theta}\\right ) \\\\\n  A^{n+1} &= A^n \\left ( 1 - I \\cfl \\sin\\theta \\right )\n\\end{align}\nso the magnitude of the amplification is\n\\begin{equation}\n \\left | \\frac{A^{n+1}}{A^n}\\right |^2 = 1 + \\cfl^2\\sin^2\\theta\n\\end{equation}\nWe see that there is no value of $\\cfl$ that can make the method stable\n($|A^{n+1}/A^n| > 1$ always).  Doing the same analysis for\nEq.~\\ref{eq:fo} would show that the upwind method is stable for $0\\le\n\\cfl \\le 1$. \n\n\\begin{exercise}[Stability of the upwind method]\n{Using the above stability analysis, considering the amplitude of a\n  single Fourier mode, show that the growth of a mode for the upwind\n  method (Eq.~\\ref{eq:fo}) is:\n  \\begin{equation}\n    \\left | \\frac{A^{n+1}}{A^n} \\right |^2 = \n           1 - 2\\cfl(1-\\cfl)(1-\\cos\\theta)\n  \\end{equation}\n  and stability requires $2\\cfl(1-\\cfl) \\ge 0$ or $0 \\le \\cfl \\le 1$.\n}\n\\end{exercise}\n\nIt is important to note that this stability analysis only works for\nlinear equations, where the different Fourier modes are decoupled,\nnevertheless, we use its ideas for nonlinear advection problems as\nwell.\n\nTruncation analysis can also help us understand stability.  The idea\nhere is to keep the higher order terms in the Taylor series to understand\nhow they modify the actual equation you are trying to solve.\n\n\\begin{exercise}[Stability analysis]\n{To get an alternate feel for stability, we can ask\n  what the terms left out by truncation look like.  That is, we can\n  begin with the discretized equation:\n\\begin{equation}\n  a_i^{n+1} - a_i^n = -\\frac{u \\Delta t}{\\Delta x} ( a_i^n - a_{i-1}^n )\n\\end{equation}\nand replace $a_i^{n+1}$ with a Taylor expansion in time, and replace\n$a_{i-1}^n$ with a Taylor expansion in space, keeping terms to\n$O(\\Delta t^3)$ and $O(\\Delta x^3)$.  Replacing $\\partial a/\\partial t$\nwith $-u \\partial a/ \\partial x$ in the higher-order terms, show \nthat our difference equation more closely corresponds to \n\\begin{eqnarray}\n\\label{eq:advect_trunc_analysis}\na_t + u a_x &=& \\frac{u \\Delta x}{2} \\left ( 1 - \\frac{\\Delta t u}{\\Delta x} \\right ) \\frac{\\partial^2 a}{\\partial x^2} \\\\\n            &=& \\frac{u \\Delta x}{2} (1 - \\cfl) \\frac{\\partial^2 a}{\\partial x^2}\n\\end{eqnarray}\n}\n\\end{exercise}\n\n\nNotice that the righthand side of Eq.~\\ref{eq:advect_trunc_analysis}\nlooks like a diffusion term, however, if $\\cfl > 1$, then the coefficient\nof the diffusion is negative---this is unphysical.  This means that\nthe diffusion would act to take smooth features and make them more\nstrongly peaked---the opposite of physical diffusion.\n\nFor FTCS, a similar truncation analysis would show that the diffusion term\nis always negative.\n\n\\subsection{Domain of dependence}\n\nAnother important view of our numerical difference scheme is to look\nat the {\\em domain of dependence}.  Figure~\\ref{fig:adv:domains}\nillustrates this for the updated point $a_i^{n+1}$ (shown at the top\nof center of the space-time diagram).  The numerical domain of\ndependence shows the points that can influence the updated value of\n$a_i$ using our difference method.  For the upwind scheme, we see that\nthis is a triangle that includes $a_{i-1}^n$ and $a_i^n$.  The\nphysical domain of dependence is shown as the orange triangle---this\nis formed by tracing backwards in time from $a_i^{n+1}$ along a\ncharacteristic, reaching out to the point $x_i - C\\Delta x = x_i -\nu\\Delta t$ over $\\Delta t$.\n\nAny stable numerical method must have a numerical domain of dependence\nthat includes the physical domain of dependence.  If it does not, then\nthe update to the solution simply does not see the points that\ncontribute to the solution over the timestep $\\Delta x$.  Notice that\nthis is a necessary, but not sufficient condition for stability.  FTCS\nhas a domain of dependence that includes the physical domain of dependence,\nbut it is not stable.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.7\\linewidth]{domains_upwind}\\\\\n\\includegraphics[width=0.7\\linewidth]{domains_downwind}\\\\\n\\includegraphics[width=0.7\\linewidth]{domains_FTCS}\n\\caption[Domain of dependence space-time diagram]{\\label{fig:adv:domains} Space-time diagrams showing the\n  numerical domain of dependence (blue region) for three different\n  difference methods.  Also show is the physical domain of\n  dependence---that formed by tracing backwards from $a_i^{n+1}$ along\n  a characteristic.}\n\\end{figure}\n\n\\section{Implicit-in-time}\n\nAn alternate approach to time-discretization is to do an implicit\ndiscretization.  Here our upwind method would appear as:\n\\begin{equation}\n\\frac{a^{n+1}_i - a^n_i}{\\Delta t} = -u \\frac{a^{n+1}_i - a^{n+1}_{i-1}}{\\Delta x}\n\\end{equation}\nThe only change here is that the righthand side is evaluated at the new timelevel,\n$n+1$. We can write this as a linear system with coupled equations:\n\\begin{equation}\n-\\cfl a^{n+1}_{i-1} + (1 + \\cfl) a^{n+1}_i = a_i^n\n\\end{equation}\n\nIf we use periodic boundary conditions, then point $0$ and $N-1$ are\nidentical, so we only need to update one of these.  Taking $a_0^{n+1} = a_{N-1}^{n+1}$, our system in matrix form appears as:\n\\begin{equation}\n\\renewcommand{\\arraystretch}{1.5}\n\\left ( \\begin{array}{ccccccc} \n1+\\cfl & & & & & & -\\cfl \\\\\n-\\cfl  & 1+\\cfl &  \\\\\n &  -\\cfl & 1+\\cfl &  \\\\\n & & -\\cfl & 1+\\cfl &  \\\\\n&&&\\ddots&\\ddots &\\\\\n&&&&-\\cfl & 1+\\cfl & \\\\\n&&&&& -\\cfl &1+\\cfl\n\\end{array}\n\\right )\n%\n\\left ( \\begin{array}{c}\na_1^{n+1} \\\\\na_2^{n+1} \\\\\na_3^{n+1} \\\\\na_4^{n+1} \\\\\n\\vdots \\\\\na_{N-2}^{n+1} \\\\\na_{N-1}^{n+1}\n\\end{array}\n\\right )\n=\n\\left ( \\begin{array}{c}\na_1^{n} \\\\\na_2^{n} \\\\\na_3^{n} \\\\\na_4^{n} \\\\\n\\vdots \\\\\na_{N-2}^{n} \\\\\na_{N-1}^{n}\n\\end{array}\n\\right )\n\\end{equation}\nThis requires a matrix solve---this makes implicit methods generally more\nexpensive than explicit methods.  However, stability analysis would show\nthat this implicit discretization is stable for any choice of $\\cfl$. (But\none must not confuse stability with accuracy---the most accurate solutions\nwith this method will still have a small $\\cfl$).  Also note that the form of \nthe matrix will change depending on the choice of boundary conditions.\nFigure~\\ref{fig:fdadvect-implicit} shows the result of solving this\nimplicit system.\n\n\\begin{figure}[t]\n\\centering\n% figure generated by hydro_examples/advection/fdadvect_implicit.py\n\\includegraphics[width=0.8\\linewidth]{fdadvect-implicit}\n\\caption[First-order implicit finite-difference solution to linear advection]\n{\\label{fig:fdadvect-implicit} Finite-difference solution to the implicit first-order\nfinite-difference upwind method for advection, using 65 points and\na variety of CFL numbers. \\\\\n\\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/advection/fdadvect_implicit.py}{fdadvect\\_implicit.py}}}\n\\end{figure}\n\n\\begin{exercise}[Implicit advection]\n{Code up the implicit advection scheme, but using outflow instead of\nperiodic boundary conditions.  This will change the form of the\nmatrix.  You can use the code from Figure~\\ref{fig:fdadvect-implicit}\nas a starting point.}\n\\end{exercise}\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Eulerian vs.\\ Lagrangian frames}\n\nIt is useful to think about how our advected quantity, $a(x,t)$, changes in\ntime.  The full time derivative is:\n\\begin{equation}\n\\frac{d a(x,t)}{dt} = \\frac{\\partial a}{\\partial t} + \\frac{\\partial a}{\\partial x}\n   \\frac{dx}{dt}\n\\end{equation}\nSo the value of this derivative depends on the path, $x(t)$, that we choose\nto follow.  \n\nConsider an observer who is stationary.  They will watch the flow move\npast them, so $dx/dt = 0$, and $da/dt = \\partial a/\\partial t$.  \nThis fixed frame is called {\\em Eulerian frame}.\n\nInstead imagine an observer who moves with the flow, at the velocity $u$.\nThis way they keep pace with an individual feature in the flow and track\nthe changes it experiences.  In this case, $dx/dt = u$, and our derivative,\ncommonly written as $D/Dt$ is:\n\\begin{equation}\n\\frac{D}{Dt} = \\frac{\\partial}{\\partial t} + u \\frac{\\partial}{\\partial x}\n\\end{equation}\nThis is the {\\em Lagrangian frame}, and the derivative, $D/Dt$ is\ncalled the {\\em Lagrangian derivative}, {\\em material derivative},\n{\\em convective derivative}, or {\\em advective\nderivative}\\footnote{and there are actually many more names...}.\n\nOur linear advection equation can be written simply as $Da/Dt = 0$.\nWe've been solving the equations in the Eulerian frame---our grid is\nfixed and the fluid moves through it.  For hydrodynamics, it will be\nuseful conceptually to consider the Lagrangian frame to understand how\nthe fluid properties change in a particular fluid element over time.\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Errors and convergence rate}\n\nFor the advection problem (with $u>0$), the analytic solution is to\nsimply propagate the initial profile to the right.  This means that\nwith periodic boundary conditions, after advecting for one period, our\nnumerical solution should be identical to the initial conditions.  Any\ndifferences are our numerical error.  We can quantify the error by\ntaking the norm of error\\footnote{see \\S~\\ref{intro:sec:norm} for the\n  definition of the norms} as:\n\\begin{equation}\n\\epsilon^\\mathrm{abs} = \\| a^\\mathrm{final} - a^\\mathrm{init} \\|_2 \\equiv\n   \\left [ \\frac{1}{N} \\sum_{i=1}^N \n   ( a_i^\\mathrm{final} - a_i^\\mathrm{init} )^2\n  \\right ]^{\\myhalf}\n\\end{equation}\nIt is sometimes useful to compare to the norm of the original solution\nto get a measure of the relative error:\n\\begin{equation}\n\\epsilon^\\mathrm{rel} \\equiv \\frac{\\| a^\\mathrm{final} - a^\\mathrm{init} \\|_2}\n   {\\| a^\\mathrm{init} \\|_2}\n\\end{equation}\nNote that for the absolute norm, it is important in these definitions\nto normalize by the number of zones, $N$, otherwise our error will be\nresolution-dependent.  For the relative norm, since we scale by a norm\non the same grid, this normalization will cancel.\n"
  },
  {
    "path": "advection/advection-higherorder.tex",
    "content": "\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Advection and the finite-volume method}\n\n\\label{ch:adv:sndorder}\n\nIn these notes, we will typically use a {\\em finite-volume} discretization.  Here we \nexplore this method for the \nadvection equation.  First we rewrite the advection equation in {\\em\n  conservation form}:\n\\begin{equation}\na_t + \\left[f(a)\\right]_x = 0\n\\label{eq:advect-cons}\n\\end{equation}\nwhere $f(a) = ua$ is the flux of the quantity $a$.  In conservation form,\nthe time derivative of a quantity is related to the divergence of \nits flux.\n\n% figure created with figures/advection/fv-ghost.py\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{fv_ghost}\n\\caption[A finite-volume grid with valid cells\n  labeled]{\\label{fig:fvghost} A finite-volume grid running from\n  $\\mathrm{lo}, \\ldots, \\mathrm{hi}$, with two ghost cells at each\n  end.}\n\\end{figure}\n\nRecall that in the finite-volume discretization, $\\langle a\\rangle_i$\nrepresents the average of $a(x,t)$ over the interval $x_{i-\\myhalf}$ to\n$x_{i+\\myhalf}$, where the half-integer indexes denote the zone edges\n(i.e.\\ $x_{i-\\myhalf} = x_i - \\Delta x/2$).  Figure~\\ref{fig:fvghost}\nshows an example of such a grid with 2 ghost cells at each end.  (For\nsimplicity of notation, we drop the $\\langle \\rangle$ going forward).\nTo discretize Eq.~\\ref{eq:advect-cons}, we integrate it over a zone,\nfrom ${x_{i-\\myhalf}}$ to ${x_{i+\\myhalf}}$, normalizing by the zone width,\n$\\Delta x$:\n\\begin{align}\n\\frac{1}{\\Delta x} \\int_{x_{i-\\myhalf}}^{x_{i+\\myhalf}} a_t \\, dx &= \n   - \\frac{1}{\\Delta x} \\int_{x_{i-\\myhalf}}^{x_{i+\\myhalf}} \\frac{\\partial}{\\partial x} f(a) \\, dx \\\\\n\\frac{\\partial}{\\partial t} a_i &= \n   - \\frac{1}{\\Delta x} \\left \\{ \\left [f(a) \\right ]_{i+\\myhalf} - \\left [f(a) \\right ]_{i-\\myhalf} \\right \\} \\label{adv:eq:fvadv}\n\\end{align}\nThis is an evolution equation for the zone-average of $a$, and shows\nthat it updates in time based on the fluxes through the boundary of\nthe zone.\n\nWe now have a choice on how to proceed with the time-discretization:\n\\begin{enumerate}\n\\item We can discretize ${\\partial a_i}/{\\partial t}$ directly as\n  $(a_i^{n+1} - a_i^n)/\\Delta t$.  Then to achieve second-order\n  accuracy, we need to evaluate the righthand side of\n  Eq.~\\ref{adv:eq:fvadv} at the midpoint in time ($n+1/2$).  This\n  gives rise to a predictor-corrector method (as described in\n  \\cite{colella:1990}.  Sometimes this is called a characteristic\n  tracing method (a name which will be more clear when we discuss the\n  Euler equations).\n\n\\item We can recognize that with the spatial discretization done, our\n  PDE has now become an ODE, and we can use standard ODE methods (like\n  Runge-Kutta) to integrate the system in time.  This is a method-of-lines\n  integration.\n\n\\end{enumerate}\n\nWe'll look at both of these in the next sections.\n\n\n\\section{Second-order predictor-corrector scheme}\n\nWe discretize Eq.~\\ref{adv:eq:fvadv} in time by evaluating the\nrighthand side at the midpoint in time---this gives a\ncentered-difference in time, which is second-order accurate:\n\\begin{equation}\n\\frac{a_i^{n+1} - a_i^n}{\\Delta t} = -\\frac{\\left [f(a) \\right ]_{i+\\myhalf}^{n+\\myhalf} - \\left [f(a) \\right ]_{i-\\myhalf}^{n+\\myhalf}}{\\Delta x}\n\\label{eq:consupdate1d}\n\\end{equation}\nTo evaluate the fluxes at the half-time, we need the state at the\nhalf-time, that is, we do :\n\\begin{equation}\n\\label{adv:eq:fluxeval}\n\\left [f(a) \\right ]_{i+\\myhalf}^{n+\\myhalf} = f(a_{i+\\myhalf}^{n+\\myhalf}) \\enskip .\n\\end{equation}\nWe construct a second-order accurate approximation to\n$a_{i+\\myhalf}^{n+\\myhalf}$ by Taylor expanding the data in the cell\nto the interface.  The construction of the interface state at the\nmidpoint in time is the prediction and the conservative update in the\ncorrection here.\n\nNotice that for each interface, there are two possible\ninterface states we can construct---one using the data to the left of\nthe interface (which we will denote with a ``L'' subscript) and the\nother using the data to the right of the interface (denoted with an\n``R'' subscript)---see Figure~\\ref{fig:riemann_adv}.  These states are:\n\\begin{eqnarray}\na_{i+\\myhalf,L}^{n+\\myhalf} &=& a_i^n + \\frac{\\Delta x}{2} \\left .\\frac{\\partial a}{\\partial x} \\right |_i + \\frac{\\Delta t}{2} \\left .\\frac{\\partial a}{\\partial t} \\right |_i + \\mathcal{O}(\\Delta x^2) + \\mathcal{O}(\\Delta t^2) \\nonumber \\\\\n    &=& a_i^n + \\frac{\\Delta x}{2} \\left .\\frac{\\partial a}{\\partial x} \\right |_i +  \\frac{\\Delta t}{2} \\left ( - u \\left .\\frac{\\partial a}{\\partial x} \\right |_i \\right ) + \\ldots \\nonumber \\\\\n    &=& a_i^n + \\frac{\\Delta x}{2} \\left ( 1 - \\frac{\\Delta t}{\\Delta x} u \\right ) \\left .\\frac{\\partial a}{\\partial x} \\right |_i +  \\ldots \\label{eq:statel}\\\\\n%%\na_{i+\\myhalf,R}^{n+\\myhalf} &=& a_{i+1}^n - \\frac{\\Delta x}{2} \\left .\\frac{\\partial a}{\\partial x} \\right |_{i+1} + \\frac{\\Delta t}{2} \\left .\\frac{\\partial a}{\\partial t} \\right |_{i+1} + \\mathcal{O}(\\Delta x^2) + \\mathcal{O}(\\Delta t^2) \\nonumber \\\\\n    &=& a_{i+1}^n - \\frac{\\Delta x}{2} \\left .\\frac{\\partial a}{\\partial x} \\right |_{i+1} +  \\frac{\\Delta t}{2} \\left ( - u \\left .\\frac{\\partial a}{\\partial x} \\right |_{i+1} \\right ) + \\ldots \\nonumber \\\\\n    &=& a_{i+1}^n - \\frac{\\Delta x}{2} \\left ( 1 + \\frac{\\Delta t}{\\Delta x} u \\right ) \\left .\\frac{\\partial a}{\\partial x} \\right |_{i+1} +  \\ldots \\label{eq:stater}\n\\end{eqnarray}\n% figure created with figures/advection/riemann.py\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=5.0in]{riemann-adv}\n\\caption[The input state to the Riemann\n  problem]{\\label{fig:riemann_adv} The left and right interface state\n  at the $i+\\myhalf$ interface.  Here, the left state,\n  $a_{i+\\myhalf,L}^{n+\\myhalf}$, was predicted to the interface from\n  the zone to the left of the interface, using $a_i$, and the right\n  state, $a_{i+\\myhalf,R}^{n+\\myhalf}$, was predicted to the interface\n  from the zone to the right, using $a_{i+1}$.}\n\\end{figure}\n\nA suitable estimate is needed for the slope of $a$ that appears in\nthese expressions (as $\\partial a/\\partial x$).  We can approximate\nthis simply as\n\\begin{equation}\n\\left . \\frac{\\partial a}{\\partial x}\\right |_i = \\frac{a_{i+1} - a_{i-1}}{2 \\Delta x} \\label{eq:slopecentered}\n\\end{equation}\nWe can think of this method as reconstructing the function form of the\ndata from the cell-average data in each cell using a piecewise linear\npolynomial.  Don't be worried that this looks like FTCS---we'll do\nupwinding next.\n\n\nWe now have two states, $a_{i+\\myhalf,L}^{n+\\myhalf}$ and\n$a_{i+\\myhalf,R}^{n+\\myhalf}$ separated by an interface---this is\ncalled the {\\em Riemann problem}.\n%\nThe solution to this will depend on the equation being solved, and\nresults in a single state at the interface:\n\\begin{equation}\na_{i+\\myhalf}^{n+\\myhalf} = \\mathcal{R}(a_{i+\\myhalf,L}^{n+\\myhalf},a_{i+\\myhalf,R}^{n+\\myhalf})\n\\end{equation}\nIn our case, the advection equation simply propagates the state to the\nright (for $u > 0$), so the solution to the Riemann problem is to take\nthe left state (this is another example of upwinding).  That is we do:\n\\begin{equation}\n\\mathcal{R}(a_{i+\\myhalf,L}^{n+\\myhalf},a_{i+\\myhalf,R}^{n+\\myhalf}) = \\left \\{ \\begin{array}{ccc} a_{i+\\myhalf,L}^{n+\\myhalf} & u > 0 \\\\[2mm] a_{i+\\myhalf,R}^{n+\\myhalf} & u < 0 \\end{array} \\right .\n\\label{eq:riemannsolve}\n\\end{equation}\nTo complete the update, we use this interface state to evaluate the\nflux and update the advected quantity via Eq.~\\ref{eq:consupdate1d}\nand \\ref{adv:eq:fluxeval}.\n\nBoundary conditions are implemented by filling the ghost cells outside\neach end of the domain based on data in the interior.  Note that at\nthe very left edge of the domain, the state\n$a^{n+\\myhalf}_{\\mathrm{lo}-\\myhalf}$ requires the construction of states on\nthe left and right.  The left state at that interface,\n$a^{n+\\myhalf}_{\\mathrm{lo}-\\myhalf,L}$ depends on the slope reconstructed in\nthe $\\mathrm{lo}-1$ ghost cell, $\\partial a/\\partial x\n|_{\\mathrm{lo}-1}$.  This in turn is constructed using a limited\ncenter-difference that will consider the value in the cell to the\nleft, $\\mathrm{lo-2}$.  Therefore, we need two ghost cells at each end\nof the domain for this method---figure~\\ref{fig:advect_ghost}\nillustrates this.  Higher-order limiters may require even more ghost\ncells.\n\n% figure from figures/advection/riemann_bc.py\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{riemann-bc}\n\\caption[Reconstruction at the domain boundary]\n        {\\label{fig:advect_ghost} Reconstruction near the boundary,\n          showing the need for two ghostcells.  Here we see the left\n          and right state at the left physical boundary of the domain\n          (marked as $\\mathrm{lo}-\\myhalf$).  The gray dotted lines\n          are the piecewise constant cell averages and the red lines\n          are the reconstructed slopes.  Note that we need the slope\n          in $\\mathrm{lo}-1$ to get the left interface state at\n          $\\mathrm{lo}-\\myhalf$, and that slope in turn needed the\n          data in zone $\\mathrm{lo}-2$ to construct a\n          centered-difference.}\n\\end{figure}\n\n\n\\begin{exercise}[A second-order finite-volume solver for linear advection]\n{\\label{adv:ex:fv} Write a second-order solver for the linear advection\n  equation.  To mimic a real hydrodynamics code, your code should have\n  routines for finding initializing the state, filling boundary conditions,\n  computing the timestep,\n  constructing the interface states, solving the Riemann problem, and\n  doing the update.  The problem flow should look like:\n  \\begin{itemize}\n    \\item set initial conditions\n    \\item main evolution loop---loop until final time reached\n    \\begin{itemize}\n      \\item fill boundary conditions\n      \\item get timestep (Eq.~\\ref{eq:timestep})\n      \\item compute interface states (Eqs.~\\ref{eq:statel} and \\ref{eq:stater})\n      \\item solve Riemann problem at all interfaces (Eq.~\\ref{eq:riemannsolve})\n      \\item do conservative update (Eqs.~\\ref{eq:consupdate1d} and \\ref{adv:eq:fluxeval})\n    \\end{itemize}\n  \\end{itemize}\nUse both the top-hat and Gaussian initial conditions and periodic boundary\nconditions and compare to the first-order method.  See Figure~\\ref{fig:fvadvect}.\n}\n\\end{exercise}\n\n\\begin{figure}\n\\centering\n% figure from codes/advection/fv_advection.py\n\\includegraphics[width=0.8\\linewidth]{fv-advect}\n\\caption[Second-order finite-volume advection]\n{\\label{fig:fvadvect} Second-order finite volume advection showing the\nresult of advecting a tophat profile through five periods with both\nunlimited and limited slopes.  This calculation used 64 zones and\n$\\cfl=0.7$. \\\\ \n\\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/advection/advection.py}{advection.py}}}\n\\end{figure}\n\n\\subsection{Limiting}\n\nThe second-order method likely showed some oscillations in the\nsolution, especially for the top-hat initial conditions.  {\\em\n  Godunov's theorem} says that any monotonic linear method for\nadvection is first-order accurate (see, e.g.,~\\cite{laney}).  In this\ncontext, monotonic means that no new minima or maxima are introduced.\nThe converse is true too, which suggests that in order to have a\nsecond-order accurate method for this linear equation, the algorithm\nitself must be nonlinear.\n\n\\begin{exercise}[Limiting and overshoots]\n{To remove the oscillations in\npractice, we limit the slopes to ensure that no new minima or maxima are\nintroduced during the advection process.  There are many choices for\nlimited slopes.  A popular one is the {\\em minmod} limiter.  Here, we\nconstruct the slopes in the interface states as:\n\\begin{equation}\n\\left . \\frac{\\partial a}{\\partial x} \\right |_i = \\mathtt{minmod} \\left (\n  \\frac{a_i - a_{i-1}}{\\Delta x}, \\frac{a_{i+1} - a_i}{\\Delta x} \\right )\n\\end{equation}\ninstead of Eq.~\\ref{eq:slopecentered}.\nwith \n\\begin{equation}\n\\mathtt{minmod}(a,b) = \\left \\{ \n    \\begin{array}{ll}\n    a & \\mathit{if~} |a| < |b| \\mathrm{~and~} a\\cdot b > 0 \\\\\n    b & \\mathit{if~} |b| < |a| \\mathrm{~and~} a\\cdot b > 0 \\\\\n    0 & \\mathit{otherwise}\n    \\end{array}\n  \\right .\n\\end{equation}\nUse this slope in your second-order advection code and notice that the\noscillations go away---see Figure~\\ref{fig:fvadvect}.}\n\\end{exercise}\n\nWe can get a feel for what happens with and without limiting\npictorially. \\MarginPar{add a discussion on how a limiter is\n  designed/constructed} Figures~\\ref{fig:limitingex} and\n\\ref{fig:limitingexb} show the evolution of an initial discontinuity\nwith and without limiting.  Without limiting, we see an overshoot\nbehind the discontinuity and an undershoot ahead of it---these develop\nafter only a single step.  With each additional step, the overshoots\nand undershoots move further away from the discontinuity.  In\ncontrast, the case with limiting shows no over- or undershoots around\nthe initial discontinuity.  By the end of the evolution, we see that\nthe profile is much narrower in the limiting case than in the case\nwithout limiting.\n\nSee the text by LeVeque~\\cite{leveque:2002} for alternate choices of\nlimiters. Note: most limiters will have some sort of test on the\nproduct of a left-sided and right-sided difference ($a\\cdot b$\nabove)---this is $< 0$ at an extremum, which is precisely where we\nwant to limit.\n\n% these two sets of figures can be created with\n% figures/advection/rea-limitex.py\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.75\\linewidth]{rea-nolimit-start_001} \\\\\n\\includegraphics[width=0.75\\linewidth]{rea-nolimit-start_002} \\\\\n\\includegraphics[width=0.75\\linewidth]{rea-nolimit-start_003} \\\\\n\\includegraphics[width=0.75\\linewidth]{rea-nolimit-start_004} \\\\\n\\includegraphics[width=0.75\\linewidth]{rea-nolimit-start_005} \\\\\n\\includegraphics[width=0.75\\linewidth]{rea-nolimit-start_006} \n%\\includegraphics[width=0.55\\linewidth]{rea-nolimit-start_007} \\\\\n%\\includegraphics[width=0.55\\linewidth]{rea-nolimit-start_008} \\\\\n\\caption[The effect of no limiting on initially discontinuous data]{\\label{fig:limitingex}Initially discontinuous data evolved for several steps with\n  no limiting.  Notice that there are overshoots and undershoots\n  surrounding the discontinuity.}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.75\\linewidth]{rea-start_001} \\\\\n\\includegraphics[width=0.75\\linewidth]{rea-start_002} \\\\\n\\includegraphics[width=0.75\\linewidth]{rea-start_003} \\\\\n\\includegraphics[width=0.75\\linewidth]{rea-start_004} \\\\\n\\includegraphics[width=0.75\\linewidth]{rea-start_005} \\\\\n\\includegraphics[width=0.75\\linewidth]{rea-start_006}\n%\\includegraphics[width=0.55\\linewidth]{rea-start_007}\n%\\includegraphics[width=0.55\\linewidth]{rea-start_008} \\\\\n\\caption[The effect of limiters on initially discontinuous\n  data]{\\label{fig:limitingexb}Initially discontinuous data evolved\n  for several steps with limiting.  Note that unlike the sequence\n  without limiting (Figure~\\ref{fig:limitingex}), the\n  discontinuity remains sharper with limiting and there are no over-\n  or undershoots.}\n\\end{figure}\n\nA slightly more complex limiter is the MC limiter (monotonized central\ndifference).  First we define an extrema test,\n\\begin{equation}\n\\xi = (a_{i+1} - a_i) \\cdot (a_i - a_{i-1})\n\\end{equation}\nThen the limited difference is\n\\begin{equation}\n\\left . \\frac{\\partial a}{\\partial x} \\right |_i = \n \\left \\{\n\\begin{array}{ll}\n\\min \\left [ \\frac{| a_{i+1} - a_{i-1} |}{2 \\Delta x},\n              2 \\frac{| a_{i+1} - a_i |}{\\Delta x},\n              2 \\frac{| a_{i} - a_{i-1} |}{\\Delta x}\n      \\right ]  \\mathrm{sign}(a_{i+1} - a_{i-1}) &  \\xi > 0 \\\\\n0 & \\mathit{otherwise}\n\\end{array}\n\\right .\n\\end{equation}\n%\nNote that a slightly different form of this limiter is presented in\n\\cite{leveque:2002}, where all quantities are in a {\\tt minmod}, which appears to limit a bit less.\nThis is second-order accurate for smooth flows.  \n\nThe main goal of a limiter is to reduce the slope near extrema.\nFigure~\\ref{fig:limit} shows a finite-volume grid with the original\ndata, cell-centered slopes, and MC limited slopes.  Note that near the\nstrong gradients is where the limiting kicks in.  The different limiters\nare all constructed by enforcing a condition requiring the method to be\n{\\em total variation diminishing}, or TVD.  More details on TVD limiters\ncan be found in \\cite{toro:1997,leveque:2002}.\n\n% this figure can be create with figures/advection/generalgrid.py\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{generalgrid}\n\\caption[Piecewise linear slopes with an without\n  limiting]{\\label{fig:limit} A finite-volume grid showing the cell\n  averages (gray, dotted, horizontal lines), unlimited center-difference slopes\n  (gray, solid) and MC limited slopes (red).  Note that in zones $i$ and\n  $i+1$, the slopes are limited slightly, so as not to overshoot or\n  undershoot the neighboring cell value.  Cell $i-1$ is not limited at\n  all, whereas cells $i-2$, and $i+2$ are fully limited---the slope is\n  set to $0$---these are extrema.}\n\\end{figure}\n\nA popular extension of the MC limiter is the $4^\\mathrm{th}$-order MC\nlimiter, which is more accurate in smooth flows (this is shown in\n\\cite{colella:1985}, Eqs. 2.5 and 2.6; and \\cite{colella:1990},\nEq. 191).\n\n\\begin{exercise}[Limiting and reduction in order-of-accuracy]\n{Show analytically that if you fully limit the slopes\n\t(i.e.\\ set $\\partial a/\\partial x |_i = 0$),then the second-order\n  method reduces to precisely our first-order finite-difference discretization,\n  Eq.~\\ref{eq:fo}.  }\n\\end{exercise}\n\nIt is common to express the slope simply as the change in the state variable:\n\\begin{equation}\n\\Delta a_i = \\left . \\frac{\\partial a}{\\partial x} \\right |_i \\Delta x\n\\end{equation}\nand to indicate the limited slope as $\\overline{\\Delta a}_i$.\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\subsection{Reconstruct-evolve-average}\n\nAnother way to think about these methods is as a reconstruction,\nevolve, and average (R-E-A) process (see Figure~\\ref{fig:rea}).  \n\nWe can write the conservative update as:\n\\begin{align}\na_i^{n+1} &= a_i^n + \\frac{\\Delta t}{\\Delta x} \n    (u a^{n+\\myhalf}_{i-\\myhalf} - u a^{n+\\myhalf}_{i+\\myhalf} ) \\\\\n          &= a_i^n + \\cfl (a^{n+\\myhalf}_{i-\\myhalf} - a^{n+\\myhalf}_{i+\\myhalf} ) \n\\end{align}\nIf we take $u > 0$, then the Riemann problem will always choose the\nleft state, so we can write this as:\n\\begin{equation}\na_i^{n+1} = a_i^n + \n     \\cfl \\biggl [\\underbrace{\\left (a_{i-1}^n + \\frac{1}{2} (1-\\cfl) \\Delta a_{i-1} \\right )}_{a_{i-\\myhalf,L}} -\n              \\underbrace{\\left (a_{i}^n + \\frac{1}{2} (1-\\cfl) \\Delta a_{i} \\right )}_{a_{i+\\myhalf,L}}\n       \\biggr ] \\label{eq:rea_orig}\n\\end{equation}\n\nIf we instead look at this via the R-E-A procedure, we write the reconstructed\n$a$ in each zone in the form of a piecewise linear polynomial\n\\begin{equation}\na_i(x) = a_i + \\frac{\\Delta a_i}{\\Delta x} (x - x_i)\n\\end{equation}\nConsider zone $i$.  \nIf we are advecting with a CFL number $\\cfl$, then that means that the fraction\n$\\cfl$ of the zone immediately to the left of the $i-\\myhalf$ interface will advect\ninto zone $i$ over the timestep.  And only the fraction $1-\\cfl$ in zone $i$\nimmediately to the right of the interface will stay in that zone.  This \nis indicated by the shaded regions in Figure~\\ref{fig:rea}. \n\nThe average of the quantity $a$ from zone $i-1$ that will advect into\nzone $i$ is \n\\begin{eqnarray}\n\\mathcal{I}_< &=& \\frac{1}{\\cfl \\Delta x} \n   \\int_{x_{i-\\myhalf} - \\cfl\\Delta x}^{x_{i-\\myhalf}} a_{i-1}(x) dx \\\\\n%\n &=& \\frac{1}{\\cfl \\Delta x} \n   \\int_{x_{i-\\myhalf} - \\cfl\\Delta x}^{x_{i-\\myhalf}}\n        \\left [ a_{i-1} + \\frac{\\Delta a_{i-1}}{\\Delta x} (x - x_{i-1} ) \\right ] dx  \\\\\n &=& a_{i-1} + \\frac{1}{2} \\Delta a_{i-1} (1-\\cfl)\n\\end{eqnarray}\n\nAnd the average of the quantity $a$ in zone $i$ that will remain in zone $i$\nis\n\\begin{eqnarray}\n\\mathcal{I}_> &=& \\frac{1}{(1-\\cfl) \\Delta x} \n   \\int_{x_{i-\\myhalf}}^{x_{i-\\myhalf} + (1-\\cfl) \\Delta x} a_{i}(x) dx \\\\\n%\n &=& \\frac{1}{(1-\\cfl) \\Delta x} \n   \\int_{x_{i-\\myhalf}}^{x_{i-\\myhalf} + (1-\\cfl)\\Delta x} \n        \\left [ a_{i} + \\frac{\\Delta a_{i}}{\\Delta x} (x - x_{i} ) \\right ] dx  \\\\\n &=& a_{i} - \\frac{1}{2} \\Delta a_{i} \\cfl\n\\end{eqnarray}\n\nThe final part of the R-E-A procedure is to average over the \nadvected profiles in the new cell.  The weighted average of the\namount brought in from the left of the interface and that remains\nin the cell is\n\\begin{align}\na_i^{n+1} &= \\cfl \\mathcal{I}_< + (1 - \\cfl) \\mathcal{I}_>  \\\\\n          &= \\cfl \\left [ a^n_{i-1} + \\frac{1}{2} \\Delta a_{i-1} (1 - \\cfl) \\right ]\n   + (1-\\cfl) \\left [ a^n_i - \\frac{1}{2} \\Delta a_i \\cfl \\right ] \\\\\n          &= a^n_i + \\cfl \\left [a^n_{i-1} + \\frac{1}{2}(1 - \\cfl) \\Delta a_{i-1} \\right ]\n           - \\cfl \\left [ a^n_i + \\frac{1}{2} (1-\\cfl) \\Delta a_i \\right ]\n\\end{align}          \nThis is identical to Eq.~\\ref{eq:rea_orig}.  This demonstrates that the\nR-E-A procedure is equivalent to our reconstruction, prediction of the\ninterface states, solving the Riemann problem, and doing the \nconservative flux update.\n\n% this figure sequence is created by figures/advection/rea.py\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{rea-start} \\\\\n\\includegraphics[width=\\linewidth]{rea-reconstruction} \\\\\n\\includegraphics[width=\\linewidth]{rea-trace} \\\\\n\\includegraphics[width=\\linewidth]{rea-evolve} \\\\\n\\includegraphics[width=\\linewidth]{rea-final}\n\\caption[The Reconstruct-Evolve-Average procedure]{\\label{fig:rea}\n  Reconstruct-Evolve-Average.  The top panel shows the original\n  cell-average data.  The second panel shows the (limited) piecewise\n  linear reconstruction of the data.  Assuming a CFL number of 0.6 and\n  advection to the right, the shaded regions in the third panel show\n  the data that will wind up in cell $i$ after advecting for a single\n  step.  The fourth panel shows the piecewise-linear data advected to\n  the right by 0.6 of a cell-width (corresponding to a CFL of 0.6).\n  The final panel shows the new averages of the data, constructed by\n  averaging the advected piecewise linear data in each cell.}\n\\end{figure}\n\n\n\\begin{exercise}[Convergence testing]\n{ Run the first-order solver for several different values of $\\Delta x$,\neach a factor of 2 smaller than the previous.  Compute $\\epsilon$ for\neach resolution and observe that it converges in a first-order fashion\n(i.e.\\ $\\epsilon$ decreases by 2 when we decrease $\\Delta x$ by a factor of 2).\n\n\\noindent Do the same with the second-order solver and observe that it\nconverges as second-order. However: you may find less than\nsecond-order if your initial conditions have discontinuities and you\nare limiting.  Figure~\\ref{fig:advnorm} shows the convergence of the\nmethod with no limiting, MC, and minmod limiting, $\\cfl = 0.8$, and a\nGaussian initial condition for 5 periods.}\n\\end{exercise}\n\n% figure from hydro_examples: advection/advection.py\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{plm-converge}\n\\caption[Convergence of second-order finite-volume advection]\n        {\\label{fig:advnorm} Convergence for the second-order\n          finite-volume method with no limiting, MC, and minmod limiting advecting a\n          Gaussian initial profile with $\\cfl = 0.8$ for 5 periods. \\\\\n        \\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/advection/advection.py}{advection.py}}}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{fv-gaussian-limiters} \\\\[1em]\n\\includegraphics[width=0.8\\linewidth]{fv-tophat-limiters} \n\\caption[Effect of different limiters on evolution]\n        {\\label{fig:limiter_panel} The effect of different limiters on the\n          evolution of a Gaussian initial profile (top) and a tophat initial\n          profile (bottom), using $\\cfl = 0.8$ and 5 periods. \\\\\n        \\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/advection/advection.py}{advection.py}}}\n\\end{figure}\n\nAs seen in figure~\\ref{fig:advnorm}, the choice of limiters can have a great\neffect on the accuracy.  Figure~\\ref{fig:limiter_panel} shows the Gaussian and\ntophat profiles with several different limiters.\n\n\\section{Method of lines approach}\n\\label{adv:sec:mol_2d}\n\nIn the above constructions, we computed a time-centered flux by\npredicting the interface state to the half-time (by Taylor-expanding\nin time through $\\Delta t /2$).  Instead, we can recognize that with\nthe spatial discretization done in Eq.~\\ref{adv:eq:fvadv}, we are left\nwith an ordinary differential equation in time that can then be solved\nusing standard ODE techniques.\n\nSubstituting in the flux as $f(a) = ua$, we have the ODE:\n\\begin{equation}\n\\ddt{a_i} = - \\frac{u a_{i+1/2} - u a_{i-1/2}}{\\Delta x}\n\\end{equation}\nNote that there is no time-level indicated on the righthand side.  We\nfind the interface values of $a$ by interpolating in space only.  For\na second-order accurate method, we can use a piecewise linear\nreconstruction (see Figure~\\ref{fig:advection:riemann-mol}), and\nevaluate the interface states as simply the point on the reconstructed\nline on the interface, e.g.,\n\\begin{align}\na_{i+\\myhalf,L} &= a_{i} + \\frac{1}{2} \\Delta a_{i} \\\\\na_{i+\\myhalf,R} &= a_{i+1} - \\frac{1}{2} \\Delta a_{i+1}\n\\end{align}\n%\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{riemann-mol}\n\\caption[Method-of-lines spatial\n  reconstruction]{\\label{fig:advection:riemann-mol} Piecewise linear\n  reconstruction of the zones showing the interface states used with\n  the method of lines approach.}\n\\end{figure}\n%\nThe same limiting ideas discussed above apply here.  As usual, we\nresolve the left-right degeneracy on the interface by solving the\nRiemann problem:\n\\begin{equation}\na_{i+\\myhalf,j} = \\mathcal{R}(a_{i+\\myhalf,j,L}, a_{i+\\myhalf,j,R})\n\\end{equation}\nOnce we have this interface state, we can integrate the ODE.  A\nRunge-Kutta method works fine here (it should be atleast second-order\nto match the spatial accuracy).\n\\section{Multi-dimensional advection}\n\nThe two-dimensional linear advection equation is:\n\\begin{equation}\na_t + u a_x + v a_y = 0\n\\label{eq:advect2d}\n\\end{equation}\nwhere $u$ is the velocity in the $x$-direction and $v$ is the velocity in\nthe $y$-direction.  We denote the average of $a(x,y,t)$ in a zone $i,j$ as\n$a_{i,j}$.  Here, $i$ is the index in the $x$-direction and $j$ is the\nindex in the $y$-direction.  A 2-d grid is shown in Figure~\\ref{fig:2dgrid}.\nJust as in the one-dimensional case, we will extend the domain with a\nperimeter of ghost cells to set the boundary conditions.\n\n% this is create by figures/advection/2dgrid.py\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=4.0in]{2dgrid}\n\\caption[A 2-d grid with zone-centered indexes]{\\label{fig:2dgrid} A\n  simple 2-d grid with the zone-centered indexes.  The $\\times$s mark\n  the left and right interface states at the upper edge of the $i,j$ zone in each\n  coordinate direction.  For a finite-volume discretization, $a_{i,j}$ \n  represents the average of $a(x,y)$ over the zone.}\n\\end{figure}\n\nTo derive the finite-volume form of the update, we start by writing\nthis in conservative form.  Since $u$ and $v$ are constant, we can\nmove them inside the divergences:\n\\begin{equation}\na_t + (u a)_x + (v a)_y = 0\n\\label{eq:advect2d-cons}\n\\end{equation}\nThis is the form we will integrate over zones.  As before, we will\ndefine the average of $a$ in a zone by integrating it over the\nvolume:\n\\begin{equation}\na_{i,j} = \\frac{1}{\\Delta x \\Delta y} \n   \\int_{x_{i-\\myhalf}}^{x_{i+\\myhalf}} \\int_{y_{j-\\myhalf}}^{y_{j+\\myhalf}} \n   a(x,y,t) \\, dx \\, dy\n\\end{equation}\nIntegrating Eq.~\\ref{eq:advect2d-cons} over $x$ and $y$, we have:\n\\begin{align}\n\\frac{1}{\\Delta x \\Delta y} \n  \\int_{x_{i-\\myhalf}}^{x_{i+\\myhalf}} \n  \\int_{y_{j-\\myhalf}}^{y_{j+\\myhalf}} a_t \\, dx \\, dy =  \n  &- \\frac{1}{\\Delta x \\Delta y}\n       \\int_{x_{i-\\myhalf}}^{x_{i+\\myhalf}} \\int_{y_{j-\\myhalf}}^{y_{j+\\myhalf}}\n      (u a)_x \\, dx \\, dy \\nonumber \\\\\n  &- \\frac{1}{\\Delta x \\Delta y}\n       \\int_{x_{i-\\myhalf}}^{x_{i+\\myhalf}} \\int_{y_{j-\\myhalf}}^{y_{j+\\myhalf}}\n      (v a)_y \\, dx \\, dy \n\\end{align}\nor using the divergence theorem,\n\\begin{align}\n\\label{eq:adv:multiddivergence}\n \\frac{\\partial a_{i,j}}{\\partial t} =\n  &- \\frac{1}{\\Delta x\\Delta y} \\int_{y_{j-\\myhalf}}^{y_{j+\\myhalf}}\n     \\left \\{ (u a)_{i+\\myhalf,j} - (u a)_{i-\\myhalf,j} \\right \\} dy \\nonumber \\\\\n  &- \\frac{1}{\\Delta x\\Delta y} \\int_{x_{i-\\myhalf}}^{x_{i+\\myhalf}}\n     \\left \\{ (v a)_{i,j+\\myhalf} - (v a)_{i,j-\\myhalf} \\right \\} dx\n\\end{align}\n\nNow we integrate over time---the left side of our expression becomes\njust the difference between the new and old state.\n\\begin{align}\n a_{i,j}^{n+1} - a_{i,j}^n = \n  &- \\frac{1}{\\Delta x\\Delta y} \\int_{t^n}^{t^{n+1}} \\int_{y_{j-\\myhalf}}^{y_{j+\\myhalf}}\n     \\left \\{ (u a)_{i+\\myhalf,j} - (u a)_{i-\\myhalf,j} \\right \\} dy dt \\nonumber \\\\\n  &- \\frac{1}{\\Delta x\\Delta y} \\int_{t^n}^{t^{n+1}} \\int_{x_{i-\\myhalf}}^{x_{i+\\myhalf}}\n     \\left \\{ (v a)_{i,j+\\myhalf} - (v a)_{i,j-\\myhalf} \\right \\} dx dt\n\\label{eq:update2du}\n\\end{align}\nWe define the flux through the interface as the average over the face\nof that interface and time: \\MarginPar{is there a way to illustrate this, as some space-time diagram?}\n\\begin{itemize}\n\\item x-face:\n\\begin{equation}\n\\langle (ua)_{i+\\myhalf,j}\\rangle_{(t)} = \\frac{1}{\\Delta y \\Delta t}\n    \\int_{t^n}^{t^{n+1}} \\int_{y_{j-\\myhalf}}^{y_{j+\\myhalf}} (ua)_{i+\\myhalf,j}\\, dy dt\n\\end{equation}\n\\item y-face\n\\begin{equation}\n\\langle (va)_{i,j+\\myhalf}\\rangle_{(t)} = \\frac{1}{\\Delta x \\Delta t}\n    \\int_{t^n}^{t^{n+1}} \\int_{x_{i-\\myhalf}}^{x_{i+\\myhalf}} (va)_{i,j+\\myhalf}\\, dx dt \n\\end{equation}\n\\end{itemize}\nwhere $\\langle . \\rangle_{(t)}$ denotes the time-average over the face.\nFor a second-order accurate method in time, we replace the average in\ntime with the flux at the midpoint in time and the average over the face\nwith the flux at the center of the face: \n\\begin{equation}\n\\langle (ua)_{i+\\myhalf,j} \\rangle_{(t)} \\approx (ua)_{i+\\myhalf,j}^{n+\\myhalf}\n\\end{equation}\nThen we have:\n\\begin{equation}\na_{i,j}^{n+1} = a_{i,j}^n - \\Delta t \\left [\n   \\frac{(ua)_{i+\\myhalf,j}^{n+\\myhalf} - (ua)_{i-\\myhalf,j}^{n+\\myhalf}}{\\Delta x} +\n   \\frac{(va)_{i,j+\\myhalf}^{n+\\myhalf} - (va)_{i,j-\\myhalf}^{n+\\myhalf}}{\\Delta y} \\right ]\n\\end{equation}\n\nFor the advection problem, since $u$ and $v$ are constant, we need\nto find the interface states of $a$ alone.\nThere are two methods for computing these interface states, \n$a_{i\\pm\\myhalf,j}^{n+\\myhalf}$ on $x$-interfaces and  $a_{i,j\\pm\\myhalf}^{n+\\myhalf}$ on $y$-interfaces:\ndimensionally split and unsplit.  Dimensionally split methods are\neasier to code, since each dimension is operated on independent of the\nothers, so you can simply call a one-dimensional method for each\ndirection.  Unsplit methods, however, are more accurate and less\nsusceptible to grid effects.  \\MarginPar{show an example + refs? Zalesak test?}\n\n\\subsection{Dimensionally split}\n\\label{adv:sec:dimensionalsplit}\n\nIn a split method, we update the state in each coordinate direction\nindependently.  This is simple and a straightforward way to use\none-dimensional methods in multi-d.  To be second-order accurate in\ntime, we do {\\em Strang splitting}~\\cite{strang}, where we alternate\nthe order of the dimensional updates each timestep.  An update through\n$\\Delta t$ consists of $x$ and $y$ sweeps and appears as:\n\\begin{eqnarray}\n \\frac{a_{i,j}^\\star - a_{i,j}^n}{\\Delta t} &=&\n  - \\frac{ u a_{i+\\myhalf,j}^{n+\\myhalf} - u a_{i-\\myhalf,j}^{n+\\myhalf} }{\\Delta x} \\label{eq:splitx}\\\\\n \\frac{a_{i,j}^{n+1} - a_{i,j}^\\star}{\\Delta t} &=&\n  - \\frac{ v a_{i,j+\\myhalf}^{\\star,n+\\myhalf} - v a_{i,j-\\myhalf}^{\\star,n+\\myhalf} }{\\Delta y} \\label{eq:splity}\n\\end{eqnarray}\nHere, Eq.~\\ref{eq:splitx} is the update in the $x$-direction.  In\nconstructing the interface states, $a_{i+\\myhalf,j}^{n+\\myhalf}$ and\n$a_{i-\\myhalf,j}^{n+\\myhalf}$, we do the exact same procedure as the\none-dimensional case, constructing the left and right states at each\ninterface and then solving the same Riemann problem to find the unique\nstate on the interface.  Each dimensional sweep is done without\nknowledge of the other dimensions.  For example, in the $x$-update, we\nare solving:\n\\begin{equation}\na_t + u a_x = 0\n\\label{eq:advect1dx}\n\\end{equation}\nand in the $y$-update, we are solving:\n\\begin{equation}\na_t + v a_y = 0\n\\end{equation}\n\nThe construction of the interface states largely mimics the one-dimensional\ncase (Eq.~\\ref{eq:statel} and \\ref{eq:stater}).  For example, the\n$a_{i+\\myhalf,j,L}^{n+\\myhalf}$ state is:\n\\begin{eqnarray}\na_{i+\\myhalf,j,L}^{n+\\myhalf} &=& a_{i,j}^n + \n  \\frac{\\Delta x}{2} \\left .\\frac{\\partial a}{\\partial x} \\right |_{i,j} + \n  \\frac{\\Delta t}{2} \\left .\\frac{\\partial a}{\\partial t} \\right |_{i,j} + \n  \\mathcal{O}(\\Delta x^2) + \\mathcal{O}(\\Delta t^2) \\nonumber \\\\\n &=& a_{i,j}^n + \n   \\frac{\\Delta x}{2} \\left .\\frac{\\partial a}{\\partial x} \\right |_{i,j} + \n   \\frac{\\Delta t}{2} \\left ( \n   - u \\left .\\frac{\\partial a}{\\partial x} \\right |_{i,j} \\right\n   ) + \\ldots \\nonumber \\\\\n    &=& a_{i,j}^n + \n   \\frac{\\Delta x}{2} \\left ( 1 - \\frac{\\Delta t}{\\Delta x} u \\right ) \n   \\left .\\frac{\\partial a}{\\partial x} \\right |_{i,j} +\n   \\ldots \\label{eq:statels}\n\\end{eqnarray}\nNotice that when substituting for $\\partial a / \\partial t$, we use\nthe one-dimensional split version of the advection equation\n(Eq.~\\ref{eq:advect1dx}) instead of the full multi-dimensional\nequation.  There are no $y$-direction terms that come into play in the\nsplit method when considering the $x$-direction.\n\nThe $x$-update\n(Eq.~\\ref{eq:splitx}) updates the state only accounting for the\n$x$-fluxes---we denote this intermediate state with the `$\\star$'\nsuperscript.  For the $y$-update, we construct our interface states in\nthe analogous way as in the $x$-direction, but begin with the `$\\star$'\nstate instead of the old-time state.  In this fashion, the $y$-update\n`sees' the result of the $x$-update and couples things together.\n\nTo achieve second-order accuracy in time, it is necessary to alternate\nthe directions of the sweeps each timestep, i.e., $x$-$y$ then $y$-$x$.\nFurthermore, this pair of sweeps should use the same timestep, $\\Delta t$.\n\n\n\\subsection{Unsplit multi-dimensional advection}\n\\label{adv:sec:unsplit_2d}\n\nThe unsplit case differs from the dimensionally split case in two\nways: (1) in predicting the interface states, we use knowledge of the\nflow in the transverse direction, and (2), only a single conservative\nupdate is done per timestep, with all directions updating\nsimultaneously.  See \\cite{colella:1990} for more details.  This idea\nis sometimes called the ``corner transport upwind'' or CTU method.\n\nThe construction of the $a_{i+\\myhalf,j,L}^{n+\\myhalf}$ interface state appears as\n\\begin{eqnarray}\na_{i+\\myhalf,j,L}^{n+\\myhalf} &=& a_{i,j}^n + \n  \\frac{\\Delta x}{2} \\left .\\frac{\\partial a}{\\partial x} \\right |_{i,j} + \n  \\frac{\\Delta t}{2} \\left .\\frac{\\partial a}{\\partial t} \\right |_{i,j} + \n  \\mathcal{O}(\\Delta x^2) + \\mathcal{O}(\\Delta t^2) \\nonumber \\\\\n &=& a_{i,j}^n + \n   \\frac{\\Delta x}{2} \\left .\\frac{\\partial a}{\\partial x} \\right |_{i,j} + \n   \\frac{\\Delta t}{2} \\left ( \n   - u \\left .\\frac{\\partial a}{\\partial x} \\right |_{i,j} \n   - v \\left .\\frac{\\partial a}{\\partial y} \\right |_{i,j} \\right\n   ) + \\ldots \\nonumber \\\\\n    &=& \\underbrace{a_{i,j}^n + \n   \\frac{\\Delta x}{2} \\left ( 1 - \\frac{\\Delta t}{\\Delta x} u \\right ) \n   \\left .\\frac{\\partial a}{\\partial x} \\right |_{i,j}}_{\\hat{a}_{i+\\myhalf,j,L}^{n+\\myhalf}} \\underbrace{-\n   \\frac{\\Delta t}{2} v \\left .\\frac{\\partial a}{\\partial y} \\right |_{i,j}}_{\\mathrm{``transverse~flux~difference''}} +\n   \\ldots \\label{eq:statelu}\n\\end{eqnarray}\nThe main difference between the split and unsplit interface states is the\nexplicitly appearance of the ``transverse flux difference'' in the unsplit\ninterface state.  We rewrite this as:\n\\begin{equation}\na_{i+\\myhalf,j,L}^{n+\\myhalf} = \\hat{a}_{i+\\myhalf,j,L}^{n+\\myhalf} \n   - \\frac{\\Delta t}{2} v \\left .\\frac{\\partial a}{\\partial y} \\right |_{i,j}\n\\end{equation}\nHere, the $\\hat{a}_{i+\\myhalf,j,L}^{n+\\myhalf}$ term is just the normal\nprediction without considering the transverse direction (e.g., Eq.~\\ref{eq:statels}).  The basic\nupdate procedure is:\n\\begin{itemize}\n\\item Construct the normal predictor term, $\\hat{a}_{i+\\myhalf,j,L}^{n+\\myhalf}$,\nin a fashion identical to the one-dimensional and split methods.  We\ncompute these one-dimensional $\\hat{a}$'s at the left and right every\ninterface in both coordinate directions.  Note that these states are\nstill one-dimensional, since we have not used any information from the\ntransverse direction in their computation.  \n\n\\item Solve a Riemann problem at each of these interfaces:\n\\begin{eqnarray}\na^T_{i+\\myhalf,j} &=& \\mathcal{R}(\\hat{a}_{i+\\myhalf,j,L}^{n+\\myhalf},\n                              \\hat{a}_{i+\\myhalf,j,R}^{n+\\myhalf}) \\\\\na^T_{i,j+\\myhalf} &=& \\mathcal{R}(\\hat{a}_{i,j+\\myhalf,L}^{n+\\myhalf},\n                              \\hat{a}_{i,j+\\myhalf,R}^{n+\\myhalf}) \\\\\n\\end{eqnarray}\nThese states are given the `$^T$' superscript since they are the states\nthat are used in computing the transverse flux difference.  \n\n\\item Correct the \npreviously computed normal interface states (the $\\hat{a}$'s) with\nthe transverse flux difference:\n\\begin{equation}\na_{i+\\myhalf,j,L}^{n+\\myhalf} = \\hat{a}_{i+\\myhalf,j,L}^{n+\\myhalf} \n   - \\frac{\\Delta t}{2} v \\frac{a^T_{i,j+\\myhalf} - a^T_{i,j-\\myhalf}}{\\Delta y}\n\\end{equation}\nNotice that the\nfluxes that are differenced for the left state are those that are\ntransverse, but to the left of the interface.  Similarly, for the\nright state, it would be those that are transverse, but to the right\nof the interface:\n\\begin{equation}\na_{i+\\myhalf,j,R}^{n+\\myhalf} = \\hat{a}_{i+\\myhalf,j,R}^{n+\\myhalf} \n   - \\frac{\\Delta t}{2} v \\frac{a^T_{i+1,j+\\myhalf} - a^T_{i+1,j-\\myhalf}}{\\Delta y}\n\\end{equation}\n\\end{itemize}\nA similar procedure happens at the $y$-interfaces.  \n\nFigure~\\ref{fig:unsplitstates} illustrates the steps involved in\nthe construction of the $a_{i+\\myhalf,j,L}^{n+\\myhalf}$ state.  \\MarginPar{more panels illustrating the sequence?}\n%\n% this is created by figures/advection/2dgrid-hat.py\n% and                                  2dgrid-transverse.py\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.7\\linewidth]{2dgrid-hat} \\\\\n\\includegraphics[width=0.7\\linewidth]{2dgrid-transverse}\n\\caption[The construction of an interface state with the transverse\n  component]{\\label{fig:unsplitstates} The construction of the\n  $a_{i+\\myhalf,j,L}^{n+\\myhalf}$ state.  Top: first we compute the\n  $\\hat{a}$'s---here we show all of the $\\hat{a}$'s that will be used\n  in computing the full left interface state at $(i+\\myhalf,j)$.  Bottom:\n  after the transverse Riemann solves, we have the two transverse\n  states ($a^T_{i,j+\\myhalf}$ and $a^T_{i,j-\\myhalf}$) that will be\n  differenced and used to correct $\\hat{a}_{i+\\myhalf,j,L}^{n+\\myhalf}$\n  (illustrated by the dotted lines) to make $a_{i+\\myhalf,j,L}^{n+\\myhalf}$.}\n\\end{figure}\n\nOnce all of the full states (normal prediction $+$ transverse flux\ndifference) are computed to the left and right of all the interfaces\n($x$ and $y$), we solve another Riemann problem to find the final \nstate on each interface.\n\\begin{equation}\na_{i+\\myhalf,j}^{n+\\myhalf} = \\mathcal{R}(a_{i+\\myhalf,j,L}^{n+\\myhalf},\n                                  a_{i+\\myhalf,j,R}^{n+\\myhalf}) \\\\\n\\end{equation}\nThe final conservative update is then done via Eq.~\\ref{eq:update2du}.\n\nSee \\cite{colella:1990} for more details on this unsplit method, \nand \\cite{saltzman:1994} for details of the extension to 3-d.\n\nFigures~\\ref{fig:adv:gaussian-2d} and \\ref{fig:adv:tophat-2d} show the\nadvection of a smooth Gaussian and a discontinuous tophat diagonally on\na coarse $32\\times 32$ zone domain.  Each show a diffusion of the\ninitial function, similar to what we saw in 1-d.  In the tophat, we\nalso see a slight undershoot on the trailing side of the tophat after\none period.\n\n\n\n\n% run as: ./pyro.py advection smooth inputs.smooth\n% visualized as: ./plot.py -W 7.7 -H 6 -o smooth_init.pdf advection smooth_0000\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.48\\linewidth]{smooth_init}\\hspace{1em}\n\\includegraphics[width=0.48\\linewidth]{smooth_final}\n\\caption[Advection of Gaussian profile in 2-d]\n  {\\label{fig:adv:gaussian-2d} Advection of a Gaussian profile\n  with $u = v = 1$ for one period on a $32\\times 32$ grid, with $\\cfl =\n  0.8$.  This was run with \\pyro\\ as {\\tt ./pyro.py advection smooth\n    inputs.smooth}}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.48\\linewidth]{tophat_init}\\hspace{1em}\n\\includegraphics[width=0.48\\linewidth]{tophat_final}\n\\caption[Advection of tophat profile in 2-d]{\\label{fig:adv:tophat-2d} Advection of a tophat function with\n  $u = v = 1$ for one period on a $32\\times 32$ grid, with $\\cfl = 0.8$.\n  This was run as {\\tt ./pyro.py advection tophat inputs.tophat}}\n\\end{figure}\n\n\n\\subsection{Timestep limiter for multi-dimensions}\n\\label{sec:adv:timestep}\n\nThe timestep criterion we used in one-dimension\n(Eq.~\\ref{eq:timestep}) needs to be generalized for multi-dimensions.\nFor the dimensionally split case, since we are basically piecing\ntogether two one-dimensional sweeps, the timestep limit is usually taken\nas:\n\\begin{equation}\n\\label{eq:adv:timestep:multid}\n\\Delta t = \\cfl \\min \\left \\{ \\frac{\\Delta x}{|u|} , \\frac{\\Delta y}{|v|} \\right \\}\n\\end{equation}\nThis is, for example, the construction that is used with the\ndimensionally-split hydrodynamics solver in the original Flash\ncode~\\cite{flash}\\footnote{although, the paper does not explicitly\n  write this out}.  \\MarginPar{does the prometheus MPA 449 write it out?}\nStability requires picking a CFL number, $\\cfl \\le 1$.\n\nA general extension of the timestep restriction for a\nfully-multi-dimensional algorithm is often written in the form (see,\ne.g., \\cite{shuosher:1989b})\n\\begin{equation}\n\\label{eq:adv:timstep:mol}\n\\Delta t = \\cfl \\left ( \\sum_{i=1}^d \\frac{|\\Ub \\cdot \\mathbf{e}_d|}{\\Delta x_d} \\right )^{-1}\n\\end{equation}\nwhere $d$ is the coordinate direction, $\\Ub = (u, v)$ is the velocity\nvector, and $\\mathbf{e}_d$ is the unit vector in direction $d$.  For\nthese methods $\\cfl \\lesssim 1$.  This is usually more restrictive than\nEq.~\\ref{eq:adv:timestep:multid}.\n\nFor the CTU method described above, \\cite{colella:1990} argues that\nthe inclusion of the transverse information removes some\nof the instability inherent in simpler schemes, allowing for a larger\ntimestep, restricted by Eq.~\\ref{eq:adv:timestep:multid}.\n\nNote, because of the different form of the timestep limiters here, it\nis not enough to simply cite a CFL number when describing results, you\nmust also explain which form of the timestep calculation\n(Eq.~\\ref{eq:adv:timestep:multid} or Eq.~\\ref{eq:adv:timstep:mol}) is\nbeing used.\n\n\\subsection{Method-of-lines in multi-dimensions}\n\nThe multidimensional version of method-of-lines integration\nfollows the ideas from the one-dimensional case.\nWe can start with Eq.~\\ref{eq:adv:multiddivergence}, and define\nthe fluxes over though a face as:\n\\begin{itemize}\n\\item x-face:\n\\begin{equation}\n\\langle (ua)_{i+\\myhalf,j} \\rangle = \\frac{1}{\\Delta y}\n    \\int_{y_{j-\\myhalf}}^{y_{j+\\myhalf}} (ua)_{i+\\myhalf,j}\\, dy \n\\end{equation}\n\\item y-face\n\\begin{equation}\n\\langle (va)_{i,j+\\myhalf} \\rangle = \\frac{1}{\\Delta x}\n    \\int_{x_{i-\\myhalf}}^{x_{i+\\myhalf}} (va)_{i,j+\\myhalf}\\, dx \n\\end{equation}\n\\end{itemize}\nThese are similar to the expressions above, except there is no\nintegral over time.  Again, to second order in space, we can just\nuse the flux evaluated at the midpoint in the face:\n\\begin{equation}\n\\langle (ua)_{i+\\myhalf,j} \\rangle \\approx (ua)_{i+\\myhalf,j}\n\\end{equation}\nThe result is the ODE:\n\\begin{equation}\n\\frac{d a_{i,j}}{d t} =\n   - \\frac{(ua)_{i+\\myhalf,j} - (ua)_{i-\\myhalf,j}}{\\Delta x}\n   - \\frac{(va)_{i,j+\\myhalf} - (va)_{i,j-\\myhalf}}{\\Delta y}\n\\end{equation}\n\nFor the interface states, we separately construct slopes in each\ndimension: $\\Delta^{(x)}a_{i,j}$ is the slope in the $x$-direction,\nand $\\Delta^{(y)}a_{i,j}$ is the slope in the $y$-direction.  Then\nthe interface states are:\n\\begin{align}\na_{i+\\myhalf,j,L} &= a_{i,j} + \\frac{1}{2} \\Delta a^{(x)}_{i,j} \\\\\na_{i+\\myhalf,j,R} &= a_{i+1,j} - \\frac{1}{2} \\Delta a^{(x)}_{i+1,j}\n\\end{align}\nand\n\\begin{align}\na_{i,j+\\myhalf,L} &= a_{i,j} + \\frac{1}{2} \\Delta a^{(y)}_{i,j} \\\\\na_{i,j+\\myhalf,R} &= a_{i,j+1} - \\frac{1}{2} \\Delta a^{(y)}_{i,j+1}\n\\end{align}\n\nWith these interface states, we have the method for evaluating the\nrighthand side of our ODE.  We can then evolve the ODE using a\nRunge-Kutta integration method.  At each stage of the RK integration,\nwe do the same construction of the interface states, solve the Riemann\nproblem, etc.  Notice that unlike the directionally-split method, we\nare doing the update in both directions at the same time---we are\nrelying on the ODE integrator to do the coupling in the different\ndirections for us.\n\nThis construction is a lot simpler than the unsplit method described\npreviously.  The cost of this simplicity is a reduced range of\nstability.\n\nPerformaing the stability analysis here is much more complicated.  For\nsimplicity, consider a simple first-order upwind scheme in\ntwo-dimensions.  Higher order time integrators (like RK4) will have\nseparate stages that look like a first-order-in-time update, so this\nstability analysis can be thought of as applying to a single stage.\nOur difference equation is:\n\\begin{equation}\n\\frac{a^{n+1}_{i,j} - a^n_{i,j}}{\\Delta t} = - u \\frac{a^n_{i,j} - a^n_{i-1,j}}{\\Delta x}\n- v \\frac{a^n_{i,j} - a^n_{i,j-1}}{\\Delta y}\n\\end{equation}\nWe introduce a single-mode Fourier function, as before, but this time\nwith a mode in each direction:\n\\begin{equation}\na^n_{i,j} = A^n e^{Ii\\theta}e^{Ij\\phi}\n\\end{equation}\nNext, to simplify things, we'll assume that $u = v$, and $\\Delta x = \\Delta y$, \nthen $\\cfl = u \\Delta t/ \\Delta x = v \\Delta t / \\Delta y$.\nInserting this into our difference equation gives:\n\\begin{equation}\n\\frac{A^{n+1}}{A^n} = 1 - \\cfl \\left (1 - e^{-I\\theta} \\right ) - \\cfl \\left (1 - e^{-I\\phi} \\right )\n\\end{equation}\nThis form looks very close to the one-dimensional case, but it is\ndifficult to analytically find the values of $\\cfl$, but we can\nnumerically find the maximum $ | {A^{n+1}}/{A^n} |^2$ for $\\theta \\in\n[0, 2\\pi]$ and $\\phi \\in [0, 2\\pi]$ as a function of $\\cfl$.  In doing\nthis, we find that this discretization is unstable ($| {A^{n+1}}/{A^n}\n|^2 > 1$) for $\\cfl > 1/2$\\footnote{See the script\n  \\href{https://github.com/zingale/hydro_examples/blob/master/compressible/cfl.py}{cfl.py}}.\n\nThis restriction is $1/d$, where $d$ is the dimensionality, and it is\nanalogous to the difference between the timestep limits for stability\nfrom the more generous Eq.~\\ref{eq:adv:timestep:multid} to the more\nrestrictive Eq.~\\ref{eq:adv:timstep:mol}.\nThis reduced $\\cfl$ for stability is discussed in\n\\cite{shuosher:1989b} (see section 4), and is also discussed in\n\\cite{titarevtoro,mccorquodalecolella}.  Different reconstruction and\nintegration methods can vary this limit some, for instance, the fourth-order\nmethod in \\cite{mccorquodalecolella} allows for $\\Delta t \\sum_{i=1}^d (|\\Ub \\cdot \\mathbf{e}_d|)/\\Delta x_d  \\lesssim 1.4$.  Total variation diminishing Runge-Kutta methods are popular\nfor this application~\\cite{gottliebshu:1996}.\n\n% run as: \n%   ./pyro.py advection_rk tophat inputs.tophat\n%   ./plot.py -W 7.7 -H 6 -o tophat_rk4_cfl08.pdf advection_rk tophat_0040\n%\n%   ./pyro.py advection_rk tophat inputs.tophat driver.cfl=0.4\n%   ./plot.py -W 7.7 -H 6 -o tophat_rk4_cfl04.pdf advection_rk tophat_0080\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.48\\linewidth]{tophat_rk4_cfl08}\\hspace{1em}\n\\includegraphics[width=0.48\\linewidth]{tophat_rk4_cfl04}\n\\caption[Advection of tophat function with method-of-lines integration]\n  {\\label{fig:adv:tophat-2d-rk4} Advection of a tophat function using\n  the method-of-lines approach with 4th-order Runge-Kutta time integration.\n  We use $u = v = 1$ for one period on a $32\\times 32$ grid.  On the left\n  is $\\cfl = 0.8$---this is clearly unstable.  On the right is $\\cfl = 0.4$,\n  which is stable.  \n  This was run as {\\tt ./pyro.py advection\\_rk tophat inputs.tophat}}\n\\end{figure}\n\n\n\n\\section{High-Order Finite difference methods}\n\\label{sec:ho-fd}\n\nChapter~\\ref{ch:advection} introduced numerical methods for the linear\nadvection equation\n\\begin{equation}\n\\label{eq:ho-advect}\na_t + u a_x = 0\n\\end{equation}\nthat were first or second order in space. The advantages of using numerical\nmethods that are higher order can be large, especially when in regions where\nthe flow is smooth. However, these must be balanced against the additional\ncomputational cost and complexity. In addition, many higher order methods do no\nbetter than second order methods in cases with discontinuities.\n\n\\subsection{The problem with higher-order finite volume methods}\n\nThe finite volume approach outlined in chapter~\\ref{ch:fv}\n%\\todo{label missing in fv tex file?}\nmatches the physics of hyperbolic balance\nlaws perfectly, by maintaining the conservation of appropriate quantities using\nthe fluxes through the surfaces of each control volume or computational cell.\nHowever, this leads to significant computational costs and complexities when\ngoing beyond second order methods.\n\nThe inter-cell fluxes are computed by integrals over the face of the\ncomputational cell. In one dimension the face is a single point and the\nintegral needs the flux evaluated at a single point, as in\nequation~\\ref{eq:consup}. In higher dimensions the face is one dimensional\n(e.g., a line) or two dimensional (e.g., a square) and so the integral must be\napproximated by evaluating the flux at \\emph{multiple} points. If we use Gauss\nquadrature, this means that a third order method would require at least two\nflux evaluations per face in two dimensions, and typically four evaluations per\nface in three dimensions.\n\n%\n% This comment about the timestep doesn't hold in general, as pointed out by\n% Mike. I think it *is* relevant for directionally split codes where the\n% transverse Riemann Problem isn't solved. That's probably too technical a\n% distinction to make this early.\n%\n%As well as the additional computational cost, there is also typically an\n%additional restriction on the timestep. Intuitively this can be seen by\n%thinking about the flux evaluations again. We fix the values either side of the\n%cell boundary in order to compute the flux at one point.  The timestep\n%restriction is given by the time it take for information to propagate from a\n%different flux evaluation point, which would change the information available.\n%As we need to compute the flux at more points within the cell faces, the\n%distance between points where the flux is evaluated goes down, reducing the\n%time it takes for information to propagate from one point to another.\n\n\\subsection{Finite differences}\n\nThe conservative finite difference method starts from the endpoint for finite\nvolume methods, by considering the update formula~\\eqref{eq:consup}\n%\\todo{is this the right equation to ref? Don't think it's labelled?}\n\\begin{equation}\n\\tag{\\ref{eq:consup}}\n\\frac{\\partial \\langle \\Uc\\rangle_{i}}{\\partial t} =\n  - \\frac{1}{\\Delta x} \\left \\{ \\Fb_{i+\\myhalf} -\n                                \\Fb_{i-\\myhalf} \\right \\}.\n\\end{equation}\nIn the finite difference method the interpretation used is different. We are now\nthinking of\n\\begin{equation}\n  \\label{eq:fd-deriv}\n  \\frac{1}{\\Delta x} \\left \\{ \\Fb_{i+\\myhalf} -\n                                \\Fb_{i-\\myhalf} \\right \\}\n\\end{equation}\nas representing a direct approximation to the derivative of the flux at $x_i$.\nIn particular, we are \\emph{not} thinking of $\\Fb_{i+\\myhalf}$ as being\ndirectly linked to the flux. Crucially the update still ensures global\nconservation, as the ``flux'' term is re-used for neighbouring points in the\nsame way as in the finite volume case.\n\nWhen we combine this viewpoint with the Method of Lines approach in\nsection~\\ref{adv:sec:mol_2d}, the implementation of a high-order finite boils\ndown to\n\\begin{itemize}\n  \\item using a high-order time integration scheme (such as a high order\n  Runge-Kutta method);\n  \\item finding a way that the finite difference in equation~\\eqref{eq:fd-deriv}\n  approximates the derivative of the flux both stably, and to a sufficiently\n  high order.\n\\end{itemize}\n\n\\subsection{WENO reconstruction}\n\\label{sec:WENO}\n\n\\begin{figure}\n  \\includegraphics[width=0.96\\linewidth]{weno-convergence}\n  \\caption[Convergence rate of high-order reconstructions]{A smooth sine function is reconstructed on the unit interval from the (integral-average) data, at the blue circles in the top plot, to the right, at the red triangles in the top plot. The convergence rate with grid resolution for the 5 point stencil of equation~\\eqref{eq:weno-advection-5pt} and the 3 point stencils of equation~\\eqref{eq:weno-advection-3pt} are shown to match expectations in the bottom plots. The error from the WENO reconstruction described in section~\\ref{sec:WENO-algorithm} is, on this smooth function, converging with the expected order but larger in magnitude.}\n  \\label{fig:weno-convergence}\n\\end{figure}\n\nFor the advection equation with constant advection speed $u$, our higher-order\nmethod requires us to find a high-order approximation to $a_{i+\\myhalf}$. If we\nconsidered the five points $a_{j}, j \\in [i-2, \\dots, i+2]$ then we could use\nthe information from all five of these points to approximate $a_{i+\\myhalf}$ as\n\\begin{equation}\n  \\label{eq:weno-advection-5pt}\n  a^{(\\text{Fifth})}_{i+\\myhalf} = \\tfrac{1}{60} \\left( 2 \\langle a \\rangle_{i-2} - 13 \\langle a \\rangle_{i-1} + 47 \\langle a \\rangle_{i} + 27 \\langle a \\rangle_{i+1} - 3 \\langle a \\rangle_{i+2} \\right) + {\\cal O}(\\Delta x^5).\n\\end{equation}\nAlternatively we could use only three of the points, in three different ways:\n\\begin{subequations}\n  \\label{eq:weno-advection-3pt}\n  \\begin{align}\n    a^{(\\text{Third}, 2)}_{i+\\myhalf} &= \\tfrac{1}{6} \\left( 2 \\langle a \\rangle_{i-2} - 7 \\langle a \\rangle_{i-1} + 11 \\langle a \\rangle_{i} \\right) + {\\cal O}(\\Delta x^3), \\\\\n    a^{(\\text{Third}, 1)}_{i+\\myhalf} &= \\tfrac{1}{6} \\left( - \\langle a \\rangle_{i-1} + 5 \\langle a \\rangle_{i} + 2 \\langle a \\rangle_{i+1} \\right) + {\\cal O}(\\Delta x^3), \\\\\n    a^{(\\text{Third}, 0)}_{i+\\myhalf} &= \\tfrac{1}{6} \\left( 2 \\langle a \\rangle_{i} + 5 \\langle a \\rangle_{i+1} - \\langle a \\rangle_{i+2} \\right) + {\\cal O}(\\Delta x^3).\n  \\end{align}\n\\end{subequations}\n\nWhich is better? The five point stencil gives higher order accuracy, but\nincludes information from both sides of the point. This ignores information from\nthe characteristics and, near discontinuities, will lead to Gibbs oscillations.\nThe three point stencils give lower accuracy. However, they give us the\nflexibility to choose a stencil to avoid discontinuities, possibly using the\ncharacteristic information. The advantages of the higher-order methods are illustrated by reconstructing a smooth function in figure~\\ref{fig:weno-convergence}. The disadvantages of using a fixed stencil are illustrated by reconstructing a non-smooth function in figure~\\ref{fig:weno-weights}.\n\n\n\\begin{exercise}[ENO stencils]\n{Check that equations~\\eqref{eq:weno-advection-5pt} and\n\\eqref{eq:weno-advection-3pt} give the claimed order of accuracy results.}\n\\end{exercise}\n\nThe method of choosing the ``best'' stencil (to avoid discontinuities as far as\npossible) is called \\emph{Essentially Non-Oscillatory} (ENO) reconstruction. It\nis not as accurate as it could be given the size of the stencil it uses, and\nthe logical branches required to find the ``best'' stencil can reduce\ncomputational performance.\n\nAn alternative that avoids the problems of ENO schemes are the \\emph{Weighted}\nENO, or WENO, schemes. These use a combination of the possible ENO stencils to\nreconstruct $a_{i+\\myhalf}$. This relies on the observation that there are\nconstants $C_k$ such that, for example,\n\\begin{equation}\n  \\label{eq:WENO-C-weights}\n  a^{(\\text{Fifth})}_{i+\\myhalf} = \\sum_{k=0}^{2} C_k a^{(\\text{Third}, r)}_{i+\\myhalf}.\n\\end{equation}\n\n\\begin{exercise}[WENO weights]\n{Check that the constants $C_0 = \\tfrac{3}{10}, C_1 = \\tfrac{3}{5}, C_2 =\n\\tfrac{1}{10}$ make equation~\\eqref{eq:WENO-C-weights} consistent with\nequations~\\eqref{eq:weno-advection-5pt} and \\eqref{eq:weno-advection-3pt}.}\n\\end{exercise}\n\nThe WENO methods work by retaining the sum over all stencils as in\nequation~\\eqref{eq:WENO-C-weights}, but adjusting the weights to avoid\noscillations. Ideally, in regions where the $a^{(0,1)}$ stencils include a\nshock but the $a^{(2)}$ stencil does not the weights should be $\\tilde{C}_0 = 0\n= \\tilde{C}_1$ and $\\tilde{C}_2 = 1$. In order for this sum to make sense we\nwill need the weights to add to $1$.\n\n\\subsubsection{WENO algorithm}\n\\label{sec:WENO-algorithm}\n\n\\begin{figure}\n  \\includegraphics[width=0.96\\linewidth]{weno-weights}\n  \\caption[WENO reconstruction and weights]{A non-smooth function is reconstructed on the unit interval. The 5 point stencil of equation~\\eqref{eq:weno-advection-5pt} and the 3 point stencils of equation~\\eqref{eq:weno-advection-3pt} both show oscillations near the discontinuity, whilst the WENO reconstruction does not. The WENO algorithm depends on measuring the smoothness of the function. The Jiang-Shu smoothness indicators $\\beta$ are mostly zero, but spike near the discontinuity. This feeds into the weight $\\omega$ that the WENO algorithm gives to each 3 point stencil it uses. Away from the discontinuity the weights revert to the optimal $C$ weights, but near the discontinuity a single $\\omega$ weight dominates meaning only a single 3 point stencil is used.}\n  \\label{fig:weno-weights}\n\\end{figure}\n\nWe can now write out the full WENO algorithm of order $(2 r - 1)$. The constant\n$r$ sets the size of the individual stencils as well as the (optimal) order of\nthe scheme. This section mostly uses the notation of Gerolymos et al.~\\cite{Gerolymos2009}, and we will drop the integral average notation (so $\\langle a \\rangle_i$ will be denoted $a_i$).\n\nFirst compute the individual stencils\n\\begin{equation}\n  \\label{eq:weno-stencils}\n  a_{r, k, i+\\myhalf} = \\sum_{\\ell=0}^{r-1} A_{r, k, \\ell} a_{i+k-\\ell}, \\quad k = 0, \\dots, r-1.\n\\end{equation}\nHere the $A_{r, k, \\ell}$ terms are constants which, for given scheme accuracy\n$r$ are needed to compute the $k^{\\text{th}}$ stencil.\n\nWe then want to combine these individual stencils to compute the WENO result.\nFor this we write\n\\begin{equation}\n  \\label{eq:weno-reconstruction}\n  a_{r, WENO, i+\\myhalf} = \\sum_{k=0}^{r-1} \\omega_{r, k, i+\\myhalf} a_{r, k, i+\\myhalf}.\n\\end{equation}\nHere the $\\omega_{r, k, i+\\myhalf}$ terms are the weights that vary from point\nto point. They must sum to one, so that we have a convex combination of the\nstencils:\n\\begin{equation}\n  \\label{eq:weno-convex}\n  \\sum_{k=1}^{r-1} \\omega_{r, k, i+\\myhalf} = 1.\n\\end{equation}\nWe also want the weights to match the \\emph{optimal} weights $C_{r, k}$ in\nsmooth regions. The optimal weights are defined such that\n\\begin{equation}\n  \\label{eq:weno-optimal}\n  a_{r, WENO, i+\\myhalf} = \\sum_{k=0}^{r-1} C_{r, k} a_{r, k, i+\\myhalf}\n\\end{equation}\nis accurate to order ${\\cal O}(\\Delta x^{(2 r - 1)})$ when $q$ is smooth.\n\nThe \\emph{choice} of how to get from the optimal weights $C_{r, k}$ to the\nnonlinear weights $\\omega_{r, k, i+\\myhalf}$ defines the WENO scheme. The\nstandard choice of Jiang and Shu is to introduce a measure of the smoothness of\nthe $k^{\\text{th}}$ stencil by computing the sum of the integral averages of\nits derivatives from order $1$ up to order $2 r - 1$. For practical\nimplementation purposes this means computing the \\emph{smoothness indicators}\n$\\beta_{r, k, i+\\myhalf}$ as\n\\begin{equation}\n  \\label{eq:weno-smoothness}\n  \\beta_{r, k, i+\\myhalf} = \\sigma_{r, k, \\ell, m} a_{i+k-\\ell} a_{i+k-m}.\n\\end{equation}\nThe terms $\\sigma_{r, k, \\ell, m}$ are pre-computed constants which give the\ncoefficients in the quadratic form for the $k^{\\text{th}}$ stencil of width $r$\nin the reconstruction. These smoothness indicators are non-negative, and will\nbe large when the derivatives are large in magnitude. The Jiang and Shu weights\nare then set by\n\\begin{subequations}\n  \\label{eq:weno-nonlinear-weights}\n  \\begin{align}\n    \\label{eq:weno-nonlinear-weights-omega}\n    \\omega_{r, k, i+\\myhalf} &= \\frac{\\alpha_{r, k, i+\\myhalf}}{\\sum_{k=0}^{r-1} \\alpha_{r, k, i+\\myhalf}}, \\\\\n      \\label{eq:weno-nonlinear-weights-alpha}\n    \\alpha_{r, k, i+\\myhalf} &= \\frac{C_{r, k}}{\\epsilon + \\beta_{r, k, i+\\myhalf}^2}.\n  \\end{align}\n\\end{subequations}\nHere $\\epsilon$ is a small number introduced to avoid division-by-zero problems.\nThe form of $\\omega_{r, k, i+\\myhalf}$ in\nequation~\\eqref{eq:weno-nonlinear-weights-omega} guarantees that the convex sum\ncondition in equation~\\eqref{eq:weno-convex} holds. The form of $\\alpha_{r, k,\ni+\\myhalf}$ ensures that when all the smoothness indicators $\\beta_{r, k,\ni+\\myhalf}$ are the same magnitude the weights $\\omega_{r, k, i+\\myhalf}$ will\nmatch the optimal weights $C_{r, k}$, but when a smoothness indicator is large\n(i.e., when the associated stencil has large derivatives, typically associated\nwith discontinuities), the contribution of its associated stencil will be small.\nAn example of this is shown in figure~\\ref{fig:weno-weights}.\n\nFinally, we note that this method has reconstructed $a_{i+\\myhalf}$ from the\nleft. For the advection equation case where the advection speed $u$ is positive\nthis is all we need: the characteristic information tells us that we should use\nthis reconstruction. In the case where the advection speed is negative we\nshould reconstruct from the right. In the implementation it is easiest to\nimplement only one reconstruction direction and pass the data in in reverse\norder.\n\n\n\\begin{exercise}[WENO reconstruction]\n{Taking the values for the $C$, $A$ and $\\sigma$ constants from Gerolymos et\nal.\\ or Shu's review~\\cite{Shu1997}, construct a WENO reconstruction method for\n$r=2$. Check your results on smooth and non-smooth functions.}\n\\end{exercise}\n\n\\subsubsection{Implementation issues with WENO schemes}\n\\label{sec:weno-implementation}\n\nA convergence test on the WENO schemes with $r=3$ and $r=5$ is shown in\nfigure~\\ref{fig:weno-converge-gaussian-rk4}. This should be compared to\nfigure~\\ref{fig:advnorm}, as in both a Gaussian profile is advected five\ntimes around a periodic domain. The WENO schemes show higher absolute accuracy\nfor moderate size grids ($N \\gtrsim 100$) and show faster convergence.\n\nHowever, it is clear that whilst the $r=3$ method converges at fifth order as\nexpected, the $r=5$ WENO method does not converge at ninth order but at fourth\norder. This is explained by the time integrator. We remember that any solver\nhas an error both from the spatial and the time discretization. In\nfigure~\\ref{fig:weno-converge-gaussian-rk4} the time integrator is the classic\nfourth order Runge-Kutta, and it is the error from this integrator that is\ndominating.\n\nTo confirm this, figure~\\ref{fig:weno-converge-gaussian} uses the eigth order\nDormand-Price Runge-Kutta method with adaptive step size control (using the\n\\texttt{scipy.integrate.ode} routine). Here we see high order convergence for\nall $r$, and in each case the convergence rate is $2 r - 2$ for \\textbf{no\nreason I can understand}.\n\n\\begin{figure}[t]\n\\centering\n% figure generated by hydro_examples/advection/weno.py\n\\includegraphics[width=0.8\\linewidth]{weno-converge-gaussian-rk4}\n\\caption[High order WENO convergence rates for linear advection]\n{\\label{fig:weno-converge-gaussian-rk4} WENO solutions for advecting a Gaussian five periods, using two different orders. A fourth order Runge-Kutta method is used for time evolution. For $r=3$ we see the expected fifth ($2 r - 1$) order convergence. For $r=5$ we see fourth order, rather than the expected ninth order, convergence. This is as the error from the time integrator dominates. \\\\\n\\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/advection/weno.py}{weno.py}}}\n\\end{figure}\n%\n\n\\begin{figure}[t]\n\\centering\n% figure generated by hydro_examples/advection/weno.py\n\\includegraphics[width=0.8\\linewidth]{weno-converge-gaussian}\n\\caption[Very high order WENO convergence rates for linear advection]\n{\\label{fig:weno-converge-gaussian} WENO solutions for advecting a Gaussian one periods, using four different orders. An eighth order Dormand-Price Runge-Kutta method is used for time evolution. This minimizes the time integrator error and we see convergence at order $2 r - 2$ for all schemes, although the time integrator error eventually shows in the highest order case. \\\\\n\\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/advection/weno.py}{weno.py}}}\n\\end{figure}\n%\n\n\n\\section{Going further}\n\n\\begin{itemize}\n\n\\item {\\em Slope limiting}: there are a wide variety of slope\n  limiters.  All of them are designed to reduce oscillations in the\n  presence of discontinuities or extrema, but some are higher-order\n  and can be less restrictive when dealing with smooth flows.  Most\n  hydro texts (e.g.\\ \\cite{leveque:2002,toro:1997}) provide an\n  introduction to the design of such limiters.\n\n\\item {\\em Multi-dimensional limiting}: the procedure described above\n  still does the limiting in each dimension independent of the other\n  when doing the unsplit reconstruction.  This can lead to overshoots/\n  undershoots.  An example of a method that considers the limiting\n  in multi-dimensions is \\cite{BDS,quadBDS}.\n\n\\item {\\em Spatially-varying velocity field}: if we consider a spatially\n  varying velocity field, $u(x,y)$ and $v(x,y)$ that is specified externally,\n  then we can describe the advection of a quantity $\\phi$ as:\n  \\begin{equation}\n  \\phi_t + (\\phi u)_x + (\\phi v)_y = 0\n  \\end{equation}\n  The solution procedure is largely the same as described above.  We write:\n\\begin{align}\n\\phi_{i+\\myhalf,j,L}^{n+\\myhalf} &= \\phi_{i,j}^n + \n   \\frac{\\Delta x}{2} \\frac{\\partial \\phi}{\\partial x} +\n   \\frac{\\Delta t}{2} \\frac{\\partial \\phi}{\\partial t} + \\ldots \\nonumber \\\\\n%\n &= \\phi_{i,j}^n + \n    \\frac{\\Delta x}{2} \\frac{\\partial \\phi}{\\partial x} +\n    \\frac{\\Delta t}{2} \\left [ -(\\phi u)_x -(\\phi v)_y \\right ]_{i,j} \\nonumber\\\\\n%\n &= \\underbrace{\\phi_{i,j}^n + \n   \\frac{\\Delta x}{2} \\left ( 1 - \\frac{\\Delta t}{\\Delta x} u_{i,j} \\right )\n        \\frac{\\partial \\phi}{\\partial x}}_{\\hat{\\phi}_{i+\\myhalf,j,L}^{n+\\myhalf}}\n   - \\frac{\\Delta t}{2} \\left [\\phi u_x \\right ]_{i,j} \n   - \\frac{\\Delta t}{2} \\left [ (\\phi v)_y \\right ]_{i,j}\n\\end{align}\n  and upwinding is used to resolve the Riemann problem for both the\n  transverse and normal interface states.  This type of construction\n  comes up in low Mach number flows, where the density can be advected\n  according to the velocity field in much the fashion shown here, and\n  is described in \\S~\\ref{sec:lm:density}.\n\n  \n  For compressible hydrodynamics, we often have density-weighted quantities\n  that we advect.  This extension is described in \\S~\\ref{sec:euler:further}.\n\n\\end{itemize}\n\n\n\\section{\\pyro\\ experimentation}\n\nTo gain some experiences with these ideas, we can use the advection \nsolver in \\pyro\\ (see Appendix~\\ref{app:pyro} to get started).\nThe \\pyro\\ advection solver implements the second-order unsplit\nadvection algorithm described in the previous sections.  To run\nthis solver on the Gaussian advection problem, do:\n\\begin{verbatim}\n./pyro.py advection smooth inputs.smooth\n\\end{verbatim}\nBy default, this will advect a Gaussian profile diagonally across the\ndomain for a single period.\n\nTo get a feel for the advection algorithm, here are some suggested\nexercises:\n\n\\begin{exercise}[Role of limiters]\n{Implement a tophat initial profile and run with and without limiters\n  (this is controlled by the {\\tt advection.limiter} runtime\n  parameter).}\n\\end{exercise}\n\n\\begin{exercise}[Grid effects]\n{Look at the solution when you advect purely in the $x$- or\n  $y$-direction and compare to the diagonal case---notice how the\n  direction affects the error in the solution.  This is the imprint\n  of the grid we discretize on.}\n\\end{exercise}\n\n\\begin{exercise}[Split vs.\\ unsplit]\n{Implement a dimensionally-split version of the advection algorithm\nand compare the results to the unsplit version.  Pick a suitable norm\nand measure the convergence of the methods.}\n\\end{exercise}\n\n"
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    "content": "\\documentclass[11pt]{article}\n\n% margins\n\\usepackage[margin=0.75in]{geometry}\n\n% figures\n\\usepackage{graphicx}\n\n% font\n\\usepackage{mathpazo}\n\n\\usepackage{helvet}\n\n% coloring\n\\usepackage{color}\n\\definecolor{mygray}{gray}{0.5}\n\n% footer\n\\usepackage{fancyhdr}\n\\pagestyle{fancy}\n\\fancyfoot[LO,LE]{\\footnotesize \\sffamily \\color{mygray} M.\\ Zingale---Notes on Advection}\n\\fancyfoot[RO,RE]{\\footnotesize \\sffamily \\color{mygray} (\\today)}\n\\fancyfoot[CO,CE]{\\thepage}\n\\fancyhead{}\n\\renewcommand{\\headrulewidth}{0.0pt}\n\\renewcommand{\\footrulewidth}{0.0pt}\n\n% captions\n\\usepackage{caption}\n\\renewcommand{\\captionfont}{\\footnotesize}\n\\renewcommand{\\captionlabelfont}{\\footnotesize}\n\\setlength{\\captionmargin}{5em}\n\n\n\\usepackage{sectsty}\n\\allsectionsfont{\\sffamily}\n\n\\usepackage{amsmath}\n\n\\newcounter{exercise}\n\\usepackage{changepage}\n\n\\newenvironment{exercise}% environment name \n{% begin code \n  \\begin{adjustwidth}{1.0cm}{0.5cm}\n  \\par\\vspace{\\baselineskip}\\noindent \n  \\refstepcounter{exercise}%\n  \\textbf{Exercise \\theexercise\\ }\\begin{itshape}% \n  %\\par\\vspace{\\baselineskip}\\noindent\\ignorespaces \n}% \n{% end code \n  \\end{itshape}\\end{adjustwidth}\\vspace{\\baselineskip}\\ignorespacesafterend \n}\n\n\\begin{document}\n\n\\begin{center}\n{\\LARGE \\textsf{\\textbf{\nNotes on Advection}}\n}\n\\end{center}\n\n\n\\input advection\n\n\\bibliographystyle{plain}\n\\bibliography{../refs}\n\n\\end{document}\n"
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  {
    "path": "advection/old-scripts/fv.py",
    "content": "import math\nimport numpy\nimport pylab\nimport grid_plot_util as gpu\n\n\nnzones = 5\n\n# data that lives on the grid\na = numpy.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.15, 0.5])\n\ngr = gpu.grid(nzones)\n\n\ngpu.drawGrid(gr)\n\ngpu.labelCenter(gr, nzones/2,   r\"$x_i$\")\ngpu.labelAvg(gr, nzones/2,   a[nzones/2], r\"$\\langle f\\rangle_i$\")\n\ngpu.labelCenter(gr, nzones/2-1, r\"$x_{i-1}$\")\ngpu.labelCenter(gr, nzones/2+1, r\"$x_{i+1}$\")\ngpu.labelCenter(gr, nzones/2-2, r\"$x_{i-2}$\")\ngpu.labelCenter(gr, nzones/2+2, r\"$x_{i+2}$\")\n\ngpu.labelEdge(gr, nzones/2,   r\"$x_{i-1/2}$\")\ngpu.labelEdge(gr, nzones/2+1, r\"$x_{i+1/2}$\")\n\nn = 0\nwhile (n < nzones):\n    gpu.drawCellAvg(gr, n, a[n])\n    n += 1\n\n\npylab.axis([gr.xmin-0.5*gr.dx,gr.xmax+0.5*gr.dx, -0.25, 1.2])\npylab.axis(\"off\")\n\npylab.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)\n\nf = pylab.gcf()\nf.set_size_inches(8.0,2.0)\n\n\npylab.savefig(\"fv.png\", dpi=200)\npylab.savefig(\"fv.eps\")\n               \n\n\n"
  },
  {
    "path": "advection/old-scripts/simplegrid.py",
    "content": "import math\nimport numpy\nimport pylab\n\ndef simplegrid():\n\n    # grid info\n    xmin = 0.0\n    xmax = 1.0\n\n    nzones = 7\n    ng = 1\n    \n    dx = (xmax - xmin)/float(nzones)\n\n    xl = (numpy.arange(2*ng+nzones) - ng)*dx\n    xr = (numpy.arange(2*ng+nzones)+1 - ng)*dx\n\n    xc = 0.5*(xl + xr)\n\n\n    #------------------------------------------------------------------------\n    # plot a domain without ghostcells\n    pylab.plot([xmin,xmax], [0,0], color=\"k\", lw=2)\n\n    # domain left edge\n    pylab.plot([xl[ng], xl[ng]], [0, 0.5], color=\"k\", lw=4)\n\n    n = ng\n    while (n < ng+nzones):\n\n        # draw right edge\n        pylab.plot([xr[n], xr[n]], [0, 0.5], color=\"k\", lw=2)        \n\n        # draw center marker\n        pylab.plot([xc[n], xc[n]], [-0.05, 0], color=\"k\")                \n        n += 1\n\n    # domain right edge\n    pylab.plot([xl[ng+nzones], xl[ng+nzones]], [0, 0.5], color=\"k\", lw=4)\n       \n\n    # label a few\n    pylab.text(xc[ng+nzones/2], -0.1, r\"$i$\", \n               horizontalalignment='center', verticalalignment='top')\n\n    pylab.text(xc[ng+nzones/2-1], -0.1, r\"$i-1$\", \n               horizontalalignment='center', verticalalignment='top')\n\n    pylab.text(xc[ng+nzones/2+1], -0.1, r\"$i+1$\", \n               horizontalalignment='center', verticalalignment='top')\n\n    pylab.xlim(xl[0]-0.5*dx,xr[2*ng+nzones-1]+0.5*dx)\n    pylab.ylim(-0.25, 0.6)\n    pylab.axis(\"off\")\n\n    pylab.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)\n\n    f = pylab.gcf()\n    f.set_size_inches(8.0,2.0)\n\n\n    pylab.savefig(\"simplegrid.png\")\n    pylab.savefig(\"simplegrid.eps\")\n               \n\n\n    #------------------------------------------------------------------------\n    # now draw one with ghostcells\n    pylab.plot([xmin-ng*dx,xmax+ng*dx], [0,0], color=\"k\", lw=2)\n\n\n    # domain (w/ ghostcells) left edge\n    pylab.plot([xl[0], xl[0]], [0, 0.5], color=\"k\", lw=2)\n\n    n = 0\n    while (n < 2*ng+nzones):\n\n        # draw right edge\n        pylab.plot([xr[n], xr[n]], [0, 0.5], color=\"k\", lw=2)        \n\n        # draw center marker\n        pylab.plot([xc[n], xc[n]], [-0.05, 0], color=\"k\")                \n        n += 1\n\n\n    # domain left edge\n    pylab.plot([xl[ng], xl[ng]], [0, 0.5], color=\"k\", lw=4)\n\n    # domain right edge\n    pylab.plot([xl[ng+nzones], xl[ng+nzones]], [0, 0.5], color=\"k\", lw=4)\n       \n\n    # label a few\n    pylab.text(xc[ng+nzones/2], -0.1, r\"$i$\", \n               horizontalalignment='center', verticalalignment='top')\n\n    pylab.text(xc[ng+nzones/2-1], -0.1, r\"$i-1$\", \n               horizontalalignment='center', verticalalignment='top')\n\n    pylab.text(xc[ng+nzones/2+1], -0.1, r\"$i+1$\", \n               horizontalalignment='center', verticalalignment='top')\n\n    pylab.text(xc[ng-1], -0.1, r\"$-1$\", \n               horizontalalignment='center', verticalalignment='top')\n\n    pylab.text(xc[ng], -0.1, r\"$0$\", \n               horizontalalignment='center', verticalalignment='top')\n\n    pylab.text(xc[ng+nzones-1], -0.1, r\"$N-1$\", \n               horizontalalignment='center', verticalalignment='top')\n\n    pylab.text(xc[ng+nzones], -0.1, r\"$N$\", \n               horizontalalignment='center', verticalalignment='top')\n\n    pylab.xlim(xl[0]-0.5*dx,xr[2*ng+nzones-1]+0.5*dx)\n    pylab.ylim(-0.25, 0.6)\n    pylab.axis(\"off\")\n\n    pylab.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)\n\n    f = pylab.gcf()\n    f.set_size_inches(8.0,2.0)\n\n\n    pylab.savefig(\"simplegrid_gc.png\")\n    pylab.savefig(\"simplegrid_gc.eps\")\n               \n\n\n\nif __name__== \"__main__\":\n    simplegrid()\n"
  },
  {
    "path": "advection/old-scripts/simplegrid2.py",
    "content": "import math\nimport numpy\nimport pylab\n\ndef simplegrid():\n\n    xmin = 0.0\n    xmax = 1.0\n\n    nzones = 7\n\n    dx = (xmax - xmin)/float(nzones)\n\n    xl = numpy.arange(nzones)*dx\n    xr = (numpy.arange(nzones)+1)*dx\n\n    xc = 0.5*(xl + xr)\n\n    pylab.plot([xmin-0.5*dx,xmax+0.5*dx], [0,0], color=\"k\", lw=2)\n\n    pylab.plot([xl[0], xl[0]], [0, 0.5], color=\"k\", lw=2)\n    n = 0\n    while (n < nzones):\n\n        # draw right edge\n        pylab.plot([xr[n], xr[n]], [0, 0.5], color=\"k\", lw=2)        \n\n        # draw center marker\n        pylab.plot([xc[n], xc[n]], [-0.05, 0], color=\"k\")                \n\n        # draw edge marker\n        if (n == 0):\n            pylab.plot([xl[0], xl[0]], [-0.05, 0], color=\"k\")                \n\n        pylab.plot([xr[n], xr[n]], [-0.05, 0], color=\"k\")                \n\n        n += 1\n\n\n    # label a few cell-centers\n    pylab.text(xc[nzones/2], -0.1, r\"$i$\", \n               horizontalalignment='center', verticalalignment='top', \n               fontsize=\"small\")\n\n    pylab.text(xc[nzones/2-1], -0.1, r\"$i-1$\", \n               horizontalalignment='center', verticalalignment='top', \n               fontsize=\"small\")\n\n    pylab.text(xc[nzones/2+1], -0.1, r\"$i+1$\", \n               horizontalalignment='center', verticalalignment='top',\n               fontsize=\"small\")\n\n\n    # label a few edges\n    pylab.text(xl[nzones/2], -0.075, r\"$i-1/2$\", \n               horizontalalignment='center', verticalalignment='top', \n               fontsize=\"small\")\n\n    pylab.text(xr[nzones/2], -0.075, r\"$i+1/2$\", \n               horizontalalignment='center', verticalalignment='top', \n               fontsize=\"small\")\n\n\n\n    pylab.axis([xmin-0.5*dx,xmax+0.5*dx, -0.25, 0.6])\n    pylab.axis(\"off\")\n\n    pylab.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)\n\n    f = pylab.gcf()\n    f.set_size_inches(8.0,2.0)\n\n\n    pylab.savefig(\"simplegrid2.png\")\n    pylab.savefig(\"simplegrid2.eps\")\n               \n\n\nif __name__== \"__main__\":\n    simplegrid()\n"
  },
  {
    "path": "burgers/burgers.tex",
    "content": "\\label{ch:burgers}\n\n\\section{Burgers' equation}\n\nThe inviscid Burgers' equation is the simplest {\\em nonlinear} hyperbolic\nequation:\n\\begin{equation}\nu_t + u u_x = 0\n\\end{equation}\nHere $u$ is both the quantity being advected and the speed at which\nit is moving.  Recall that for the linear advection equation, we saw\nthat the solution was constant along lines $x = ut + x_0$, which are\nparallel, since $u$ is spatially constant.  For Burgers' equation,\nthis is no longer the case, and the characteristic lines are now\ngiven by $dx/dt = u$, with $x(0) = x_0$.  Since $u = u(t)$, we cannot\nintegrate this directly.\n\n\\begin{exercise}[Burgers' characteristics]\n{To find the lines along which the solution to Burgers' equation is\nconstant, consider the full change in $u$:\n\\begin{equation}\ndu = \\left . \\frac{\\partial u}{\\partial t} \\right |_x dt +\n     \\left . \\frac{\\partial u}{\\partial x} \\right |_t dx = 0\n\\end{equation}\nHere, we seek $u = \\mathrm{constant}$, so we set $du = 0$.  Now,\nusing the Burgers' equation itself, show that this implies that\nthe lines along which the solution is constant are:\n\\begin{equation}\n\\frac{dx}{dt} = u(x,t)\n\\end{equation}\n}\n\\end{exercise}\n\nIf we take $u_0 = u(t=0)$, then we can look at how the characteristic\nbehave over a small time interval (before $u(x,t)$ changes\nsignificantly).  Figure~\\ref{fig:burgers_char} shows the behavior for\nan initial velocity with sinusoidal profile.  We see that after a\nshort period of time, the characteristics intersect.  At the point,\n$(x_s, t_s)$ where they intersect, there is no way to trace backwards\nalong the characteristics to find a unique initial state.  This\nmerging of the characteristics in the $x$-$t$ plane is a {\\em shock}, and\nrepresents just one way that nonlinear problems can differ from linear\nones.\n\nAnother type of wave not present in a linear system is a {\\em\n  rarefaction}.  Figure~\\ref{fig:burgers_char_rare} shows initial\nconditions of slower velocity to the left of faster velocity.  We see\nthat the characteristics diverge in this case, and we will be left\nwith having to fill in the solution inbetween as some intermediate\nstate.\n\n\n% figure created with figures/burgers/characteristics.py\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.6\\linewidth]{burgers-characteristics}\n\\caption[Characteristics for shock initial conditions]\n{\\label{fig:burgers_char} (top) Initially sinusoidal\nvelocity distribution (bottom) Approximate characteristic\nstructure for Burgers' equation, using $u_0 = u(t)$.  Note that\nafter a short period of time, the characteristics intersect, signaling\nthe formation of a shock.}\n\\end{figure}\n\n% figure created with figures/burgers/characteristics.py\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.6\\linewidth]{burgers-characteristics-rare}\n\\caption[Characteristics for rarefaction initial conditions]\n        {\\label{fig:burgers_char_rare} (top) Initially discontinuous\n          velocity distribution with low velocity left of high\n          velocity (bottom) Approximate characteristic structure for\n          Burgers' equation, using $u_0 = u(t)$.  Note that after a\n          short period of time, the characteristics diverge---this is \n          a rarefaction.}\n\\end{figure}\n\n\n\nIn conservative form, Burgers' equation appears as:\n\\begin{equation}\nu_t + \\left [\\tfrac{1}{2} u^2 \\right ]_x = 0\n\\end{equation}\nThe solution of this follows the same methodology as described in\nChapter~\\ref{ch:advection}.  The interface states are predicted as to\nsecond order by Taylor expanding in space and time:\n\\begin{eqnarray}\nu^{n+1}_{i+\\myhalf,L}\n &=& u^n_i + \\frac{\\Delta x}{2} \\frac{\\partial u}{\\partial x}\n    + \\frac{\\Delta t}{2} \\left . \\frac{\\partial u}{\\partial t} \\right |_i\n    + \\ldots \\\\\n &=& u^n_i + \\frac{\\Delta x}{2} \\frac{\\partial u}{\\partial x}\n    + \\frac{\\Delta t}{2} \\left . \\left (-u_i \\frac{\\partial u}{\\partial x}\n         \\right ) \\right |_i\n    + \\ldots \\\\\n &=& u^n_i + \\frac{\\Delta x}{2}\n   \\left ( 1 - \\frac{\\Delta t}{\\Delta x} u_i \\right )\n   \\left . \\frac{\\partial u}{\\partial x} \\right |_i + \\ldots\n\\end{eqnarray}\nThe only difference with the linear advection equation is that now\n$u_i \\Delta t/\\Delta x$ varies from zone to zone, whereas with linear\nadvection, it is the constant $C$.  The slopes are computed using\nthe same limiters as with linear advection.\n\nThe Riemann problem differs from linear advection.  As we saw above,\nthe characteristic curves can intersect in the $x$-$t$ plane, and it\nis not possible to trace backward from time to learn where the flow\noriginated.  This is the condition for a {\\em shock}.\n\n% figure made by figures/burgers/rh.py\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=4in]{rh}\n\\caption[Rankine-Hugoniot conditions]{\\label{fig:rh} A rightward moving shock in the $x$-$t$\n   plane separating two states: $u_l$ and $u_r$.}\n\\end{figure}\n\nThe shock speed is computed through the {\\em Rankine-Hugoniot} jump\nconditions.  For a scalar equation, these are easy to construct.\nWe'll follow the method of \\cite{leveque:2002}.  Figure~\\ref{fig:rh}\nshows two states separated by a rightward moving shock in the $x$-$t$\nplane.  At time $t^n$, the state in our interval ($x \\in [x_l,x_r]$)\nis entirely $u_r$.  As time evolves, we imagine our interval $[x_l,x_r]$ moving\nvertically upwards in the diagram, and we see that it contains a mix of states $u_l$ and $u_r$.  Finally, at time, $t^{n+1}$ it is entirely\n$u_l$.  The shock moves with a speed $S = \\Delta x/\\Delta t$ in this\nfigure.  To determine the speed, we integrate our conservation law over\nboth space and time (and normalize by $\\Delta x = x_r - x_l$):\n\\begin{equation}\n\\frac{1}{\\Delta x} \\int_{x_l}^{x_r} dx \\int_{t^n}^{t^{n+1}} dt\\, u_t =\n  - \\frac{1}{\\Delta x} \\int_{x_l}^{x_r} dx \\int_{t^n}^{t^{n+1}} dt \\left [ f(u) \\right ]_x\n\\end{equation}\nDoing the $t$ integral on the left and $x$ integral on the right, we have\n\\begin{equation}\n\\frac{1}{\\Delta x} \\int_{x_l}^{x_r}\\left \\{ u(t^{n+1}) - u(t^n) \\right \\} dx =\n  - \\frac{1}{\\Delta x} \\int_{t^n}^{t^{n+1}} \\left \\{ f(u) |_{x=x_r} - f(u) |_{x=x_l} \\right \\} dt\n\\end{equation}\nRecognizing that at $t=t^n$, $u = u_r$ and at $t=t^{n+1}$, $u = u_l$,\nin the left side becomes\n\\begin{equation}\n\\frac{1}{\\Delta x} \\int_{x_l}^{x_r}\\left \\{ u(t^{n+1}) - u(t^n) \\right \\} dx = \\{ u(t^{n+1}) - u(t^n) \\} =  u_l -u_r \\enskip .\n\\end{equation}\nFor the right side, we see that all along $x=x_l$ the flux is $f =\nf(u_l)$ for $t\\in [t^n, t^{n+1}]$.  Likewise, all along $x=x_r$, the\nflux is $f = f(u_r)$ in the same time interval (see the figure).\nTherefore, our expression becomes:\n\\begin{equation}\n(u_l - u_r) = -\\frac{\\Delta t}{\\Delta x} \\left [ f(u_r) - f(u_l)\\right ]\n\\end{equation}\nand using $S = \\Delta x/\\Delta t$, we see\n\\begin{equation}\nS = \\frac{f(u_r) - f(u_l)}{u_r - u_l}\n\\end{equation}\n\nFor Burgers' equation, substituting in $f(u) = u^2/2$, we get\n\\begin{equation}\nS = \\frac{1}{2}(u_l + u_r)\n\\end{equation}\n\nWith the shock speed known, the Riemann problem is straightforward.  If there\nis a shock (compression, so $u_l > u_r$) then we compute the shock speed and\ncheck whether the shock is moving to the left or right, and then use the appropriate\nstate.  If there is no shock, then we can simply use upwinding, as there is no\nambiguity as to how to trace backwards in time to the correct state.\nPutting this together, we have:\n\\begin{eqnarray}\n\\mathrm{if~} \\underset{\\text{(shock)}}{u_l > u_r}:&& u_s = \\left \\{ \\begin{array}{cl}\n                u_l & \\mathrm{if~} S > 0 \\\\\n                u_r & \\mathrm{if~} S < 0 \\\\\n                0   & \\mathrm{if~} S = 0 \\end{array} \\right .   \\\\[1em]\n%\n\\mathrm{otherwise:}&& u_s = \\left \\{ \\begin{array}{clc}\n                u_l & \\mathrm{if~} u_l > 0 \\\\\n                u_r & \\mathrm{if~} u_r < 0 \\\\\n                0   & \\mathrm{otherwise} \\end{array} \\right .\n\\end{eqnarray}\n\\MarginPar{other forms of this from incompressible?}\n\nOnce the interface states are constructed, the flux is calculated as:\n\\begin{equation}\nF^{n+\\myhalf}_{i+\\myhalf} = \\frac{1}{2} \\left (u_{i+\\myhalf}^{n+\\myhalf} \\right )^2\n\\end{equation}\nand the conservative update is\n\\begin{equation}\nu_i^{n+1} = u_i^n + \\frac{\\Delta t}{\\Delta x}\n   \\left ( F_{i-\\myhalf}^{n+\\myhalf} - F_{i+\\myhalf}^{n+\\myhalf} \\right )\n\\end{equation}\n\nThe timestep constraint now must consider the most restrictive Courant\ncondition over all the zones:\n\\begin{equation}\n\\Delta t = \\min_i \\left \\{ \\Delta x / u_i \\right \\}\n\\end{equation}\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{fv-burger-rarefaction}\n\\caption[Rarefaction solution to the inviscid Burgers'\n  equation]{\\label{fig:burgers-rarefaction} Solution to the inviscid\n  Burgers' equation with 256 zones and a Courant number, $C = 0.8$ for\n  initial conditions that generate a rarefaction: the left half of the\n  domain was initialized with $u = 1$ and the right half with $u = 2$.\n  This initial velocity state creates a divergent flow.  The curves\n  are shown 0.02~s apart, with the darker grayscale representing later\n  in\n  time. \\\\ \\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/burgers/burgers.py}{burgers.py}}}\n\\end{figure}\n\n\nFigure~\\ref{fig:burgers-rarefaction} shows the solution to Burgers'\nequation using the 2nd-order piecewise linear method described here,\nwith the MC limiter.  The initial conditions chosen are all positive\nvelocity, with a lower velocity to the left of the higher velocity.\nAs the solution evolves, the state on the right will rush away from\nthe state on the left, and spread out like a fan.  This is called a\n{\\em rarefaction wave} or simply a {\\em rarefaction}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{fv-burger-sine}\n\\caption[Shock solutions to the inviscid Burgers'\n  equation]{\\label{fig:burgers-shock} Solution to the inviscid\n  Burgers' equation with 256 zones and a Courant number, $C = 0.8$.\n  The initial conditions here are sinusoidal and the solution quickly\n  steepens into a shock.The curves are shown 0.02~s apart, with the\n  darker grayscale representing later in\n  time. \\\\ \\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/burgers/burgers.py}{burgers.py}}}\n\\end{figure}\n\nFigure~\\ref{fig:burgers-shock} shows the solution to Burgers'\nequation with initially sinusoidal data:\n\\begin{equation}\n\\label{eq:burgers:shockic}\n\\renewcommand{\\arraystretch}{1.75}\nu(x, t=0) = \\left \\{ \\begin{array}{cc}\n    1   & x < 1/3 \\\\\n    1 + \\frac{1}{2} \\sin\\left (\\frac{2\\pi (x - 1/3)}{1/3}\\right ) & 1/3 \\le x \\le 2/3 \\\\\n    1   & x > 2/3\n\\end{array}\n\\right .\n\\renewcommand{\\arraystretch}{1.0}\n\\end{equation}\nThis is analogous to the case shown in\nFigure~\\ref{fig:burgers_char}---we see the solution steepen and form a\nshock which propagates to the right.  The shock is practically infinitesimally thin\nhere, since there is no explicitly viscosity to smear it out\\footnote{Numerical diffusion,\nin the form of truncation error of our method, will smear things a little.}.\n\n\\begin{exercise}[Simple Burgers' solver]\n{Extend your 1-d finite-volume solver for advection (from\n  Exercise~\\ref{adv:ex:fv}) to solve Burgers' equation.  You will\n  need to change the Riemann solver and use the local velocity in the\n  construction of the interface states.  Run the examples shown in\n  Figures~\\ref{fig:burgers-rarefaction} and \\ref{fig:burgers-shock}}.\n\\end{exercise}\n\nAs we'll see shortly, these two types of waves can also appear in the Euler equations\nfor hydrodynamics.\n\nA final thing to note is that we solved Burgers' equation in conservative form.\nFor shock solutions, this is essential, since as we noted earlier, that finite-volume\nmethod relates to the integral form of the PDEs, the discontinuity is nicely handled\nby the Riemann solver.  We could also imagine differencing Burgers' equation in\nnon-conservative form, i.e., starting with:\n\\begin{equation}\nu_t + u u_x = 0\n\\end{equation}\nIf you did this (for instance, using a finite difference scheme and an\nupwind difference for $u_x$), you would find that you get the wrong\nshock speed.\n\n\\begin{exercise}[Conservative form of Burgers' equation]\n{\nUsing a simple first-order finite-difference method like we described\nin Ch.~\\ref{ch:advection} for linear advection, difference the\nconservative and non-conservation formulations of Burgers' equation\nas:\n\\begin{equation}\n\\frac{u^{n+1}_i - u^n_i}{\\Delta t} = \n   -\\frac{1}{2} \\frac{(u^n_i)^2 - (u^n_{i-1})^2}{\\Delta x}\n\\end{equation}\nand\n\\begin{equation}\n  \\frac{u_{i}^{n+1} - u_i^n}{\\Delta t} = - \n      \\frac{u_i^n (u_i^n - u_{i-1}^n)}{\\Delta x}\n\\end{equation}\n(Note: these discretizations are upwind so long as $u > 0$).\n\nRun these with the shock initial Riemann conditions:\n\\begin{equation}\n\\renewcommand{\\arraystretch}{1.75}\nu(x, t=0) = \\left \\{ \\begin{array}{cc}\n    2   & x < 1/2 \\\\\n    1   & x > 1/2\n\\end{array}\n\\right .\n\\renewcommand{\\arraystretch}{1.0}\n\\end{equation}\nand measure the shock speed from your solution by simply differencing\nthe location of the discontinuity at two different times.  Compare to\nthe analytic solution for a shock for the Riemann problem.}\n\\end{exercise}\n\n\\section{Characteristic tracing}\n\nA concept that will be useful in the next section is {\\em\n  characteristic tracing}.  The idea is that we only include\ninformation in the interface states if the characteristic that carries\nthat information is moving toward the interface.  For Burgers equation,\nthis is simple, since there is only a single characteristic---the velocity.\nSo for the left state on an interface, we'd only add the change if\nthe velocity $u_i > 0$ (moving toward the interface $i+\\myhalf$):\n\\begin{equation}\nu^{n+1}_{i+\\myhalf,L}\n = u^n_i + \\frac{\\Delta x}{2}\n   \\left ( 1 - \\frac{\\Delta t}{\\Delta x} \\max(0, u_i) \\right )\n   \\left . \\frac{\\partial u}{\\partial x} \\right |_i + \\ldots\n\\end{equation}\nNotice that the effect of this is to set the interface state simply to\nthe value given by the piecewise linear reconstruction on the interface\nif the wave isn't moving to the interface.\n\n\\section{Going further}\n\n\\begin{itemize}\n\\item The equation we've been dealing with here is the {\\em inviscid}\n  Burgers' equation.  The full Burgers' equation includes viscosity (a\n  velocity diffusion):\n  \\begin{equation}\n    u_t + u u_x = \\epsilon u_{xx}\n  \\end{equation}\n  To solve this, we need to first learn about techniques for\n  diffusion, and then how to solve equations that span multiple PDE\n  types.  This will be described in \\S\n  \\ref{ch:multiphysics:sec:adburgers}.\n\n\\item Aside from pedagogical interest, Burgers' equation can be used\n  as a simple model of traffic flow (where shocks can arise from\n  people slamming on the brakes).  Many sources discuss this\n  application, including the text by~\\cite{leveque:2002}.\n\\MarginPar{more refs?}\n\n\\end{itemize}\n"
  },
  {
    "path": "codes/intro/sin-converge.py",
    "content": "import math\n\nX = [0.5, 0.25, 0.125, 0.0625]\n\nfor x in X:\n    print x, math.sin(x) - (x - x**3/6.0)\n\n"
  },
  {
    "path": "diffusion/GNUmakefile",
    "content": "EPStoPDF = epstopdf\n\nALL: diffusion.pdf\n\n\ndiffusion.pdf: main.tex diffusion.tex\n\tpdflatex -jobname=diffusion main.tex < /dev/null\n\tbibtex diffusion.aux\n\tpdflatex -jobname=diffusion main.tex < /dev/null\n\tpdflatex -jobname=diffusion main.tex < /dev/null\n\n\nclean:\n\t$(RM) *.aux *.log *.dvi *.bbl *.blg \n\t$(RM) *~\n\n.PHONY: clean\n"
  },
  {
    "path": "diffusion/diffusion.tex",
    "content": "\\label{ch:diffusion}\n\n\\section{Diffusion}\n\n\\label{sec:diffusion}\n\nPhysically, a diffusive process obeys Fick's law---the quantity that is\ndiffusing, $\\phi$, moves from higher to lower concentration at a rate\nproportional to the gradient,\n\\begin{equation}\n  F = - k \\nabla \\phi\n\\end{equation}\nIf we think of this as a diffusive flux, then we can write the time-rate-of-change of $\\phi$ as a conservation law:\n\\begin{equation}\n  \\phi_t + \\nabla\\cdot F(\\phi) = 0\n\\end{equation}\nThis gives rise to the diffusion equation:\n\\begin{equation}\n\\phi_t = \\nabla \\cdot (k \\nabla \\phi)\n\\end{equation}\n\nDiffusion has a time-dependence like the advection equations we\nalready saw, but also has a global character like an elliptic problem.\nUnlike advection, there is no sense of upwinding in diffusion.  As we\nwill see, we can cast our differenced equations in a form reminiscent\nof an elliptic problem.\n\nIn one-dimension, and assuming that the diffusion coefficient, $k$, is \nconstant, we have\n\\begin{equation}\n\\frac{\\partial \\phi}{\\partial t} = \n  \\frac{\\partial }{\\partial x} \n  \\left ( k \\frac{\\partial \\phi}{\\partial x} \\right )\n\\end{equation}\n\n\nThe diffusion equation can describe thermal diffusion (for example, as\npart of the energy equation in compressible flow), species/mass\ndiffusion for multi-species flows, or the viscous terms in\nincompressible flows.  In this form, the diffusion coefficient (or\nconductivity), $k$, can be a function of $x$, or even $\\phi$.  \n\n\nWe will consider a constant diffusion coefficient as our model problem:\n\\begin{equation}\n\\frac{\\partial \\phi}{\\partial t} = k \\frac{\\partial^2 \\phi}{\\partial x^2}\n\\end{equation}\nThe diffusion equation is the prototypical parabolic PDE.\nThe basic behavior of the diffusion equation is to take strongly peaked\nconcentrations of $\\phi$ and smooth them out with time.\n\n\\section{Explicit differencing}\n\nA nice overview of the discretizations for diffusion and their stability\nis given in \\cite{richtmyermorton}.  \nThe simplest way to difference this equation is {\\em explicit} in time\n(i.e., the righthand side depends only on the old state):\n\\begin{equation}\n\\label{eq:diff:explicitdiff}\n\\frac{\\phi_i^{n+1} - \\phi_i^n}{\\Delta t} = \n  k \\frac{\\phi_{i+1}^n - 2\\phi_i^n + \\phi_{i-1}^n}{\\Delta x^2}\n\\end{equation}\nThis is second-order accurate in space, but only first order accurate in\ntime (since the righthand side is not centered in time).\n\nAs with the advection equation, when differenced explicitly, there is\na constraint on the timestep required for stability.  Looking at the\ngrowth of a single Fourier mode, $\\phi^n_i = A^n e^{i\\imag\\theta}$ with $\\imag =\n\\sqrt{-1}$, we find:\n\\begin{equation}\n\\label{eq:diff:amplitude}\n\\frac{A^{n+1}}{A^n} = 1 + 2 \\frac{k \\Delta t}{\\Delta x^2} ( \\cos\\theta - 1)\n\\end{equation}\nStability requires that $|A^{n+1}/A^n| \\le 1$, which can only be true\nif $2k\\Delta t/\\Delta x^2 \\le 1$.  Therefore, our timestep\nconstraint in this case is\n\\begin{equation}\n\\label{eq:diff:dt}\n\\Delta t < \\frac{1}{2} \\frac{\\Delta x^2}{k}\n\\end{equation}\nAs with advection, we often write the timestep as\n\\begin{equation}\n\\Delta t = \\frac{C}{2} \\frac{\\Delta x^2}{k}\n\\end{equation}\nwhere $C$ is a constant, $C < 1$. \nNote the $\\Delta x^2$ dependence---this constraint can become really\nrestrictive.\n\n\\begin{exercise}\n[Explicit diffusion stability condition]\nDerive Eq.~\\ref{eq:diff:amplitude} by inserting $\\phi^n_i = A^n e^{ij\\theta}$\ninto Eq.~\\ref{eq:diff:explicitdiff}\n\\end{exercise}\n\nTo complete the solution, we need boundary conditions at the left\n($x_l$) and right ($x_r$) boundaries.  These are typically either\nDirichlet:\n\\begin{eqnarray}\n\\phi |_{x=x_l} &=& \\phi_l\\\\\n\\phi |_{x=x_r} &=& \\phi_r\n\\end{eqnarray}\nor Neumann:\n\\begin{eqnarray}\n\\phi_x |_{x=x_l} &=& \\phi_x |_l\\\\\n\\phi_x |_{x=x_r} &=& \\phi_x |_r\n\\end{eqnarray}\nPhysically, a Dirichlet BC means fixing the value of $\\phi$ (e.g., temperature)\non the boundary.  A Neumann BC means fixing the flux on the boundary.\n\nLike with advection, it helps to have a good test problem to evaluate our\nmethods for diffusion.  We can use the fact that the diffusion of a Gaussian\nis a Gaussian with a smaller amplitude and greaterwidth.\n\n\\begin{exercise}[1-d explicit diffusion]\n{Write a one-dimensional explicit diffusion solver for the\n  domain $[0,1]$ with Neumann boundary conditions at each end and $k = 1$,\n  using the discretization of Eq.~\\ref{eq:diff:explicitdiff}.\n\n  If we begin with a Gaussian, the resulting solution is also a Gaussian,\n  giving a solution\\footnote{Note: the 2- and 3-d solutions are slightly different than this 1-d solution}:\n  \\begin{equation}\n   \\phi(x,t) = (\\phi_2 - \\phi_1) \\sqrt{\\frac{t_0}{t + t_0}} e^{-\\frac{1}{4}(x - x_c)^2/k(t+t_0)} + \\phi_1\n  \\end{equation}\n  Initialize our problem with $t = 0$, and take\n  $t_0 = 0.001$, $\\phi_1 = 1$, and $\\phi_2 = 2$, and $x_c$ is the\n  coordinate of the center of the domain.  Run until $t = 0.01$ and\n  compare to the analytic solution above.}\n\\end{exercise}\n\n\nFigure~\\ref{fig:diff_explicit} shows the solution for 64 zones\nand a value of $C = 0.8$.  We see good agreement with the\nanalytic solution.  Note that if the initial conditions are\nnot well resolve initially, then the solution will be bad\n(see Figure~\\ref{fig:diff_explicit_res}).\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{diff-explicit-64}\n\\caption[Explicit diffusion of a Gaussian]{\\label{fig:diff_explicit}\nDiffusion of a Gaussian using the explicit differencing of\nEq.~\\ref{eq:diff:explicitdiff} with 64 zones and C = 0.8, shown at\nseveral times.  The dotted line is the analytic\nsolution.  \\\\ \\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/diffusion/diffusion_explicit.py}{diffusion\\_explicit.py}}}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{diffexplicit-res}\n\\caption[Underresolved explicit diffusion of a Gaussian]{\\label{fig:diff_explicit_res}\nDiffusion of a Gaussian using the explicit differencing of\nEq.~\\ref{eq:diff:explicitdiff} with different resolutions.\nseveral times.  The dotted line is the analytic\nsolution.  \\\\ \\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/diffusion/diffusion_explicit.py}{diffusion\\_explicit.py}}}\n\\end{figure}\n\nAs with advection, if we exceed the timestep limit ($C > 1$), then\nthe solution is unstable.  This is shown in Figure~\\ref{fig:diff_explicit_bad}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{diff-explicit-64-bad}\n\\caption[Unstable explicit diffusion]{\\label{fig:diff_explicit_bad}\nDiffusion of a Gaussian using the explicit differencing of\nEq.~\\ref{eq:diff:explicitdiff} with 64 zones, but a timestep\nwith $C > 1$, showing that the solution is unstable.\n \\\\ \\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/diffusion/diffusion_explicit.py}{diffusion\\_explicit.py}}}\n\\end{figure}\n\nOur spatial order-of-accuracy is second-order.\nFigure~\\ref{fig:diffexplicit_converge} shows the error as a function\nof number of zones, using the $L_2$ norm of the solution with respect\nto the analytic solution.  Notice that at the coarsest resolution the\nerror is very high---we are not resolving the initial conditions well\nenough to have a meaningful solution.  At higher resolutions, we\nconverge as $O(\\Delta x^2)$.  Recall though that our method has a\ntruncation error that is $O(\\Delta t) + O(\\Delta x^2)$, but we don't\nsee the first-order in time scaling.  This is because of the timestep\nrestriction.  Since $\\Delta t \\sim \\Delta x^2/k$, as we cut $\\Delta x$\nby 2, $\\Delta t$ drops by 4, so the timestep choice makes our\n$O(\\Delta t)$ truncation error go as $O(\\Delta x^2)$.\n\nNote, this is not the case for advection, where $\\Delta t \\sim \\Delta x$, so\nfor our advection discretizations we will want the same order of accuracy\nin our spatial and temporal discretizations.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{diffexplicit-converge-0_8}\n\\caption[Error convergence of explicit diffusion]{\\label{fig:diffexplicit_converge}\nError convergence with resolution of the explicit diffusion using the differencing of \nEq.~\\ref{eq:diff:explicitdiff}\n \\\\ \\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/diffusion/diffusion_explicit.py}{diffusion\\_explicit.py}}}\n\\end{figure}\n\n\\section{Implicit with direct solve}\n\nRecall that an implicit discretization of advection did not have a\ntimestep restriction for stability.  The same holds true for\ndiffusion.  A backward-Euler implicit discretization would be:\n\\begin{equation}\n\\label{eq:diff:foimplicit}\n\\frac{\\phi_i^{n+1} - \\phi_i^n}{\\Delta t} = \n  k \\frac{\\phi_{i+1}^{n+1} - 2\\phi_i^{n+1} + \\phi_{i-1}^{n+1}}{\\Delta x^2}\n\\end{equation}\nThe only difference with Eq.~\\ref{eq:diff:explicitdiff} is the\ntime-level of $\\phi$ on the righthand side.  Defining:\n\\begin{equation}\n\\alpha \\equiv k \\frac{\\Delta t}{\\Delta x^2}\n\\end{equation}\nThis is still first-order in time.  We can do the stability analysis\nto see the growth of a mode, giving,\n\\begin{equation}\n\\frac{A^{n+1}}{A^n} = \\frac{1}{1 - 2\\alpha (\\cos\\theta - 1)}\n\\end{equation}\nWe see that $|A^{n+1}/A^n| \\le 1$ for all $\\theta$, $\\alpha$, so this\ndiscretization is unconditionally stable.  However, the timestep will\nstill determine the accuracy.\n\nWe can write Eq.~\\ref{eq:diff:foimplicit} as:\n\\begin{equation}\n\\label{eq:diff:implicit}\n-\\alpha \\phi_{i+1}^{n+1} + (1 + 2\\alpha) \\phi_{i}^{n+1} - \\alpha \\phi_{i-1}^{n+1} = \\phi_i^n\n\\end{equation}\nThis is a set of coupled algebraic equations.  We can write this in\nmatrix form.  Using a cell-centered grid, we will solve for the values\n$[\\mathrm{lo},\\mathrm{hi}]$.  \nThe implicit method can use any $C > 0$.\n\nWe specify boundary conditions by modifying the stencil\n(Eq.~\\ref{eq:diff:implicit}) for the updates to $\\mathrm{lo}$ and\n$\\mathrm{hi}$\\footnote{Here, we use the $\\mathrm{lo}$, $\\mathrm{hi}$\n  notation for grid indices from \\S~\\ref{sec:fv:bcs}}.  For example, homogeneous\nNeumann BCs on the left mean:\n\\begin{equation}\n\\phi_\\mathrm{lo-1} = \\phi_\\mathrm{lo}\n\\end{equation}\nand substituting this into Eq~\\ref{eq:diff:implicit}, the update for the leftmost cell is:\n\\begin{equation}\n (1 + \\alpha) \\phi_\\mathrm{lo}^{n+1} -\\alpha \\phi_\\mathrm{lo+1}^{n+1}  = \n  \\phi_\\mathrm{lo}^n\n\\end{equation}\nIf we choose Dirichlet BCs on the right ($\\phi |_{x=x_l} = A$), then:\n\\begin{equation}\n\\phi_\\mathrm{hi+1} = 2 A - \\phi_\\mathrm{hi}\n\\end{equation}\nSubstituting this into Eq~\\ref{eq:diff:implicit} the update for the rightmost cell is:\n\\begin{equation}\n- \\alpha \\phi_\\mathrm{hi-1}^{n+1} + (1 + 3\\alpha) \\phi_\\mathrm{hi}^{n+1}  =\n  \\phi_\\mathrm{hi}^n + \\alpha 2 A\n\\end{equation}\nFor all other interior cells, the stencil is unchanged.  The resulting\nsystem appears as a {\\em tridiagonal} matrix.\n\\begin{equation}\n\\renewcommand\\arraystretch{1.5}\n\\left (\n\\begin{array}{ccccccc}\n1+\\alpha &   -\\alpha &           &        &         &           &          \\\\\n-\\alpha  & 1+2\\alpha & -\\alpha   &        &         &           &          \\\\\n         & -\\alpha   & 1+2\\alpha & -\\alpha&         &           &          \\\\\n         &           & \\ddots    & \\ddots & \\ddots  &           &          \\\\\n         &           &           & \\ddots & \\ddots  & \\ddots    &          \\\\\n         &           &           &        & -\\alpha & 1+2\\alpha &-\\alpha   \\\\\n         &           &           &        &         & -\\alpha   &1+3\\alpha \\\\\n\\end{array}\n\\right )\n\\left (\n\\begin{array}{c}\n\\phi_\\mathrm{lo}^{n+1} \\\\\n\\phi_\\mathrm{lo+1}^{n+1} \\\\\n\\phi_\\mathrm{lo+2}^{n+1} \\\\\n\\vdots \\\\\n\\vdots \\\\\n\\phi_\\mathrm{hi-1}^{n+1} \\\\\n\\phi_\\mathrm{hi}^{n+1} \\\\\n\\end{array}\n\\right )\n=\n\\left (\n\\begin{array}{c}\n\\phi_\\mathrm{lo}^{n} \\\\\n\\phi_\\mathrm{lo+1}^{n} \\\\\n\\phi_\\mathrm{lo+2}^{n} \\\\\n\\vdots \\\\\n\\vdots \\\\\n\\phi_\\mathrm{hi-1}^{n} \\\\\n\\phi_\\mathrm{hi}^{n} + \\alpha 2 A\\\\\n\\end{array}\n\\right )\n\\end{equation}\nThis can be solved by standard matrix operations, using a tridiagonal\nsolvers (for example).  Notice that the ghost cells do not appear in this\nlinear system---we are only updating the interior points.\n\n\\subsubsection{Crank-Nicolson time discretization}\n\nA second-order in time discretization requires us to center the\nrighthand side in time.  We do this as:\n\\begin{equation}\n\\frac{\\phi_i^{n+1} - \\phi_i^n}{\\Delta t} = \n  \\frac{k}{2} \\left ( \\frac{\\phi_{i+1}^{n} - 2\\phi_i^{n} + \\phi_{i-1}^{n}}{\\Delta x^2} +\n                      \\frac{\\phi_{i+1}^{n+1} - 2\\phi_i^{n+1} + \\phi_{i-1}^{n+1}}{\\Delta x^2} \\right )\n\\end{equation}\nThis looks like the average of the explicit and implicit systems we just saw.\nThis time-discretization is called {\\em Crank-Nicolson}.  Again, using\n$\\alpha \\equiv k\\Delta t / \\Delta x^2$, and grouping all the $n+1$\nterms on the left we have:\n\\begin{equation}\n\\phi^{n+1}_i - \\frac{\\alpha}{2} \\left ( \\phi^{n+1}_{i+1} - 2\\phi^{n+1}_i + \\phi^{n+1}_{i-1} \\right )\n  = \\phi^n_i + \\frac{\\alpha}{2} \\left ( \\phi^{n}_{i+1} - 2\\phi^{n}_i + \\phi^{n}_{i-1} \\right )\n\\end{equation}\nand grouping together the the $n+1$ terms by zone, we have:\n\\begin{equation}\n-\\frac{\\alpha}{2} \\phi^{n+1}_{i+1} + (1 + \\alpha)\\phi^{n+1}_i - \\frac{\\alpha}{2} \\phi^{n+1}_{i-1}\n  = \\phi^n_i + \\frac{\\alpha}{2} \\left ( \\phi^n_{i+1} - 2 \\phi^n_i + \\phi^n_{i-1} \\right )\n\\end{equation}\n\nConsidering homogeneous Neumann boundary conditions on the left and\nright, we again have $\\phi^{n+1}_\\mathrm{lo-1} =\n\\phi^{n+1}_\\mathrm{lo}$ and $\\phi^{n+1}_\\mathrm{hi+1} =\n\\phi^{n+1}_\\mathrm{hi}$, and our stencil at the boundary becomes\n\\begin{equation}\n-\\frac{\\alpha}{2} \\phi^{n+1}_\\mathrm{lo+1} + \\left (1 + \\frac{\\alpha}{2} \\right ) \\phi^{n+1}_\\mathrm{lo} =\n    \\phi^n_\\mathrm{lo} + \\frac{\\alpha}{2} \\left ( \\phi^{n}_\\mathrm{lo+1} - 2\\phi^{n}_\\mathrm{lo} + \\phi^{n}_\\mathrm{lo-1} \\right )\n\\end{equation}\n\nThe matrix form of this system is:\n\\newcommand{\\atwo}{\\frac{\\alpha}{2}}\n\\begin{equation}\n\\label{eq:diff:cnmatrix}\n\\renewcommand\\arraystretch{1.5}\n\\left (\n\\begin{array}{ccccccc}\n1+\\atwo &   -\\atwo &          &        &        &           &          \\\\\n-\\atwo  & 1+\\alpha & -\\atwo   &        &        &           &          \\\\\n        &   -\\atwo & 1+\\alpha & -\\atwo &        &           &          \\\\\n        &          & \\ddots   & \\ddots & \\ddots &           &          \\\\\n        &          &          & \\ddots & \\ddots & \\ddots    &          \\\\\n        &          &          &        & -\\atwo & 1+\\alpha  &-\\atwo   \\\\\n        &          &          &        &        & -\\atwo    &1+\\atwo \\\\\n\\end{array}\n\\right )\n\\left (\n\\begin{array}{c}\n\\phi_\\mathrm{lo}^{n+1} \\\\\n\\phi_\\mathrm{lo+1}^{n+1} \\\\\n\\phi_\\mathrm{lo+2}^{n+1} \\\\\n\\vdots \\\\\n\\vdots \\\\\n\\phi_\\mathrm{hi-1}^{n+1} \\\\\n\\phi_\\mathrm{hi}^{n+1} \\\\\n\\end{array}\n\\right )\n=\n\\left (\n\\begin{array}{c}\n\\phi_\\mathrm{lo}^{n}  + \\frac{k\\Delta t}{2} [\\nabla^2 \\phi]^n_\\mathrm{lo}\\\\\n\\phi_\\mathrm{lo+1}^{n} + \\frac{k\\Delta t}{2} [\\nabla^2 \\phi]^n_\\mathrm{lo+1} \\\\\n\\phi_\\mathrm{lo+2}^{n} + \\frac{k\\Delta t}{2} [\\nabla^2 \\phi]^n_\\mathrm{lo+2}\\\\\n\\vdots \\\\\n\\vdots \\\\\n\\phi_\\mathrm{hi-1}^{n} + \\frac{k\\Delta t}{2} [\\nabla^2 \\phi]^n_\\mathrm{hi-1}\\\\\n\\phi_\\mathrm{hi}^{n} + \\frac{k\\Delta t}{2} [\\nabla^2 \\phi]^n_\\mathrm{hi}\\\\\n\\end{array}\n\\right )\n\\end{equation}\n\nFigure~\\ref{fig:diffuse} shows the result of using $\\alpha = 0.8$ and $\\alpha = 8.0$.  We \nsee that they are both stable, but that the smaller timestep is closer to the analytic\nsolution (especially at early times).\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.75\\linewidth]{diff-implicit-128-CFL_0_8}\\\\\n\\includegraphics[width=0.75\\linewidth]{diff-implicit-128-CFL_8_0}\n\\caption[Implicit diffusion of a Gaussian]{\\label{fig:diffuse}\n  Implicit diffusion of a Gaussian (with Crank-Nicolson\n  discretization) with $C = 0.8$ and $C = 8.0$.  The exact solution at\n  each time is shown as the dotted\n  line. \\\\ \\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/diffusion/diffusion_implicit.py}{diffusion\\_implicit.py}}}\n\\end{figure}\n\n\\begin{exercise}[1-d implicit diffusion]\n{Write a one-dimensional implicit diffusion solver for the\n  domain $[0,1]$ with Neumann boundary conditions at each end and $k = 1$.\n  Your solver should use a tridiagonal solver and initialize a matrix like\n  that above.  Use a timestep close to the explicit step, a grid with\n  N = 128 zones.\n\n  Use a Gaussian for your initial conditions (as you did for the \n  explicit problem.}\n\\end{exercise}\n\n\n\\section{Implicit multi-dimensional diffusion via multigrid}\n\\label{diff:sec:implicit_mg}\n\nInstead of doing a direct solve of the matrix form of the system, we \ncan use multigrid techniques.  Consider the Crank-Nicolson system we just \nlooked at:\n\\begin{equation}\n\\frac{\\phi^{n+1}_i - \\phi^n_i}{\\Delta t} = \n   \\frac{1}{2} \\left ( k \\nabla^2 \\phi^n_i + k \\nabla^2 \\phi^{n+1}_i \\right )\n\\end{equation}\nGrouping all the \n$n+1$ terms on the left, we find:\n\\begin{equation}\n\\phi^{n+1}_i - \\frac{\\Delta t}{2} k \\nabla^2 \\phi^{n+1}_i = \n    \\phi^n_i + \\frac{\\Delta t}{2} k \\nabla^2 \\phi^n_i\n\\end{equation}\nThis is in the form of a constant-coefficient Helmholtz equation,\n\\begin{equation}\n\\label{eq:diff:helmholtz}\n(\\alpha - \\beta \\nabla^2) \\phi^{n+1} = f\n\\end{equation}\nwith\n\\begin{eqnarray}\n\\alpha &=& 1 \\\\\n\\beta &=& \\frac{\\Delta t}{2} k \\\\\nf &=& \\phi^n + \\frac{\\Delta t}{2} k \\nabla^2 \\phi^n\n\\end{eqnarray}\nThis can be solved using multigrid techniques with a Helmholtz\noperator.  The same boundary conditions discussed in\nChapter~\\ref{ch:multigrid} apply here.  The main difference between\nthe multigrid technique for the Poisson problem and the Helmholtz\nproblem is the form of the smoother.  In 1-d, we discretize\nEq.~\\ref{eq:diff:helmholtz} with a second-order difference expression\nfor the second derivative, and isolate $\\phi_i$, giving a smoothing\noperation of the form:\n\\begin{equation}\n\\label{eq:diff:discrete1d}\n\\phi_{i} \\leftarrow\n \\left .    \\left ( f_{i} + \\frac{\\beta}{\\Delta x^2} [\\phi_{i+1}\n                             + \\phi_{i-1}] \\right ) \\middle / \n\\left ( \\alpha + \\frac{2 \\beta}{\\Delta x^2}  \\right )  \\right .\n\\end{equation}\nNote that when you take $\\alpha = 0$ and $\\beta = -1$ in\nEq.~\\ref{eq:diff:discrete1d}, the smoothing operation reduces to the\nform that we saw in Chapter~\\ref{ch:multigrid} for just the Poisson\nequation.\n\nRecall, when using multigrid, you do not need to actually construct the\nmatrix.  This is usually the most efficient way to implement diffusion\nin a multi-dimensional simulation code, especially when distributing\nthe grid across parallel processors.  Since the discretization is the\nsame as the direct matrix solve, Eq.~\\ref{eq:diff:cnmatrix}, the\nresult will be exactly the same (to the tolerance of the multigrid\nsolver).\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.48\\linewidth]{gauss_diff_start}\n\\includegraphics[width=0.48\\linewidth]{gauss_diff_end}\n\\caption[2-d diffusion of a Gaussian]{\\label{fig:diff:twodcompare}\n  Diffusion of a Gaussian in 2-d with $128^2$ zones using $k = 1.0$\n  and $\\cfl = 2.0$.  This used Crank-Nicolson time-discretization and\n  multigrid.  This can be run in \\pyro\\ as {\\tt ./pyro diffusion\n    gaussian inputs.diffusion}.}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{gauss_diffusion_compare}\n\\caption[Comparison of 2-d implicit diffusion with analytic\n  solution]{\\label{fig:diffusion:multid} A comparison of 2-d implicit\n  diffusion from Figure~\\ref{fig:diff:twodcompare} with the analytic solution at several times.\n  The 2-d solution was averaged over angles to yield a profile as a \n  function of radius, using the {\\tt\n    gauss\\_diffusion\\_compare.py} in \\pyro.}\n\\end{figure}\n\nFigure~\\ref{fig:diffusion:multid} shows the result of diffusing a\nGaussian profile on a 2-d grid using Crank-Nicolson\ntime-discretization, a constant coefficient, and multigrid to solve\nthe discretized system, using \\pyro.  There is a good agreement\nbetween the analytic and numerical solution, showing that this scheme\nmodels the diffusion of an initially resolved Gaussian well.\n\n\n\\begin{exercise}[Implicit multi-dimensional diffusion]\n{The diffusion solver in \\pyro\\ uses Crank-Nicolson differencing in time.  \nModify the solver to do first-order backward Euler.  This will change\nthe form of the linear system (coefficients and righthand side), but\nshould not require any changes to the multigrid solver itself. \n\nCompare the solution with the Crank-Nicolson solution for a very coarsely-resolved\nGaussian and a finely-resolved Gaussian.\n}\n\\end{exercise}\n\n\n\\subsection{Convergence}\n\nOne needs to be careful with the Crank-Nicolson discretization.  If\nthe initial data is under-resolved and you are taking a big timestep\n($C \\gg 1$), then the method can be unstable.\nFigure~\\ref{fig:diff:cnunstable} shows such a case for the Gaussian\ndiffusion with Crank-Nicolson discretization and 64 zones with $C =\n10$.  For this reason, simulation codes often drop down to the\nsimple backwards difference time-discretization for implicit diffusion\nof under-resolved flows.  A good discussion of the different types\nof stability can be found in \\cite{leveque:fd}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{diff-implicit-64-CFL_10_0}\n\\caption[Under-resolved Crank-Nicolson diffusion]\n{\\label{fig:diff:cnunstable} Crank-Nicolson diffusion of a Gaussian\non under-resolved initial data with a large timestep.  Here we use\n$64$ zones and $C= 10$.  \\\\ \\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/diffusion/diffusion_implicit.py}{diffusion\\_implicit.py}}}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.75\\linewidth]{diffimplicit-converge-0_8} \\\\\n\\includegraphics[width=0.75\\linewidth]{diffimplicit-converge-8_0} \n\\caption[Convergence of diffusion methods]\n{\\label{fig:diff:convergence} Convergence of the explicit, backward-difference,\nand Crank-Nicolson diffusion methods for $C = 0.8$ (top) and $C = 8.0$ (bottom).\nFor the latter case, the explicit method is not valid and is not shown.\nWe see that the Crank-Nicolson method has the lowest error, and when resolving\nthe data, has second-order convergence.\\\\\n\\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/diffusion/diff_converge.py}{diff\\_converge.py}}}\n\\end{figure}\n\n\nFigure~\\ref{fig:diff:convergence} shows the convergence of several of\nthe methods discussed here on the diffusion of a Gaussian, for two\ndifferent Courant numbers, 0.8 and 8.0.  For the lower case, we can do\nthe diffusion explicitly as well as implicitly, and we see that all\nmethods show the same trends, with the Crank-Nicolson method being the\nmost accurate (since it is the only second-order-in-time method\nshown).  For the larger timestep, we only run the implicit methods\n(the explicit case is unstable).  We see that at coarse resolution,\nthe errors in the methods are similar---this is because the solution\nis unresolved.  At higher resolution, the error for the Crank-Nicolson\nmethod is an order of magnitude lower.\n\n\n\n\\section{Non-constant Conductivity}\n\nFor a non-constant conductivity, our equation has the form:\n\\begin{equation}\n\\frac{\\partial \\phi}{\\partial t} = \\frac{\\partial}{\\partial x} \\left (k \\frac{\\partial \\phi}{\\partial x} \\right )\n\\end{equation}\n\nFor the case where $k = k(x)$,\n  we discretize as:\n  \\begin{equation}\n  \\frac{\\phi_i^{n+1} - \\phi_i^n}{\\Delta t} = \n        \\frac{ \\{ k \\nabla \\phi \\}_{i+\\myhalf} -\n               \\{ k \\nabla \\phi \\}_{i-\\myhalf}}{\\Delta x}\n  \\end{equation}\n Here we need the values of $k$ at the interfaces, $k_{i-\\myhalf}$ and\n $k_{i+\\myhalf}$.  We can get these from the cell-centered values in a\n variety of ways including straight-averaging:\n \\begin{equation}\n k_{i+\\myhalf} = \\frac{1}{2} (k_i + k_{i+1})\n \\end{equation}\n or averaging the inverses:\n \\begin{equation}\n \\frac{1}{k_{i+\\myhalf}} = \\frac{1}{2} \\left (\\frac{1}{k_i} + \\frac{1}{k_{i+1}} \\right )\n \\end{equation}\nThe actual form should be motivated by the physics\n \nSlightly more complicated are state-dependent transport\ncoefficients---the transport coefficients themselves depend on the\nquantity being diffused:\n  \\begin{equation}\n  \\frac{\\phi_i^{n+1} - \\phi_i^n}{\\Delta t} = \n        \\frac{1}{2} \\left \\{\n               \\nabla \\cdot [ k(\\phi^n) \\nabla \\phi^n ]_i +\n               \\nabla \\cdot [ k(\\phi^{n+1}) \\nabla \\phi^{n+1} ]_i \n               \\right \\}\n  \\end{equation}\n  (for example, with thermal diffusion, the conductivity can\n  be temperature dependent).  In this case, we can achieve second-order\n  accuracy by doing a predictor-corrector.  First we diffuse with\n  the transport coefficients evaluated at the old time, giving a provisional\n  state, $\\phi^\\star$:\n  \\begin{equation}\n  \\frac{\\phi_i^\\star - \\phi_i^n}{\\Delta t} = \n        \\frac{1}{2} \\left \\{\n               \\nabla \\cdot [ k(\\phi^n) \\nabla \\phi^n ] +\n               \\nabla \\cdot [ k(\\phi^n) \\nabla \\phi^\\star ] \n               \\right \\}\n  \\end{equation}\n  Then we redo the diffusion, evaluating $k$ with $\\phi^\\star$ to\n  center the righthand side in time, giving the new state, $\\phi^{n+1}$:\n  \\begin{equation}\n  \\frac{\\phi_i^{n+1} - \\phi_i^n}{\\Delta t} = \n        \\frac{1}{2} \\left \\{\n               \\nabla \\cdot [ k(\\phi^n) \\nabla \\phi^n ] +\n               \\nabla \\cdot [ k(\\phi^\\star) \\nabla \\phi^{n+1} ] \n               \\right \\}\n  \\end{equation}\n  This is the approach used, for example, in \\cite{SNpaper}.\n\n\n\\section{Diffusion in Hydrodynamics} \n\nOften we find diffusion represented as one of many physical processes\nin a single equation.  For example, consider the internal energy\nequation with both reactions and diffusion:\n \\begin{equation}\n \\rho\\frac{\\partial e}{\\partial t} + \\rho U \\cdot \\nabla e + p \\nabla \\cdot U = \\nabla \\cdot k \\nabla T + \\rho S\n \\end{equation}\n This can be solved via an explicit-implicit discretization.  First\n the advection terms are computed as:\n \\begin{equation}\n A = \\rho U \\cdot \\nabla e + p \\nabla \\cdot U\n \\end{equation}\n Then the advective-diffusive part is solved implicitly.  Expressing\n $e = e(\\rho, T)$, and using the chain rule,\n \\begin{equation}\n   \\nabla e = e_T \\nabla T + e_\\rho \\nabla \\rho\n \\end{equation}\n where $e_T = \\partial e / \\partial T |_\\rho \\equiv c_v$ is the\n specific heat at constant volume and $e_\\rho = \\partial e / \\partial\n \\rho |_T$.  Rewriting, we have:\n \\begin{equation}\n \\nabla T = (\\nabla e - e_\\rho \\nabla \\rho)/c_v\n \\end{equation}\n  and then\n \\begin{equation}\n \\rho \\frac{\\partial e}{\\partial t} = \\nabla \\cdot (k/c_v) \\nabla e - \\nabla \\cdot (k e_\\rho/c_v) \\nabla \\rho -A + \\rho S\n \\end{equation}\n This is now a diffusion equation for $e$, which can be solved by the\n techniques described above.  Note: if the transport coefficients\n (including $c_v$, $e_\\rho$) are dependent on $e$, then we still need\n to do a predictor-corrector method here.  This is discussed, for\n example, in \\cite{SNpaper,malone:2011}.  A simple case for this type of \n advection-diffusion is also shown in \\S~\\ref{sec:multiphys:diffreact}.\n\n"
  },
  {
    "path": "figures/Euler/multiple_interfaces.py",
    "content": "import numpy as np\nimport matplotlib.pyplot as plt\nimport matplotlib as mpl\nimport random\n\nimport grid_plot as gp\n\nmpl.rcParams['mathtext.fontset'] = 'cm'\nmpl.rcParams['mathtext.rm'] = 'serif'\n\n# font sizes\nmpl.rcParams['font.size'] = 12\nmpl.rcParams['legend.fontsize'] = 'large'\nmpl.rcParams['figure.titlesize'] = 'medium'\n\n\nclass RiemannWaves(object):\n    \"\"\"draw a diagram with a the three waves \"\"\"\n\n    def __init__(self, x0, dx):\n        \"\"\" create a random collection of Riemann waves \"\"\"\n\n        states = [\"shock\", \"rarefaction\"]\n\n        self.x0 = x0\n        self.dx = dx\n\n        # first pick if the angle of the contact, from a normalize distribution centered on 0\n        _a = np.degrees(0.4*np.random.randn())\n        while _a > 45:\n            _a = np.degrees(0.4*np.random.randn())\n        self.contact_pos = _a\n        print(_a)\n\n        # now the left state -- first set how far it is from the contact\n        _a = np.degrees(0.4*np.random.randn())\n        while _a > 45:\n            _a = np.degrees(0.4*np.random.randn())\n        print(_a)\n        self.left_pos = self.contact_pos - np.abs(_a)\n        self.left_type = random.choice(states)\n\n        # and the right\n        _a = np.degrees(0.4*np.random.randn())\n        while _a > 45:\n            _a = np.degrees(0.4*np.random.randn())\n        print(_a)\n        self.right_pos = self.contact_pos + np.abs(_a)\n        self.right_type = random.choice(states)\n\n    def draw(self, output=\"plot.png\", label=None):\n        \"\"\" draw the wave structure \"\"\"\n\n        print(self.left_pos, self.contact_pos, self.right_pos)\n\n        # each wave has a point on the origin.  The maximum height we\n        # want to draw to is self.dx -- we'll do the center and right\n        x1 = self.x0 + self.dx * np.tan(np.radians(self.left_pos))\n        xc = self.x0 + self.dx * np.tan(np.radians(self.contact_pos))\n        x3 = self.x0 + self.dx * np.tan(np.radians(self.right_pos))\n\n        # draw the waves\n        if self.left_type == \"shock\":\n            plt.plot([self.x0, x1], [0, self.dx], color=\"C0\", lw=3)\n        else:\n            for n in range(5):\n                x1 = self.x0 + self.dx * np.tan(np.radians(self.left_pos - 2*n))\n                plt.plot([self.x0, x1], [0, self.dx], color=\"C0\", lw=1)\n\n        plt.plot([self.x0, xc], [0, self.dx], color=\"C1\")\n\n        if self.right_type == \"shock\":\n            plt.plot([self.x0, x3], [0, self.dx], color=\"C0\", lw=3)\n        else:\n            for n in range(5):\n                x3 = self.x0 + self.dx * np.tan(np.radians(self.right_pos + 2*n))\n                plt.plot([self.x0, x3], [0, self.dx], color=\"C0\", lw=1)\n\n\nif __name__ == \"__main__\":\n\n\n    xmin = 0.0\n    xmax = 8.0\n\n    nzones = 4\n    ng = 0\n\n    gr = gp.FVGrid(nzones, xmin=xmin, xmax=xmax)\n\n    gr.draw_grid()\n\n    gr.label_edge(0, r\"$i-\\mythreehalf$\", fontsize=\"medium\")\n    gr.label_edge(1, r\"$i-\\myhalf$\", fontsize=\"medium\")\n    gr.label_edge(2, r\"$i+\\myhalf$\", fontsize=\"medium\")\n    gr.label_edge(3, r\"$i+\\mythreehalf$\", fontsize=\"medium\")\n    gr.label_edge(3, r\"$i+\\myfivehalf$\", fontsize=\"medium\", right_edge=True)\n\n    gr.label_center(0, r\"$i-1$\", fontsize=\"medium\")\n    gr.label_center(1, r\"$i$\", fontsize=\"medium\")\n    gr.label_center(2, r\"$i+1$\", fontsize=\"medium\")\n    gr.label_center(3, r\"$i+2$\", fontsize=\"medium\")\n\n    riemann = []\n    for n in range(nzones):\n        riemann.append(RiemannWaves(gr.xl[n], 0.5*gr.dx))\n    riemann.append(RiemannWaves(gr.xr[nzones-1], 0.5*gr.dx))\n\n    # draw\n    for r in riemann:\n        r.draw()\n\n    gr.clean_axes()\n\n    f = plt.gcf()\n    f.set_size_inches(8.0, 2.5)\n\n    plt.tight_layout()\n\n    plt.savefig(\"multiple_interfaces.pdf\", dpi=150)\n"
  },
  {
    "path": "figures/Euler/ppm-sequence.py",
    "content": "import math\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport grid_plot as gp\nimport riemann\n\nclass RiemannProblem(object):\n    def __init__(self, q_l, q_r, gamma=5./3., N=3):\n\n        self.q_l = q_l\n        self.q_r = q_r\n        self.nx = 2*N\n\n        self.gr = gp. FVGrid(self.nx)\n\n        # allocate space -- we just care about N zones to the left and\n        # right of the interface\n        self.vars = {}\n\n        rho = self.gr.scratch_array()\n        rho[0:N] = q_l.rho\n        rho[N:2**N] = q_r.rho\n\n        u = self.gr.scratch_array()\n        u[0:N] = q_l.u\n        u[N:2*N] = q_r.u\n\n        p = self.gr.scratch_array()\n        p[0:N] = q_l.p\n        p[N:2*N] = q_r.p\n\n        rscale = max(q_l.rho, q_r.rho)\n        uscale = max(q_l.u, q_r.u)\n        pscale = max(q_l.p, q_r.p)\n\n        # do the ppm reconstruction on all of these variables\n        self.vars[\"rho\"] = gp.PiecewiseParabolic(self.gr, rho, scale=rscale)\n        self.vars[\"u\"] = gp.PiecewiseParabolic(self.gr, u, scale=uscale)\n        self.vars[\"p\"] = gp.PiecewiseParabolic(self.gr, p, scale=pscale)\n\n    def draw_cubic_points(self, var, color=\"r\"):\n        # these are defined on the i+1/2 interface\n        # note that we require 2 zones on either side of the interface, so\n        # we cannot draw points for the first 2 or last 2 zones\n        aint = self.vars[var].aint\n        plt.scatter(self.gr.xr[1:-2], aint[1:-2], marker=\"x\", s=40, color=color, zorder=10)\n\n    def draw_parabola(self, var):\n        ap, am = self.vars[var].ap, self.vars[var].am\n        for n in range(2,self.nx-2):\n            self.vars[var].draw_parabola(n)\n\n    def draw_grid(self):\n        self.gr.draw_grid()\n\n        self.gr.label_center(self.nx/2,   r\"$i$\")\n        self.gr.label_center(self.nx/2-1, r\"$i-1$\")\n        self.gr.label_center(self.nx/2+1, r\"$i+1$\")\n        self.gr.label_center(self.nx/2-2, r\"$i-2$\")\n        self.gr.label_center(self.nx/2+2, r\"$i+2$\")\n\n    def draw_var_avg(self, var, scale=1.0):\n        for n in range(self.gr.nx):\n            self.vars[var].draw_cell_avg(n, color=\"r\")\n\n\n#-----------------------------------------------------------------------------\n\n\n\n# multipage PDF\n# http://matplotlib.org/examples/pylab_examples/multipage_pdf.html\n\n\n# left state\nq_l = riemann.State(rho=1.0, u=0.0, p=1.0)\n\n# right state\nq_r = riemann.State(rho=0.125, u=0.0, p=0.1)\n\nr = RiemannProblem(q_l, q_r)\n\n\nsubidx = [311, 312, 313]\nstates = [\"rho\", \"u\", \"p\"]\nstate_labels = [r\"$\\rho$\", r\"$u$\", r\"$p$\"]\n\n\n#-----------------------------------------------------------------------------\n#  plot 1: initial state\nplt.clf()\n\nfor n, s in enumerate(states):\n\n    plt.subplot(subidx[n])\n    r.draw_grid()\n    r.draw_var_avg(s)\n    r.gr.clean_axes()\n\n    plt.title(state_labels[n])\n\nf = plt.gcf()\nf.set_size_inches(8.0,7.0)\nplt.tight_layout()\n\nplt.savefig(\"ppm-seq-1.png\")\n\n\n#-----------------------------------------------------------------------------\n#  plot 2: cubic points (three vertical)\nplt.clf()\n\nfor n, s in enumerate(states):\n\n    plt.subplot(subidx[n])\n    r.draw_grid()\n    r.draw_var_avg(s)\n    r.draw_cubic_points(s)\n\n    r.gr.clean_axes()\n\n    plt.title(state_labels[n])\n\nf = plt.gcf()\nf.set_size_inches(8.0,7.0)\nplt.tight_layout()\n\nplt.savefig(\"ppm-seq-2.png\")\n\n\n#-----------------------------------------------------------------------------\n#  plot 3: parabola (three vertical)\nplt.clf()\n\nfor n, s in enumerate(states):\n\n    plt.subplot(subidx[n])\n    r.draw_grid()\n    r.draw_var_avg(s)\n    r.draw_cubic_points(s)\n    r.draw_parabola(s)\n    r.gr.clean_axes()\n\n    plt.title(state_labels[n])\n\nf = plt.gcf()\nf.set_size_inches(8.0,7.0)\nplt.tight_layout()\n\nplt.savefig(\"ppm-seq-3.png\")\n\n\n\n#-----------------------------------------------------------------------------\n#  plot 4: cell-center Riemann waves to show what hits the interface\n\n\n#-----------------------------------------------------------------------------\n#  plot 5: tracing (zoom in, 3 horizontal)\n\n\n\n#-----------------------------------------------------------------------------\n#  plot 6: final interface state (zoom in, 3 horizontal)\n\n\n\n#-----------------------------------------------------------------------------\n#  plot 7: Riemann phase\n\n\n\n#-----------------------------------------------------------------------------\n#  plot 8: Riemann solution\n\n\n\n"
  },
  {
    "path": "figures/Euler/ppm-trace.py",
    "content": "import numpy as np\nimport matplotlib.pyplot as plt\nimport grid_plot as gp\n\n\n#-----------------------------------------------------------------------------\n\nnzones = 8\n\n# data that lives on the grid\n#a = np.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.15, 0.5, 0.55])\na = np.array([0.3, 1.0, 1.0, 0.8, 0.2, 0.15, 0.5, 0.55])\n\ngr = gp.FVGrid(nzones)\n\n\nplt.clf()\n\ngr.draw_grid(center_only=1)\n\ngr.label_center(nzones//2-1,   r\"$i$\")\ngr.label_center(nzones//2, r\"$i+1$\")\n\n\nppm = gp.PiecewiseParabolic(gr, a)\n\nfor n in range(gr.nx//2-1, gr.nx//2+1):\n    ppm.draw_parabola(n, color=\"r\")\n\nnn = gr.nx//2-1\nsigma = 0.6\nppm.ppm_trace_left(nn, sigma, color=\"0.75\")\n\nplt.plot([gr.xr[gr.nx//2-1]-sigma*gr.dx, \n          gr.xr[gr.nx//2-1]], [1.2, 1.2], color=\"k\")\nplt.plot([gr.xr[gr.nx//2-1]-sigma*gr.dx, \n          gr.xr[gr.nx//2-1]-sigma*gr.dx], [1.15, 1.25], color=\"k\")\nplt.plot([gr.xr[gr.nx//2-1], \n          gr.xr[gr.nx//2-1]], [1.15, 1.25], color=\"k\")\n\nplt.text(gr.xr[gr.nx//2-1]-0.5*sigma*gr.dx, 1.3, \n         r\"$\\sigma_{i}^{(\\nu)} \\Delta x$\", \n         horizontalalignment=\"center\")\n\nplt.text(gr.xr[gr.nx//2-1]-0.5*sigma*gr.dx, 0.4, \n         r\"$\\mathcal{I}_+^{(\\nu)}$\",\n           color=\"r\", horizontalalignment=\"center\")\n\nplt.axis([gr.xl[gr.nx//2-1]-0.5*gr.dx,gr.xr[gr.nx//2]+0.5*gr.dx, -0.25, 1.4])\nplt.axis(\"off\")\n\nplt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)\n\nf = plt.gcf()\nf.set_size_inches(3.0,2.0)\n\nplt.savefig(\"ppm-trace.pdf\")\nplt.savefig(\"ppm-trace.png\")\n               \n\n\n"
  },
  {
    "path": "figures/Euler/ppm.py",
    "content": "import math\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport grid_plot as gp\n\n\n#-----------------------------------------------------------------------------\n\nnzones = 8\n\n# data that lives on the grid\na = np.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.1, 0.5, 0.55])\n\ngr = gp.FVGrid(nzones)\n\nplt.clf()\n\ngr.draw_grid()\n\ngr.label_center(nzones//2,   r\"$i$\")\ngr.label_center(nzones//2-1, r\"$i-1$\")\ngr.label_center(nzones//2+1, r\"$i+1$\")\ngr.label_center(nzones//2-2, r\"$i-2$\")\ngr.label_center(nzones//2+2, r\"$i+2$\")\n\n\npc = gp.PiecewiseConstant(gr, a)\n\nfor n in range(nzones):\n    pc.draw_cell_avg(n, color=\"r\")\n\n\ngr.clean_axes()\nplt.ylim(-0.25, 1.2)\n\nplt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)\n\nf = plt.gcf()\nf.set_size_inches(8.0,2.0)\n\nplt.savefig(\"piecewise-constant.pdf\")\nplt.savefig(\"piecewise-constant.png\")\n\n\n#------------- PLM -------------\nplt.clf()\n\ngr.draw_grid()\n\ngr.label_center(nzones//2,   r\"$i$\")\ngr.label_center(nzones//2-1, r\"$i-1$\")\ngr.label_center(nzones//2+1, r\"$i+1$\")\ngr.label_center(nzones//2-2, r\"$i-2$\")\ngr.label_center(nzones//2+2, r\"$i+2$\")\n\n\n# not limited and limited\npl_n = gp.PiecewiseLinear(gr, a, nolimit=1)\npl_y = gp.PiecewiseLinear(gr, a)\n\nfor n in range(nzones):\n    pc.draw_cell_avg(n, color=\"0.5\")\n\nfor n in range(2, nzones-2):\n    pl_n.draw_slope(n, color=\"r\", ls=\":\")\n    pl_y.draw_slope(n, color=\"r\")\n\n\ngr.clean_axes()\nplt.ylim(-0.25, 1.2)\n\nplt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)\n\nf = plt.gcf()\nf.set_size_inches(8.0,2.0)\n\nplt.savefig(\"piecewise-linear.pdf\")\nplt.savefig(\"piecewise-linear.png\")\n\n\n#------------- PPM -------------\nplt.clf()\n\ngr.draw_grid()\n\ngr.label_center(nzones//2,   r\"$i$\")\ngr.label_center(nzones//2-1, r\"$i-1$\")\ngr.label_center(nzones//2+1, r\"$i+1$\")\ngr.label_center(nzones//2-2, r\"$i-2$\")\ngr.label_center(nzones//2+2, r\"$i+2$\")\n\nfor n in range(nzones):\n    pc.draw_cell_avg(n, color=\"0.5\")\n\n\n# compute the parabolic coefficients -- limited and not limited\npm_n = gp.PiecewiseParabolic(gr, a, nolimit=1)\npm_y = gp.PiecewiseParabolic(gr, a)\n\nfor n in range(2, nzones-2):\n    pm_n.draw_parabola(n, color=\"r\", ls=\":\")\n    pm_y.draw_parabola(n, color=\"r\")\n\n\ngr.clean_axes()\nplt.ylim(-0.25, 1.2)\n\nplt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)\n\nf = plt.gcf()\nf.set_size_inches(8.0,2.0)\n\n\nplt.savefig(\"piecewise-parabolic.pdf\")\nplt.savefig(\"piecewise-parabolic.png\")\n\n               \n\n\n"
  },
  {
    "path": "figures/Euler/rarefaction_cartoon.py",
    "content": "import numpy as np\nimport matplotlib.pyplot as plt\nimport matplotlib as mpl\n\nmpl.rcParams['mathtext.fontset'] = 'cm'\nmpl.rcParams['mathtext.rm'] = 'serif'\n\n# font sizes\nmpl.rcParams['font.size'] = 12\nmpl.rcParams['legend.fontsize'] = 'large'\nmpl.rcParams['figure.titlesize'] = 'medium'\n\n\nclass RiemannWaves(object):\n    \"\"\"draw a diagram with a left rarefaction and 2 other waves \"\"\"\n\n    def __init__(self, rpos=-1):\n        \"\"\"rpos is position of rarefaction, -1 is left of the interface, +1 is\n\tright of the interface\n\n        \"\"\"\n\n        self.rpos = rpos\n\n        # these control the spacing of the waves in the figure\n        self.h = 1.0\n\n        # this controls the spacing of the structure in the rarefaction\n        self.rare_theta = np.radians(2.5)\n\n        if rpos < 0:\n            self.head = np.radians(-45)\n            self.center = np.radians(5)\n            self.right = np.radians(40)\n        elif rpos == 0:\n            self.head = 0 - 2.5*self.rare_theta\n            self.center = np.radians(30)\n            self.right = np.radians(45)\n        else:\n            self.head = np.radians(5)\n            self.center = np.radians(35)\n            self.right = np.radians(50)\n\n        self.tail = self.head + 5*self.rare_theta\n\n    def draw(self, output=\"plot.png\", label=None):\n        \"\"\" draw the wave structure \"\"\"\n\n        # each wave has a point on the origin.  The maximum height we\n        # want to draw to is self.h -- we'll do the center and right\n        xc = self.h*np.tan(self.center)\n        x3 = self.h*np.tan(self.right)\n\n        plt.clf()\n\n        # draw the waves\n        plt.plot([0, xc], [0, self.h], color=\"C0\")\n        plt.plot([0, x3], [0, self.h], color=\"C0\")\n\n        # rarefaction\n        for t in np.arange(self.head, self.tail+self.rare_theta, \n                           self.rare_theta):\n            xr = self.h*np.tan(t)\n            plt.plot([0, xr], [0, self.h], color=\"C0\")\n\n        xhead = self.h*np.tan(self.head)\n        xtail = self.h*np.tan(self.tail) #+self.rare_theta)\n\n        # draw the grid\n        L = 1.1*max(abs(xhead), abs(x3))\n\n        plt.plot([-L, L], [0, 0], color=\"k\", zorder=-100)\n        plt.plot([0, 0], [0, 1.2*self.h], color=\"k\", zorder=-100)\n\n        plt.text(xhead, self.h, r\"$\\lambda_\\mathrm{head}$\", \n                 horizontalalignment=\"right\", color=\"C0\")\n\n        plt.text(xtail, self.h, r\"$\\lambda_\\mathrm{tail}$\", \n                 horizontalalignment=\"left\", color=\"C0\")\n\n\n        dx = abs(0.75*self.h*np.tan(0.5*(self.head + self.tail)))\n\n        if self.rpos == -1:\n            fac = 0.4\n        elif self.rpos == 0:\n            fac = 1.0\n        else:\n            fac = 1.0\n\n        xh = 0.75*self.h*np.tan(self.head)\n        xt = 0.75*self.h*np.tan(self.tail)\n        xc = 0.75*self.h*np.tan(self.center)\n        xr = 0.75*self.h*np.tan(self.right)\n\n        if dx == 0:\n            dx = abs(xh)\n\n        xL = xh - fac*dx\n        xLstar = 0.5*(xt + xc)\n        xRstar = 0.5*(xc + xr)\n        xR = xr + fac*dx\n\n        plt.text(xL, 0.65*self.h, r\"$L$\", horizontalalignment=\"right\", color=\"C0\")\n        plt.text(xLstar, 0.75*self.h, r\"$L_\\star$\", horizontalalignment=\"center\", color=\"C0\")\n        plt.text(xRstar, 0.75*self.h, r\"$R_\\star$\", horizontalalignment=\"center\", color=\"C0\")\n        plt.text(xR, 0.65*self.h, r\"$R$\", horizontalalignment=\"left\", color=\"C0\")\n\n\n        f = plt.gcf()\n\n        # label?\n        if label is not None:\n            plt.text(0.1, 0.9, label, transform=f.transFigure)\n\n        plt.axis(\"off\")\n        plt.subplots_adjust(left=0.02, right=0.95, bottom=0.05, top=0.95)\n\n        f.set_size_inches(5.0, 3.5)\n\n        plt.tight_layout()\n        plt.savefig(output, bbox_inches=\"tight\", pad_inches=0)\n\n\nif __name__ == \"__main__\":\n\n    rw = RiemannWaves(rpos=-1)\n    rw.draw(\"rarefaction_left.pdf\", label=\"(a)\")\n\n    rw = RiemannWaves(rpos=0)\n    rw.draw(\"rarefaction_center.pdf\", label=\"(b)\")\n\n    rw = RiemannWaves(rpos=1)\n    rw.draw(\"rarefaction_right.pdf\", label=\"(c)\")\n"
  },
  {
    "path": "figures/Euler/riemann-states-q.py",
    "content": "import numpy as np\nimport matplotlib.pyplot as plt\nimport grid_plot as gp\n\ndef riemann(with_time=True):\n\n    # grid info\n    xmin = 0.0\n    xmax = 1.0\n\n    nzones = 2\n    ng = 0\n\n    gr = gp.FVGrid(nzones, xmin=xmin, xmax=xmax)\n\n    plt.clf()\n\n    #------------------------------------------------------------------------\n    # plot a domain without ghostcells\n    gr.draw_grid(emphasize_end=True)\n\n    # label a few\n    gr.label_center(0, r\"$i$\", fontsize=\"medium\")\n    gr.label_center(1, r\"$i+1$\", fontsize=\"medium\")\n    gr.label_edge(1, r\"$i+1/2$\", fontsize=\"medium\")\n\n    if with_time:\n        gr.mark_cell_left_state(1, r\"$q_{i+1/2,R}^{n+1/2}$\", fontsize=\"large\",\n                                color=\"b\")\n        gr.mark_cell_right_state(0, r\"$q_{i+1/2,L}^{n+1/2}$\", fontsize=\"large\",\n                                 color=\"b\")\n    else:\n        gr.mark_cell_left_state(1, r\"$q_{i+1/2,R}$\", fontsize=\"large\",\n                                color=\"b\")\n        gr.mark_cell_right_state(0, r\"$q_{i+1/2,L}$\", fontsize=\"large\",\n                                 color=\"b\")\n\n    gr.label_cell_center(0, r\"$q_i$\")\n    gr.label_cell_center(1, r\"$q_{i+1}$\")\n\n\n\n    # flux\n    plt.arrow(gr.xl[ng+nzones//2]-0.25*gr.dx, 1.05, 0.5*gr.dx, 0,\n                shape='full', head_width=0.075, head_length=0.05,\n                lw=1, width=0.03,\n                edgecolor=\"none\", facecolor=\"red\",\n                length_includes_head=True, zorder=100)\n\n    if with_time:\n        plt.text(gr.xl[ng+nzones//2], 1.15, r\"$F(U(q_{i+1/2}^{n+1/2}))$\", color=\"red\",\n                 horizontalalignment=\"center\")\n    else:\n        plt.text(gr.xl[ng+nzones//2], 1.15, r\"$F(U(q_{i+1/2}))$\", color=\"red\",\n                 horizontalalignment=\"center\")\n\n    gr.clean_axes(padding=False)\n    plt.ylim(-0.25, 1.25)\n\n    plt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)\n\n    f = plt.gcf()\n    f.set_size_inches(6.0,2.25)\n\n\n    plt.tight_layout()\n\n    if with_time:\n        plt.savefig(\"riemann_comp_q.pdf\")\n    else:\n        plt.savefig(\"riemann_comp_q_mol.pdf\")\n\nif __name__== \"__main__\":\n    riemann()\n    riemann(with_time=False)\n\n"
  },
  {
    "path": "figures/Euler/riemann-states.py",
    "content": "import numpy as np\nimport matplotlib.pyplot as plt\nimport grid_plot as gp\n\ndef riemann(with_time=True):\n\n    # grid info\n    xmin = 0.0\n    xmax = 1.0\n\n    nzones = 2\n    ng = 0\n\n    gr = gp.FVGrid(nzones, xmin=xmin, xmax=xmax)\n\n    plt.clf()\n\n    #------------------------------------------------------------------------\n    # plot a domain without ghostcells\n    gr.draw_grid(emphasize_end=True)\n\n    # label a few\n    gr.label_center(0, r\"$i$\", fontsize=\"medium\")\n    gr.label_center(1, r\"$i+1$\", fontsize=\"medium\")\n    gr.label_edge(1, r\"$i+\\myhalf$\", fontsize=\"medium\")\n\n    if with_time:\n        gr.mark_cell_left_state(1, r\"$U_{i+\\myhalf,R}^{n+\\myhalf}$\", fontsize=\"large\",\n                                color=\"b\")\n        gr.mark_cell_right_state(0, r\"$U_{i+\\myhalf,L}^{n+\\myhalf}$\", fontsize=\"large\",\n                                 color=\"b\")\n    else:\n        gr.mark_cell_left_state(1, r\"$U_{i+\\myhalf,R}$\", fontsize=\"large\",\n                                color=\"b\")\n        gr.mark_cell_right_state(0, r\"$U_{i+\\myhalf,L}$\", fontsize=\"large\",\n                                 color=\"b\")\n\n    gr.label_cell_center(0, r\"$U_i$\")\n    gr.label_cell_center(1, r\"$U_{i+1}$\")\n\n\n\n    # flux\n    plt.arrow(gr.xl[ng+nzones//2]-0.25*gr.dx, 1.05, 0.5*gr.dx, 0, \n                shape='full', head_width=0.075, head_length=0.05, \n                lw=1, width=0.03,\n                edgecolor=\"none\", facecolor=\"red\",\n                length_includes_head=True, zorder=100)\n    \n    if with_time:\n        plt.text(gr.xl[ng+nzones//2], 1.15, r\"$F(U_{i+\\myhalf}^{n+\\myhalf})$\", color=\"red\",\n                 horizontalalignment=\"center\")\n    else:\n        plt.text(gr.xl[ng+nzones//2], 1.15, r\"$F(U_{i+\\myhalf})$\", color=\"red\",\n                 horizontalalignment=\"center\")\n\n    gr.clean_axes(padding=False)\n    plt.ylim(-0.25, 1.25)\n\n    plt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)\n\n    f = plt.gcf()\n    f.set_size_inches(6.0,2.25)\n\n    plt.tight_layout()\n\n    if with_time:\n        plt.savefig(\"riemann_comp.pdf\")\n    else:\n        plt.savefig(\"riemann_comp_mol.pdf\")\n\nif __name__== \"__main__\":\n    riemann()\n    riemann(with_time=False)\n"
  },
  {
    "path": "figures/Euler/riemann-waves-jump.py",
    "content": "import matplotlib.pyplot as plt\nimport grid_plot as gp\n\ndef simplegrid():\n\n    # grid info\n    xmin = 0.0\n    xmax = 1.0\n\n    nzones = 2\n    ng = 0\n\n    gr = gp.FVGrid(nzones, xmin=xmin, xmax=xmax)\n\n\n    #------------------------------------------------------------------------\n    # plot a domain without ghostcells\n    gr.draw_grid(draw_end=False, edge_ticks=False)\n\n    gr.label_center(0, r\"$i$\")\n    gr.label_center(1, r\"$i+1$\")\n\n    gr.label_edge(1, r\"$q_{i+\\myhalf}$\")\n\n\n    # draw waves\n    # u - c\n    plt.plot([gr.xr[0], gr.xr[0]-0.75*gr.dx], [0,1.0], color=\"C0\", ls=\"-\")\n    plt.text(gr.xr[0]-0.75*gr.dx, 1.0+0.05, \"$\\lambda^{(-)} =\\, u - c$\", \n               horizontalalignment=\"center\")\n\n    # u\n    plt.plot([gr.xr[0], gr.xr[0]-0.2*gr.dx], [0,1.0], color=\"C0\", ls=\"-\")\n    plt.text(gr.xr[0]-0.2*gr.dx, 1.0+0.05, \"$\\lambda^{(\\circ)} =\\, u$\", \n               horizontalalignment=\"center\")\n\n    # u + c\n    plt.plot([gr.xr[0], gr.xr[0]+0.4*gr.dx], [0,1.0], color=\"C0\", ls=\"-\")\n    plt.text(gr.xr[0]+0.4*gr.dx, 1.0+0.05, \"$\\lambda^{(+)} =\\, u + c$\", \n               horizontalalignment=\"center\")\n\n\n    plt.plot([gr.xl[0], gr.xr[0]], [0.3, 0.3], color=\"C1\", linewidth=2)\n    plt.text(gr.xc[0], 0.33, r\"$\\langle q \\rangle_i$\", color=\"C1\")\n    \n    plt.plot([gr.xl[1], gr.xr[1]], [0.6, 0.6], color=\"C1\", linewidth=2)\n    plt.text(gr.xc[1], 0.63, r\"$\\langle q \\rangle_{i+1}$\", color=\"C1\")\n\n    gr.clean_axes(padding=False)\n    plt.ylim(-0.2,1.2)\n\n    plt.tight_layout()\n\n    f = plt.gcf()\n    f.set_size_inches(6, 3.5)\n\n\n    plt.savefig(\"riemann-waves-jump.png\", dpi=150)\n\nif __name__== \"__main__\":\n    simplegrid()\n"
  },
  {
    "path": "figures/Euler/riemann-waves.py",
    "content": "import matplotlib.pyplot as plt\nimport grid_plot as gp\n\ndef simplegrid():\n\n    # grid info\n    xmin = 0.0\n    xmax = 1.0\n\n    nzones = 2\n    ng = 0\n\n    gr = gp.FVGrid(nzones, xmin=xmin, xmax=xmax)\n\n\n    #------------------------------------------------------------------------\n    # plot a domain without ghostcells\n    gr.draw_grid(draw_end=False, edge_ticks=False)\n\n    gr.label_center(0, r\"$i$\")\n    gr.label_center(1, r\"$i+1$\")\n\n    gr.label_edge(1, r\"$q_{i+\\myhalf}$\")\n\n\n    # draw waves\n    # u - c\n    plt.plot([gr.xr[0], gr.xr[0]-0.75*gr.dx], [0,1.0], color=\"C0\", ls=\"-\")\n    plt.text(gr.xr[0]-0.75*gr.dx, 1.0+0.05, \"$\\lambda^{(-)} =\\, u - c$\", \n               horizontalalignment=\"center\")\n\n    # u\n    plt.plot([gr.xr[0], gr.xr[0]-0.2*gr.dx], [0,1.0], color=\"C0\", ls=\"-\")\n    plt.text(gr.xr[0]-0.2*gr.dx, 1.0+0.05, \"$\\lambda^{(\\circ)} =\\, u$\", \n               horizontalalignment=\"center\")\n\n    # u + c\n    plt.plot([gr.xr[0], gr.xr[0]+0.4*gr.dx], [0,1.0], color=\"C0\", ls=\"-\")\n    plt.text(gr.xr[0]+0.4*gr.dx, 1.0+0.05, \"$\\lambda^{(+)} =\\, u + c$\", \n               horizontalalignment=\"center\")\n\n    # label regions\n    plt.text(gr.xr[0]-0.5*gr.dx, 0.3, r\"$L$\", horizontalalignment=\"center\", color=\"C1\")\n    plt.text(gr.xr[0]-0.5*gr.dx, 0.22, r\"$(\\rho_L, u_L, p_L)$\",\n             horizontalalignment=\"center\", color=\"C1\", fontsize=\"small\")\n\n    plt.text(gr.xr[0]-0.33*gr.dx, 0.75, r\"$L_*$\",\n             horizontalalignment=\"center\", color=\"C1\")\n    plt.text(gr.xr[0]-0.33*gr.dx, 0.67, r\"$(\\rho_{\\star L}, u_\\star, p_\\star)$\",\n             horizontalalignment=\"center\", color=\"C1\", fontsize=\"small\")\n\n    plt.text(gr.xr[0]+0.1*gr.dx, 0.75, r\"$R_*$\", horizontalalignment=\"center\", color=\"C1\")\n    plt.text(gr.xr[0]+0.1*gr.dx, 0.67, r\"$(\\rho_{\\star R}, u_\\star, p_\\star)$\",\n             horizontalalignment=\"center\", color=\"C1\", fontsize=\"small\", zorder=1000)\n\n    plt.text(gr.xr[0]+0.3*gr.dx, 0.3, r\"$R$\", horizontalalignment=\"center\", color=\"C1\")\n    plt.text(gr.xr[0]+0.3*gr.dx, 0.22, r\"$(\\rho_R, u_R, p_R)$\",\n             horizontalalignment=\"center\", color=\"C1\", fontsize=\"small\")\n\n\n\n    gr.clean_axes(padding=False)\n    plt.ylim(-0.2,1.2)\n\n    plt.tight_layout()\n\n    f = plt.gcf()\n    f.set_size_inches(6,3.5)\n\n\n    plt.savefig(\"riemann-waves.pdf\")\n\nif __name__== \"__main__\":\n    simplegrid()\n"
  },
  {
    "path": "figures/Euler/riemann_cartoon.py",
    "content": "import numpy as np\nimport matplotlib.pyplot as plt\nimport matplotlib as mpl\n\nmpl.rcParams['mathtext.fontset'] = 'cm'\nmpl.rcParams['mathtext.rm'] = 'serif'\n\n# font sizes\nmpl.rcParams['font.size'] = 12\nmpl.rcParams['legend.fontsize'] = 'large'\nmpl.rcParams['figure.titlesize'] = 'medium'\n\n\nclass RiemannWaves(object):\n    \"\"\"draw a diagram with the 3 waves for the Euler Riemann problem\"\"\"\n\n    def __init__(self, left=-1, contact=-1, right=1):\n        \"\"\"for each of the waves, -1 is left of the interface, +1 is right of\n\tthe interface\"\"\"\n\n        self.left = left\n        self.contact = contact\n        self.right = right\n\n        # these control the spacing of the waves in the figure\n        self.theta = np.radians(25)\n        self.h = 1.0\n\n    def draw(self, output=\"plot.png\", label=None):\n        \"\"\" draw the wave structure \"\"\"\n\n        # by default, we'll make the waves theta degrees apart\n        if self.right < 0:\n            angle3 = -0.75*self.theta\n        elif self.right > 0 and self.contact < 0:\n            angle3 = 0.534*self.theta\n        elif self.right > 0 and self.contact > 0 and self.left < 0:\n            angle3 = 0.966*self.theta\n        else:\n            angle3 = 2.25*self.theta\n\n        anglec = angle3 - 0.75*self.theta\n        angle1 = anglec - 0.75*self.theta\n\n        print(np.degrees(angle1), np.degrees(anglec), np.degrees(angle3))\n\n        # each wave has a point on the origin.  The maximum height we want to draw to is self.h\n        x1 = self.h*np.tan(angle1)\n        xc = self.h*np.tan(anglec)\n        x3 = self.h*np.tan(angle3)\n\n        print(x1, xc, x3)\n\n        plt.clf()\n\n        # draw the waves\n        plt.plot([0, x1], [0, self.h], color=\"C0\")\n        plt.plot([0, xc], [0, self.h], color=\"C0\")\n        plt.plot([0, x3], [0, self.h], color=\"C0\")\n\n        # label regions\n        x1h = 0.75*self.h*np.tan(angle1)\n        xch = 0.75*self.h*np.tan(anglec)\n        x3h = 0.75*self.h*np.tan(angle3)\n\n        dx = max(xch - x1h, x3h - xch)\n        xL = x1h - 0.6*dx\n        xLstar = 0.5*(x1h + xch)\n        xRstar = 0.5*(xch + x3h)\n        xR = x3h + 0.6*dx\n\n        plt.text(xL, 0.65*self.h, r\"$L$\", horizontalalignment=\"right\", color=\"C0\")\n        plt.text(xLstar, 0.75*self.h, r\"$L_\\star$\", horizontalalignment=\"center\", color=\"C0\")\n        plt.text(xRstar, 0.75*self.h, r\"$R_\\star$\", horizontalalignment=\"center\", color=\"C0\")\n        plt.text(xR, 0.65*self.h, r\"$R$\", horizontalalignment=\"left\", color=\"C0\")\n\n        # draw the grid\n        L = 1.1*max(abs(x1), abs(x3))\n\n        plt.plot([-L, L], [0, 0], color=\"k\")\n        plt.plot([0, 0], [0, 1.2*self.h], color=\"k\")\n\n        f = plt.gcf()\n\n        # label?\n        if label is not None:\n            plt.text(0.1, 0.9, label, transform=f.transFigure)\n\n        plt.axis(\"off\")\n        plt.subplots_adjust(left=0.02, right=0.95, bottom=0.05, top=0.95)\n\n\n        f.set_size_inches(5.0, 3.5)\n\n        plt.tight_layout()\n        plt.savefig(output, bbox_inches=\"tight\", pad_inches=0)\n\n\nif __name__ == \"__main__\":\n\n    rw = RiemannWaves(left=-1, contact=-1, right=-1)\n    rw.draw(\"riemann_waves_ifc_R.pdf\", label=\"(a)\")\n\n    rw = RiemannWaves(left=-1, contact=-1, right=1)\n    rw.draw(\"riemann_waves_ifc_Rstar.pdf\", label=\"(b)\")\n\n    rw = RiemannWaves(left=-1, contact=1, right=1)\n    rw.draw(\"riemann_waves_ifc_Lstar.pdf\", label=\"(c)\")\n\n    rw = RiemannWaves(left=1, contact=1, right=1)\n    rw.draw(\"riemann_waves_ifc_L.pdf\", label=\"(d)\")\n"
  },
  {
    "path": "figures/Euler/states.py",
    "content": "import math\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport grid_plot as gp\n\ndef riemann():\n\n    # grid info\n    xmin = 0.0\n    xmax = 1.0\n\n    nzones = 1\n    ng = 0\n\n    gr = gp.FVGrid(nzones, xmin=xmin, xmax=xmax)\n\n\n    #------------------------------------------------------------------------\n    # plot a domain without ghostcells\n    gr.draw_grid()\n\n    gr.label_center(0, r\"$i$\")\n\n    gr.label_cell_center(0, r\"$q_i$\")\n\n    gr.mark_cell_left_state(0, r\"$q_{i-1/2,R}^{n+1/2}$\", color=\"r\")\n    gr.mark_cell_right_state(0, r\"$q_{i+1/2,L}^{n+1/2}$\", color=\"r\")\n\n\n    plt.arrow(gr.xc[0]-0.05*gr.dx, 0.5, -0.13*gr.dx, 0,\n                shape='full', head_width=0.075, head_length=0.05,\n                lw=1, width=0.01,\n                edgecolor=\"none\", facecolor=\"r\",\n                length_includes_head=True, zorder=100)\n\n    plt.arrow(gr.xc[0]+0.05*gr.dx, 0.5, 0.13*gr.dx, 0,\n                shape='full', head_width=0.075, head_length=0.05,\n                lw=1, width=0.01,\n                edgecolor=\"none\", facecolor=\"r\",\n                length_includes_head=True, zorder=100)\n\n\n    plt.xlim(gr.xl[0]-0.25*gr.dx,gr.xr[2*ng+nzones-1]+0.25*gr.dx)\n    # plt.ylim(-0.25, 0.75)\n    plt.axis(\"off\")\n\n    plt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)\n\n    f = plt.gcf()\n    f.set_size_inches(4.0,2.5)\n\n\n    plt.savefig(\"states.png\")\n    plt.savefig(\"states.pdf\")\n\n\nif __name__== \"__main__\":\n    riemann()\n"
  },
  {
    "path": "figures/Euler/update.sh",
    "content": "#!/bin/sh\n\nDEST=../../Euler\n\n# riemann-states.py makes riemann_comp.pdf\npython3 riemann-states.py\ncp -f riemann_comp.pdf ${DEST}\n\n\n# ppm.py makes piecewise-constant.pdf piecewise-linear.pdf piecewise-parabolic.pdf\npython3 ppm.py\ncp -f piecewise-constant.pdf piecewise-linear.pdf piecewise-parabolic.pdf ${DEST}\n\n\n# states.py makes states.pdf\npython3 states.py\ncp -f states.pdf ${DEST}\n\n\n"
  },
  {
    "path": "figures/advection/2dFD.py",
    "content": "import math\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport grid_plot as gp\n\ndef simplegrid():\n\n    nzones = 5\n    gr = gp.FDGrid2d(nzones, nzones, ng=0)\n\n    # plot a domain without ghostcells\n    gr.draw_grid()\n\n    #------------------------------------------------------------------------\n    # label\n    gr.label_cell_center(nzones/2, nzones/2, r\"$a_{i,j}$\", \n                         fontsize=\"x-large\", color=\"r\")\n\n    # grid labels\n    gr.label_center_x(nzones/2-2, r\"$i-2$\")\n    gr.label_center_x(nzones/2-1, r\"$i-1$\")\n    gr.label_center_x(nzones/2, r\"$i$\")\n    gr.label_center_x(nzones/2+1, r\"$i+1$\")\n    gr.label_center_x(nzones/2+2, r\"$i+2$\")\n\n    gr.label_center_y(nzones/2-2, r\"$j-2$\")\n    gr.label_center_y(nzones/2-1, r\"$j-1$\")\n    gr.label_center_y(nzones/2, r\"$j$\")\n    gr.label_center_y(nzones/2+1, r\"$j+1$\")\n    gr.label_center_y(nzones/2+2, r\"$j+2$\")\n\n    # axes\n    gr.clean_axes()\n    plt.subplots_adjust(left=0.025,right=0.98,bottom=0.1,top=0.98)\n\n    f = plt.gcf()\n    f.set_size_inches(7.0,7.0)\n\n    plt.savefig(\"2dFD.png\")\n\nif __name__== \"__main__\":\n    simplegrid()\n"
  },
  {
    "path": "figures/advection/2dFV.py",
    "content": "import math\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport grid_plot as gp\n\ndef simplegrid():\n\n    nzones = 5\n    gr = gp.FVGrid2d(nzones, nzones, ng=0)\n\n    # plot a domain without ghostcells\n    gr.draw_grid()\n\n    #------------------------------------------------------------------------\n    # label\n    gr.label_cell_center(nzones/2, nzones/2, r\"$a_{i,j}$\", \n                         fontsize=\"x-large\", color=\"r\")\n    gr.shade_cell(nzones/2, nzones/2)\n\n    # grid labels\n    gr.label_center_x(nzones/2-2, r\"$i-2$\")\n    gr.label_center_x(nzones/2-1, r\"$i-1$\")\n    gr.label_center_x(nzones/2, r\"$i$\")\n    gr.label_center_x(nzones/2+1, r\"$i+1$\")\n    gr.label_center_x(nzones/2+2, r\"$i+2$\")\n\n    gr.label_center_y(nzones/2-2, r\"$j-2$\")\n    gr.label_center_y(nzones/2-1, r\"$j-1$\")\n    gr.label_center_y(nzones/2, r\"$j$\")\n    gr.label_center_y(nzones/2+1, r\"$j+1$\")\n    gr.label_center_y(nzones/2+2, r\"$j+2$\")\n\n    # axes\n    gr.clean_axes()\n    plt.subplots_adjust(left=0.025,right=0.98,bottom=0.1,top=0.98)\n\n    f = plt.gcf()\n    f.set_size_inches(7.0,7.0)\n\n    plt.savefig(\"2dFV.png\")\n\nif __name__== \"__main__\":\n    simplegrid()\n"
  },
  {
    "path": "figures/advection/2dgrid.py",
    "content": "import numpy as np\nimport matplotlib.pyplot as plt\nimport grid_plot as gp\n\ndef simplegrid():\n\n    nzones = 3\n    gr = gp.FVGrid2d(nzones, nzones, ng=0)\n\n    # plot a domain without ghostcells\n    gr.draw_grid()\n    \n\n    #------------------------------------------------------------------------\n    # label\n    gr.label_cell_center(nzones//2, nzones//2, r\"$a_{i,j}$\", fontsize=\"large\")\n    gr.label_cell_center(nzones//2+1, nzones//2, r\"$a_{i+1,j}$\", fontsize=\"large\")\n    gr.label_cell_center(nzones//2, nzones//2+1, r\"$a_{i,j+1}$\", fontsize=\"large\")\n\n    # i+1/2,j interface\n    gr.mark_cell_left_state_x(nzones//2, nzones//2, r\"$a^{n+\\myhalf}_{i+\\myhalf,j,L}$\",\n                              color=\"b\")\n    gr.mark_cell_right_state_x(nzones//2+1, nzones//2, r\"$a^{n+\\myhalf}_{i+\\myhalf,j,R}$\",\n                               color=\"b\")\n\n\n    # i,j+1/2 interface\n    gr.mark_cell_left_state_y(nzones//2, nzones//2, r\"$a^{n+\\myhalf}_{i,j+\\myhalf,L}$\",\n                              color=\"b\")\n    gr.mark_cell_right_state_y(nzones//2, nzones//2+1, r\"$a^{n+\\myhalf}_{i,j+\\myhalf,R}$\",\n                               color=\"b\")\n\n    # grid labels\n    gr.label_center_x(nzones//2-1, r\"$i-1$\")\n    gr.label_center_x(nzones//2, r\"$i$\")\n    gr.label_center_x(nzones//2+1, r\"$i+1$\")\n\n    gr.label_center_y(nzones//2-1, r\"$j-1$\")\n    gr.label_center_y(nzones//2, r\"$j$\")\n    gr.label_center_y(nzones//2+1, r\"$j+1$\")\n\n\n    # axes\n    gr.clean_axes()\n    plt.subplots_adjust(left=0.02,right=0.98,bottom=0.02,top=0.98)\n\n    f = plt.gcf()\n    f.set_size_inches(7.0,7.0)\n\n    plt.savefig(\"2dgrid.pdf\")\n\nif __name__== \"__main__\":\n    simplegrid()\n"
  },
  {
    "path": "figures/advection/2dgrid_hat.py",
    "content": "import numpy\nimport matplotlib.pyplot as plt\nimport grid_plot as gp\n\ndef simplegrid():\n\n    nzones = 3\n    gr = gp.FVGrid2d(nzones, nzones, ng=0)\n\n\n    # plot a domain without ghostcells\n    gr.draw_grid()\n\n\n    #------------------------------------------------------------------------\n    # label\n    gr.label_cell_center(nzones//2, nzones//2, r\"$a_{i,j}$\", fontsize=\"large\")\n    gr.label_cell_center(nzones//2+1, nzones//2, r\"$a_{i+1,j}$\", fontsize=\"large\")\n    gr.label_cell_center(nzones//2, nzones//2+1, r\"$a_{i,j+1}$\", fontsize=\"large\")\n    gr.label_cell_center(nzones//2, nzones//2-1, r\"$a_{i,j-1}$\", fontsize=\"large\")\n\n    # i+1/2,j interface\n    gr.mark_cell_left_state_x(nzones//2, nzones//2, r\"$\\hat{a}^{n+\\myhalf}_{i+\\myhalf,j,L}$\", color=\"b\")\n\n    # i,j+1/2 interface\n    gr.mark_cell_left_state_y(nzones//2, nzones//2, r\"$\\hat{a}^{n+\\myhalf}_{i,j+\\myhalf,L}$\", color=\"b\")\n    gr.mark_cell_right_state_y(nzones//2, nzones//2+1, r\"$\\hat{a}^{n+\\myhalf}_{i,j+\\myhalf,R}$\", color=\"b\")\n\n    # i,j-1/2 interface\n    gr.mark_cell_left_state_y(nzones//2, nzones//2-1, r\"$\\hat{a}^{n+\\myhalf}_{i,j-\\myhalf,L}$\", color=\"b\")\n    gr.mark_cell_right_state_y(nzones//2, nzones//2, r\"$\\hat{a}^{n+\\myhalf}_{i,j-\\myhalf,R}$\", color=\"b\")\n\n    # grid labels\n    gr.label_center_x(nzones//2-1, r\"$i-1$\")\n    gr.label_center_x(nzones//2, r\"$i$\")\n    gr.label_center_x(nzones//2+1, r\"$i+1$\")\n\n    gr.label_center_y(nzones//2-1, r\"$j-1$\")\n    gr.label_center_y(nzones//2, r\"$j$\")\n    gr.label_center_y(nzones//2+1, r\"$j+1$\")\n\n    # axes\n    gr.clean_axes()\n    plt.subplots_adjust(left=0.02,right=0.98,bottom=0.02,top=0.98)\n\n    f = plt.gcf()\n    f.set_size_inches(7.0,7.0)\n\n    plt.savefig(\"2dgrid-hat.pdf\")\n\nif __name__== \"__main__\":\n    simplegrid()\n"
  },
  {
    "path": "figures/advection/2dgrid_transverse.py",
    "content": "import math\nimport numpy\nimport matplotlib.pyplot as plt\nimport grid_plot as gp\n\ndef simplegrid():\n\n    nzones = 3\n    gr = gp.FVGrid2d(nzones, nzones, ng=0)\n\n\n    # plot a domain without ghostcells\n    gr.draw_grid()\n\n    #------------------------------------------------------------------------\n    # label\n    gr.label_cell_center(nzones//2, nzones//2, r\"$a_{i,j}$\", fontsize=\"large\")\n    gr.label_cell_center(nzones//2+1, nzones//2, r\"$a_{i+1,j}$\", fontsize=\"large\")\n    gr.label_cell_center(nzones//2, nzones//2+1, r\"$a_{i,j+1}$\", fontsize=\"large\")\n    gr.label_cell_center(nzones//2, nzones//2-1, r\"$a_{i,j-1}$\", fontsize=\"large\")\n\n    # i+1/2,j interface\n    gr.mark_cell_left_state_x(nzones//2, nzones//2, r\"$\\hat{a}^{n+\\myhalf}_{i+\\myhalf,j,L}$\", color=\"b\")\n\n    # i,j+1/2 interface\n    gr.mark_cell_state_y(nzones//2, nzones//2, r\"${a}^T_{i,j+\\myhalf}$\")\n\n    # i,j-1/2 interface\n    gr.mark_cell_state_y(nzones//2, nzones//2-1, r\"${a}^T_{i,j-\\myhalf}$\", off_sign=-1)\n\n    # helpful line showing the transverse bits\n    plt.plot([gr.xc[nzones//2], gr.xc[nzones//2]], \n               [gr.yl[nzones//2]+0.025*gr.dy, gr.yr[nzones//2]-0.025*gr.dy], linestyle=\":\",\n               color=\"0.5\", zorder=1000)\n\n    plt.plot([gr.xc[nzones//2], gr.xr[nzones//2]-0.26*gr.dx],\n               [gr.yc[nzones//2], gr.yc[nzones//2]], linestyle=\":\",\n               color=\"0.5\", zorder=1000)\n\n    plt.plot([gr.xr[nzones//2]-0.34*gr.dx, gr.xr[nzones//2]-0.26*gr.dx],\n               [gr.yc[nzones//2]+0.04*gr.dy, gr.yc[nzones//2]], linestyle=\":\",\n               color=\"0.5\", zorder=1000)\n\n    plt.plot([gr.xr[nzones//2]-0.34*gr.dx, gr.xr[nzones//2]-0.26*gr.dx],\n               [gr.yc[nzones//2]-0.04*gr.dy, gr.yc[nzones//2]], linestyle=\":\",\n               color=\"0.5\", zorder=1000)\n\n\n    # grid labels\n    gr.label_center_x(nzones//2-1, r\"$i-1$\")\n    gr.label_center_x(nzones//2, r\"$i$\")\n    gr.label_center_x(nzones//2+1, r\"$i+1$\")\n\n    gr.label_center_y(nzones//2-1, r\"$j-1$\")\n    gr.label_center_y(nzones//2, r\"$j$\")\n    gr.label_center_y(nzones//2+1, r\"$j+1$\")\n\n\n    # axes\n    gr.clean_axes()\n\n    plt.subplots_adjust(left=0.02,right=0.98,bottom=0.02,top=0.98)\n\n    f = plt.gcf()\n    f.set_size_inches(7.0,7.0)\n\n    plt.savefig(\"2dgrid-transverse.pdf\")\n               \n\nif __name__== \"__main__\":\n    simplegrid()\n"
  },
  {
    "path": "figures/advection/characteristics.py",
    "content": "import numpy as np\nimport matplotlib as mpl\nimport matplotlib.pyplot as plt\n\nmpl.rcParams['mathtext.fontset'] = 'cm'\nmpl.rcParams['mathtext.rm'] = 'serif'\n\n\ndef fun(x):\n    state = np.zeros_like(xplot)\n    state[x < 0.5] = 1.0\n    return state\n\ndef funs(x):\n    state = 0.5 + 0.25*np.sin(2.0*np.pi*x)\n    return state\n\nnpts_plot = 1000\nxplot = np.linspace(0.0, 1.0, npts_plot)\n\nnchar = 20\nxchar = np.linspace(0.0, 1.0, nchar)\n\nstate = funs(xplot)\nprint(type(state))\n\nplt.subplot(211)\n\nplt.plot(xplot, state, lw=2)\n\n\nax = plt.gca()\nax.spines['right'].set_visible(False)\nax.spines['top'].set_visible(False)\nax.spines['right'].set_visible(False)\nax.spines['top'].set_visible(False)\nax.spines['right'].set_visible(False)\nax.spines['top'].set_visible(False)\nax.xaxis.set_ticks_position('bottom')\nax.yaxis.set_ticks_position('left')\n\nplt.ylim(-0.1, 1.1)\n\nplt.xlabel(r\"$x$\", fontsize=\"large\")\nplt.ylabel(r\"$a$\", fontsize=\"large\")\n\n\n\nplt.subplot(212)\n\nuchar = 1.0\nt = np.linspace(0.0, 1.0, 100)\nfor n in range(nchar):\n    xc = xchar[n] + uchar*t\n    plt.plot(xc, t, color=\"0.5\")\n\nax = plt.gca()\nax.spines['right'].set_visible(False)\nax.spines['top'].set_visible(False)\nax.spines['right'].set_visible(False)\nax.spines['top'].set_visible(False)\nax.spines['right'].set_visible(False)\nax.spines['top'].set_visible(False)\nax.xaxis.set_ticks_position('bottom')\nax.yaxis.set_ticks_position('left')\n\nplt.xlabel(r\"$x$\", fontsize=\"large\")\nplt.ylabel(r\"$t$\", fontsize=\"large\")\n\nplt.xlim(0.0, 1.0)\nplt.ylim(0.0, 1.1)\n\nf = plt.gcf()\nf.set_size_inches(6.0, 8.0)\n\nplt.tight_layout()\n\nplt.savefig(\"advection-characteristics.png\")\n\n"
  },
  {
    "path": "figures/advection/domains.py",
    "content": "import numpy as np\nimport matplotlib.pylab as plt\nimport matplotlib as mpl\n\nmpl.rcParams['mathtext.fontset'] = 'cm'\nmpl.rcParams['mathtext.rm'] = 'serif'\n\nmpl.rcParams['font.size'] = 12\nmpl.rcParams['legend.fontsize'] = 'large'\nmpl.rcParams['figure.titlesize'] = 'medium'\n\ndef domains(method=\"FTCS\"):\n\n    # grid info\n    xl = 0.0\n    xc = 1.0\n    xr = 2.0\n\n    dx = xr - xc\n\n    t0 = 0.0\n    t1 = 1.0\n\n    dt = t1 - t0\n\n    C = 0.6\n    xp = xc - C*dx\n\n    plt.clf()\n\n    # plot a square representing [x, x+dx] x [t, t+dt]\n\n    # x-axis\n    plt.arrow(0, 0, 2.5*dx, 0,\n              shape=\"full\", head_width=0.04, head_length=0.06,\n              lw=1, width=0.005,\n              facecolor=\"k\",\n              length_includes_head=True, zorder=100)\n\n    plt.text(2.55*dx, 0, r\"$x$\", fontsize=16, verticalalignment=\"center\")\n\n    # x points labeled\n    for p, l in [((xc, 0), r\"$x_i$\"), ((xl, 0), r\"$x_{i-1}$\"), ((xr, 0), r\"$x_{i+1}$\")]:\n        plt.text(p[0], p[1]-0.07, l, fontsize=16,\n             horizontalalignment=\"center\", verticalalignment=\"top\")\n        plt.plot([p[0], p[0]], [p[1], p[1]-0.04], color=\"k\")\n\n    plt.text(xp, -0.07, r\"$x_i - C \\Delta x$\", fontsize=16,\n             horizontalalignment=\"center\", verticalalignment=\"top\")\n    plt.plot([xp, xp], [0, -0.04], color=\"k\")\n\n    # points\n    plt.scatter([xl, xc, xr, xc], [0, 0, 0, dt], color=\"C3\",\n                marker=\"o\", s=40, zorder=1000, alpha=1.0)\n\n    # data points\n    eps = 0.02*dx\n    for p, l in [((xl, 0), r\"$a_{i-1}^n$\"), ((xc, 0), r\"$a_{i}^n$\"),\n                 ((xr, 0), r\"$a_{i+1}^n$\"), ((xc, dt), r\"$a_{i}^{n+1}$\")]:\n        plt.text(p[0]+eps, p[1]+eps, l,\n                 horizontalalignment=\"left\", verticalalignment=\"bottom\",\n                 fontsize=16, color=\"C3\", zorder=1000)\n\n    # time axis\n    plt.arrow(0, 0, 0, 1.3*dt,\n              shape=\"full\", head_width=0.04, head_length=0.06,\n              lw=1, width=0.005,\n              facecolor=\"k\",\n              length_includes_head=True, zorder=100)\n\n    plt.text(0, 1.35*dt, r\"$t$\", fontsize=16, horizontalalignment=\"center\")\n\n    plt.text(-0.25, dt, r\"$t^{n+1}$\", fontsize=16)\n    plt.plot([-0.04, 0], [dt, dt], color=\"k\")\n\n    plt.text(-0.25, 0, r\"$t^n$\", fontsize=16)\n    plt.plot([-0.04, 0], [0, 0], color=\"k\")\n\n    plt.plot([0, 2.0*dx], [dt, dt], ls=\":\", color=\"0.5\")\n    plt.plot([dx, dx], [0, dt], ls=\":\", color=\"0.5\")\n    plt.plot([2.0*dx, 2.0*dx], [0, dt], ls=\":\", color=\"0.5\")\n\n    # domain of dependence\n    if method == \"upwind\":\n        plt.fill([0, xc, xc, 0], [0, dt, 0, 0], color=\"C0\", lw=2, alpha=0.5)\n    elif method == \"FTCS\":\n        plt.fill([0, xc, xr, 0], [0, dt, 0, 0], color=\"C0\", lw=2, alpha=0.5)\n    elif method == \"downwind\":\n        plt.fill([xc, xc, xr, 0], [0, dt, 0, 0], color=\"C0\", lw=2, alpha=0.5)\n\n    # true domain of dependence\n    plt.fill([xp, xc, xc, 0], [0, dt, 0, 0], color=\"C1\", lw=2, alpha=0.5)\n\n\n    # label\n    plt.text(xr+0.1*dx, 0.5*dt, method, fontsize=16,\n             horizontalalignment=\"left\")\n\n    plt.axis(\"off\")\n\n    plt.subplots_adjust(left=0.05, right=0.95, bottom=0.05, top=0.95)\n\n    plt.xlim(-0.4,2.8*dx)\n    plt.ylim(-0.1,1.4*dx)\n\n    f = plt.gcf()\n    f.set_size_inches(9.0,5.0)\n\n    plt.tight_layout()\n    plt.savefig(f\"domains_{method}.pdf\", bbox_inches=\"tight\")\n\n\nif __name__== \"__main__\":\n    domains(method=\"upwind\")\n    domains(method=\"FTCS\")\n    domains(method=\"downwind\")\n"
  },
  {
    "path": "figures/advection/fd_ghost.py",
    "content": "import grid_plot as gp\nimport numpy as np\n\n# plot a simple finite-difference grid\n\n#-----------------------------------------------------------------------------\n\nnzones = 9\n\n# data that lives on the grid\n#a = np.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.15, 0.5, 0.55])\na = np.array([0.55, 0.3, 1.0, 0.9, 0.8, 0.25, 0.1, 0.5, 0.55])\n\ngr = gp.FDGrid(nzones, ng=1)\n\ngr.draw_grid(draw_ghost=1)\n\nlabels = [\"-1\", \"0\", \"1\", \"\", \"i-1\", \"i\", \"i+1\", \"\", \"N-2\", \"N-1\", \"N\"]\n\nfor i in range(gr.ilo-gr.ng, gr.ng+gr.nx+1):\n    if not labels[i] == \"\":\n        gr.label_node(i, r\"$%s$\" % (labels[i]), fontsize=\"medium\")\n\n# draw the data\nfor i in range(gr.ilo, gr.ihi+1):\n    gr.draw_data(i, a[i-gr.ng], color=\"r\")\n\n\ngr.label_value(gr.ilo+4, a[gr.ilo+4-gr.ng], r\"$a_i$\", color=\"r\")\n\n# label dx\ngr.label_dx(gr.ng+nzones//2)\n\ngr.clean_axes(pad_fac=0.1, show_ghost=True, ylim=(-0.5, 1.3))\n\nfig = gr.fig\nfig.set_size_inches(10.0,3.0)\n\n\nfig.savefig(\"fd_ghost.pdf\")\n"
  },
  {
    "path": "figures/advection/fv_ghost.py",
    "content": "import numpy as np\nimport matplotlib.pyplot as plt\nimport grid_plot as gp\n\n# plot a simple finite-difference grid\n\n#-----------------------------------------------------------------------------\n\nnzones = 7\nng = 2\n\n# data that lives on the grid\n#a = np.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.15, 0.5, 0.55])\n#a = np.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.1, 0.5, 0.55])\na = np.array([1.0, 0.9, 0.8, 0.25, 0.1, 0.5, 0.55])\n\ngr = gp.FVGrid(nzones, ng)\n\naa = gr.scratch_array()\naa[gr.ilo:gr.ihi+1] = a\n\ncc = gp.PiecewiseConstant(gr, aa)\n\nplt.clf()\n\ngr.draw_grid(draw_ghost=1, emphasize_end=1)\n\ngr.label_center(ng+nzones//2,   r\"$i$\")\ngr.label_center(ng+nzones//2-1, r\"$i-1$\")\ngr.label_center(ng+nzones//2+1, r\"$i+1$\")\n\ngr.label_center(gr.ilo, r\"$\\mathrm{lo}$\")\ngr.label_center(gr.ilo-1, r\"$\\mathrm{lo-1}$\")\ngr.label_center(gr.ilo-2, r\"$\\mathrm{lo-2}$\")\n\ngr.label_center(gr.ihi, r\"$\\mathrm{hi}$\")\ngr.label_center(gr.ihi+1, r\"$\\mathrm{hi+1}$\")\ngr.label_center(gr.ihi+2, r\"$\\mathrm{hi+2}$\")\n\ngr.label_edge(ng+nzones//2,   r\"$i-\\sfrac{1}{2}$\")\ngr.label_edge(ng+nzones//2+1,   r\"$i+\\sfrac{1}{2}$\")\n\n# draw the data\nfor i in range(nzones):\n    cc.draw_cell_avg(ng+i, color=\"r\")    \n    \ncc.label_cell_avg(ng+nzones//2, r\"$\\langle a\\rangle_i$\", color=\"r\")\n\n# label dx\ngr.label_dx(gr.ng+nzones//2)\n\ngr.clean_axes(show_ghost=True, pad_fac=0.02, ylim=(-0.5, 1.6))\n\nf = plt.gcf()\nf.set_size_inches(10.0,2.5)\n\nplt.savefig(\"fv_ghost.pdf\")\n"
  },
  {
    "path": "figures/advection/generalgrid.py",
    "content": "import math\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport grid_plot as gp\n\n\nnzones = 7\n\n# data that lives on the grid\nainit = np.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.15, 0.5])\n\ngr = gp.FVGrid(nzones)\na = gr.scratch_array()\na[gr.ilo:gr.ihi+1] = ainit[:]\n\npl_nolim = gp.PiecewiseLinear(gr, a, nolimit=1)\npl = gp.PiecewiseLinear(gr, a)\n\ngr.draw_grid()\n\ngr.label_center(nzones//2,   r\"$i$\")\ngr.label_center(nzones//2-1, r\"$i-1$\")\ngr.label_center(nzones//2+1, r\"$i+1$\")\ngr.label_center(nzones//2-2, r\"$i-2$\")\ngr.label_center(nzones//2+2, r\"$i+2$\")\n\n#labelEdge(gr, nzones//2,   r\"$i-1/2$\")\n#labelEdge(gr, nzones//2+1, r\"$i+1/2$\")\n\nfor n in range(nzones):\n    pl.draw_cell_avg(n, ls=\":\", color=\"0.5\")\n    pl_nolim.draw_slope(n, color=\"0.5\")\n    pl.draw_slope(n, color=\"r\")\n\ngr.clean_axes(ylim=(-0.25, 1.2))\n\nf = plt.gcf()\nf.set_size_inches(8.0,2.0)\n\nplt.savefig(\"generalgrid.pdf\")\n               \n\n\n"
  },
  {
    "path": "figures/advection/rea-limitex.py",
    "content": "# reconstruct - evolve - average: demonstrate what happens when we don't\n# limit\n\n\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport grid_plot as gp\n\n#-----------------------------------------------------------------------------\ndef evolve(pl, C, num, nolimit=1):\n    \n    # first frame -- the original cell-averages\n\n    plt.clf()\n\n    pl.gr.draw_grid()\n\n    pl.gr.label_center(pl.gr.ng + pl.gr.nx//2,   r\"$i$\", fontsize=\"medium\")\n    pl.gr.label_center(pl.gr.ng + pl.gr.nx//2-1, r\"$i-1$\", fontsize=\"medium\")\n    pl.gr.label_center(pl.gr.ng + pl.gr.nx//2+1, r\"$i+1$\", fontsize=\"medium\")\n    pl.gr.label_center(pl.gr.ng + pl.gr.nx//2-2, r\"$i-2$\", fontsize=\"medium\")\n    pl.gr.label_center(pl.gr.ng + pl.gr.nx//2+2, r\"$i+2$\", fontsize=\"medium\")\n\n    # draw cell averages\n    for n in range(pl.gr.ilo, gr.ihi+1):\n        pl.draw_cell_avg(n, color=\"r\")\n\n    pl.gr.clean_axes(ylim=(-0.25, 1.2))\n\n    f = plt.gcf()\n    f.set_size_inches(8.0,2.0)\n\n    if nolimit:\n        plt.savefig(\"rea-nolimit-start_%3.3d.pdf\" % (num))\n    else:\n        plt.savefig(\"rea-start_%3.3d.pdf\" % (num))\n\n\n    # second frame -- reconstruction\n\n    # draw\n    plt.clf()\n\n    pl.gr.draw_grid()\n\n    pl.gr.label_center(pl.gr.ng + pl.gr.nx//2,   r\"$i$\", fontsize=\"medium\")\n    pl.gr.label_center(pl.gr.ng + pl.gr.nx//2-1, r\"$i-1$\", fontsize=\"medium\")\n    pl.gr.label_center(pl.gr.ng + pl.gr.nx//2+1, r\"$i+1$\", fontsize=\"medium\")\n    pl.gr.label_center(pl.gr.ng + pl.gr.nx//2-2, r\"$i-2$\", fontsize=\"medium\")\n    pl.gr.label_center(pl.gr.ng + pl.gr.nx//2+2, r\"$i+2$\", fontsize=\"medium\")\n\n    # draw cell averages and slopes\n    for n in range(pl.gr.ilo, gr.ihi+1):\n        pl.draw_cell_avg(n, color=\"0.5\", ls=\":\")\n        pl.draw_slope(n, color=\"r\")\n\n    plt.axis([pl.gr.xmin-0.5*pl.gr.dx,pl.gr.xmax+0.5*pl.gr.dx, -0.25, 1.2])\n    plt.axis(\"off\")\n\n    plt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)\n\n    f = plt.gcf()\n    f.set_size_inches(8.0,2.0)\n\n    if nolimit:\n        plt.savefig(\"rea-nolimit-reconstruction_%3.3d.pdf\" % (num))\n    else:\n        plt.savefig(\"rea-reconstruction_%3.3d.pdf\" % (num))\n\n\n    # third frame -- evolve\n\n    # draw\n    plt.clf()\n\n    pl.gr.draw_grid()\n\n    pl.gr.label_center(pl.gr.ng + pl.gr.nx//2,   r\"$i$\", fontsize=\"medium\")\n    pl.gr.label_center(pl.gr.ng + pl.gr.nx//2-1, r\"$i-1$\", fontsize=\"medium\")\n    pl.gr.label_center(pl.gr.ng + pl.gr.nx//2+1, r\"$i+1$\", fontsize=\"medium\")\n    pl.gr.label_center(pl.gr.ng + pl.gr.nx//2-2, r\"$i-2$\", fontsize=\"medium\")\n    pl.gr.label_center(pl.gr.ng + pl.gr.nx//2+2, r\"$i+2$\", fontsize=\"medium\")\n\n    # draw cell averages and slopes\n    for n in range(pl.gr.ilo, gr.ihi+1):\n        #pl.draw_cell_avg(n, color=\"r\")\n        pl.draw_slope(n, color=\"0.5\", ls=\":\")\n\n\n    # evolve\n    for n in range(pl.gr.ilo, pl.gr.ihi+1):\n        pl.evolve_to_right(n, C, color=\"r\")\n\n    pl.gr.clean_axes(ylim=(-0.25, 1.2))\n\n    f = plt.gcf()\n    f.set_size_inches(8.0,2.0)\n\n    if nolimit:\n        plt.savefig(\"rea-nolimit-evolve_%3.3d.pdf\" % (num))\n    else:\n        plt.savefig(\"rea-evolve_%3.3d.pdf\" % (num))\n\n\n    #-------------------------------------------------------------------------\n    # fourth frame -- re-average\n\n    # left states (we don't need the right state when u > 0)\n    al = pl.gr.scratch_array()\n\n    for n in range(pl.gr.ilo, pl.gr.ihi+2):\n        al[n] = pl.a[n-1] + 0.5*(1.0 - C)*pl.slope[n-1]\n\n    # the Riemann problem just picks the left state.  Do a conservative\n    # update\n    anew = pl.gr.scratch_array()\n\n    anew[pl.gr.ilo:pl.gr.ihi+1] = pl.a[pl.gr.ilo:pl.gr.ihi+1] + \\\n            C*(al[pl.gr.ilo:pl.gr.ihi+1] - al[pl.gr.ilo+1:pl.gr.ihi+2])\n\n\n    plt.clf()\n\n    pl.gr.draw_grid()\n\n    pl.gr.label_center(pl.gr.ng + pl.gr.nx//2,   r\"$i$\", fontsize=\"medium\")\n    pl.gr.label_center(pl.gr.ng + pl.gr.nx//2-1, r\"$i-1$\", fontsize=\"medium\")\n    pl.gr.label_center(pl.gr.ng + pl.gr.nx//2+1, r\"$i+1$\", fontsize=\"medium\")\n    pl.gr.label_center(pl.gr.ng + pl.gr.nx//2-2, r\"$i-2$\", fontsize=\"medium\")\n    pl.gr.label_center(pl.gr.ng + pl.gr.nx//2+2, r\"$i+2$\", fontsize=\"medium\")\n\n\n    # show old evolved profiles and new averages\n    for n in range(pl.gr.ilo, pl.gr.ihi+1):\n        pl.evolve_to_right(n, C, color=\"0.5\", ls=\":\")\n\n    # draw new averages\n    pl.a[:] = anew[:]\n    pl.fill_zero_gradient()\n    pl.calculate_slopes()\n    for n in range(pl.gr.ilo, pl.gr.ihi+1):\n        pl.draw_cell_avg(n, color=\"r\")\n\n    pl.gr.clean_axes(ylim=(-0.25, 1.2))\n\n    f = plt.gcf()\n    f.set_size_inches(8.0,2.0)\n\n    if nolimit:\n        plt.savefig(\"rea-nolimit-final_%3.3d.pdf\" % (num))\n    else:\n        plt.savefig(\"rea-final_%3.3d.pdf\" % (num))\n\n    return anew\n\n\n\n#-----------------------------------------------------------------------------\nainit = np.array([1.0, 1.0, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1])\nnzones = len(ainit)\n\n# CFL number\nC = 0.7\nfor nolimit in range(2):\n\n    gr = gp.FVGrid(nzones, ng=4)\n\n    a = gr.scratch_array()\n    a[gr.ilo:gr.ihi+1] = ainit[:]\n\n    pl = gp.PiecewiseLinear(gr, a, nolimit=nolimit)\n\n    # loop\n    for i in range(1,9):\n\n        pl.fill_zero_gradient()\n        print(i, pl.a[:])\n        evolve(pl, C, i, nolimit=nolimit)\n\n"
  },
  {
    "path": "figures/advection/rea.py",
    "content": "# reconstruct - evolve - average\n\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport grid_plot as gp\n\n\n#-----------------------------------------------------------------------------\n\natemp = np.array([1.0, 1.0, 0.9, 0.8, 0.25, 0.1, 0.1, 0.1])\nnzones = len(atemp)\n\n# CFL number\nC = 0.6\n\n\ngr = gp.FVGrid(nzones, ng=4)\n\na = gr.scratch_array()\na[gr.ilo:gr.ihi+1] = atemp[:]\n\npl = gp.PiecewiseLinear(gr, a)\npl.fill_zero_gradient()\n\n\n#-----------------------------------------------------------------------------\n# first frame -- the original cell-averages\n\nplt.clf()\n\ngr.draw_grid()\n\nlabels = [\"$i-2$\", \"$i-1$\", \"$i$\", \"$i+1$\", \"$i+2$\"]\nindices = [gr.ng+nzones//2-2, gr.ng+nzones//2-1, \n           gr.ng+nzones//2, gr.ng+nzones//2+1, gr.ng+nzones//2+2]\n\nfor i, l in zip(indices, labels):\n    gr.label_center(i, l)\n\n# draw cell averages\nfor n in range(gr.ilo, gr.ihi+1):\n    pl.draw_cell_avg(n, color=\"r\")\n\ngr.clean_axes(ylim=(-0.25, 1.2))\n\nf = plt.gcf()\nf.set_size_inches(8.0,2.0)\n\nplt.savefig(\"rea-start.pdf\")\n\n\n#-----------------------------------------------------------------------------\n# second frame -- reconstruction\n\nplt.clf()\n\ngr.draw_grid()\n\nfor i, l in zip(indices, labels):\n    gr.label_center(i, l)\n\n# draw cell averages and slopes\nfor n in range(gr.ilo, gr.ihi+1):\n    pl.draw_cell_avg(n, color=\"0.5\", ls=\":\")\n    pl.draw_slope(n, color=\"r\")\n\n\ngr.clean_axes(ylim=(-0.25, 1.2))\n\nf = plt.gcf()\nf.set_size_inches(8.0,2.0)\n\nplt.savefig(\"rea-reconstruction.pdf\")\n\n\n#-----------------------------------------------------------------------------\n# third frame -- shade the portion that will come into the cell\n\nplt.clf()\n\ngr.draw_grid()\n\nfor i, l in zip(indices, labels):\n    gr.label_center(i, l)\n\n# draw cell averages and slopes\nfor n in range(gr.ilo, gr.ihi+1):\n    pl.draw_slope(n, color=\"r\")\n\n\n# shade regions\nii = gr.ng + nzones//2-1\npl.slope_trace_left(ii, C, color=\"0.75\")\npl.slope_trace_right(ii+1, 1.0-C, color=\"0.75\")\n\ngr.clean_axes(ylim=(-0.25, 1.2))\n\nf = plt.gcf()\nf.set_size_inches(8.0,2.0)\n\nplt.savefig(\"rea-trace.pdf\")\n\n\n#-----------------------------------------------------------------------------\n# fourth frame -- evolve\n\nplt.clf()\n\ngr.draw_grid()\n\nfor i, l in zip(indices, labels):\n    gr.label_center(i, l)\n\n# draw cell averages and slopes\nfor n in range(gr.ilo, gr.ihi+1):\n    pl.draw_slope(n, color=\"0.75\", ls=\":\")\n    pl.evolve_to_right(n, C, color=\"r\")\n\ngr.clean_axes(ylim=(-0.25, 1.2))\n\nf = plt.gcf()\nf.set_size_inches(8.0,2.0)\n\nplt.savefig(\"rea-evolve.pdf\")\n\n\n#-----------------------------------------------------------------------------\n# fifth frame -- re-average\n\n# left states (we don't need the right state when u > 0)\nal = np.zeros(2*gr.ng + gr.nx, dtype=np.float64)\n\nfor n in range(gr.ilo, gr.ihi+2):\n    al[n] = a[n-1] + 0.5*(1 - C)*pl.slope[n-1]\n\n# the Riemann problem just picks the right state.  Do a conservative\n# update\nanew = gr.scratch_array()\n\nanew[gr.ilo:gr.ihi+1] = a[gr.ilo:gr.ihi+1] + \\\n    C*(al[gr.ilo:gr.ihi+1] - al[gr.ilo+1:gr.ihi+2])\n\n\nplt.clf()\n\ngr.draw_grid()\n\nfor i, l in zip(indices, labels):\n    gr.label_center(i, l)\n\n\n# show the evolved profiles from the old time\nfor n in range(gr.ilo, gr.ihi+1):\n    pl.evolve_to_right(n, C, color=\"0.5\", ls=\":\")\n\n\npl.a[:] = anew[:]\npl.fill_zero_gradient()\npl.calculate_slopes()\n\nfor n in range(pl.gr.ilo, pl.gr.ihi+1):\n    pl.draw_cell_avg(n, color=\"r\")\n\n\ngr.clean_axes(ylim=(-0.25, 1.2))\n\nf = plt.gcf()\nf.set_size_inches(8.0,2.0)\n\nplt.savefig(\"rea-final.pdf\")\n\n\n\n"
  },
  {
    "path": "figures/advection/riemann.py",
    "content": "import numpy as np\nimport matplotlib.pylab as plt\nimport grid_plot as gp\n\ndef riemann(with_time=True):\n\n    # grid info\n    xmin = 0.0\n    xmax = 1.0\n\n    nzones = 2\n    ng = 0\n\n    gr = gp.FVGrid(nzones, xmin=xmin, xmax=xmax)\n\n    plt.clf()\n\n    #------------------------------------------------------------------------\n    # plot a domain without ghostcells\n    gr.draw_grid()\n\n    gr.label_center(0, r\"$i$\", fontsize=\"medium\")\n    gr.label_center(1, r\"$i+1$\", fontsize=\"medium\")\n\n    gr.label_edge(1, r\"$i+\\myhalf$\", fontsize=\"medium\")\n\n\n    plt.arrow(gr.xc[0]+0.05*gr.dx, 0.5, 0.12*gr.dx, 0,\n                shape='full', head_width=0.05, head_length=0.025,\n                lw=1, width=0.02,\n                edgecolor=\"none\", facecolor=\"r\",\n                length_includes_head=True, zorder=100)\n\n    plt.arrow(gr.xc[1]-0.1*gr.dx, 0.5, -0.12*gr.dx, 0,\n                shape='full', head_width=0.05, head_length=0.025,\n                lw=1, width=0.02,\n                edgecolor=\"none\", facecolor=\"r\",\n                length_includes_head=True, zorder=100)\n\n    if with_time:\n        gr.mark_cell_left_state(1, r\"$a_{i+\\myhalf,R}^{n+\\myhalf}$\", fontsize=\"large\",\n                                color=\"b\")\n        gr.mark_cell_right_state(0, r\"$a_{i+\\myhalf,L}^{n+\\myhalf}$\", fontsize=\"large\",\n                                 color=\"b\")\n    else:\n        gr.mark_cell_left_state(1, r\"$a_{i+\\myhalf,R}$\", fontsize=\"large\",\n                                color=\"b\")\n        gr.mark_cell_right_state(0, r\"$a_{i+\\myhalf,L}$\", fontsize=\"large\",\n                                 color=\"b\")\n\n    gr.label_cell_center(0, r\"$a_i$\")\n    gr.label_cell_center(1, r\"$a_{i+1}$\")\n\n\n    gr.clean_axes(pad_fac=0.125, ylim=(-0.25, 1.0))\n\n    f = plt.gcf()\n    f.set_size_inches(7.0,2.0)\n\n    plt.tight_layout()\n\n    if with_time:\n        plt.savefig(\"riemann-adv.pdf\")\n    else:\n        plt.savefig(\"riemann-adv-mol.pdf\")\n\nif __name__== \"__main__\":\n    riemann()\n    riemann(with_time=False)\n"
  },
  {
    "path": "figures/advection/riemann_bc.py",
    "content": "import numpy as np\nimport matplotlib.pyplot as plt\nimport grid_plot as gp\n\ndef riemann():\n\n    # grid info\n    xmin = 0.0\n    xmax = 1.0\n\n    nzones = 4\n    ng = 2\n\n    gr = gp.FVGrid(nzones, ng=ng, xmin=xmin, xmax=xmax)\n    \n    # interior and ghost cell initialization\n    a = gr.scratch_array()\n\n    a[gr.ilo:gr.ihi+1] = np.array([0.8, 0.4, 0.3, 0.5])\n    a[0:gr.ilo] = a[gr.ihi-1:gr.ihi+1]\n    a[gr.ihi:2*gr.ng+gr.nx] = a[gr.ihi]\n\n    pc = gp.PiecewiseConstant(gr, a)\n    pl = gp.PiecewiseLinear(gr, a, nolimit=1)\n\n\n\n    #------------------------------------------------------------------------\n    # plot a domain without ghostcells\n    gr.draw_grid(draw_ghost=1, emphasize_end=1)\n\n    gr.label_center(gr.ng-2, r\"$\\mathrm{lo-2}$\", fontsize=\"medium\")\n    gr.label_center(gr.ng-1, r\"$\\mathrm{lo-1}$\", fontsize=\"medium\")\n    gr.label_center(gr.ng, r\"$\\mathrm{lo}$\", fontsize=\"medium\")\n    gr.label_center(gr.ng+1, r\"$\\mathrm{lo+1}$\", fontsize=\"medium\")\n\n    gr.label_edge(gr.ng, r\"$\\mathrm{lo}-\\myhalf$\", fontsize=\"medium\")\n    \n    # draw cell averages\n    for n in range(0, gr.ng+gr.nx-1):\n        pc.draw_cell_avg(n, color=\"0.5\", ls=\":\")\n\n\n    # draw slopes\n    for n in range(gr.ilo-1, gr.ihi):\n        pl.draw_slope(n, color=\"r\")\n\n\n    # compute the states to the left and right of lo-1/2\n    C = 0.5 # CFL\n    print(a)\n    print(pl.slope)\n    al = a[gr.ilo-1] + 0.5*(1.0 - C)*pl.slope[gr.ilo-1]\n    ar = a[gr.ilo] - 0.5*(1.0 + C)*pl.slope[gr.ilo]\n\n    # L\n    gr.mark_cell_right_state(ng-1, r\"$a_{\\mathrm{lo}-\\myhalf,L}^{n+\\myhalf}$\", \n                             value=al, vertical=\"center\", color=\"b\")\n\n    # R\n    gr.mark_cell_left_state(ng, r\"$a_{\\mathrm{lo}-\\myhalf,R}^{n+\\myhalf}$\", \n                            value=ar, vertical=\"top\", color=\"b\")\n\n\n    plt.xlim(gr.xl[0]-0.025*gr.dx,gr.xr[ng+1]+0.15*gr.dx)\n    plt.ylim(-0.25, 1.1)\n    plt.axis(\"off\")\n\n    plt.subplots_adjust(left=0.025,right=0.95,bottom=0.05,top=0.95)\n\n    f = plt.gcf()\n    f.set_size_inches(8.0,2.0)\n\n    plt.tight_layout()\n\n    plt.savefig(\"riemann-bc.pdf\")\n\nif __name__== \"__main__\":\n    riemann()\n"
  },
  {
    "path": "figures/advection/riemann_mol.py",
    "content": "import numpy as np\nimport matplotlib.pyplot as plt\nimport grid_plot as gp\n\ndef riemann():\n\n    # grid info\n    xmin = 0.0\n    xmax = 1.0\n\n    nzones = 4\n    ng = 2\n\n    gr = gp.FVGrid(nzones, ng=ng, xmin=xmin, xmax=xmax)\n    \n    # interior and ghost cell initialization\n    a = gr.scratch_array()\n\n    a[gr.ilo:gr.ihi+1] = np.array([0.8, 0.7, 0.4, 0.5])\n    a[0:gr.ilo] = a[gr.ihi-1:gr.ihi+1]\n    a[gr.ihi:2*gr.ng+gr.nx] = a[gr.ihi]\n\n    pc = gp.PiecewiseConstant(gr, a)\n    pl = gp.PiecewiseLinear(gr, a, nolimit=1)\n\n\n\n    #------------------------------------------------------------------------\n    # plot a domain without ghostcells\n    gr.draw_grid(draw_ghost=0, emphasize_end=0)\n\n    zi = gr.ilo+1\n\n    gr.label_center(zi-1, r\"${i-1}$\", fontsize=\"medium\")\n    gr.label_center(zi, r\"${i}$\", fontsize=\"medium\")\n    gr.label_center(zi+1, r\"${i+1}$\", fontsize=\"medium\")\n\n    gr.label_edge(zi+1, r\"${i}+\\myhalf$\", fontsize=\"medium\")\n\n    # draw cell averages\n    for n in range(gr.ilo, gr.ihi+1):\n        pc.draw_cell_avg(n, color=\"0.5\", ls=\":\")\n\n\n    # draw slopes\n    for n in range(gr.ilo, gr.ihi+1):\n        pl.draw_slope(n, color=\"r\")\n\n    # compute the states to the left and right of lo-1/2\n    al = a[zi] + 0.5*pl.slope[zi]\n    ar = a[zi+1] - 0.5*pl.slope[zi+1]\n\n    # L\n    gr.mark_cell_right_state(zi, r\"$q_{i+\\myhalf,L}$\",\n                             value=al, vertical=\"top\", color=\"b\", zorder=100)\n\n    # R\n    gr.mark_cell_left_state(zi+1, r\"$q_{i+\\myhalf,R}$\",\n                            value=ar, vertical=\"top\", color=\"b\", zorder=100)\n\n\n    plt.xlim(gr.xl[gr.ilo]-0.025*gr.dx, gr.xr[gr.ihi]+0.15*gr.dx)\n    plt.ylim(-0.25, 1.1)\n    plt.axis(\"off\")\n\n    plt.subplots_adjust(left=0.025,right=0.95,bottom=0.05,top=0.95)\n\n    f = plt.gcf()\n    f.set_size_inches(8.0,2.0)\n\n    plt.tight_layout()\n\n    plt.savefig(\"riemann-mol.pdf\")\n\nif __name__== \"__main__\":\n    riemann()\n"
  },
  {
    "path": "figures/advection/update.sh",
    "content": "#!/bin/sh\n\nDEST=../../advection\n\n# characteristics.py makes advection-characteristics.pdf\npython3 characteristics.py\ncp -f advection-characteristics.pdf ${DEST}\n\n# fd_ghost.py makes fd_ghost.pdf\npython3 fd_ghost.py\ncp -f fd_ghost.pdf ${DEST}\n\n# hydro_examples: fdadvect.py makes fdadvect-upwind.pdf\n\n\n# hydro_examples: fdadvect.py makes fdadvect-FTCS-C0_1.pdf, fdadvect-FTCS-C0_5.pdf\n\n\n# hydro_examples: fdadvect_implicit.py makes fdadvect-implicit.pdf\n\n\n# fv_ghost.py makes fv_ghost.pdf\npython3 fv_ghost.py\ncp -f fv_ghost.pdf ${DEST}\n\n# riemann.py makes riemann-adv.pdf\npython3 riemann.py\ncp -f riemann-adv.pdf ${DEST}\n\n# riemann_bc.py makes riemamm-bc.pdf\npython3 riemann_bc.py\ncp -f riemann-bc.pdf ${DEST}\n\n# hydro_examples: fv-advection.py makes fv-advect.pdf\n\n\n# rea-limitex.py makes rea*.pdf\npython3 rea-limitex.py\ncp -f rea*start_???.pdf ${DEST}\n\n# generalgrid.py makes generalgrid.pdf\npython3 generalgrid.py\ncp -f generalgrid.pdf ${DEST}\n\n# rea.py makes rea-*.pdf\npython3 rea.py\ncp -f rea-final.pdf rea-evolve.pdf rea-trace.pdf rea-reconstruction.pdf rea-start.pdf ${DEST}\n\n# hydro_examples: advection.py makes plm-converge.pdf, fv-gaussian-limiters.pdf fv-tophat-limiters.pdf\n\n\n# 2dgrid.py makes 2dgrid.pdf\npython3 2dgrid.py\ncp -f 2dgrid.pdf ${DEST}\n\n# 2dgrid_hat.py makes 2dgrid-hat.pdf\npython3 2dgrid_hat.py\ncp -f 2dgrid-hat.pdf ${DEST}\n\n# 2dgrid_transverse.py makes 2dgrid-transverse.pdf\npython3 2dgrid_transverse.py\ncp -f 2dgrid-transverse.pdf ${DEST}\n\n# pyro makes smooth_init.pdf smooth_final.pdf tophat_init.pdf tophat_final.pdf\n# tophat_rk4_cfl*.pdf\n\n"
  },
  {
    "path": "figures/burgers/characteristics.py",
    "content": "import numpy as np\nimport matplotlib as mpl\nimport matplotlib.pyplot as plt\n\nmpl.rcParams['mathtext.fontset'] = 'cm'\nmpl.rcParams['mathtext.rm'] = 'serif'\n\n\ndef fun(x):\n    state = np.zeros_like(x)\n    state[x < 0.5] = 1.0\n    return state\n\ndef funs(x):\n    state = 0.5 + 0.25*np.sin(2.0*np.pi*x)\n    return state\n\ndef funr(x):\n    state = np.ones_like(x)\n    state[x < 0.5] = 0.2\n    return state\n\ndef make_plot(icfun=None, label=\"\"):\n\n    npts_plot = 1000\n    xplot = np.linspace(0.0, 1.0, npts_plot)\n\n    nchar = 20\n    xchar = np.linspace(0.0, 1.0, nchar)\n\n    state = icfun(xplot)\n    print(type(state))\n\n    plt.clf()\n\n    plt.subplot(211)\n\n    plt.plot(xplot, state, lw=2)\n\n\n    ax = plt.gca()\n    ax.spines['right'].set_visible(False)\n    ax.spines['top'].set_visible(False)\n    ax.spines['right'].set_visible(False)\n    ax.spines['top'].set_visible(False)\n    ax.spines['right'].set_visible(False)\n    ax.spines['top'].set_visible(False)\n    ax.xaxis.set_ticks_position('bottom')\n    ax.yaxis.set_ticks_position('left')\n\n    plt.ylim(-0.1, 1.1)\n\n    plt.xlabel(r\"$x$\", fontsize=\"large\")\n    plt.ylabel(r\"$u$\", fontsize=\"large\")\n\n\n\n    plt.subplot(212)\n\n    uchar = icfun(xchar)\n    t = np.linspace(0.0, 1.0, 100)\n    for n in range(nchar):\n        xc = xchar[n] + uchar[n]*t\n        plt.plot(xc, t, color=\"0.5\")\n\n    ax = plt.gca()\n    ax.spines['right'].set_visible(False)\n    ax.spines['top'].set_visible(False)\n    ax.spines['right'].set_visible(False)\n    ax.spines['top'].set_visible(False)\n    ax.spines['right'].set_visible(False)\n    ax.spines['top'].set_visible(False)\n    ax.xaxis.set_ticks_position('bottom')\n    ax.yaxis.set_ticks_position('left')\n\n    plt.xlabel(r\"$x$\", fontsize=\"large\")\n    plt.ylabel(r\"$t$\", fontsize=\"large\")\n\n    plt.xlim(0.0, 1.0)\n    plt.ylim(0.0, 1.1)\n\n    f = plt.gcf()\n    f.set_size_inches(6.0, 8.0)\n\n    plt.tight_layout()\n\n    plt.savefig(f\"burgers-characteristics-{label}.png\")\n\nif __name__ == \"__main__\":\n\n    make_plot(funr, \"rare\")\n    make_plot(funs, \"shock\")\n"
  },
  {
    "path": "figures/burgers/rh.py",
    "content": "import numpy as np\nimport matplotlib.pylab as plt\nimport matplotlib as mpl\n\nmpl.rcParams['mathtext.fontset'] = 'cm'\nmpl.rcParams['mathtext.rm'] = 'serif'\n\nmpl.rcParams['font.size'] = 12\nmpl.rcParams['legend.fontsize'] = 'large'\nmpl.rcParams['figure.titlesize'] = 'medium'\n\ndef rh():\n\n    # grid info\n    xl = 0.0\n    xr = 1.0\n\n    dx = xr - xl\n\n    xc = 0.5*(xl + xr)\n\n    t0 = 0.0\n    t1 = 1.0\n\n    dt = t1 - t0\n\n    # plot a square representing [x, x+dx] x [t, t+dt] \n\n    # x-axis\n    plt.arrow(0, 0, 1.3*dx, 0,\n              shape=\"full\", head_width=0.04, head_length=0.06,\n              lw=1, width=0.005,\n              facecolor=\"k\",\n              length_includes_head=True, zorder=100)\n\n    plt.text(1.35*dx, 0, r\"$x$\", fontsize=20, verticalalignment=\"center\")\n\n    plt.text(dx, -0.1, r\"$x_r$\", fontsize=20)\n    plt.text(0, -0.1, r\"$x_l$\", fontsize=20)\n\n    # time axis\n    plt.arrow(0, 0, 0, 1.3*dt,\n              shape=\"full\", head_width=0.04, head_length=0.06,\n              lw=1, width=0.005,\n              facecolor=\"k\",\n              length_includes_head=True, zorder=100)\n\n    plt.text(0, 1.35*dt, r\"$t$\", fontsize=20, horizontalalignment=\"center\")\n\n    plt.text(-0.17, dt, r\"$t^{n+1}$\", fontsize=20)\n    plt.text(-0.17, 0, r\"$t^n$\", fontsize=20)\n\n    \n    # space-time volume\n    plt.plot([dx,dx], [0,dt], ls=\"--\", color=\"0.5\", lw=2)\n    plt.plot([0,dx], [dt,dt], ls=\"--\", color=\"0.5\", lw=2)\n\n\n    # diagonal representing S\n    plt.plot([0,dx], [0,dt], color=\"k\", lw=5, solid_capstyle=\"butt\")\n    \n    plt.annotate(r\"shock: $S = \\Delta x / \\Delta t$\", xy=(0.74*dx,0.76*dt),\n                 xytext=(0.2*dx, 1.2*dt), textcoords=\"data\",\n                 fontsize=18,\n                 arrowprops=dict(arrowstyle=\"->\", \n                                 connectionstyle=\"arc3,rad=.2\"))\n\n    # states\n    plt.text(0.66*dx, 0.33*dt, r\"$u_r$\", color=\"k\", fontsize=20)\n    plt.text(0.33*dx, 0.66*dt, r\"$u_l$\", color=\"k\", fontsize=20)\n\n    \n    # fluxes\n    plt.arrow(-0.1*dx, 0.5*dt, 0.2*dx, 0,\n              shape=\"full\", head_width=0.04, head_length=0.06,\n              lw=1, width=0.005,\n              edgecolor=\"r\", facecolor=\"r\",\n              length_includes_head=True, zorder=100)\n\n    plt.text(-0.12*dx, 0.5*dt, r\"$f = f(u_l)$\",\n             horizontalalignment=\"right\",\n             verticalalignment=\"center\", color=\"r\", fontsize=18)\n\n    plt.arrow(0.9*dx, 0.5*dt, 0.2*dx, 0,\n              shape=\"full\", head_width=0.04, head_length=0.06,\n              lw=1, width=0.005, \n              edgecolor=\"r\", facecolor=\"r\",\n              length_includes_head=True, zorder=100)\n\n    plt.text(1.12*dx, 0.5*dt, r\"$f = f(u_r)$\",\n             horizontalalignment=\"left\",\n             verticalalignment=\"center\", color=\"r\", fontsize=18)\n\n\n    plt.axis(\"off\")\n\n    plt.subplots_adjust(left=0.05, right=0.95, bottom=0.05, top=0.95)\n\n    plt.xlim(-0.4,1.4*dx)\n    plt.ylim(-0.1,1.4*dx)\n\n    f = plt.gcf()\n    f.set_size_inches(7.0,6.0)\n\n    plt.savefig(\"rh.pdf\")\n\n\nif __name__== \"__main__\":\n    rh()\n"
  },
  {
    "path": "figures/burgers/update.sh",
    "content": "#!/bin/sh\n\nDEST=../../burgers\n\n# characteristics.py makes burgers-characteristics.pdf\npython3 characteristics.py\ncp -f burgers-characteristics.pdf ${DEST}\n\n# rh.py makes rh.pdf\npython3 rh.py\ncp -f rh.pdf ${DEST}\n\n# hydro_examples: burgers.py makes fv-burger-rarefaction.pdf fv-burger-sine.pdf \n\n"
  },
  {
    "path": "figures/finite-volume/ccfd.py",
    "content": "import math\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport grid_plot as gp\n\n# plot a simple finite-difference grid\n\n#-----------------------------------------------------------------------------\n\nnzones = 8\n\n# data that lives on the grid\n#a = np.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.15, 0.5, 0.55])\na = np.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.1, 0.5, 0.55])\n\ngr = gp.FVGrid(nzones)\n\n\nplt.clf()\n\ngr.draw_grid()\n\ngr.label_center(nzones//2,   r\"$i$\", fontsize=\"medium\")\ngr.label_center(nzones//2-1, r\"$i-1$\", fontsize=\"medium\")\ngr.label_center(nzones//2+1, r\"$i+1$\", fontsize=\"medium\")\ngr.label_center(nzones//2-2, r\"$i-2$\", fontsize=\"medium\")\ngr.label_center(nzones//2+2, r\"$i+2$\", fontsize=\"medium\")\n\n\ncc = gp.CellCentered(gr, a)\n\n# draw the data\nfor i in range(nzones):\n    cc.draw_data_point(i, color=\"r\")\n\n\ncc.label_data_point(nzones//2, r\"$f_i$\", color=\"r\")\n\n# label dx\ngr.label_center_dx(gr.ng+nzones//2)\n\ngr.clean_axes(ylim=(-0.5, 1.2))\n\nf = plt.gcf()\nf.set_size_inches(10.0,3.0)\n\nplt.savefig(\"ccfd_grid.pdf\")\n"
  },
  {
    "path": "figures/finite-volume/domain.py",
    "content": "import matplotlib.pyplot as plt\n\ndef draw_box(ll, uu, nx, ny, gridColor=\"0.5\", ng=0):\n\n    # draw the frame\n    plt.plot([ll[0], ll[0], uu[0], uu[0], ll[0]],\n             [ll[1], uu[1], uu[1], ll[1], ll[1]], color=\"k\", lw=2)\n\n    # draw the x grid lines\n    dx = (uu[0] - ll[0])/nx\n    for n in range(1, nx):\n        plt.plot([ll[0]+n*dx, ll[0]+n*dx],\n                   [ll[1], uu[1]], color=gridColor, ls=\":\", lw=1)\n\n    # draw the y grid lines\n    dy = (uu[1] - ll[1])/ny\n    for n in range(1, ny):\n        plt.plot([ll[0], uu[0]],\n                   [ll[1]+n*dy, ll[1]+n*dy], color=gridColor, ls=\":\", lw=1)\n    \n    # ghostcells?  \n    if ng > 0:\n        xmin = ll[0]-ng*dx\n        xmax = uu[0]+ng*dx\n        ymin = ll[1]-ng*dy\n        ymax = uu[1]+ng*dy\n        plt.plot([xmin, xmin, xmax, xmax, xmin], \n                 [ymin, ymax, ymax, ymin, ymin], \n                 ls=\"--\", color=\"r\")\n\n\nplt.clf()\n\ndraw_box([0., 0.], [1., 1.], 5, 5, ng=1)\ndraw_box([1., 0.], [2., 1.], 5, 5)\ndraw_box([2., 0.], [3., 1.], 5, 5)\n\ndraw_box([0., 1.], [1., 2.], 5, 5)\ndraw_box([1., 1.], [2., 2.], 5, 5)\ndraw_box([2., 1.], [3., 2.], 5, 5)\n\nplt.xlim(-0.25,3.25)\nplt.ylim(-0.25,2.25)\n\na = plt.gca()\na.set_aspect(\"equal\", \"datalim\")\nplt.axis(\"off\")\n\nf = plt.gcf()\nf.set_size_inches(7.0,4.25)\n\n\nplt.savefig(\"domain.pdf\", bbox_inches=\"tight\")\n\n\n"
  },
  {
    "path": "figures/finite-volume/fd.py",
    "content": "\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport grid_plot as gp\n\n# plot a simple finite-difference grid\n\n#-----------------------------------------------------------------------------\n\nnzones = 8\n\n# data that lives on the grid\n#a = np.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.15, 0.5, 0.55])\na = np.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.1, 0.5, 0.55])\n\ngr = gp.FDGrid(nzones)\n\n\nplt.clf()\n\ngr.draw_grid()\n\ngr.label_node(nzones//2,   r\"$i$\",   fontsize=\"medium\")\ngr.label_node(nzones//2-1, r\"$i-1$\", fontsize=\"medium\")\ngr.label_node(nzones//2+1, r\"$i+1$\", fontsize=\"medium\")\ngr.label_node(nzones//2-2, r\"$i-2$\", fontsize=\"medium\")\ngr.label_node(nzones//2+2, r\"$i+2$\", fontsize=\"medium\")\n\n\n# draw the data\nfor i in range(nzones):\n    gr.draw_data(i, a[i], color=\"r\")\n\n\ngr.label_value(nzones//2, a[nzones//2], r\"$f_i$\", color=\"r\")\n\n# label dx\ngr.label_dx(gr.ng+nzones//2)\n\ngr.clean_axes(ylim=(-0.5, 1.2))\n\nf = plt.gcf()\nf.set_size_inches(10.0,3.0)\n\nplt.savefig(\"fd_grid.pdf\")\n"
  },
  {
    "path": "figures/finite-volume/fv.py",
    "content": "import numpy as np\nimport matplotlib.pyplot as plt\nimport grid_plot as gp\n\n\n\n# plot a simple finite-difference grid\n\n#-----------------------------------------------------------------------------\n\nnzones = 8\n\n# data that lives on the grid\n#a = np.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.15, 0.5, 0.55])\na = np.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.1, 0.5, 0.55])\n\ngr = gp.FVGrid(nzones)\n\npc = gp.PiecewiseConstant(gr, a)\n\nplt.clf()\n\ngr.draw_grid()\n\ngr.label_center(nzones//2,   r\"$i$\", fontsize=\"medium\")\ngr.label_center(nzones//2-1, r\"$i-1$\", fontsize=\"medium\")\ngr.label_center(nzones//2+1, r\"$i+1$\", fontsize=\"medium\")\ngr.label_center(nzones//2-2, r\"$i-2$\", fontsize=\"medium\")\ngr.label_center(nzones//2+2, r\"$i+2$\", fontsize=\"medium\")\n\ngr.label_edge(nzones//2,   r\"$i-\\sfrac{1}{2}$\", fontsize=\"small\")\ngr.label_edge(nzones//2+1, r\"$i+\\sfrac{1}{2}$\", fontsize=\"small\")\n\n# draw the data\nfor i in range(nzones):\n    pc.draw_cell_avg(i, color=\"r\", filled=True)\n\npc.label_cell_avg(nzones//2, r\"$\\langle f\\rangle_i$\", color=\"r\")\n\n# label dx\ngr.label_dx(gr.ng+nzones//2)\n\ngr.clean_axes(ylim=(-0.5, 1.2))\n\nf = plt.gcf()\nf.set_size_inches(10.0,3.0)\n\nplt.savefig(\"fv_grid.pdf\")\n"
  },
  {
    "path": "figures/finite-volume/simplegrid2.py",
    "content": "import math\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport grid_plot as gp\n\ndef simplegrid():\n\n    nzones = 7\n\n    gr = gp.FVGrid(nzones, xmin=0, xmax=1)\n\n    gr.draw_grid(edge_ticks=0)\n\n    # label a few cell-centers\n    gr.label_center(nzones/2, r\"$i$\")\n    gr.label_center(nzones/2-1, r\"$i-1$\")\n    gr.label_center(nzones/2+1, r\"$i+1$\")\n\n    # label a few edges\n    gr.label_edge(nzones/2, r\"$i-1/2$\")\n    gr.label_edge(nzones/2+1, r\"$i+1/2$\")\n\n    # sample data\n    A = 0.4\n    a = A*np.ones(nzones, dtype=np.float64)\n\n    pc = gp.PiecewiseConstant(gr, a)\n\n    # draw an average quantity\n    pc.draw_cell_avg(nzones/2, color=\"r\")\n    pc.label_cell_avg(nzones/2, r\"$\\,\\langle a \\rangle_i$\", color=\"r\")\n\n    gr.clean_axes()\n    plt.ylim(-0.25, 1.5)\n\n    plt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)\n\n    f = plt.gcf()\n    f.set_size_inches(10.0,2.5)\n\n    plt.savefig(\"simplegrid2.png\")\n    plt.savefig(\"simplegrid2.pdf\")\n\nif __name__== \"__main__\":\n    simplegrid()\n"
  },
  {
    "path": "figures/finite-volume/simplegrid_gc.py",
    "content": "import numpy as np\nimport matplotlib.pyplot as plt\nimport grid_plot as gp\n\ndef simplegrid():\n\n    # grid info\n    nzones = 7\n    ng = 1\n\n    gr = gp.FVGrid(nzones, ng, xmin=0.0, xmax=1.0)\n\n    gr.draw_grid(emphasize_end=1, edge_ticks=0, draw_ghost=1)\n\n    # label a few\n    gr.label_center(ng+nzones//2, r\"$i$\", fontsize=\"medium\")\n    gr.label_center(ng+nzones//2-1, r\"$i-1$\", fontsize=\"medium\")\n    gr.label_center(ng+nzones//2+1, r\"$i+1$\", fontsize=\"medium\")\n\n    gr.label_center(ng-1, r\"$\\mathrm{lo}-1$\", fontsize=\"medium\")\n    gr.label_center(ng, r\"$\\mathrm{lo}$\", fontsize=\"medium\")\n    gr.label_center(ng+nzones-1, r\"$\\mathrm{hi}$\", fontsize=\"medium\")\n    gr.label_center(ng+nzones, r\"$\\mathrm{hi+1}$\", fontsize=\"medium\")\n\n\n    # label dx\n    gr.label_dx(ng+nzones//2-2)\n\n\n    gr.clean_axes(show_ghost=True, ylim=(-0.5, 1.5), padding=True, pad_fac=0.05)\n    #plt.xlim(gr.xl[0]-0.05*gr.dx,gr.xr[2*ng+nzones-1]+0.25*gr.dx)\n    #plt.ylim(-0.5, 1.5)\n    #plt.axis(\"off\")\n\n    #plt.subplots_adjust(left=0.025,right=0.95,bottom=0.05,top=0.95)\n\n    f = plt.gcf()\n    f.set_size_inches(10.0,2.5)\n\n    plt.tight_layout()\n\n    plt.savefig(\"simplegrid_gc.pdf\", bbox_inches=\"tight\")\n               \n\nif __name__== \"__main__\":\n    simplegrid()\n"
  },
  {
    "path": "figures/finite-volume/update.sh",
    "content": "#!/bin/sh\n\n# fd.py creates fd_grid.pdf\npython3 fd.py\ncp -f fd_grid.pdf ../../finite-volume/\n\n# ccfd.py creates ccfd_grid.pdf\npython3 ccfd.py\ncp -f ccfd_grid.pdf ../../finite-volume/\n\n# fv.py creates fv_grid.pdf\npython3 fv.py\ncp -f fv_grid.pdf ../../finite-volume/\n\n# simplegrid_gc.py creates simplegrid_gc.pdf\npython3 simplegrid_gc.py\ncp -f simplegrid_gc.pdf ../../finite-volume/\n\n# domain.py creates domain.pdf\npython3 domain.py\ncp -r domain.pdf ../../finite-volume/\n"
  },
  {
    "path": "figures/gridFigs/fdrestrict.py",
    "content": "import numpy as np\nimport matplotlib.pyplot as plt\nimport grid_plot as gp\n\n# plot two stacked fv grids of different (2x) resolution to show prolongation\n\n#-----------------------------------------------------------------------------\n\nnf = 5\nnc = 3\n\ngrf = gp.FDGrid(nf, voff=2.0)\ngrc = gp.FDGrid(nc)\n\nplt.clf()\n\ngrf.draw_grid()\ngrc.draw_grid()\n\ngrf.label_node(nf/2-2, r\"$i-2$\")\ngrf.label_node(nf/2-1, r\"$i-1$\")\ngrf.label_node(nf/2,   r\"$i$\")\ngrf.label_node(nf/2+1, r\"$i+1$\")\ngrf.label_node(nf/2+2, r\"$i+2$\")\n\n\ngrc.label_node(nc/2-1, r\"$j-1$\")\ngrc.label_node(nc/2,   r\"$j$\")\ngrc.label_node(nc/2+1, r\"$j+1$\")\n\ngrf.label_node_data(nf/2-2, r\"$\\phi_{i-2}^h$\")\ngrf.label_node_data(nf/2-1, r\"$\\phi_{i-1}^h$\")\ngrf.label_node_data(nf/2,   r\"$\\phi_i^h$\")\ngrf.label_node_data(nf/2+1, r\"$\\phi_{i+1}^h$\")\ngrf.label_node_data(nf/2+2, r\"$\\phi_{i+2}^h$\")\n\ngrc.label_node_data(nc/2-1, r\"$\\phi_{j-1}^{2h}$\")\ngrc.label_node_data(nc/2,   r\"$\\phi_{j}^{2h}$\")\ngrc.label_node_data(nc/2+1, r\"$\\phi_{j+1}^{2h}$\")\n    \n\n# connect the dots...\n\nplt.plot([grf.xc[nf/2-2], grf.xc[nf/2-2]], [-0.25, 3.25], \":\", color=\"0.5\")\nplt.plot([grf.xc[nf/2],   grf.xc[nf/2]],   [-0.25, 3.25], \":\", color=\"0.5\")\nplt.plot([grf.xc[nf/2+2], grf.xc[nf/2+2]], [-0.25, 3.25], \":\", color=\"0.5\")\n\n\nplt.axis([grf.xmin-0.5*grf.dx,grf.xmax+0.5*grf.dx, -0.5, 3.5])\nplt.axis(\"off\")\n\nplt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)\n\nf = plt.gcf()\nf.set_size_inches(6.0,5.0)\n\nplt.savefig(\"fdrestrict.pdf\")\n\n\n"
  },
  {
    "path": "figures/gridFigs/fvprolong.py",
    "content": "import matplotlib.pyplot as plt\nimport grid_plot as gp\n\n# plot two stacked fv grids of different (2x) resolution to show prolongation\n\n#-----------------------------------------------------------------------------\n\nnf = 4\nnc = nf/2\n\ngrf = gp.FVGrid(nf)\ngrc = gp.FVGrid(nc, voff=2.0)\n\n\nplt.clf()\n\ngrf.draw_grid()\ngrc.draw_grid()\n\ngrf.label_center(nf/2-2, r\"$i-2$\")\ngrf.label_center(nf/2-1, r\"$i-1$\")\ngrf.label_center(nf/2,   r\"$i$\")\ngrf.label_center(nf/2+1, r\"$i+1$\")\n\ngrc.label_center(nc/2-1, r\"$j-1$\")\ngrc.label_center(nc/2,   r\"$j$\")\n\ngrf.label_cell_center(nf/2-2, r\"$\\phi_{i-2}^h$\")\ngrf.label_cell_center(nf/2-1, r\"$\\phi_{i-1}^h$\")\ngrf.label_cell_center(nf/2,   r\"$\\phi_i^h$\")\ngrf.label_cell_center(nf/2+1, r\"$\\phi_{i+1}^h$\")\n\ngrc.label_cell_center(nc/2-1, r\"$\\phi_{j-1}^{2h}$\")\ngrc.label_cell_center(nc/2,   r\"$\\phi_{j}^{2h}$\")\n    \n\n# connect the dots...\n\nplt.plot([grf.xl[nf/2-2], grf.xl[nf/2-2]], [-0.25, 3.25], \":\", color=\"0.5\")\nplt.plot([grf.xl[nf/2], grf.xl[nf/2]], [-0.25, 3.25], \":\", color=\"0.5\")\nplt.plot([grf.xr[nf/2+1], grf.xr[nf/2+1]], [-0.25, 3.25], \":\", color=\"0.5\")\n\n\nplt.axis([grf.xmin-0.5*grf.dx,grf.xmax+0.5*grf.dx, -0.5, 3.5])\nplt.axis(\"off\")\n\nplt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)\n\nf = plt.gcf()\nf.set_size_inches(6.0,5.0)\n\nplt.savefig(\"fvprolong.pdf\")\n\n"
  },
  {
    "path": "figures/gridFigs/fvrestrict.py",
    "content": "import numpy as np\nimport matplotlib.pyplot as plt\nimport grid_plot as gp\n\n# plot two stacked fv grids of different (2x) resolution to show prolongation\n\n#-----------------------------------------------------------------------------\n\nnf = 4\nnc = nf/2\n\ngrf = gp.FVGrid(nf, voff=2.0)\ngrc = gp.FVGrid(nc)\n\n\nplt.clf()\n\ngrf.draw_grid()\ngrc.draw_grid()\n\ngrf.label_center(nf/2-2, r\"$i-2$\")\ngrf.label_center(nf/2-1, r\"$i-1$\")\ngrf.label_center(nf/2,   r\"$i$\")\ngrf.label_center(nf/2+1, r\"$i+1$\")\n\ngrc.label_center(nc/2-1, r\"$j-1$\")\ngrc.label_center(nc/2,   r\"$j$\")\n\ngrf.label_cell_center(nf/2-2, r\"$\\phi_{i-2}^h$\")\ngrf.label_cell_center(nf/2-1, r\"$\\phi_{i-1}^h$\")\ngrf.label_cell_center(nf/2,   r\"$\\phi_i^h$\")\ngrf.label_cell_center(nf/2+1, r\"$\\phi_{i+1}^h$\")\n\ngrc.label_cell_center(nc/2-1, r\"$\\phi_{j-1}^{2h}$\")\ngrc.label_cell_center(nc/2,   r\"$\\phi_{j}^{2h}$\")\n    \n\n# connect the dots...\n\nplt.plot([grf.xl[nf/2-2], grf.xl[nf/2-2]], [-0.25, 3.25], \":\", color=\"0.5\")\nplt.plot([grf.xl[nf/2], grf.xl[nf/2]], [-0.25, 3.25], \":\", color=\"0.5\")\nplt.plot([grf.xr[nf/2+1], grf.xr[nf/2+1]], [-0.25, 3.25], \":\", color=\"0.5\")\n\n\nplt.axis([grf.xmin-0.5*grf.dx,grf.xmax+0.5*grf.dx, -0.5, 3.5])\nplt.axis(\"off\")\n\nplt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)\n\nf = plt.gcf()\nf.set_size_inches(6.0,5.0)\n\nplt.savefig(\"fvrestrict.pdf\")\n\n"
  },
  {
    "path": "figures/gridFigs/nested.py",
    "content": "import matplotlib.pyplot as plt\nimport grid_plot as gp\n\n# plot two stacked fv grids of different (2x) resolution to show prolongation\n\n#-----------------------------------------------------------------------------\n\ngr = []\n\nnf = 8\ngr.append(gp.FVGrid(nf, ng=1, voff=0.0))\ngr.append(gp.FVGrid(nf, ng=1, voff=0.0, xmin=0.25, xmax=0.75))\n    \n\nplt.clf()\n\ngr[0].draw_grid(emphasize_end=1, draw_ghost=0, edge_ticks=0, color=\"0.75\")\ngr[1].draw_grid(emphasize_end=1, draw_ghost=0, edge_ticks=0)\n\nf = plt.gcf()\nf.set_size_inches(7.0,1.0)\n\n\ngrf = gr[0]\nplt.xlim(grf.xmin-0.75*grf.dx,grf.xmax+0.25*grf.dx)\n\nplt.axis(\"off\")\nplt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)\n\nplt.savefig(\"nested1.pdf\")\n\n#-----------------------------------------------------------------------------\n\nplt.clf()\n\ngr = []\ngr.append(gp.FVGrid(nf, ng=1, voff=0.0))\ngr.append(gp.FVGrid(nf, ng=1, voff=4.0))\n\ngr[0].draw_grid(emphasize_end=1, draw_ghost=0, edge_ticks=0)\ngr[1].draw_grid(emphasize_end=1, draw_ghost=0, edge_ticks=0)\n\n\n# labels\nplt.text(gr[0].xmin-0.7*gr[0].dx, gr[0].voff+0.5, r\"$\\phi^{c,n}$\", fontsize=\"medium\")\nplt.text(gr[1].xmin-0.7*gr[1].dx, gr[1].voff+0.5, r\"$\\phi^{c,n+1}$\", fontsize=\"medium\")\n\nplt.arrow(gr[0].xmin-0.5*gr[0].dx, 0.5+3.0*gr[0].dx, 0.001, gr[1].voff-4.0*gr[0].dx,\n            shape='full', head_width=0.025, head_length=0.25, \n            lw=1, width=0.01,\n            edgecolor=\"none\", facecolor=\"r\",\n            length_includes_head=True, zorder=100)\n\n\nf = plt.gcf()\nf.set_size_inches(7.0,5.0)\n\n\ngrf = gr[0]\nplt.xlim(grf.xmin-0.75*grf.dx,grf.xmax+0.25*grf.dx)\n\nplt.axis(\"off\")\n\nplt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)\n\nplt.savefig(\"nested2.pdf\")\n\n\n#-----------------------------------------------------------------------------\n\nplt.clf()\n\ngr1 = []\ngr1.append(gp.FVGrid(nf, ng=1, voff=0.0))\ngr1.append(gp.FVGrid(nf, ng=1, voff=0.0, xmin=0.25, xmax=0.75))\n\ngr2 = []\ngr2.append(gp.FVGrid(nf, ng=1, voff=2.0))\ngr2.append(gp.FVGrid(nf, ng=1, voff=2.0, xmin=0.25, xmax=0.75))\n\ngr3 = []\ngr3.append(gp.FVGrid(nf, ng=1, voff=4.0))\ngr3.append(gp.FVGrid(nf, ng=1, voff=4.0, xmin=0.25, xmax=0.75))\n\n\ngr1[0].draw_grid(emphasize_end=1, draw_ghost=0, edge_ticks=0, color=\"0.75\")\ngr1[1].draw_grid(emphasize_end=1, draw_ghost=0, edge_ticks=0)\n\ngr2[0].draw_grid(emphasize_end=1, draw_ghost=0, edge_ticks=0, color=\"0.75\")\ngr2[1].draw_grid(emphasize_end=1, draw_ghost=0, edge_ticks=0)\n\ngr3[0].draw_grid(emphasize_end=1, draw_ghost=0, edge_ticks=0, color=\"0.75\")\ngr3[1].draw_grid(emphasize_end=1, draw_ghost=0, edge_ticks=0)\n\nplt.text(gr1[1].xmin-0.95*gr1[0].dx, gr1[1].voff+0.5, r\"$\\phi^{f,n}$\", \n         fontsize=\"11\", horizontalalignment=\"left\")\nplt.text(gr2[1].xmin-0.95*gr2[0].dx, gr2[1].voff+0.5, r\"$\\phi^{f,n+1/2}$\", \n         fontsize=\"11\", horizontalalignment=\"left\")\nplt.text(gr3[1].xmin-0.95*gr3[0].dx, gr3[1].voff+0.5, r\"$\\phi^{f,n+1}$\", \n         fontsize=\"11\", horizontalalignment=\"left\")\n\n\nplt.annotate(r\"Dirichlet BCs $\\phi^{c,n}$\", xy=(gr1[1].xmin, gr1[1].voff), xycoords='data',\n               xytext=(30,-25), textcoords=\"offset points\",\n               arrowprops=dict(arrowstyle=\"->\", \n                               connectionstyle=\"angle,angleA=180,angleB=-90,rad=10\", \n                               ec=\"0.5\"),\n               fontsize=\"small\",color=\"0.5\")\n\nplt.annotate(r\"Dirichlet BCs $(\\phi^{c,n} + \\phi^{c,n+1})/2$\", xy=(gr2[1].xmin, gr2[1].voff), xycoords='data',\n               xytext=(30,-25), textcoords=\"offset points\",\n               arrowprops=dict(arrowstyle=\"->\", \n                               connectionstyle=\"angle,angleA=180,angleB=-90,rad=10\", \n                               ec=\"0.5\"),\n               fontsize=\"small\",color=\"0.5\")\n\nplt.annotate(r\"Dirichlet BCs $\\phi^{c,n+1}$\", xy=(gr3[1].xmin, gr3[1].voff), xycoords='data',\n               xytext=(30,-25), textcoords=\"offset points\",\n               arrowprops=dict(arrowstyle=\"->\", \n                               connectionstyle=\"angle,angleA=180,angleB=-90,rad=10\", \n                               ec=\"0.5\"),\n               fontsize=\"small\",color=\"0.5\")\n\n\nplt.arrow(gr1[1].xmin-0.75*gr1[0].dx, 0.5+3.0*gr1[0].dx, 0.001, gr2[0].voff-4.0*gr2[0].dx,\n            shape='full', head_width=0.025, head_length=0.25, \n            lw=1, width=0.01,\n            edgecolor=\"none\", facecolor=\"r\",\n            length_includes_head=True, zorder=100)\n\nplt.arrow(gr2[1].xmin-0.75*gr2[0].dx, gr2[1].voff+0.5+3.0*gr2[0].dx, 0.001, gr3[1].voff-gr2[1].voff-4.0*gr3[0].dx,\n            shape='full', head_width=0.025, head_length=0.25, \n            lw=1, width=0.01,\n            edgecolor=\"none\", facecolor=\"r\",\n            length_includes_head=True, zorder=100)\n\nf = plt.gcf()\nf.set_size_inches(7.0,5.0)\n\n\ngrf = gr[0]\nplt.xlim(grf.xmin-0.75*grf.dx,grf.xmax+0.25*grf.dx)\n\nplt.axis(\"off\")\n\nplt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)\n\nplt.savefig(\"nested3.pdf\")\n\n\n\n"
  },
  {
    "path": "figures/incompressible/MAC-solve.py",
    "content": "import math\nimport numpy\nimport pylab\n\n\n\nclass Grid2d:\n\n    def __init__(self, nx, ny,\n                 xmin=0.0, xmax=1.0, ymin=0.0, ymax=1.0):\n                 \n        self.xmin = xmin\n        self.xmax = xmax\n\n        self.ymin = ymin\n        self.ymax = ymax\n\n        self.nx = nx\n        self.ny = ny\n    \n        self.dx = (self.xmax - self.xmin)/float(self.nx)\n        self.dy = (self.ymax - self.ymin)/float(self.ny)\n\n        self.xl = (numpy.arange(self.nx)  )*self.dx\n        self.xr = (numpy.arange(self.nx)+1)*self.dx\n\n        self.xc = 0.5*(self.xl + self.xr)\n\n        self.yl = (numpy.arange(self.ny)  )*self.dy\n        self.yr = (numpy.arange(self.ny)+1)*self.dy\n\n        self.yc = 0.5*(self.yl + self.yr)\n\n\n    def draw_grid(self):\n        \"\"\" plot a domain without ghostcells \"\"\"\n    \n        # lines parallel to the x-axis\n        n = 0\n        while n < self.ny:\n            pylab.plot([self.xmin-0.25*self.dx,self.xmax+0.25*self.dx], \n                       [self.yl[n],self.yl[n]], color=\"k\", lw=2)\n            n += 1\n        \n        pylab.plot([self.xmin-0.25*self.dx,self.xmax+0.25*self.dx], \n                   [self.yr[self.ny-1],self.yr[self.ny-1]], color=\"k\", lw=2)\n\n\n        # lines parallel to the y-axis\n        n = 0\n        while n < self.nx:\n            pylab.plot([self.xl[n],self.xl[n]], \n                       [self.ymin-0.25*self.dy,self.ymax+0.25*self.dy], color=\"k\", lw=2)\n            n += 1\n        \n        pylab.plot([self.xr[self.nx-1],self.xr[self.nx-1]], \n                   [self.ymin-0.25*self.dy,self.ymax+0.25*self.dy], color=\"k\", lw=2)\n\n\n        # grid labels\n        pylab.text(self.xc[self.nx/2], self.yl[0]-0.35*self.dy, r\"$i$\",\n                   horizontalalignment='center', fontsize=\"16\")\n\n        if self.nx/2-1 >= 0:\n            pylab.text(self.xc[self.nx/2-1], self.yl[0]-0.35*self.dy, r\"$i-1$\",\n                       horizontalalignment='center', fontsize=\"16\")\n        \n        if self.nx/2+1 < self.nx:\n            pylab.text(self.xc[self.nx/2+1], self.yl[0]-0.35*self.dy, r\"$i+1$\",\n                       horizontalalignment='center', fontsize=\"16\")\n\n\n        pylab.text(self.xl[0]-0.35*self.dx, self.yc[self.ny/2], r\"$j$\",\n                   verticalalignment='center', fontsize=\"16\")\n\n        if self.ny/2-1 >= 0:\n            pylab.text(self.xl[0]-0.35*self.dx, self.yc[self.ny/2-1], r\"$j-1$\",\n                       verticalalignment='center', fontsize=\"16\")\n\n        if self.ny/2+1 < self.ny:\n            pylab.text(self.xl[0]-0.35*self.dx, self.yc[self.ny/2+1], r\"$j+1$\",\n                       verticalalignment='center', fontsize=\"16\")\n\n\n\ndef simplegrid():\n\n    g = Grid2d(3,3)\n\n    pylab.subplot(121)\n\n    g.draw_grid()\n\n    #------------------------------------------------------------------------\n    # label\n    pylab.text(g.xc[g.nx/2], g.yc[g.ny/2], \n               r\"$\\phi_{i,j}$\", fontsize=\"15\",\n               horizontalalignment='center', verticalalignment='center')\n\n\n    # phi\n    pylab.text(g.xc[g.nx/2+1], g.yc[g.ny/2]+0.05*g.dy, \n               r\"$\\phi_{i+1,j}$\", \n               fontsize=\"15\",\n               horizontalalignment='center', verticalalignment='center')\n\n    pylab.text(g.xc[g.nx/2-1], g.yc[g.ny/2]+0.05*g.dy, \n               r\"$\\phi_{i-1,j}$\", \n               fontsize=\"15\",\n               horizontalalignment='center', verticalalignment='center')\n\n    pylab.text(g.xc[g.nx/2], g.yc[g.ny/2+1]+0.05*g.dy, \n               r\"$\\phi_{i,j+1}$\", \n               fontsize=\"15\",\n               horizontalalignment='center', verticalalignment='center')\n\n    pylab.text(g.xc[g.nx/2], g.yc[g.ny/2-1]+0.05*g.dy, \n               r\"$\\phi_{i,j-1}$\", \n               fontsize=\"15\",\n               horizontalalignment='center', verticalalignment='center')\n\n\n    # MAC velocities\n\n    # i+1/2,j interface\n    pylab.scatter(g.xr[g.nx/2], g.yc[g.ny/2], marker=\"x\", s=50)\n    pylab.text(g.xr[g.nx/2]+0.05*g.dx, g.yc[g.ny/2], \n               r\"${u}^\\mathrm{adv}_{i+1/2,j}$\", \n               fontsize=\"15\", rotation=270,\n               horizontalalignment='left', verticalalignment='center')\n\n    # i-1/2,j interface\n    pylab.scatter(g.xr[g.nx/2-1], g.yc[g.ny/2], marker=\"x\", s=50)\n    pylab.text(g.xr[g.nx/2-1]-0.05*g.dx, g.yc[g.ny/2], \n               r\"${u}^\\mathrm{adv}_{i-1/2,j}$\", \n               fontsize=\"15\", rotation=270, \n               horizontalalignment='right', verticalalignment='center')\n\n    # i,j+1/2 interface\n    pylab.scatter(g.xc[g.nx/2], g.yr[g.ny/2], marker=\"x\", s=50)\n    pylab.text(g.xc[g.nx/2], g.yr[g.ny/2]+0.05*g.dy, \n               r\"${v}^\\mathrm{adv}_{i,j+1/2}$\", \n               fontsize=\"15\", \n               horizontalalignment='center', verticalalignment='bottom')\n\n    # i,j-1/2 interface\n    pylab.scatter(g.xc[g.nx/2], g.yr[g.ny/2-1], marker=\"x\", s=50)\n    pylab.text(g.xc[g.nx/2], g.yr[g.ny/2-1]-0.05*g.dy, \n               r\"${v}^\\mathrm{adv}_{i,j-1/2}$\", \n               fontsize=\"15\", \n               horizontalalignment='center', verticalalignment='top')\n\n\n\n    # axes\n    pylab.xlim(g.xl[0]-0.5*g.dx,g.xr[g.nx-1]+0.25*g.dx)\n    pylab.ylim(g.yl[0]-0.5*g.dy,g.yr[g.ny-1]+0.25*g.dy)\n    pylab.axis(\"off\")\n\n    pylab.subplots_adjust(left=0.02,right=0.98,bottom=0.02,top=0.98)\n\n    ax = pylab.gca()\n    ax.set_aspect(\"equal\", \"datalim\")\n\n\n\n    pylab.subplot(122)\n\n    g.draw_grid()\n\n    #------------------------------------------------------------------------\n    # label\n    pylab.text(g.xc[g.nx/2], g.yc[g.ny/2]+0.1*g.dy, \n               r\"$(DG\\phi)_{i,j}$\", fontsize=\"15\",\n               horizontalalignment='center', verticalalignment='center')\n\n    pylab.text(g.xc[g.nx/2], g.yc[g.ny/2]-0.1*g.dy, \n               r\"$(DU^\\mathrm{adv})_{i,j}$\", fontsize=\"15\",\n               horizontalalignment='center', verticalalignment='center')\n\n\n\n    # Gphi velocities\n\n    # i+1/2,j interface\n    pylab.scatter(g.xr[g.nx/2], g.yc[g.ny/2], marker=\"x\", s=50)\n    pylab.text(g.xr[g.nx/2]+0.05*g.dx, g.yc[g.ny/2], \n               r\"${(G\\phi)}_{i+1/2,j}$\", \n               fontsize=\"15\", rotation=270,\n               horizontalalignment='left', verticalalignment='center')\n\n    # i-1/2,j interface\n    pylab.scatter(g.xr[g.nx/2-1], g.yc[g.ny/2], marker=\"x\", s=50)\n    pylab.text(g.xr[g.nx/2-1]-0.05*g.dx, g.yc[g.ny/2], \n               r\"${(G\\phi)}_{i-1/2,j}$\", \n               fontsize=\"15\", rotation=270, \n               horizontalalignment='right', verticalalignment='center')\n\n    # i,j+1/2 interface\n    pylab.scatter(g.xc[g.nx/2], g.yr[g.ny/2], marker=\"x\", s=50)\n    pylab.text(g.xc[g.nx/2], g.yr[g.ny/2]+0.05*g.dy, \n               r\"${(G\\phi)}_{i,j+1/2}$\", \n               fontsize=\"15\", \n               horizontalalignment='center', verticalalignment='bottom')\n\n    # i,j-1/2 interface\n    pylab.scatter(g.xc[g.nx/2], g.yr[g.ny/2-1], marker=\"x\", s=50)\n    pylab.text(g.xc[g.nx/2], g.yr[g.ny/2-1]-0.05*g.dy, \n               r\"${(G\\phi)}_{i,j-1/2}$\", \n               fontsize=\"15\", \n               horizontalalignment='center', verticalalignment='top')\n\n\n\n    # axes\n    pylab.xlim(g.xl[0]-0.5*g.dx,g.xr[g.nx-1]+0.25*g.dx)\n    pylab.ylim(g.yl[0]-0.5*g.dy,g.yr[g.ny-1]+0.25*g.dy)\n    pylab.axis(\"off\")\n\n    pylab.subplots_adjust(left=0.02,right=0.98,bottom=0.02,top=0.98)\n\n    ax = pylab.gca()\n    ax.set_aspect(\"equal\", \"datalim\")\n\n\n    f = pylab.gcf()\n    f.set_size_inches(12.0,6.0)\n\n    pylab.tight_layout()\n\n    pylab.savefig(\"MAC_solve.png\")\n    pylab.savefig(\"MAC_solve.eps\", bbox_inches=\"tight\")\n               \n\nif __name__== \"__main__\":\n    simplegrid()\n"
  },
  {
    "path": "figures/incompressible/MAC.py",
    "content": "import math\nimport numpy\nimport pylab\n\ndef simplegrid():\n\n    # grid info\n    xmin = 0.0\n    xmax = 1.0\n\n    ymin = 0.0\n    ymax = 1.0\n\n    nzones = 3\n    ng = 0\n    \n    dx = (xmax - xmin)/float(nzones)\n    dy = (ymax - ymin)/float(nzones)\n\n    xl = (numpy.arange(2*ng+nzones) - ng)*dx\n    xr = (numpy.arange(2*ng+nzones)+1 - ng)*dx\n\n    xc = 0.5*(xl + xr)\n\n    yl = (numpy.arange(2*ng+nzones) - ng)*dy\n    yr = (numpy.arange(2*ng+nzones)+1 - ng)*dy\n\n    yc = 0.5*(yl + yr)\n\n\n    #------------------------------------------------------------------------\n    # plot a domain without ghostcells\n    \n    # x lines\n    n = 0\n    while (n < nzones):\n        pylab.plot([xmin-0.25*dx,xmax+0.25*dx], [yl[n],yl[n]], color=\"k\", lw=2)\n        n += 1\n        \n    pylab.plot([xmin-0.25*dx,xmax+0.25*dx], [yr[nzones-1],yr[nzones-1]], color=\"k\", lw=2)\n\n\n    # y lines\n    n = 0\n    while (n < nzones):\n        pylab.plot([xl[n],xl[n]], [ymin-0.25*dy,ymax+0.25*dy], color=\"k\", lw=2)\n        n += 1\n        \n    pylab.plot([xr[nzones-1],xr[nzones-1]], [ymin-0.25*dy,ymax+0.25*dy], color=\"k\", lw=2)\n\n\n    #------------------------------------------------------------------------\n    # label\n    pylab.text(xc[nzones/2], yc[nzones/2]+0.05*dy, r\"$u_{i,j}$\", fontsize=\"15\",\n               horizontalalignment='center', verticalalignment='center')\n\n    pylab.text(xc[nzones/2], yc[nzones/2]-0.05*dy, r\"$v_{i,j}$\", fontsize=\"15\",\n               horizontalalignment='center', verticalalignment='center')\n\n\n    pylab.text(xc[nzones/2+1], yc[nzones/2]+0.05*dy, r\"$u_{i+1,j}$\", \n               fontsize=\"15\",\n               horizontalalignment='center', verticalalignment='center')\n\n    pylab.text(xc[nzones/2+1], yc[nzones/2]-0.05*dy, r\"$v_{i+1,j}$\", \n               fontsize=\"15\",\n               horizontalalignment='center', verticalalignment='center')\n\n\n    pylab.text(xc[nzones/2-1], yc[nzones/2]+0.05*dy, r\"$u_{i-1,j}$\", \n               fontsize=\"15\",\n               horizontalalignment='center', verticalalignment='center')\n\n    pylab.text(xc[nzones/2-1], yc[nzones/2]-0.05*dy, r\"$v_{i-1,j}$\", \n               fontsize=\"15\",\n               horizontalalignment='center', verticalalignment='center')\n\n\n    pylab.text(xc[nzones/2], yc[nzones/2+1]+0.05*dy, r\"$u_{i,j+1}$\", \n               fontsize=\"15\",\n               horizontalalignment='center', verticalalignment='center')\n\n    pylab.text(xc[nzones/2], yc[nzones/2+1]-0.05*dy, r\"$v_{i,j+1}$\", \n               fontsize=\"15\",\n               horizontalalignment='center', verticalalignment='center')\n\n\n    pylab.text(xc[nzones/2], yc[nzones/2-1]+0.05*dy, r\"$u_{i,j-1}$\", \n               fontsize=\"15\",\n               horizontalalignment='center', verticalalignment='center')\n\n    pylab.text(xc[nzones/2], yc[nzones/2-1]-0.05*dy, r\"$v_{i,j-1}$\", \n               fontsize=\"15\",\n               horizontalalignment='center', verticalalignment='center')\n\n\n    # i+1/2,j interface\n    pylab.scatter(xr[nzones/2], yc[nzones/2], marker=\"x\", s=50)\n    pylab.text(xr[nzones/2]+0.05*dx, yc[nzones/2], \n               r\"${u}^\\mathrm{adv}_{i+1/2,j}$\", \n               fontsize=\"15\", rotation=270,\n               horizontalalignment='left', verticalalignment='center')\n\n    # i-1/2,j interface\n    pylab.scatter(xr[nzones/2-1], yc[nzones/2], marker=\"x\", s=50)\n    pylab.text(xr[nzones/2-1]-0.05*dx, yc[nzones/2], \n               r\"${u}^\\mathrm{adv}_{i-1/2,j}$\", \n               fontsize=\"15\", rotation=270, \n               horizontalalignment='right', verticalalignment='center')\n\n\n\n    # i,j+1/2 interface\n    pylab.scatter(xc[nzones/2], yr[nzones/2], marker=\"x\", s=50)\n    pylab.text(xc[nzones/2], yr[nzones/2]+0.05*dy, \n               r\"${v}^\\mathrm{adv}_{i,j+1/2}$\", \n               fontsize=\"15\", \n               horizontalalignment='center', verticalalignment='bottom')\n\n\n    # i,j-1/2 interface\n    pylab.scatter(xc[nzones/2], yr[nzones/2-1], marker=\"x\", s=50)\n    pylab.text(xc[nzones/2], yr[nzones/2-1]-0.05*dy, \n               r\"${v}^\\mathrm{adv}_{i,j-1/2}$\", \n               fontsize=\"15\", \n               horizontalalignment='center', verticalalignment='top')\n\n\n\n\n    # grid labels\n    pylab.text(xc[nzones/2-1], yl[0]-0.35*dy, r\"$i-1$\",\n               horizontalalignment='center', fontsize=\"16\")\n\n    pylab.text(xc[nzones/2], yl[0]-0.35*dy, r\"$i$\",\n               horizontalalignment='center', fontsize=\"16\")\n\n    pylab.text(xc[nzones/2+1], yl[0]-0.35*dy, r\"$i+1$\",\n               horizontalalignment='center', fontsize=\"16\")\n\n\n    pylab.text(xl[0]-0.35*dx, yc[nzones/2-1], r\"$j-1$\",\n               verticalalignment='center', fontsize=\"16\")\n\n    pylab.text(xl[0]-0.35*dx, yc[nzones/2], r\"$j$\",\n               verticalalignment='center', fontsize=\"16\")\n\n    pylab.text(xl[0]-0.35*dx, yc[nzones/2+1], r\"$j+1$\",\n               verticalalignment='center', fontsize=\"16\")\n\n\n\n    # axes\n    pylab.xlim(xl[0]-0.5*dx,xr[2*ng+nzones-1]+0.25*dx)\n    pylab.ylim(yl[0]-0.5*dy,yr[2*ng+nzones-1]+0.25*dy)\n    pylab.axis(\"off\")\n\n    pylab.subplots_adjust(left=0.02,right=0.98,bottom=0.02,top=0.98)\n\n    f = pylab.gcf()\n    f.set_size_inches(8.0,8.0)\n\n\n    pylab.savefig(\"MAC.png\")\n    pylab.savefig(\"MAC.eps\")\n               \n\nif __name__== \"__main__\":\n    simplegrid()\n"
  },
  {
    "path": "figures/intro/fft_simple_examples.py",
    "content": "from __future__ import print_function\n\nimport matplotlib.pyplot as plt\nimport matplotlib as mpl\nimport numpy as np\nimport math\n\nmpl.rcParams['mathtext.fontset'] = 'cm'\nmpl.rcParams['mathtext.rm'] = 'serif'\n\n\n# see: http://glowingpython.blogspot.com/2011/08/how-to-plot-frequency-spectrum-with.html\n# and\n# http://docs.scipy.org/doc/numpy/reference/routines.fft.html\n\n# Since our input data is real, the negative frequency components\n# don't include any new information, and are not interesting to us.\n# The rfft routines understand this, and rfft takes n real points and\n# returns n/2+1 complex output points.  The corresponding inverse\n# knows this, and acts accordingly.\n#\n# these are the routines we want for real valued data\n#\n# note that the scipy version of rfft returns that data differently\n#\n# M. Zingale (2013-03-03)\n\n\ndef single_freq_sine(npts, xmax, f_0):\n\n    # a pure sine with no phase shift will result in pure imaginary\n    # signal\n\n    xx = np.linspace(0.0, xmax, npts, endpoint=False)\n    return xx, np.sin(2.0*math.pi*f_0*xx)\n\ndef single_freq_sine_plus_shift(npts, xmax, f_0):\n\n    # a pure sine with no phase shift will result in pure imaginary\n    # signal\n    xx = np.linspace(0.0, xmax, npts, endpoint=False)\n    return xx, np.sin(2.0*math.pi*f_0*xx + math.pi/4)\n\ndef two_freq_sine(npts, xmax, f_0, f_1):\n\n    # a pure sine with no phase shift will result in pure imaginary\n    # signal\n    xx = np.linspace(0.0, xmax, npts, endpoint=False)\n    f = 0.5*(np.sin(2.0*math.pi*f_0*xx) + np.sin(2.0*math.pi*f_1*xx))\n    return xx, f\n\ndef single_freq_cosine(npts, xmax, f_0):\n\n    # a pure cosine with no phase shift will result in pure real\n    # signal\n    xx = np.linspace(0.0, xmax, npts, endpoint=False)\n    f = np.cos(2.0*math.pi*f_0*xx)\n    return xx, f\n\ndef plot_FFT(xx, xmax, f, outfile):\n\n    plt.clf()\n\n    plt.rc(\"font\", size=10)\n\n    npts = len(xx)\n\n    # Forward transform: f(x) -> F(k)\n    fk = np.fft.rfft(f)\n\n    # Normalization -- the '2' here comes from the fact that we are\n    # neglecting the negative portion of the frequency space, since\n    # the FFT of a real function contains redundant information, so \n    # we are only dealing with 1/2 of the frequency space.\n    #\n    # technically, we should only scale the 0 bin by N, since k=0 is\n    # not duplicated -- we won't worry about that for these plots\n    norm = 2.0/npts\n\n    fk = fk*norm\n\n    fk_r = fk.real\n    fk_i = fk.imag\n\n    # the fftfreq returns the postive and negative (and 0) frequencies\n    # the newer versions of numpy (>=1.8) have an rfftfreq() function\n    # that really does what we want -- we'll use that here.\n    k = np.fft.rfftfreq(npts)\n\n    # to make these dimensional, we need to divide by dx.  Note that\n    # max(xx) is not the true length, since we didn't have a point\n    # at the endpoint of the domain.\n    kfreq = k*npts/(max(xx) + xx[1])\n\n    # Inverse transform: F(k) -> f(x) -- without the normalization\n    fkinv = np.fft.irfft(fk/norm)\n\n    plt.subplot(411)\n\n    plt.plot(xx, f)\n    plt.xlabel(\"x\")\n    plt.ylabel(\"$f(x)$\")\n\n    plt.xlim(0, xmax)\n\n    plt.subplot(412)\n\n    plt.plot(kfreq, fk_r, label=r\"Re($\\mathcal{F}$)\")\n    plt.plot(kfreq, fk_i, ls=\":\", label=r\"Im($\\mathcal{F}$)\")\n    plt.xlabel(r\"$k$\")\n    plt.ylabel(\"$\\mathcal{F}_k$\")\n\n    plt.legend(fontsize=\"small\", frameon=False, ncol=2, loc=\"upper right\")\n\n    plt.subplot(413)\n\n    plt.plot(kfreq, np.abs(fk))\n    plt.xlabel(r\"$k$\")\n    plt.ylabel(r\"$|\\mathcal{F}_k|$\")\n\n\n    plt.subplot(414)\n\n    plt.plot(xx, fkinv.real)\n    plt.xlabel(r\"$x$\")\n    plt.ylabel(r\"$\\mathcal{F}^{-1}(\\mathcal{F}_k)$\")\n\n    plt.xlim(0, xmax)\n\n    plt.tight_layout()\n\n    plt.savefig(outfile)\n\n\n\n#-----------------------------------------------------------------------------\n\nnpts = 256  #64  #256\n\nf_0 = 0.2\n\nxmax = 10.0/f_0\n    \n# FFT of sine\nxx, f = single_freq_sine(npts, xmax, f_0)\nplot_FFT(xx, xmax, f, \"fft-sine.pdf\")\n\n\n# FFT of cosine\nxx, f = single_freq_cosine(npts, xmax, f_0)\nplot_FFT(xx, xmax, f, \"fft-cosine.pdf\")\n\n# FFT of sine with pi/4 phase\nxx, f = single_freq_sine_plus_shift(npts, xmax, f_0)\nplot_FFT(xx, xmax, f, \"fft-sine-phase.pdf\")\n\n# FFT of two sines\nf_1 = 0.5\nxx, f = two_freq_sine(npts, xmax, f_0, f_1)\nplot_FFT(xx, xmax, f, \"fft-two-sines.pdf\")\n\n"
  },
  {
    "path": "figures/intro/integrals.py",
    "content": "import numpy as np\nimport matplotlib as mpl\nimport matplotlib.pyplot as plt\nfrom matplotlib.patches import Polygon\n\nmpl.rcParams['mathtext.fontset'] = 'cm'\nmpl.rcParams['mathtext.rm'] = 'serif'\n\n\ndef f(x):\n    \"\"\" the function we are integrating \"\"\"\n    #return 1.0 + x*0.25*np.sin(np.pi*x)\n    return (1.0 + x*0.25*np.sin(5*np.pi*x)) * np.exp(-(x - 1.0)**2/0.5)\n\n\nplt.rcParams.update({'xtick.labelsize': 18,\n                     'ytick.labelsize': 18,\n                     'font.size': 18})\n\n\n\ndef plot_base(xp, fp, xfine, a, b, label_mid=False):\n\n    fmax = fp.max()\n\n    for xl in xp:\n        plt.plot([xl,xl], [0.0, 1.2*fmax], ls=\"--\", color=\"0.5\", zorder=-1)\n\n    plt.scatter(xp, fp, marker=\"o\", color=\"r\", zorder=100)\n\n    plt.figtext(0.9, 0.05, '$x$', fontsize=20)\n    plt.figtext(0.1, 0.9, '$y$', fontsize=20)\n\n    ax = plt.gca()\n\n    ax.spines['right'].set_visible(False)\n    ax.spines['top'].set_visible(False)\n    ax.xaxis.set_ticks_position('bottom')\n\n    if label_mid:\n        ax.set_xticks((a, (a+b)/2, b))\n        ax.set_xticklabels(('$a$', r'$\\frac{(a+b)}{2}$', '$b$'))\n    else:\n        ax.set_xticks((a, b))\n        ax.set_xticklabels(('$a$', '$b$'))\n\n    ax.set_yticks([])\n\n    plt.plot(xfine, f(xfine), \"r\", linewidth=2)\n\n    plt.xlim(np.min(xfine), 1.05*np.max(xfine))\n    plt.ylim(ymin = 0)\n\n\n\ndef rectangle(xp, fp, a, b):\n\n    ax = plt.gca()\n\n    integral = 0.0\n\n    for n in range(len(xp)-1):\n\n        xl = xp[n]\n        xr = xp[n+1]\n\n        fl = fp[n]\n\n        # shade region\n        verts = [(xl, 0), (xl, fl), (xr, fl), (xr, 0)]\n        ax.add_patch(Polygon(verts, facecolor=\"0.8\", edgecolor=\"k\"))\n\n        # and bonus! actually compute the integral in this approximation\n        integral += (xr - xl) * fl\n\n    return integral\n\n\ndef trapezoid(xp, fp, a, b):\n\n    ax = plt.gca()\n\n    integral = 0.0\n\n    for n in range(len(xp)-1):\n\n        xl = xp[n]\n        xr = xp[n+1]\n\n        # shade region\n        fl = f(xl)\n        fr = f(xr)\n\n        verts = [(xl, 0), (xl, fl), (xr, fr), (xr, 0)]\n        ax.add_patch(Polygon(verts, facecolor=\"0.8\", edgecolor=\"k\"))\n\n        integral += 0.5 * (xr - xl) * (fl + fr)\n\n    return integral\n\ndef simpsons(xp, fp, a, b):\n\n    ax = plt.gca()\n\n    integral = 0.0\n\n    for n in range(0, len(xp)-1, 2):\n\n        # we need to handle the 1 bin case specially\n\n        if len(xp) == 2:\n\n            xl = xp[0]\n            xr = xp[1]\n            xm = 0.5 * (xl + xr)\n\n            f0 = f(xl)\n            f1 = f(xm)\n            f2 = f(xr)\n\n        else:\n            f0 = fp[n]\n            f1 = fp[n+1]\n            f2 = fp[n+2]\n\n            xl = xp[n]\n            xr = xp[n+2]\n\n        delta = 0.5*(xr - xl)\n\n        A = (f0 - 2*f1 + f2)/(2*delta**2)\n        B = -(f2 - 4*f1 + 3*f0)/(2*delta)\n        C = f0\n\n        xsimp = np.linspace(xl, xr, 100)\n        fsimp = A * (xsimp - xl)**2  + B * (xsimp - xl) + C\n\n        simpvert = list(zip(xsimp, fsimp))\n\n        verts = [(xl, 0)] + simpvert + [(xr, 0)]\n        ax.add_patch(Polygon(verts, facecolor=\"0.8\", edgecolor=\"k\"))\n\n        integral += (xr - xl) / 6.0 * (f0 + 4 * f1 + f2)\n\n    return integral\n\ndef main():\n\n    N_fine = 200\n\n    # the number of bins to divide [a, b]\n    N_bins = 16\n\n    xmin = 0.5\n    xmax = 1.5\n\n    dx_extra  = 0.5\n\n    # add a bin on each end of the domain outside of the integral\n    xmin_plot = xmin - dx_extra\n    xmax_plot = xmax + dx_extra\n\n    xfine = np.linspace(xmin_plot, xmax_plot, N_fine+2)\n\n    xp = np.linspace(xmin, xmax, N_bins+1)\n\n    # integral range\n    a = xmin\n    b = xmax\n\n    # function points\n    fp = f(xp)\n\n\n    # rectangle method\n\n    plt.clf()\n\n    plot_base(xp, fp, xfine, a, b)\n    I_r = rectangle(xp, fp, a, b)\n\n    plt.savefig(f\"rectangle_N{N_bins}.png\", bbox_inches=\"tight\")\n\n    # trapezoid method\n\n    plt.clf()\n\n    plot_base(xp, fp, xfine, a, b)\n    I_t = trapezoid(xp, fp, a, b)\n\n    plt.savefig(f\"trapezoid_N{N_bins}.png\", bbox_inches=\"tight\")\n\n\n    # simpsons method\n\n    plt.clf()\n\n    xp_tmp = list(xp)\n    fp_tmp = list(fp)\n    label_mid = False\n\n    # if N_bins is 1, we need an extra point for Simpsons\n    if N_bins == 1:\n        xp_tmp.append((a + b)/2)\n        fp_tmp.append(f((a + b)/2))\n        label_mid = True\n\n    plot_base(np.array(xp_tmp), np.array(fp_tmp), xfine, a, b,\n              label_mid=label_mid)\n\n\n    I_s = simpsons(xp, fp, a, b)\n\n    plt.savefig(f\"simpsons_N{N_bins}.png\", bbox_inches=\"tight\")\n\n    print(f\"integral approximations: {I_r}, {I_t}, {I_s}\")\n\nif __name__ == \"__main__\":\n    main()\n\n\n\n"
  },
  {
    "path": "figures/intro/rk4_plot.py",
    "content": "# make a plot of what the 4th-order Runge-Kutta is doing using the\n# example dy/dt = -y (this comes from Garcia).\n\nfrom __future__ import print_function\n\n\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport matplotlib as mpl\n\nmpl.rcParams['mathtext.fontset'] = 'cm'\nmpl.rcParams['mathtext.rm'] = 'serif'\n\n\ndef rhs(y):\n    \"\"\" return dy/dt \"\"\"\n    return -y\n\ndef exact(t):\n    \"\"\" analytic solution \"\"\"\n    return np.exp(-t)\n\n\n# frame 1 -- show the initial condition (y^n)\n\ny0 = 1.0\nt0 = 0.0\n\ndt = 0.75\n\ntt = np.linspace(t0, t0+2.0*dt, 100)\n\n\ndef start():\n    \"\"\" default starting point \"\"\"\n\n    plt.plot(tt, exact(tt), label=\"analytic solution\", color=\"k\")\n\n    # draw the timeline\n    plt.plot(tt, 0*tt, color=\"k\")\n\n    # label the current point\n    plt.scatter([t0], [y0], color=\"r\")\n    plt.text(t0, y0+0.03, r\"$y^n$\", \n               horizontalalignment=\"left\", color=\"r\", fontsize=20) \n\n    plt.plot([t0,t0], [0, y0], ls=\":\", color=\"0.5\")\n    plt.text(t0, -0.05, r\"$t^n$\", \n               verticalalignment=\"top\", horizontalalignment=\"center\",\n               fontsize=20)\n\n\n    # illustrate where t^n+1 is\n    plt.plot([t0+dt,t0+dt], [0, y0], ls=\":\", color=\"0.5\")\n    plt.text(t0+dt, -0.05, r\"$t^{n+1}$\", \n               verticalalignment=\"top\", horizontalalignment=\"center\",\n               fontsize=20)\n\n\nstart()\n\nplt.axis(\"off\")\n\nleg = plt.legend()\nltext = leg.get_texts()\nplt.setp(ltext, fontsize='large')\nleg.draw_frame(0)\n\nplt.xlim(t0-0.05*dt, t0+1.5*dt)\n\nplt.tight_layout()\nplt.savefig(\"rk4_initial.pdf\", bbox_inches='tight')\n\n\n# now draw the solution Euler would get\nslope = rhs(y0)\ntEuler = np.linspace(t0, t0+dt, 2)\nyEuler = y0 + slope*(tEuler-t0)\n\nplt.plot(tEuler, yEuler, label=\"Euler step\")\n\nplt.scatter([tEuler[1]], [yEuler[1]], color=\"r\")\nplt.text(tEuler[1]+0.015, yEuler[1], r\"$y^{n+1}$\", \n           horizontalalignment=\"left\", verticalalignment=\"center\",\n           color=\"r\", fontsize=20) \n\n\nleg = plt.legend()\nltext = leg.get_texts()\nplt.setp(ltext, fontsize='large')\nleg.draw_frame(0)\n\nplt.xlim(t0-0.25*dt, t0+1.5*dt)\n\nplt.tight_layout()\nplt.savefig(\"rk4_Euler.pdf\", bbox_inches='tight')\n\n\n#----------------------------------------------------------------------------\n# show k1 slope \nplt.clf()\nstart()\n\nplt.axis(\"off\")\n\nk1 = rhs(y0)\ntSlope = np.linspace(t0-0.2*dt, t0+0.2*dt, 2)\nySlope = y0 + k1*(tSlope - t0)\n\nplt.plot(tSlope, ySlope, label=r\"slope\", color=\"g\", ls=\"--\", lw=2, zorder=10)\nplt.text(t0-0.1*dt, y0, r\"$k_1$\", color=\"g\", fontsize=\"large\")\n\nleg = plt.legend()\nltext = leg.get_texts()\nplt.setp(ltext, fontsize='large')\nleg.draw_frame(0)\n\nplt.xlim(t0-0.25*dt, t0+1.5*dt)\n\nplt.tight_layout()\nplt.savefig(\"rk4_k1.pdf\", bbox_inches='tight')\n\n\n#----------------------------------------------------------------------------\n# follow k1 to define k2\n\nytmp = y0 + k1*0.5*dt\nk2 = rhs(ytmp)\n\n# draw the k1 half step\nplt.plot([t0, t0+0.5*dt], [y0, ytmp], color=\"b\", label=\"half-dt k1 step\")\nplt.scatter(t0+0.5*dt, ytmp, color=\"b\")\n\n# draw slope there\n\ntSlope = np.linspace(t0+0.5*dt-0.2*dt, t0+0.5*dt+0.2*dt, 2)\nySlope = ytmp + k2*(tSlope - (t0 + 0.5*dt))\n\nplt.plot(tSlope, ySlope, color=\"g\", ls=\"--\", lw=2)\nplt.text(t0+0.5*dt-0.1*dt, ytmp, r\"$k_2$\", color=\"g\", \n         verticalalignment=\"top\", fontsize=\"large\")\n\nleg = plt.legend()\nltext = leg.get_texts()\nplt.setp(ltext, fontsize='large')\nleg.draw_frame(0)\n\nplt.xlim(t0-0.25*dt, t0+1.5*dt)\n\nplt.tight_layout()\nplt.savefig(\"rk4_k2.pdf\", bbox_inches='tight')\n\n\n#----------------------------------------------------------------------------\n# follow k2 to define k3\nytmp = y0 + k2*0.5*dt\nk3 = rhs(ytmp)\n\n# draw k2 half step\nplt.plot([t0, t0+0.5*dt], [y0, ytmp], color=\"c\", label=\"half-dt k2 step\")\nplt.scatter(t0+0.5*dt, ytmp, color=\"c\")\n\n# draw slope there\nySlope = ytmp + k3*(tSlope - (t0 + 0.5*dt))\nplt.plot(tSlope, ySlope, color=\"g\", ls=\"--\", lw=2)\nplt.text(t0+0.5*dt+0.05*dt, ytmp, r\"$k_3$\", color=\"g\", \n         verticalalignment=\"bottom\", fontsize=\"large\")\n\nleg = plt.legend()\nltext = leg.get_texts()\nplt.setp(ltext, fontsize='large')\nleg.draw_frame(0)\n\nplt.xlim(t0-0.25*dt, t0+1.5*dt)\n\nplt.tight_layout()\nplt.savefig(\"rk4_k3.pdf\", bbox_inches='tight')\n\n\n#----------------------------------------------------------------------------\n# follow k3 to define k4\nytmp = y0 + k3*dt\nk4 = rhs(ytmp)\n\n# draw k3 full step\nplt.plot([t0, t0+dt], [y0, ytmp], color=\"0.5\", label=\"full-dt k3 step\")\nplt.scatter(t0+dt, ytmp, color=\"0.5\")\n\n# draw slope there\ntSlope2 = np.linspace(t0+dt-0.2*dt, t0+dt+0.2*dt, 2)\nySlope = ytmp + k4*(tSlope2 - (t0 + dt))\nplt.plot(tSlope2, ySlope, color=\"g\", ls=\"--\", lw=2)\nplt.text(t0+dt-0.1*dt, ytmp, r\"$k_4$\", color=\"g\", \n         verticalalignment=\"top\", fontsize=\"large\")\n\nleg = plt.legend()\nltext = leg.get_texts()\nplt.setp(ltext, fontsize='large')\nleg.draw_frame(0)\n\nplt.xlim(t0-0.25*dt, t0+1.5*dt)\n\nplt.tight_layout()\nplt.savefig(\"rk4_k4.pdf\", bbox_inches='tight')\n\n\n\n#----------------------------------------------------------------------------\n# final RK-4 step\nynew = y0 + (dt/6.0)*(k1 + 2.0*k2 + 2.0*k3 + k4)\n\n\n# draw full RK-4 step\nplt.plot([t0, t0+dt], [y0, ynew], color=\"r\", label=\"full 4th-order RK step\")\nplt.scatter(t0+dt, ynew, color=\"r\")\nplt.text(t0+1.05*dt, ynew+0.015, r\"$y^{n+1}$\", color = \"r\", \n           horizontalalignment=\"left\", fontsize=20)\n\n\nleg = plt.legend()\nltext = leg.get_texts()\nplt.setp(ltext, fontsize='large')\nleg.draw_frame(0)\n\nplt.xlim(t0-0.25*dt, t0+1.5*dt)\n\nplt.tight_layout()\nplt.savefig(\"rk4_final.pdf\", bbox_inches='tight')\n\n"
  },
  {
    "path": "figures/intro/update.sh",
    "content": "#!/bin/sh\n\n# integrals.py creates rectange.pdf trapezoid.pdf simpsons.pdf\npython3 integrals.py \ncp -f rectangle.pdf trapezoid.pdf simpsons.pdf ../../intro/\n\n# rk4_plot.py creates rk4_k[1-4].pdf, rk4_final.pdf\npython3 rk4_plot.py\ncp -f rk4_k[1-4].pdf rk4_final.pdf ../../intro\n\n"
  },
  {
    "path": "figures/multigrid/2dgrid-mg.py",
    "content": "import math\nimport numpy\nimport pylab\n\ndef simplegrid():\n\n    #-------------------------------------------------------------------------\n    # prolongation\n    #-------------------------------------------------------------------------\n\n    # grid info\n    xmin = 0.0\n    xmax = 1.0\n\n    ymin = 0.0\n    ymax = 1.0\n\n    nzones = 3\n    ng = 0\n    \n    dx = (xmax - xmin)/float(nzones)\n    dy = (ymax - ymin)/float(nzones)\n\n    xl = (numpy.arange(2*ng+nzones) - ng)*dx\n    xr = (numpy.arange(2*ng+nzones)+1 - ng)*dx\n\n    xc = 0.5*(xl + xr)\n\n    yl = (numpy.arange(2*ng+nzones) - ng)*dy\n    yr = (numpy.arange(2*ng+nzones)+1 - ng)*dy\n\n    yc = 0.5*(yl + yr)\n\n\n    #------------------------------------------------------------------------\n    # plot a domain without ghostcells\n    \n    # x lines\n    n = 0\n    while (n < nzones):\n        pylab.plot([xmin-0.25*dx,xmax+0.25*dx], [yl[n],yl[n]], color=\"k\", lw=2)\n        n += 1\n        \n    pylab.plot([xmin-0.25*dx,xmax+0.25*dx], [yr[nzones-1],yr[nzones-1]], color=\"k\", lw=2)\n\n\n    # y lines\n    n = 0\n    while (n < nzones):\n        pylab.plot([xl[n],xl[n]], [ymin-0.25*dy,ymax+0.25*dy], color=\"k\", lw=2)\n        n += 1\n        \n    pylab.plot([xr[nzones-1],xr[nzones-1]], [ymin-0.25*dy,ymax+0.25*dy], color=\"k\", lw=2)\n\n\n    #------------------------------------------------------------------------\n    # label\n    pylab.text(xc[nzones/2], yc[nzones/2], r\"$\\phi_{i,j}^c$\", fontsize=\"18\",\n               horizontalalignment='center', verticalalignment='center', \n               zorder=100, color=\"b\")\n\n    pylab.text(xc[nzones/2+1], yc[nzones/2], r\"$\\phi_{i+1,j}^c$\", fontsize=\"18\",\n               horizontalalignment='center', verticalalignment='center',\n               color=\"b\")\n\n    pylab.text(xc[nzones/2], yc[nzones/2+1], r\"$\\phi_{i,j+1}^c$\", fontsize=\"18\",\n               horizontalalignment='center', verticalalignment='center',\n               color=\"b\")\n\n    pylab.text(xc[nzones/2-1], yc[nzones/2], r\"$\\phi_{i-1,j}^c$\", fontsize=\"18\",\n               horizontalalignment='center', verticalalignment='center',\n               color=\"b\")\n\n    pylab.text(xc[nzones/2], yc[nzones/2-1], r\"$\\phi_{i,j-1}^c$\", fontsize=\"18\",\n               horizontalalignment='center', verticalalignment='center',\n               color=\"b\")\n\n\n    # shading\n    ii = nzones/2; jj = nzones/2\n    pylab.fill([xl[ii], xl[ii], xr[ii], xr[ii], xl[ii]],\n               [yl[jj], yr[jj], yr[jj], yl[jj], yl[jj]], color=\"0.85\", \n               zorder=-1)\n\n    ii = nzones/2+1; jj = nzones/2\n    pylab.fill([xl[ii], xl[ii], xr[ii], xr[ii], xl[ii]],\n               [yl[jj], yr[jj], yr[jj], yl[jj], yl[jj]], color=\"0.85\", \n               zorder=-1)\n\n    ii = nzones/2-1; jj = nzones/2\n    pylab.fill([xl[ii], xl[ii], xr[ii], xr[ii], xl[ii]],\n               [yl[jj], yr[jj], yr[jj], yl[jj], yl[jj]], color=\"0.85\", \n               zorder=-1)\n\n    ii = nzones/2; jj = nzones/2+1\n    pylab.fill([xl[ii], xl[ii], xr[ii], xr[ii], xl[ii]],\n               [yl[jj], yr[jj], yr[jj], yl[jj], yl[jj]], color=\"0.85\", \n               zorder=-1)\n\n    ii = nzones/2; jj = nzones/2-1\n    pylab.fill([xl[ii], xl[ii], xr[ii], xr[ii], xl[ii]],\n               [yl[jj], yr[jj], yr[jj], yl[jj], yl[jj]], color=\"0.85\", \n               zorder=-1)\n\n\n    # fine cells\n    ii = nzones/2; jj = nzones/2\n    pylab.plot([xc[ii], xc[ii]], [yl[ii], yr[ii]], linestyle=\"--\", color=\"0.3\")\n    pylab.plot([xl[ii], xr[ii]], [yc[ii], yc[ii]], linestyle=\"--\", color=\"0.3\")\n\n\n    pylab.text(xc[ii]-dx/4, yc[jj]-dy/4, r\"$\\phi_{--}^{f}$\", fontsize=\"18\",\n               horizontalalignment='center', verticalalignment='center', \n               zorder=100, color=\"r\")\n\n    pylab.text(xc[ii]-dx/4, yc[jj]+dy/4, r\"$\\phi_{-+}^{f}$\", fontsize=\"18\",\n               horizontalalignment='center', verticalalignment='center', \n               zorder=100, color=\"r\")\n\n    pylab.text(xc[ii]+dx/4, yc[jj]-dy/4, r\"$\\phi_{+-}^{f}$\", fontsize=\"18\",\n               horizontalalignment='center', verticalalignment='center', \n               zorder=100, color=\"r\")\n\n    pylab.text(xc[ii]+dx/4, yc[jj]+dy/4, r\"$\\phi_{++}^{f}$\", fontsize=\"18\",\n               horizontalalignment='center', verticalalignment='center', \n               zorder=100, color=\"r\")\n\n\n    # grid labels\n    pylab.text(xc[nzones/2-1], yl[0]-0.35*dy, r\"$i-1$\",\n               horizontalalignment='center', fontsize=\"16\")\n\n    pylab.text(xc[nzones/2], yl[0]-0.35*dy, r\"$i$\",\n               horizontalalignment='center', fontsize=\"16\")\n\n    pylab.text(xc[nzones/2+1], yl[0]-0.35*dy, r\"$i+1$\",\n               horizontalalignment='center', fontsize=\"16\")\n\n\n    pylab.text(xl[0]-0.35*dx, yc[nzones/2-1], r\"$j-1$\",\n               verticalalignment='center', fontsize=\"16\")\n\n    pylab.text(xl[0]-0.35*dx, yc[nzones/2], r\"$j$\",\n               verticalalignment='center', fontsize=\"16\")\n\n    pylab.text(xl[0]-0.35*dx, yc[nzones/2+1], r\"$j+1$\",\n               verticalalignment='center', fontsize=\"16\")\n\n\n\n    # axes\n    pylab.xlim(xl[0]-0.5*dx,xr[2*ng+nzones-1]+0.25*dx)\n    pylab.ylim(yl[0]-0.5*dy,yr[2*ng+nzones-1]+0.25*dy)\n    pylab.axis(\"off\")\n\n    pylab.subplots_adjust(left=0.02,right=0.98,bottom=0.02,top=0.98)\n\n    f = pylab.gcf()\n    f.set_size_inches(8.0,8.0)\n\n\n    pylab.savefig(\"2dgrid-prolong.png\")\n    pylab.savefig(\"2dgrid-prolong.eps\")\n\n\n               \n\nif __name__== \"__main__\":\n    simplegrid()\n"
  },
  {
    "path": "figures/multigrid/fv-fd_bnd.py",
    "content": "import matplotlib.pyplot as plt\nimport numpy as np\nimport grid_plot as gp\n\n# plot a simple finite-difference grid showing the boundary data\n# and a finite-volume grid with the same data, showing it's boundary\n\n#-----------------------------------------------------------------------------\n\n\n# finite-difference\n\n# data that lives on the grid\n#a = np.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.15, 0.5, 0.55])\na = np.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.1, 0.2, 0.5, 0.55])\nnzones = len(a)\n\nvoff = 2.0\ngr = gp.FDGrid(nzones, voff=voff)\n\ngr.draw_grid(emphasize_end=True)\n\ngr.label_node(nzones/2,   r\"$i$\",   fontsize=\"large\")\ngr.label_node(nzones/2-1, r\"$i-1$\", fontsize=\"large\")\ngr.label_node(nzones/2+1, r\"$i+1$\", fontsize=\"large\")\ngr.label_node(nzones/2-2, r\"$i-2$\", fontsize=\"large\")\ngr.label_node(nzones/2+2, r\"$i+2$\", fontsize=\"large\")\n\n# draw the data\nfor i in range(nzones):\n    gr.draw_data(i, a[i], color=\"r\")\n\n\ngr.label_value(nzones/2, a[nzones/2], r\"$f_i$\", color=\"r\")\n\n# label dx\nplt.plot([gr.xc[gr.ng+nzones/2-1], gr.xc[gr.ng+nzones/2-1]], [-0.35+voff,-0.25+voff], color=\"k\")\nplt.plot([gr.xc[gr.ng+nzones/2], gr.xc[gr.ng+nzones/2]], [-0.35+voff,-0.25+voff], color=\"k\")\nplt.plot([gr.xc[gr.ng+nzones/2-1], gr.xc[gr.ng+nzones/2]], [-0.3+voff,-0.3+voff], color=\"k\")\nplt.text(0.5*(gr.xc[gr.ng+nzones/2-1] + gr.xc[gr.ng+nzones/2]), -0.45+voff, r\"$\\Delta x$\",\n           horizontalalignment=\"center\")\n\n\n\n# finite-volume\nav = 0.5*(a[0:nzones-1] + a[1:])\nnzones = len(av)\nng = 1\n\ngr = gp.FVGrid(nzones, ng=1)\n\ngr.draw_grid(emphasize_end=True, draw_ghost=True)\n\ngr.label_center(ng+nzones/2,   r\"$i$\",   fontsize=\"large\")\ngr.label_center(ng+nzones/2-1, r\"$i-1$\", fontsize=\"large\")\ngr.label_center(ng+nzones/2+1, r\"$i+1$\", fontsize=\"large\")\n\ngr.label_center(ng+nzones-1, r\"$\\mathrm{hi}$\", fontsize=\"large\")\ngr.label_center(ng+nzones, r\"$\\mathrm{hi+1}$\", fontsize=\"large\")\n\ngr.label_center(gr.ilo, r\"$\\mathrm{lo}$\", fontsize=\"large\")\ngr.label_center(gr.ilo-1, r\"$\\mathrm{lo-1}$\", fontsize=\"large\")\n\na = gr.scratch_array()\na[gr.ilo:gr.ihi+1] = av\n\ncc = gp.CellCentered(gr, a)\n\n# draw the data\nfor i in range(gr.ilo, gr.ihi+1):\n    cc.draw_data_point(i, color=\"r\")\n\ncc.label_data_point(ng+nzones/2, r\"$f_i$\", color=\"r\")\n\n# label dx\nplt.plot([gr.xl[gr.ng+nzones/2], gr.xl[gr.ng+nzones/2]], [-0.35,-0.25], color=\"k\")\nplt.plot([gr.xr[gr.ng+nzones/2], gr.xr[gr.ng+nzones/2]], [-0.35,-0.25], color=\"k\")\nplt.plot([gr.xl[gr.ng+nzones/2], gr.xr[gr.ng+nzones/2]], [-0.3,-0.3], color=\"k\")\nplt.text(gr.xc[gr.ng+nzones/2], -0.45, r\"$\\Delta x$\",\n           horizontalalignment=\"center\")\n\n\n\n# illustrate the boundaries\nplt.plot([gr.xl[gr.ilo], gr.xl[gr.ilo]], [-0.5, 2.0], ls=\":\", color=\"0.5\")\nplt.plot([gr.xr[gr.ihi], gr.xr[gr.ihi]], [-0.5, 2.0], ls=\":\", color=\"0.5\")\n\nplt.text(gr.xl[gr.ilo], -0.5, \"left BC\", horizontalalignment=\"center\",\n         verticalalignment=\"top\")\n\nplt.text(gr.xr[gr.ihi], -0.5, \"right BC\", horizontalalignment=\"center\",\n         verticalalignment=\"top\")\n\nplt.axis([gr.xmin-gr.dx,gr.xmax+gr.dx, -0.5, 3.2])\nplt.axis(\"off\")\n\nplt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)\n\nf = plt.gcf()\nf.set_size_inches(10.0,6.0)\n\nplt.savefig(\"fv-fd_grid_bc.pdf\")\n"
  },
  {
    "path": "figures/multigrid/fvrestrict.py",
    "content": "import numpy as np\nimport matplotlib.pyplot as plt\nimport grid_plot as gp\n\n# plot two stacked fv grids of different (2x) resolution to show prolongation\n\n#-----------------------------------------------------------------------------\n\nnf = 4\nnc = nf//2\n\ngrf = gp.FVGrid(nf, voff=2.0)\ngrc = gp.FVGrid(nc)\n\n\nplt.clf()\n\ngrf.draw_grid()\ngrc.draw_grid()\n\ngrf.label_center(nf//2-2, r\"$i-2$\")\ngrf.label_center(nf//2-1, r\"$i-1$\")\ngrf.label_center(nf//2,   r\"$i$\")\ngrf.label_center(nf//2+1, r\"$i+1$\")\n\ngrc.label_center(nc//2-1, r\"$j-1$\")\ngrc.label_center(nc//2,   r\"$j$\")\n\ngrf.label_cell_center(nf//2-2, r\"$\\phi_{i-2}^f$\")\ngrf.label_cell_center(nf//2-1, r\"$\\phi_{i-1}^f$\")\ngrf.label_cell_center(nf//2,   r\"$\\phi_i^f$\")\ngrf.label_cell_center(nf//2+1, r\"$\\phi_{i+1}^f$\")\n\ngrc.label_cell_center(nc//2-1, r\"$\\phi_{j-1}^{c}$\")\ngrc.label_cell_center(nc//2,   r\"$\\phi_{j}^{c}$\")\n    \n\n# connect the dots...\n\nplt.plot([grf.xl[nf//2-2], grf.xl[nf//2-2]], [-0.25, 3.25], \":\", color=\"0.5\")\nplt.plot([grf.xl[nf//2], grf.xl[nf//2]], [-0.25, 3.25], \":\", color=\"0.5\")\nplt.plot([grf.xr[nf//2+1], grf.xr[nf//2+1]], [-0.25, 3.25], \":\", color=\"0.5\")\n\n\nplt.axis([grf.xmin-0.5*grf.dx,grf.xmax+0.5*grf.dx, -0.5, 3.5])\nplt.axis(\"off\")\n\nplt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)\n\nf = plt.gcf()\nf.set_size_inches(6.0,5.0)\n\nplt.savefig(\"fvrestrict.pdf\")\n\n"
  },
  {
    "path": "figures/multigrid/laplacian.py",
    "content": "import numpy as np\nimport matplotlib.pyplot as plt\nimport grid_plot as gp\n\ndef laplace():\n\n    # grid info\n    xmin = 0.0\n    xmax = 1.0\n\n    nzones = 3\n    ng = 0\n\n    gr = gp.FVGrid(nzones, xmin=xmin, xmax=xmax)\n\n\n    #------------------------------------------------------------------------\n    # plot a domain without ghostcells\n    gr.draw_grid(emphasize_end=True)\n\n    # label a few\n    gr.label_center(0, r\"$i-1$\", fontsize=\"medium\")\n    gr.label_center(1, r\"$i$\", fontsize=\"medium\")\n    gr.label_center(2, r\"$i+1$\", fontsize=\"medium\")\n    gr.label_edge(1, r\"$i-\\myhalf$\", fontsize=\"medium\")\n    gr.label_edge(2, r\"$i+\\myhalf$\", fontsize=\"medium\")\n\n    gr.label_cell_center(0, r\"$\\phi_{i-1}$\")\n    gr.label_cell_center(1, r\"$\\phi_{i}$\", value=0.7)\n    gr.label_cell_center(2, r\"$\\phi_{i+1}$\")\n\n    gr.mark_cell_edge(1, r\"$\\left .\\frac{d\\phi}{dx} \\right |_{i-\\myhalf}$\", color=\"C0\")\n    gr.mark_cell_edge(2, r\"$\\left .\\frac{d\\phi}{dx} \\right |_{i+\\myhalf}$\", color=\"C0\")\n    \n    gr.label_cell_center(1, r\"$\\left .\\frac{d^2\\phi}{dx^2} \\right |_{i}$\", \n                         value=0.3, color=\"C1\")\n\n    gr.clean_axes(padding=False)\n    plt.ylim(-0.25, 1.25)\n\n    plt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)\n\n    f = plt.gcf()\n    f.set_size_inches(6.0,2.25)\n\n\n    plt.tight_layout()\n\n    plt.savefig(\"laplacian.pdf\")\n\n\nif __name__== \"__main__\":\n    laplace()\n"
  },
  {
    "path": "figures/multigrid/mgtower.py",
    "content": "import matplotlib.pyplot as plt\nimport grid_plot as gp\n\n# plot two stacked fv grids of different (2x) resolution to show prolongation\n\n#-----------------------------------------------------------------------------\n\ngr = []\n\nnzones = [2, 4, 8, 16]\nfor nf in nzones:\n    gr.append(gp.FVGrid(nf, ng=1, voff=2.0*len(gr)))\n\n\nplt.clf()\n\nfor g in gr:\n    g.draw_grid(emphasize_end=1, draw_ghost=1, edge_ticks=0)\n\nf = plt.gcf()\nf.set_size_inches(7.0,5.0)\n\ngrf = gr[0]\nplt.xlim(grf.xmin-1.1*grf.dx,grf.xmax+1.1*grf.dx)\n\nplt.axis(\"off\")\n\nplt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)\n\nplt.savefig(\"mgtower.pdf\")\n\n"
  },
  {
    "path": "figures/multigrid/red_black.py",
    "content": "import math\nimport numpy\nimport pylab\nimport random\n\n# red black gauss seidel pattern\n\nclass marker:\n\n    def __init__(self, xc, yc, color):\n\n        # a marker is indicated by its center (xc,yc).\n        self.xc = xc\n        self.yc = yc\n        \n        # keep track of the color\n        self.color = color\n\n        \ndef RB():\n\n    # define the number of markers in x and y\n    nx = 10\n    ny = 10\n\n    # define the length of a marker side\n    L = 0.8\n\n    # create a list of marker objects, one at each grid location\n    markers = []\n    \n\n    color = 0\n    while (color <= 1):\n\n        j = 0\n        while (j < ny):\n\n            if (color == 0):\n                ioff = j % 2\n            else:\n                ioff = 1 - (j % 2)\n                \n            i = ioff\n            while (i < nx):\n                markers.append(marker(i, j, color))\n                i += 2\n\n            j += 1\n\n        color += 1\n\n\n    pylab.clf()\n\n    # the margins are funny -- we pick them to ensure that the\n    # plot size is an integer multiple of the number of markers in\n    # each dimension\n    pylab.subplots_adjust(left=0.1,  right=0.9,\n                          bottom=0.1,top=0.9)\n\n    # draw the current state\n    n = 0\n    while (n < len(markers)):\n\n        if (markers[n].color == 1):\n            c = \"r\"\n        else:\n            c = \"k\"\n\n        pylab.fill([markers[n].xc-L/2, markers[n].xc-L/2,\n                    markers[n].xc+L/2, markers[n].xc+L/2,\n                    markers[n].xc-L/2],\n                   [markers[n].yc-L/2, markers[n].yc+L/2,\n                    markers[n].yc+L/2, markers[n].yc-L/2,\n                    markers[n].yc-L/2], \n                   c)\n\n        n += 1\n\n            \n    ax = pylab.axis([-0.5,nx+0.5,-0.5,ny+0.5])\n    pylab.axis(\"off\")\n\n    f = pylab.gcf()\n    f.set_size_inches(6.0,6.0)\n\n    pylab.tight_layout()\n\n    pylab.savefig(\"rb.png\")\n    pylab.savefig(\"rb.eps\", bbox_inches=\"tight\", pad_inches=0)\n\n\nif __name__== \"__main__\":\n    RB()\n\n\n    \n        \n"
  },
  {
    "path": "figures/multigrid/smooth-separate.py",
    "content": "#!/usr/bin/env python\n\n\"\"\"\n\nan example of solving Poisson's equation via smoothing only.  Here, we\nsolve\n\nu_xx = sin(x)\nu = 0 on the boundary [0,1]\n\nThe analytic solution is u(x) = -sin(x) + x sin(1)\n\nThis version (separate) differs from smooth.py in that we implement the\nsmoothing here directly, instead of using the MG solver.\n\nM. Zingale (2013-03-31)\n\n\"\"\"\n#from io import *\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport sys\n\ndef true(x):\n    # the analytic solution\n    return -np.sin(x) + x*np.sin(1.0)\n\n\ndef error(ilo, ihi, dx, r):\n    # L2 norm of elements in r, multiplied by dx to normalize\n    return np.sqrt(dx*np.sum((r[ilo:ihi+1]**2)))\n\n\ndef f(x):\n    # the righthand side\n    return np.sin(x)\n\n\ndef compute_residual(ilo, ihi, dx, phi, frhs):\n    # compute r = f - L phi\n    r = np.zeros(len(phi))\n    r[ilo:ihi+1] = frhs[ilo:ihi+1] - \\\n        (phi[ilo+1:ihi+2] - 2.0*phi[ilo:ihi+1] + phi[ilo-1:ihi])/dx**2\n    return r\n\n\ndef smooth_run(nx, method=\"GS\"):\n\n    xmin = 0.0\n    xmax = 1.0\n\n    ng = 1\n\n    # initialize the solution to zero.  Put one ghost cell on either end\n    phi = np.zeros(nx + 2*ng, dtype=np.float64)\n    phinew = np.zeros_like(phi)\n\n    ilo = ng\n    ihi = ng + nx - 1\n\n    # coordinates of centers\n    dx = (xmax - xmin)/nx\n    x = (np.arange(nx+2*ng) - ng + 0.5)*dx + xmin\n\n    # initialize the RHS using the function f\n    frhs = f(x)\n\n    # smooth \n    n = np.arange(20000) + 1\n    e = []\n    r = []\n\n    print(\"source norm: \", error(ilo, ihi, dx, frhs))\n    print(np.sum(frhs[ilo:ihi+1]))\n\n    for i in n:\n\n        # fill the ghost cells\n        phi[ilo-1] = -phi[ilo]\n        phi[ihi+1] = -phi[ihi]\n\n        if method == \"Jacobi\":\n            phinew[ilo:ihi+1] = \\\n                (-dx*dx*frhs[ilo:ihi+1] + phi[ilo+1:ihi+2] + phi[ilo-1:ihi])/2.0\n            phi[:] = phinew[:]\n\n        elif method == \"GS\":\n\n            # red-black Gauss-Seidel -- first do the odd, then even points\n            phi[ilo:ihi+1:2] = \\\n                0.5*(-dx*dx*frhs[ilo:ihi+1:2] + \n                     phi[ilo+1:ihi+2:2] + phi[ilo-1:ihi:2])\n\n            # fill the ghost cells between red and black\n            phi[ilo-1] = -phi[ilo]\n            phi[ihi+1] = -phi[ihi]\n\n            phi[ilo+1:ihi+1:2] = \\\n                0.5*(-dx*dx*frhs[ilo+1:ihi+1:2] + \\\n                     phi[ilo+2:ihi+2:2] + phi[ilo:ihi:2])\n\n        else:\n            sys.exit(\"invalid method\")\n\n        # compute the true error (wrt the analytic solution) and residual\n        e.append(error(ilo, ihi, dx, phi - true(x)))\n        \n        # compute the residual\n        resid = compute_residual(ilo, ihi, dx, phi, frhs)\n        r.append(error(ilo, ihi, dx, resid))\n\n    return n, np.array(r), np.array(e)\n\n\n# test the multigrid solver\nN = [16, 32, 64]\n\nc = [\"r\", \"g\", \"b\"]\n\nfor nx in N:\n\n    n, r, e = smooth_run(nx)\n    color = c.pop()\n    plt.plot(n, e, color=color, label = str(nx))\n    plt.plot(n, r, color=color, ls=\":\")\n\nax = plt.gca()\nax.set_xscale('log')\nax.set_yscale('log')\n\nplt.xlabel(\"# of iterations\")\nplt.ylabel(\"L2 norm of true error (solid) and residual (dotted)\")\nplt.legend(frameon=False, fontsize=\"small\")\n\nplt.savefig(\"smooth-error.png\")\n\n\n"
  },
  {
    "path": "figures/weno/weno.py",
    "content": "import numpy\nfrom matplotlib import pyplot\n\nng = 2\n\ndef stencil_5_pts(q):\n    \"\"\"\n    Do reconstruction to q_{i+1/2} using five point stencil.\n    \n    Parameters\n    ----------\n    \n    q : numpy array\n        Scalar data to reconstruct\n        \n    Returns\n    -------\n    \n    qL : numpy array\n        Reconstructed data - boundary points are zero\n    \"\"\"\n    qL = numpy.zeros_like(q)\n    np = len(q) - 2 * ng\n    for i in range(ng, np+ng):\n        qL[i] = ( 2 * q[i-2] - 13 * q[i-1] + 47 * q[i] + \n                  27 * q[i+1] - 3 * q[i+2] ) / 60\n    return qL\n\ndef stencils_3_pts(q):\n    \"\"\"\n    Do reconstruction to q_{i+1/2} using three point stencils.\n    \n    Parameters\n    ----------\n    \n    q : numpy array\n        Scalar data to reconstruct\n        \n    Returns\n    -------\n    \n    qL : numpy array\n        Reconstructed data - boundary points are zero\n    \"\"\"\n    qL = numpy.zeros((3, len(q)))\n    np = len(q) - 2 * ng\n    for i in range(ng, np+ng):\n        qL[2, i] = ( 2 * q[i-2] - 7 * q[i-1] + 11 * q[i  ]) / 6\n        qL[1, i] = (-1 * q[i-1] + 5 * q[i  ] +  2 * q[i+1]) / 6\n        qL[0, i] = ( 2 * q[i  ] + 5 * q[i+1] -  1 * q[i+2]) / 6\n    return qL\n\ndef weno3(q):\n    \"\"\"\n    Do WENO reconstruction, r=3\n    \n    Parameters\n    ----------\n    \n    q : numpy array\n        Scalar data to reconstruct\n        \n    Returns\n    -------\n    \n    qL : numpy array\n        Reconstructed data - boundary points are zero\n    \"\"\"\n    order = 3\n    C = numpy.array([1, 6, 3]) / 10\n    a = numpy.array([[11, -7, 2], [2, 5, -1], [-1, 5, 2]]) / 6\n    sigma = numpy.array([[[40, 0, 0],\n                            [-124, 100, 0],\n                            [44, -76, 16] ],\n                           [[16, 0, 0],\n                            [-52, 52, 0],\n                            [20, -52, 16] ],\n                           [[16, 0, 0],\n                            [-76, 100, 0],\n                            [44, -124, 40] ] ]) / 12\n\n    qL = numpy.zeros_like(q)\n    beta = numpy.zeros((3, len(q)))\n    w = numpy.zeros_like(beta)\n    np = len(q) - 2 * ng\n    epsilon = 1e-16\n    for i in range(ng, np+ng):\n        q_stencils = numpy.zeros(order)\n        alpha = numpy.zeros(order)\n        for k in range(order):\n            for l in range(order):\n                for m in range(l+1):\n                    beta[k, i] += sigma[k, l, m] * q[i+k-l] * q[i+k-m]\n            alpha[k] = C[k] / (epsilon + beta[k, i]**2)\n            for l in range(order):\n                q_stencils[k] += a[k, l] * q[i+k-l]\n        w[:, i] = alpha / numpy.sum(alpha)\n        qL[i] = numpy.dot(w[:, i], q_stencils)\n    \n    return qL, beta, w\n\n# First, use a smooth sine function to check convergence of the\n# various reconstructions\n\nf_smooth = lambda x: numpy.sin(2 * numpy.pi * x)\nf_antiderivative = lambda x: -numpy.cos(2 * numpy.pi * x) / (2 * numpy.pi)\n\nx_plot = numpy.linspace(0, 1, 200)\nx_recon, dx = numpy.linspace(0, 1, 20, retstep=True)\nx_recon_interior = x_recon[ng:-ng] + dx / 2\ns_plot = f_smooth(x_plot)\ns_recon = f_smooth(x_recon)\nrecon_exact = f_smooth(x_recon_interior)\ns_5 = stencil_5_pts(s_recon)\ns_3 = stencils_3_pts(s_recon)\n\n# The fifth order case\npyplot.figure()\npyplot.plot(x_plot, s_plot, 'k--', lw=2)\npyplot.plot(x_recon, s_recon, 'bo', ms=5)\npyplot.plot(x_recon_interior, s_5[ng:-ng], 'r^', ms=5)\npyplot.xlabel(r\"$x$\")\npyplot.xlim(0, 1)\n\n# The third order cases\nfig, axes = pyplot.subplots(1, 3)\nfor r in range(3):\n    axes[r].plot(x_plot, s_plot, 'k--', lw=2)\n    axes[r].plot(x_recon, s_recon, 'bo', ms=5)\n    axes[r].plot(x_recon_interior, s_3[r, ng:-ng], 'r^', ms=5)\n    axes[r].set_xlabel(r\"$x$\")\n    axes[r].set_xlim(0, 1)\nfig.tight_layout()\n\n# Convergence check, using the lazy fix to get error norms.\npoints = 20 * 2**numpy.arange(1, 7)\nerror_1norm_5 = numpy.zeros((len(points),))\nerror_1norm_3 = numpy.zeros((3, len(points)))\nerror_1norm_weno = numpy.zeros((len(points)))\nerror_2norm_5 = numpy.zeros((len(points),))\nerror_2norm_3 = numpy.zeros((3, len(points)))\nerror_2norm_weno = numpy.zeros((len(points),))\nfor i, pts in enumerate(points):\n    x_recon, dx = numpy.linspace(0, 1, pts, retstep=True)\n    x_recon_interior = x_recon[ng:-ng] + dx / 2\n    s_recon = (f_antiderivative(x_recon + dx/2) -\n               f_antiderivative(x_recon - dx/2)) / dx\n    recon_exact = f_smooth(x_recon_interior)\n    s_5 = stencil_5_pts(s_recon)\n    s_3 = stencils_3_pts(s_recon)\n    s_weno, beta, w = weno3(s_recon)\n    error_1norm_5[i] = numpy.linalg.norm(s_5[ng:-ng] - recon_exact, 1) / pts\n    error_2norm_5[i] = numpy.linalg.norm(s_5[ng:-ng] - recon_exact, 2) / numpy.sqrt(pts)\n    error_1norm_weno[i] = numpy.linalg.norm(s_weno[ng:-ng] - recon_exact, 1) / pts\n    error_2norm_weno[i] = numpy.linalg.norm(s_weno[ng:-ng] - recon_exact, 2) / numpy.sqrt(pts)\n    for r in range(3):\n        error_1norm_3[r, i] = numpy.linalg.norm(s_3[r, ng:-ng] - recon_exact, 1) / pts\n        error_2norm_3[r, i] = numpy.linalg.norm(s_3[r, ng:-ng] - recon_exact, 2) / numpy.sqrt(pts)\n\n\n# Plot everything\nx_plot = numpy.linspace(0, 1, 200)\nx_recon, dx = numpy.linspace(0, 1, 20, retstep=True)\nx_recon_interior = x_recon[ng:-ng] + dx / 2\ns_plot = f_smooth(x_plot)\ns_recon = f_smooth(x_recon)\nrecon_exact = f_smooth(x_recon_interior)\ns_5 = stencil_5_pts(s_recon)\ns_3 = stencils_3_pts(s_recon)\ns_weno, beta, w = weno3(s_recon)\nfig = pyplot.figure(figsize=(10,8))\nax1 = pyplot.subplot2grid((2,2),(0,0),colspan=2)\nax2 = pyplot.subplot2grid((2,2),(1,0))\nax3 = pyplot.subplot2grid((2,2),(1,1))\nax1.plot(x_plot, s_plot, 'k--', lw=2)\nax1.plot(x_recon, s_recon, 'bo', ms=7)\nax1.plot(x_recon_interior, s_5[ng:-ng], 'r^', ms=7)\nax1.set_xlabel(r\"$x$\")\nax1.set_xlim(0, 1)\nax1.set_ylabel(\"Smooth function\")\nax2.loglog(1/points, error_1norm_5, 'k^', ms=5, label='stencil 5')\nax2.loglog(1/points, error_1norm_weno, 'r+', ms=5, label='WENO')\nfor r in range(3):\n    ax2.loglog(1/points, error_1norm_3[r,:], 'bo', ms=3, \n                  label=\"stencil 3 ({})\".format(r))\nax2.loglog(1/points, error_1norm_3[0,0]*(points[0]/points)**3, \n              label=r\"$\\propto \\Delta x^3$\")\nax2.loglog(1/points, error_1norm_5[0]*(points[0]/points)**5, \n              label=r\"$\\propto \\Delta x^5$\")\nax2.legend()\nax2.set_xlabel(r\"$\\Delta x$\")\nax2.set_ylabel(r\"$|$Error$|_1$\")\n\nax3.loglog(1/points, error_2norm_5, 'k^', ms=5, label='stencil 5')\nax3.loglog(1/points, error_2norm_weno, 'r+', ms=5, label='WENO')\nfor r in range(3):\n    ax3.loglog(1/points, error_2norm_3[r,:], 'bo', ms=3, \n                  label=\"stencil 3 ({})\".format(r))\nax3.loglog(1/points, error_2norm_3[0,0]*(points[0]/points)**3, \n              label=r\"$\\propto \\Delta x^3$\")\nax3.loglog(1/points, error_2norm_5[0]*(points[0]/points)**5, \n              label=r\"$\\propto \\Delta x^5$\")\nax3.legend()\nax3.set_xlabel(r\"$\\Delta x$\")\nax3.set_ylabel(r\"$|$Error$|_2$\")\nfig.tight_layout()\npyplot.savefig(\"weno-convergence.pdf\")\n\n\n# Just the convergence plots\nfig, axes = pyplot.subplots(1, 2)\naxes[0].loglog(1/points, error_1norm_5, 'k^', ms=5, label='stencil 5')\naxes[0].loglog(1/points, error_1norm_weno, 'r+', ms=5, label='WENO')\nfor r in range(3):\n    axes[0].loglog(1/points, error_1norm_3[r,:], 'bo', ms=3, \n                  label=\"stencil 3 ({})\".format(r))\naxes[0].loglog(1/points, error_1norm_3[0,0]*(points[0]/points)**3, \n              label=r\"$\\propto \\Delta x^3$\")\naxes[0].loglog(1/points, error_1norm_5[0]*(points[0]/points)**5, \n              label=r\"$\\propto \\Delta x^5$\")\naxes[0].legend()\naxes[0].set_xlabel(r\"$\\Delta x$\")\naxes[0].set_ylabel(r\"$|$Error$|_1$\")\n\naxes[1].loglog(1/points, error_2norm_5, 'k^', ms=5, label='stencil 5')\naxes[1].loglog(1/points, error_2norm_weno, 'r+', ms=5, label='WENO')\nfor r in range(3):\n    axes[1].loglog(1/points, error_2norm_3[r,:], 'bo', ms=3, \n                  label=\"stencil 3 ({})\".format(r))\naxes[1].loglog(1/points, error_2norm_3[0,0]*(points[0]/points)**3, \n              label=r\"$\\propto \\Delta x^3$\")\naxes[1].loglog(1/points, error_2norm_5[0]*(points[0]/points)**5, \n              label=r\"$\\propto \\Delta x^5$\")\naxes[1].legend()\naxes[1].set_xlabel(r\"$\\Delta x$\")\naxes[1].set_ylabel(r\"$|$Error$|_2$\")\nfig.tight_layout()    \n\n# Now we work with a non-smooth function\n\nf_nonsmooth = lambda x: numpy.where(x<0.5, numpy.sin(2 * numpy.pi * x),\n                                           numpy.cos(2 * numpy.pi * x))\n\nx_plot = numpy.linspace(0, 1, 200)\nx_recon, dx = numpy.linspace(0, 1, 20, retstep=True)\nx_recon_interior = x_recon[ng:-ng] + dx / 2\ns_plot = f_nonsmooth(x_plot)\ns_recon = f_nonsmooth(x_recon)\nrecon_exact = f_nonsmooth(x_recon_interior)\ns_5 = stencil_5_pts(s_recon)\ns_3 = stencils_3_pts(s_recon)\ns_weno, beta, w = weno3(s_recon)\n\n# 5 point stencil\npyplot.figure()\npyplot.plot(x_plot, s_plot, 'k--', lw=2)\npyplot.plot(x_recon, s_recon, 'bo', ms=5)\npyplot.plot(x_recon_interior, s_5[ng:-ng], 'r^', ms=5)\npyplot.xlabel(r\"$x$\")\npyplot.xlim(0, 1)\n\n# 3 point stencils\nfig, axes = pyplot.subplots(1, 3)\nfor r in range(3):\n    axes[r].plot(x_plot, s_plot, 'k--', lw=2)\n    axes[r].plot(x_recon, s_recon, 'bo', ms=5)\n    axes[r].plot(x_recon_interior, s_3[r, ng:-ng], 'r^', ms=5)\n    axes[r].set_xlabel(r\"$x$\")\n    axes[r].set_xlim(0, 1)\nfig.tight_layout()\n\n# WENO reconstruction\npyplot.figure()\npyplot.plot(x_plot, s_plot, 'k--', lw=2)\npyplot.plot(x_recon, s_recon, 'bo', ms=5)\npyplot.plot(x_recon_interior, s_weno[ng:-ng], 'r^', ms=5)\npyplot.xlabel(r\"$x$\")\npyplot.xlim(0, 1)\n\n# The smoothness indicators and the weights\ncolors=['r','b','g']\nmarkers=['o','^','s']\noptimal_C = numpy.array([1, 6, 3]) / 10\nfig, axes = pyplot.subplots(2, 1, sharex=True, figsize=(6,10))\nfig.subplots_adjust(hspace=0)\nfor r, (color, marker) in enumerate(zip(colors, markers)):\n    axes[0].scatter(x_recon_interior, beta[r, ng:-ng], \n        marker=marker, color=color,\n        label=r\"Jiang-Shu $\\beta_{}$\".format(r))\n    axes[1].scatter(x_recon_interior, w[r, ng:-ng], \n        marker=marker, color=color,\n        label=r\"WENO weight $\\omega_{}$\".format(r))\n    axes[1].plot(x_recon, optimal_C[r] * numpy.ones_like(x_recon),\n        linestyle='--', color=color, lw=2,\n        label=r\"Optimal weight $C_{}$\".format(r))\naxes[1].set_xlabel(r\"$x$\")\naxes[0].set_ylabel(r\"$\\beta$\")\naxes[0].set_xlim(0,1)\naxes[1].set_xlim(0,1)\naxes[0].legend()\naxes[1].legend()\nfig.tight_layout()\n\n\n# Plot everything\nfig, axes = pyplot.subplots(3, 2, sharex = True, figsize=(12,14))\nfig.subplots_adjust(hspace=0)\n# 5 point stencil\naxes[0,0].plot(x_plot, s_plot, 'k--', lw=2)\naxes[0,0].plot(x_recon, s_recon, 'kx', ms=7)\naxes[0,0].plot(x_recon_interior, s_5[ng:-ng], 'r^', ms=7)\naxes[0,0].set_xlim(0, 1)\naxes[0,0].set_ylabel('5 point stencil')\n\n# 3 point stencils\ncolors=['r','b','g']\nmarkers=['o','^','s']\naxes[1,0].plot(x_plot, s_plot, 'k--', lw=2)\naxes[1,0].plot(x_recon, s_recon, 'kx', ms=7)\nfor r, (color, marker) in enumerate(zip(colors, markers)):\n    axes[1,0].plot(x_recon_interior, s_3[r, ng:-ng], \n                   '{}{}'.format(color, marker), ms=7)\naxes[1,0].set_xlim(0, 1)\naxes[1,0].set_ylabel('3 point stencils')\n\n# WENO reconstruction\naxes[2,0].plot(x_plot, s_plot, 'k--', lw=2)\naxes[2,0].plot(x_recon, s_recon, 'kx', ms=7)\naxes[2,0].plot(x_recon_interior, s_weno[ng:-ng], 'r^', ms=7)\naxes[2,0].set_xlabel(r\"$x$\")\naxes[2,0].set_xlim(0, 1)\naxes[2,0].set_ylabel('WENO stencils')\n\n# Shift the right axes\nfor r in range(3):\n    axes[r,1].yaxis.tick_right()\n    axes[r,1].yaxis.set_label_position(\"right\")\n\n# The function again\naxes[0,1].plot(x_plot, s_plot, 'k-', lw=2)\naxes[0,1].set_ylabel('Non-smooth function')\n\n# The smoothness indicators and the weights\ncolors=['r','b','g']\nmarkers=['o','^','s']\noptimal_C = numpy.array([1, 6, 3]) / 10\nfor r, (color, marker) in enumerate(zip(colors, markers)):\n    axes[1,1].scatter(x_recon_interior, beta[r, ng:-ng], \n        marker=marker, color=color,\n        label=r\"Jiang-Shu $\\beta_{}$\".format(r))\n    axes[2,1].scatter(x_recon_interior, w[r, ng:-ng], \n        marker=marker, color=color,\n        label=r\"WENO weight $\\omega_{}$\".format(r))\n    axes[2,1].plot(x_recon, optimal_C[r] * numpy.ones_like(x_recon),\n        linestyle='--', color=color, lw=2,\n        label=r\"Optimal weight $C_{}$\".format(r))\naxes[2,1].set_xlabel(r\"$x$\")\naxes[1,1].set_ylabel(r\"$\\beta$\")\naxes[2,1].set_ylabel(r\"$C$, $\\omega$\")\naxes[1,1].set_xlim(0,1)\naxes[2,1].set_xlim(0,1)\naxes[1,1].legend()\naxes[2,1].legend()\nfig.tight_layout()\npyplot.savefig(\"weno-weights.pdf\")"
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  {
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1426 -417 ct \n1456 -393 1476 -358 1484 -314 ct 1411 -302 l  1404 -332 1392 -355 1374 -370 ct \n1357 -385 1335 -392 1310 -392 ct 1273 -392 1242 -379 1218 -351 ct 1194 -324 1183 -281 1183 -222 ct \n1183 -163 1194 -119 1217 -92 ct 1240 -65 1269 -52 1306 -52 ct 1336 -52 1360 -61 1380 -79 ct \n1400 -97 1412 -125 1418 -163 ct p ef\n1547 0 m  1547 -444 l  1614 -444 l  1614 -377 l  1631 -408 1647 -429 1662 -439 ct \n1677 -449 1693 -454 1710 -454 ct 1736 -454 1761 -446 1787 -430 ct 1762 -360 l \n1743 -371 1725 -376 1706 -376 ct 1690 -376 1675 -371 1662 -361 ct 1649 -352 1640 -338 1634 -320 ct \n1626 -293 1622 -264 1622 -232 ct 1622 0 l  1547 0 l  p ef\n2130 -143 m  2208 -134 l  2195 -88 2173 -53 2139 -28 ct 2106 -3 2064 10 2013 10 ct \n1948 10 1896 -10 1858 -50 ct 1820 -90 1801 -146 1801 -218 ct 1801 -293 1821 -351 1859 -392 ct \n1897 -433 1947 -454 2008 -454 ct 2068 -454 2116 -434 2154 -393 ct 2191 -353 2210 -296 2210 -223 ct \n2210 -219 2210 -212 2210 -203 ct 1879 -203 l  1882 -154 1896 -117 1920 -91 ct \n1945 -65 1976 -52 2013 -52 ct 2041 -52 2064 -59 2084 -74 ct 2103 -88 2118 -111 2130 -143 ct \np\n1883 -265 m  2131 -265 l  2127 -302 2118 -330 2102 -349 ct 2078 -378 2047 -392 2009 -392 ct \n1975 -392 1946 -381 1922 -358 ct 1899 -334 1886 -304 1883 -265 ct p ef\n2454 -67 m  2465 -1 l  2444 4 2425 6 2409 6 ct 2381 6 2360 2 2345 -7 ct 2330 -16 2319 -27 2313 -41 ct \n2307 -55 2304 -85 2304 -130 ct 2304 -385 l  2249 -385 l  2249 -444 l  2304 -444 l \n2304 -554 l  2379 -599 l  2379 -444 l  2454 -444 l  2454 -385 l  2379 -385 l \n2379 -126 l  2379 -104 2380 -90 2383 -84 ct 2385 -78 2390 -73 2396 -70 ct 2402 -66 2410 -64 2421 -64 ct \n2430 -64 2441 -65 2454 -67 ct p ef\n2524 -526 m  2524 -613 l  2599 -613 l  2599 -526 l  2524 -526 l  p\n2524 0 m  2524 -444 l  2599 -444 l  2599 0 l  2524 0 l  p ef\n2671 0 m  2671 -61 l  2953 -384 l  2921 -384 2893 -384 2868 -384 ct 2687 -384 l \n2687 -444 l  3050 -444 l  3050 -394 l  2810 -112 l  2763 -61 l  2797 -64 2829 -65 2858 -65 ct \n3063 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3184 -119 ct 3184 -99 3192 -82 3207 -69 ct \n3222 -56 3244 -49 3273 -49 ct 3302 -49 3328 -55 3351 -68 ct 3373 -81 3390 -98 3400 -120 ct \n3408 -137 3412 -162 3412 -195 ct 3412 -223 l  p ef\n3758 -67 m  3769 -1 l  3748 4 3729 6 3713 6 ct 3685 6 3664 2 3649 -7 ct 3634 -16 3623 -27 3617 -41 ct \n3611 -55 3608 -85 3608 -130 ct 3608 -385 l  3553 -385 l  3553 -444 l  3608 -444 l \n3608 -554 l  3683 -599 l  3683 -444 l  3758 -444 l  3758 -385 l  3683 -385 l \n3683 -126 l  3683 -104 3684 -90 3687 -84 ct 3689 -78 3694 -73 3700 -70 ct 3706 -66 3714 -64 3725 -64 ct \n3734 -64 3745 -65 3758 -67 ct p ef\n3828 -526 m  3828 -613 l  3903 -613 l  3903 -526 l  3828 -526 l  p\n3828 0 m  3828 -444 l  3903 -444 l  3903 0 l  3828 0 l  p ef\n3986 -222 m  3986 -304 4009 -365 4055 -405 ct 4093 -438 4139 -454 4194 -454 ct \n4255 -454 4305 -434 4344 -394 ct 4382 -354 4402 -299 4402 -228 ct 4402 -171 4393 -126 4376 -93 ct \n4359 -61 4334 -35 4301 -17 ct 4269 1 4233 10 4194 10 ct 4132 10 4082 -10 4044 -50 ct 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135 86 158 86 ct 184 86 203 81 217 71 ct \n231 60 240 46 245 28 ct 248 17 249 -7 249 -42 ct 225 -14 195 0 159 0 ct 115 0 81 -16 56 -48 ct \n32 -80 20 -118 20 -163 ct 20 -194 26 -222 37 -248 ct 48 -274 64 -295 85 -309 ct \n106 -323 131 -330 160 -330 ct 198 -330 229 -315 254 -284 ct 254 -323 l  304 -323 l \n304 -44 l  304 7 299 42 289 63 ct 279 84 263 101 240 113 ct 218 125 191 131 159 131 ct \n120 131 89 122 65 105 ct 42 88 30 62 31 27 ct p\n76 -167 m  76 -125 85 -94 101 -74 ct 118 -55 139 -45 165 -45 ct 190 -45 211 -55 228 -74 ct \n245 -94 253 -124 253 -165 ct 253 -205 245 -235 227 -255 ct 210 -275 188 -285 164 -285 ct \n139 -285 119 -275 102 -255 ct 85 -235 76 -206 76 -167 ct p ef\n377 0 m  377 -323 l  427 -323 l  427 -274 l  439 -297 451 -312 461 -319 ct \n472 -326 484 -330 496 -330 ct 515 -330 534 -324 553 -312 ct 534 -262 l  520 -270 507 -273 494 -273 ct \n482 -273 471 -270 462 -263 ct 452 -256 445 -246 441 -233 ct 435 -213 432 -192 432 -169 ct \n432 0 l  377 0 l  p ef\n580 -383 m  580 -446 l  635 -446 l  635 -383 l  580 -383 l  p\n580 0 m  580 -323 l  635 -323 l  635 0 l  580 0 l  p ef\n925 -1 m  925 -41 l  905 -9 875 7 835 7 ct 809 7 786 0 764 -14 ct 743 -28 726 -48 714 -74 ct \n702 -99 696 -128 696 -161 ct 696 -194 702 -223 712 -249 ct 723 -275 739 -295 761 -309 ct \n782 -323 806 -330 833 -330 ct 852 -330 869 -326 885 -318 ct 900 -310 912 -299 922 -286 ct \n922 -446 l  976 -446 l  976 -1 l  925 -1 l  p\n752 -161 m  752 -120 761 -89 779 -69 ct 796 -48 817 -38 840 -38 ct 864 -38 884 -48 901 -67 ct \n918 -87 926 -117 926 -157 ct 926 -201 918 -233 901 -254 ct 884 -275 863 -285 838 -285 ct \n814 -285 793 -275 777 -255 ct 761 -235 752 -204 752 -161 ct p ef\n1272 -1 m  1222 -1 l  1222 -446 l  1276 -446 l  1276 -287 l  1299 -316 1329 -330 1365 -330 ct \n1385 -330 1403 -326 1421 -318 ct 1439 -310 1453 -299 1465 -284 ct 1476 -270 1485 -252 1492 -232 ct \n1498 -211 1501 -190 1501 -166 ct 1501 -111 1488 -68 1460 -38 ct 1433 -8 1400 7 1362 7 ct \n1324 7 1294 -9 1272 -41 ct 1272 -1 l  p\n1272 -164 m  1272 -125 1277 -97 1288 -80 ct 1305 -52 1328 -38 1357 -38 ct 1381 -38 1402 -48 1419 -69 ct \n1437 -90 1446 -121 1446 -162 ct 1446 -204 1437 -235 1420 -255 ct 1404 -275 1384 -285 1360 -285 ct \n1336 -285 1315 -275 1298 -254 ct 1281 -233 1272 -203 1272 -164 ct p ef\n1770 -40 m  1750 -23 1731 -11 1712 -4 ct 1693 3 1673 7 1652 7 ct 1616 7 1589 -2 1570 -19 ct \n1551 -36 1541 -58 1541 -85 ct 1541 -101 1545 -116 1552 -129 ct 1559 -142 1569 -152 1581 -160 ct \n1592 -168 1605 -174 1620 -178 ct 1631 -181 1647 -184 1668 -186 ct 1713 -192 1745 -198 1766 -205 ct \n1766 -213 1766 -217 1766 -219 ct 1766 -242 1761 -257 1751 -266 ct 1737 -279 1716 -285 1688 -285 ct \n1663 -285 1644 -281 1632 -271 ct 1619 -262 1610 -247 1604 -224 ct 1551 -231 l \n1556 -254 1564 -272 1575 -286 ct 1586 -300 1602 -311 1623 -319 ct 1644 -326 1669 -330 1696 -330 ct \n1724 -330 1746 -327 1763 -321 ct 1781 -314 1793 -306 1801 -296 ct 1810 -286 1815 -274 1818 -259 ct \n1820 -250 1821 -233 1821 -208 ct 1821 -136 l  1821 -85 1822 -53 1825 -39 ct \n1827 -26 1832 -13 1839 0 ct 1781 0 l  1776 -12 1772 -25 1770 -40 ct p\n1766 -162 m  1746 -154 1716 -147 1677 -142 ct 1654 -138 1638 -135 1629 -131 ct \n1620 -127 1612 -121 1607 -113 ct 1602 -105 1600 -96 1600 -87 ct 1600 -72 1605 -60 1616 -50 ct \n1627 -41 1644 -36 1665 -36 ct 1686 -36 1705 -40 1721 -50 ct 1737 -59 1749 -72 1757 -88 ct \n1763 -100 1766 -118 1766 -142 ct 1766 -162 l  p ef\n1875 -97 m  1929 -105 l  1932 -83 1941 -67 1955 -55 ct 1968 -44 1988 -38 2013 -38 ct \n2038 -38 2056 -43 2069 -53 ct 2081 -64 2087 -76 2087 -89 ct 2087 -102 2081 -111 2071 -118 ct \n2063 -123 2045 -129 2015 -137 ct 1975 -147 1947 -156 1931 -163 ct 1916 -171 1904 -181 1896 -194 ct \n1888 -207 1884 -221 1884 -237 ct 1884 -251 1888 -265 1894 -277 ct 1901 -289 1910 -299 1921 -308 ct \n1930 -314 1941 -319 1956 -323 ct 1970 -328 1986 -330 2003 -330 ct 2028 -330 2051 -326 2070 -319 ct \n2089 -312 2103 -302 2112 -289 ct 2121 -277 2127 -260 2131 -240 ct 2077 -232 l 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999 -60 1003 -57 ct 1008 -54 1015 -52 1024 -52 ct \n1031 -52 1040 -53 1051 -55 ct p ef\n1357 -117 m  1420 -109 l  1410 -72 1392 -43 1364 -23 ct 1337 -2 1303 8 1261 8 ct \n1208 8 1165 -8 1134 -41 ct 1103 -74 1088 -119 1088 -178 ct 1088 -239 1103 -287 1135 -320 ct \n1166 -354 1207 -371 1257 -371 ct 1306 -371 1345 -355 1376 -322 ct 1407 -289 1422 -242 1422 -182 ct \n1422 -179 1422 -173 1422 -166 ct 1151 -166 l  1153 -126 1165 -96 1185 -74 ct \n1205 -53 1231 -43 1261 -43 ct 1283 -43 1303 -48 1319 -60 ct 1335 -72 1347 -91 1357 -117 ct \np\n1155 -217 m  1357 -217 l  1355 -247 1347 -270 1334 -285 ct 1314 -309 1289 -320 1258 -320 ct \n1230 -320 1206 -311 1187 -292 ct 1167 -273 1157 -248 1155 -217 ct p ef\n1463 -150 m  1463 -212 l  1652 -212 l  1652 -150 l  1463 -150 l  p ef\n1815 0 m  1677 -363 l  1742 -363 l  1820 -146 l  1828 -122 1836 -98 1843 -72 ct \n1848 -92 1856 -115 1866 -142 ct 1947 -363 l  2010 -363 l  1872 0 l  1815 0 l \np ef\n2034 -182 m  2034 -249 2053 -298 2090 -331 ct 2122 -358 2160 -371 2204 -371 ct \n2254 -371 2295 -355 2327 -322 ct 2358 -289 2374 -244 2374 -187 ct 2374 -140 2367 -103 2353 -77 ct \n2339 -50 2319 -29 2292 -14 ct 2265 1 2236 8 2204 8 ct 2154 8 2113 -8 2081 -41 ct \n2050 -73 2034 -120 2034 -182 ct p\n2097 -182 m  2097 -135 2108 -100 2128 -77 ct 2148 -54 2174 -43 2204 -43 ct 2235 -43 2260 -54 2281 -77 ct \n2301 -101 2311 -136 2311 -184 ct 2311 -228 2301 -262 2281 -285 ct 2260 -309 2235 -320 2204 -320 ct \n2174 -320 2148 -309 2128 -286 ct 2108 -263 2097 -228 2097 -182 ct p ef\n2434 0 m  2434 -501 l  2495 -501 l  2495 0 l  2434 0 l  p ef\n2824 0 m  2824 -54 l  2796 -13 2757 8 2709 8 ct 2687 8 2667 4 2649 -4 ct 2630 -13 2617 -23 2608 -35 ct \n2599 -48 2592 -63 2589 -81 ct 2586 -93 2585 -112 2585 -138 ct 2585 -363 l  2646 -363 l \n2646 -162 l  2646 -130 2648 -108 2650 -97 ct 2654 -81 2662 -68 2675 -59 ct 2687 -50 2703 -45 2721 -45 ct \n2740 -45 2757 -50 2773 -59 ct 2789 -69 2801 -81 2807 -98 ct 2814 -114 2818 -138 2818 -169 ct \n2818 -363 l  2879 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704 -184 ct 704 -147 698 -114 684 -84 ct 671 -54 652 -32 626 -16 ct \n601 0 574 8 546 8 ct 526 8 507 4 491 -5 ct 475 -14 461 -25 451 -38 ct 451 139 l \n389 139 l  p\n445 -179 m  445 -133 454 -98 473 -76 ct 492 -54 515 -43 542 -43 ct 569 -43 593 -54 612 -77 ct \n632 -100 641 -136 641 -185 ct 641 -231 632 -265 613 -288 ct 594 -311 571 -323 545 -323 ct \n518 -323 495 -311 475 -286 ct 455 -262 445 -226 445 -179 ct p ef\n1016 -117 m  1079 -109 l  1069 -72 1051 -43 1023 -23 ct 996 -2 962 8 920 8 ct \n867 8 824 -8 793 -41 ct 762 -74 747 -119 747 -178 ct 747 -239 762 -287 794 -320 ct \n825 -354 866 -371 916 -371 ct 965 -371 1004 -355 1035 -322 ct 1066 -289 1081 -242 1081 -182 ct \n1081 -179 1081 -173 1081 -166 ct 810 -166 l  812 -126 824 -96 844 -74 ct 864 -53 890 -43 920 -43 ct \n942 -43 962 -48 978 -60 ct 994 -72 1006 -91 1016 -117 ct p\n814 -217 m  1016 -217 l  1014 -247 1006 -270 993 -285 ct 973 -309 948 -320 917 -320 ct \n889 -320 865 -311 846 -292 ct 826 -273 816 -248 814 -217 ct p 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    "content": "\\label{ch:fv}\n\n\\section{Discretization}\n\nThe physical systems we model are described by continuous mathematical\nfunctions, $f(x,t)$ and their derivatives in space and time.  To\nrepresent this continuous system on a computer we must discretize\nit---convert the continuous function into a discrete number of points\nin space at one or more discrete instances in time.\nThere are many different discretization methods used throughout the\nphysical sciences, engineering, and applied mathematics fields, each\nwith their own strengths and weaknesses.  Broadly speaking, we can\ndivide these methods into grid-based and gridless methods.\n\n%% \\begin{figure}[h]\n%% \\centering\n%% \\includegraphics[width=\\linewidth]{discretizations}\n%% \\caption[Taxonomy of different discretizations]\n%%    {\\label{fig:disc} Taxonomy of different discretizations.}\n%% \\end{figure}\n\nGridless methods include those which represent the function as a\nsuperposition of continuous basis functions (e.g.\\ sines and cosines).\nThis is the fundamental idea behind {\\em spectral methods}.  A different\nclass of methods are those that use discrete particles to represent the\nmass distribution and produce continuous functions by integrating\nover these particles with a suitable kernel---this is the basis of\n{\\em smoothed particle hydrodynamics} (SPH)~\\cite{SPH}.  SPH is a very popular\nmethod in astrophysics.\n\nFor grid-based methods, we talk about both the style of the grid\n(structured vs.\\ unstructured) and the discretization method, e.g.\\ the\nfinite-difference, finite-volume, and finite-element methods.\n\nStructured grids are logically Cartesian.  This means that you can\nreference the location of any cell in the computational domain via an\ninteger index in each spatial dimension.  From a programming\nstandpoint, the grid structure can be represented exactly by a\nmulti-dimensional array.  Unstructured grids don't have this simple\npattern.  A popular type of unstructured grid is created using\ntriangular cells (in 2-d) or tetrahedra (in 3-d).  \\MarginPar{add\n  figure} The main advantage of these grids is that you can easily\nrepresent irregularly-shaped domains.  The disadvantage is that the\ndata structures required to describe the grid are more complicated\nthan a simple array (and tend to have more inefficient memory access).\n\n\\ifdefined\\debugmode\nFinite element methods are\nwidely used in engineering together with unstructured grids.  A type\nof finite-element method called a {\\em Galerkin method} can be formed\nby integrating the continuous function against a compact basis set\nrepresented by tetrahedra.  These methods work well with the irregular\ngeometries that arise in engineering.\n\\fi\n\nOnce a grid is established, the system of PDEs is converted into a\nsystem of discrete equations on the grid.  Finite-difference and\nfinite-volume methods can both be applied to {\\em structured grids}.\nThe main difference between these methods is that the\nfinite-difference methods build from the differential form of PDEs\nwhile the finite-volume methods build from the integral form of the\nPDEs.  The attractiveness of finite-volume methods is that\nconservation is a natural consequence of the discretization---this is\nwhy they are popular in astrophysics.\n\nIn these notes, we will focus on finite-volume techniques on\nstructured grids.\n\n\n\\section{Grid basics}\n\nThe grid is the fundamental object for representing continuous\nfunctions in a discretized fashion, making them amenable to\ncomputation.  In astrophysics, we typically use structured\ngrids---these are logically Cartesian, meaning that the position of a\nquantity on the grid can be specified by a single integer index in\neach dimension.  This works for our types of problems because\nwe don't have irregular geometries---we typically use boxes, disks,\nor spheres.\n\nWe represent derivatives numerically by discretizing the domain into\na finite number of zones (a numerical grid).\nThis converts a continuous derivative into a difference of discrete data.\nDifferent approximations have different levels of accuracy.\n\nThere are two main types of structured grids used in astrophysics:\n{\\em finite-difference} and {\\em finite-volume}.  These differ in way\nthe data is represented.  On a finite-difference grid, the discrete\ndata is associated with a specific point in space.  On a\nfinite-volume grid, the discrete data is represented by averages over\na control volume.  Nevertheless, these methods can often lead to very\nsimilar looking discrete equations.\n\nConsider the set of grids show in Figure~\\ref{fig:grids}.  On the top\nis a classic finite-difference grid.  The discrete data, $f_i$, are\nstored as points regularly spaced in $x$.  With this discretization,\nthe spatial locations of the points are simply $x_i = i \\Delta x$,\nwhere $i = 0, \\ldots, N$\\footnote{When you see $f_{i+1}$, you can\n  think of this as meaning $f((i+1)\\Delta x)$ or $f(x_i + \\Delta x)$}.\nNote that for a finite-sized domain, we would put a grid point\nprecisely on the physical boundary at each end.\n\nThe middle grid is also finite-difference, but now we imagine first\ndividing the domain into $N$ cells or zones, and we store the discrete\ndata, $f_i$, at the center of the zone.  This is often called a {\\em\n  cell-centered finite-difference} grid.  The physical coordinate of\nthe zone centers (where the data lives) are: $x_i = (i + \\myhalf)\\Delta\nx$, where $i = 0, \\ldots, N-1$.  Note that now for a finite-sized\ndomain, the left edge of the first cell will be on the boundary and\nthe first data value will be associated at a point $\\Delta x/2$ inside\nthe boundary.  A similar situation arises at the right physical\nboundary.  Some finite-difference schemes stagger the variables,\ne.g., putting velocity on the boundaries and density at the center.\n\nFinally, the bottom grid is a finite-volume grid.  The layout looks\nidentical to the cell-centered finite difference grid, except now\ninstead of the discrete data being associated at a single point in\nspace, keep track of the total amount of $f$ in the zone (indicated as the shaded regions).  Since we generally don't know how $f$ varies in the zone,\nwe will typically talk about the average of $f$, $\\favg{f}_i$, over\nthe zone, and represent this by a horizontal.  The total amount of $f$ in\nthe zone is then simply $\\Delta x \\favg{f}_i$. \nWe label the left and right edges of a zone with half-integer indices\n$i-\\myhalf$ and $i+\\myhalf$.  The physical coordinate of the center of the zone\nis the same as in the cell-centered finite-difference case.\n\n\n% the scripts to generate these figures are in figures/finite-volume\n% as fd.py, ccfd.py, and fv.py\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{fd_grid} \\\\\n\\includegraphics[width=\\linewidth]{ccfd_grid} \\\\\n\\includegraphics[width=\\linewidth]{fv_grid}\n\\caption[Types of structured grids]{\\label{fig:grids} Different types of structured grids showing\n  the same data.  Top:\n  a finite-difference grid---the discrete data are associated with a\n  specific point in space.  Middle: a cell-centered finite-difference\n  grid---again the data is at a specific point, but now we imagine the\n  domain divided into zone with the data living at the center of each\n  zone.  Bottom: a finite-volume grid---here the domain is divided\n  into zones and we store the average value of the function within\n  each zone.}\n\\end{figure}\n\nIn all cases, for a {\\em regular} structured grid, we take $\\Delta x$\nto be constant.  For the finite-difference grids, the discrete value at\neach point is obtained from the continuous function $f(x)$ as:\n\\begin{equation}\nf_i = f(x_i)\n\\end{equation}\n\n\n\\section{Finite-volume grids}\n\nIn the finite-volume discretization, $f_i$ represents the average of\n$f(x,t)$ over the interval $x_{i-\\myhalf}$ to $x_{i+\\myhalf}$, where\nthe half-integer indices denote the zone edges (i.e.\\ $x_{i-\\myhalf} =\nx_i - \\Delta x/2$):\n\\begin{equation}\n\\label{eq:fv_cc-a}\n\\favg{f}_i = \\frac{1}{\\Delta x} \\int_{x_{i-\\myhalf}}^{x_{i+\\myhalf}} f(x) dx\n\\end{equation}\nThe lower panel of Figure~\\ref{fig:grids} shows a finite-volume grid,\nwith the half-integer indices bounding zone $i$ marked.\n%\n% figure created with figures/finite-volume/simplegrid2.py\n%% \\begin{figure}[t]\n%% \\centering\n%% \\includegraphics[width=\\linewidth]{simplegrid2}\n%% \\caption[A simple 1-d finite-volume grid]{\\label{fig:simplefv} A simple 1-d finite-volume\n%%   grid with the zone-centered and zone-edge indexes.  Here we show\n%%   only the data associated with zone $i$.}\n%% \\end{figure}\n%\nHere we've drawn $\\favg{f}_i$ as a horizontal line spanning the entire zone---this\nis to represent that it is an average within the volume defined by the zone\nedges.\n%\nTo second-order accuracy,\n\\begin{equation}\n\\label{eq:fv_cc}\n\\favg{f}_i = \\frac{1}{\\Delta x} \\int_{x_{i-\\myhalf}}^{x_{i+\\myhalf}} f(x) dx \\sim f(x_i)\n\\end{equation}\n\n\\begin{exercise}[Finite-volume vs.\\ finite-difference centering]\n{Show that Eq.~\\ref{eq:fv_cc} is true to $O(\\Delta x^2)$ by expanding\n$f(x)$ as a Taylor series in the integral, e.g., as:\n\\begin{align}\nf(x) = f(x_i) &+ f^\\prime(x_i) (x - x_i) + \\frac{1}{2} f^{\\prime\\prime}(x_i) (x - x_i)^2 \\nonumber \\\\\n%\n &+ \\frac{1}{6} f^{\\prime\\prime\\prime}(x_i) (x - x_i)^3 + \\ldots\n\\end{align}\n}\n\\end{exercise}\n\nThis means that we can treat the average of $f$ over a zone as simply\n$f(x)$ evaluated at the zone center if we only want second-order\naccuracy.  Using the subscript notation, we can express the average of\nthe zone to the right as $\\favg{f}_{i+1}$.\n\nWhen we want to interpolate data on a finite-volume grid, we want to\nconstruct an interpolating polynomial that is conservative.  A {\\em\n  conservative interpolant} reconstructs a continuous functional form,\n$f(x)$, from a collection of cell-averages subject to the requirement\nthat when $f(x)$ is averaged over a cell, it returns the original\ncell-average.\n\n\\begin{exercise}[Conservative interpolation]\n{\\label{ex:consinterp}\n\nConsider three cell averages: $\\favg{f}_{i-1}$, $\\favg{f}_{i}$, $\\favg{f}_{i+1}$.  Fit a quadratic polynomial through these points,\n \\begin{equation}\n f(x) = A (x - x_i)^2 + B (x - x_i) + C\n \\end{equation}\n where the coefficients, $A$, $B$, and $C$ are found by applying the constraints:\n \\begin{align}\n \\favg{f}_{i-1} &= \\frac{1}{\\Delta x}\n      \\int_{x_{i-\\mythreehalf}}^{x_{i-\\myhalf}} f(x) dx \\\\\n \\favg{f}_{i} &= \\frac{1}{\\Delta x}\n      \\int_{x_{i-\\myhalf}}^{x_{i+\\myhalf}} f(x) dx \\\\\n \\favg{f}_{i+1} &= \\frac{1}{\\Delta x}\n      \\int_{x_{i+\\myhalf}}^{x_{i+\\mythreehalf}} f(x) dx\n \\end{align}\n\nShow that the conservative interpolant is:\n\\begin{align}\nf(x) = &\\frac{\\favg{f}_{i-1} - 2 \\favg{f}_i +\n             \\favg{f}_{i+1}}{2\\Delta x^2} (x-x_i)^2 + \\nonumber \\\\\n       &\\frac{\\favg{f}_{i+1} - \\favg{f}_{i-1}}\n            {2\\Delta x} (x-x_i) + \\nonumber \\\\\n       &\\frac{-\\favg{f}_{i-1} + 26 \\favg{f}_i\n             -\\favg{f}_{i+1}}{24}\n\\end{align}\n}\n\\end{exercise}\n\nThe {\\sf Jupyter} notebook\n\\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/finite-volume/conservative-interpolation.ipynb}{conservative-interpolation.ipynb}}\nshows how to derive these interpolants using {\\sf SymPy}, and gives\nhigher-order interpolants.\n\n\n\\subsection{Differences and order of accuracy}\n\nIn practice, when computing derivatives in a finite-volume\ndiscretization, we can use the second-order centered difference from\n\\S~\\ref{ch:intro:diff} treating the finite-volume data as living at the\ncell-centers and still be second-order accurate.  For higher accuracy,\nwe can fit a {\\em conservative interpolant} (as illustrated in\nexercise~\\ref{ex:consinterp}) to a collection of points and then\ndifferentiate the interpolant itself.  \\MarginPar{show convergence plot}\n\nNotice that the righthand side of all derivative approximations involve\nsome linear combinations of $f_i$'s.  We call this the {\\em stencil}.\nThe {\\em width} of the stencil is a measure of how many zones on\neither side of zone $i$ we reach when computing our approximation.\n\nFor example, a second derivative can be discretized as:\n\\begin{equation}\n\\left . \\frac{d^2 f}{dx^2} \\right |_i = \\frac{f_{i+1} - 2 f_i + f_{i-1}}{\\Delta x^2}\n\\end{equation}\nThe stencil on the righthand side encompasses 3 zones.\n\n\\subsection{Conservation}\n\nThe finite-volume grid is useful when dealing with conservation laws.\nConsider the following system:\n\\begin{equation}\n\\frac{\\partial \\Uc}{\\partial t} + \\nabla \\cdot \\Fb(\\Uc) = 0\n\\end{equation}\nwhere $\\Uc$ is a vector of conserved quantities and $\\Fb(\\Uc)$ is the flux\nof each quantity.  This type of system arises, for example, in fluid\nflow, where the system will represent conservation of mass, momentum,\nand energy.\n\nConsider one-dimension.\nIntegrating this system over a zone, and normalizing by $\\Delta x$, we have:\n\\begin{equation}\n\\frac{1}{\\Delta x} \\int_{x_{i-\\myhalf}}^{x_{i+\\myhalf}} \\ddt{\\Uc} dx =\n  -\\frac{1}{\\Delta x} \\int_{x_{i-\\myhalf}}^{x_{i+\\myhalf}} \\frac{\\partial \\Fb}{\\partial x} dx\n\\end{equation}\nOn the left, we can take the time derivative out of the integral, and\nwe are left with the definition of a zone average, so this becomes\nsimply $\\partial \\favg{\\Uc}_i/\\partial t$.  On the right, we\napply the divergence theorem, giving:\n\\begin{equation}\n\\frac{\\partial \\favg{\\Uc}_i}{\\partial t} =\n  - \\frac{1}{\\Delta x} \\left \\{ \\left . \\Fb(\\Uc) \\right |_{x_{i+\\myhalf}} -\n                                \\left . \\Fb(\\Uc) \\right |_{x_{i-\\myhalf}} \\right \\}\n\\end{equation}\nIndependent of how we discretize in time, notice that we have the cell-average\non the left and that the righthand side\nis simply a difference of fluxes through the interfaces of the zone.\nFor the $i+1$ zone, the update would be:\n\\begin{equation}\n\\label{eq:consup}\n\\frac{\\partial \\favg{\\Uc}_{i+1}}{\\partial t} =\n  - \\frac{1}{\\Delta x} \\left \\{ \\left . \\Fb(\\Uc) \\right |_{x_{i+\\mythreehalf}} -\n                                \\left . \\Fb(\\Uc) \\right |_{x_{i+\\myhalf}} \\right \\}\n\\end{equation}\nNotice that this shares the flux at the $x_{i+\\myhalf}$ interface, but with the\nopposite sign.  \\MarginPar{illustrate this} When all of the updates are done, the flux through each\nboundary adds to one zone and subtracts from its neighbor, exactly conserving\n(to round-off error) the quantity $\\Uc$.  This cancellation of the sums\nis an example of a {\\em telescoping property}, and is the main reason\nwhy finite-volume methods are attractive---conserved quantities are\nconserved to machine (roundoff) precision.\n\nNote that conservation is not the same as accuracy.  We can construct\nthe fluxes for our discretized equation by calling a random number\ngenerator and we'd still be conservative, but not at all accurate.\n\n\\subsection{Boundary conditions with finite-volume grids}\n\n\\label{sec:fv:bcs}\n\nImagine that we wish to compute the derivative in every zone using:\n\\begin{equation}\n\\left . \\frac{\\partial f}{\\partial x} \\right |_i = \\frac{f_i - f_{i-1}}{\\Delta x} \\enskip .\n\\label{eq:onesideddiff}\n\\end{equation}\nIf we denote the index corresponding to the leftmost zone in our\ndomain as `lo', then when we try to compute ${\\partial f}/{\\partial x}\n|_\\mathrm{lo}$, we will ``fall-off'' the grid, i.e., we need a value\nof $f$ for zone $\\mathrm{lo}-1$, which is outside the domain.  This is\nwhere boundary conditions for our grid come into play.\n\nWe implement boundary conditions by extending the computational grid\nbeyond the physical domain (see Figure~\\ref{fig:fv_gc}).  These\nadditional zones are called {\\em ghost cells}.  They exist solely to\nhandle the boundary conditions and allow us to use the same update\nequation (e.g.\\ Eq.~\\ref{eq:onesideddiff}) for all zones in the\ndomain.\n\n% figure created with simplegrid-gc.py\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{simplegrid_gc}\n\\caption[A simple 1-d finite-volume grid with ghost cells]\n        {\\label{fig:fv_gc} A simple 1-d finite-volume grid with a\n          single ghost cell at each end of the domain.  The domain\n          boundaries are indicated by the heavy vertical lines.  Here\n          there are $\\mathrm{hi}-\\mathrm{lo}+1$ zones used in the\n          discretization of the domain, with the first zone in the\n          domain labeled $\\mathrm{lo}$ and the last in the domain\n          labeled $\\mathrm{hi}$.}\n\\end{figure}\n\nThe number of ghostcells needed for the grid depends on how wide the\nstencils used in the computation are.  The wider the stencil, the more\nghostcells are needed.\n\nPeriodic boundary conditions would be implemented as:\n\\begin{eqnarray}\nf_{\\mathrm{hi}+1} &=& f_\\mathrm{lo} \\\\\nf_{\\mathrm{lo}-1} &=& f_\\mathrm{hi}\n\\end{eqnarray}\nA simple outflow (zero-gradient) boundary condition would be implemented as:\n\\begin{eqnarray}\nf_{\\mathrm{hi}+1} &=& f_\\mathrm{hi} \\\\\nf_{\\mathrm{lo}-1} &=& f_\\mathrm{lo}\n\\end{eqnarray}\n\n\n\\section{Numerical implementation details}\n\nThe computational grids used for finite-volume techniques maps nicely\nonto multidimensional arrays.  For the cell-average data, a typical\narray would be dimensioned as\n\\begin{lstlisting}[language=Python]\n  ilo = ng\n  ihi = nx + ng - 1\n  state = np.zeros((nx + 2*ng, nvar))\n\\end{lstlisting}\nin python (using NumPy), where {\\tt nx} is the number of zones in the \ndomain and {\\tt ng} is the number of ghost cells.  For some problems,\nthere might be more than one state variable, so {\\tt nvar}, is the number\nof state variables that live in each zone.  We would then loop from {\\tt ilo}\nto {\\tt ihi} to access the state data in the grid interior.\n\nFluxes are also stored in arrays, but since we cannot index an array with\nhalf-integer indices, the standard convention is that the index {\\tt i} in an\narray refers to the flux on the {\\em left} edge of that zone.  I.e.,\n\\begin{equation*}\n\\mathtt{flux[i]} \\quad \\longleftrightarrow \\quad F_{i-1/2}\n\\end{equation*}\n\n\\section{Going further}\n\n\\begin{itemize}\n\n\\item {\\em Domain decomposition}: when running on a parallel computer,\n the work is divided up across processor using domain decomposition.\n Here, we break the computational domain into smaller sub-domains, and\n put one (or more) sub-domains on each processor.  Each sub-domain\n has its own perimeter of ghost cells.  These are now filled by copying\n information from the neighboring sub-domains or using the physical\n boundary conditions for the full domain, depending on where the\n ghost cells lie.  Figure~\\ref{fig:domain} shows a simple decomposition of\n a domain into 6 sub-domains.\n\n % create with figures/finite-volume/domain.py\n \\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{domain}\n \\caption[Domain decomposition example] {\\label{fig:domain} Domain\n   decomposition of the computational domain into 6 separate\n   sub-domains.  Each sub-domain here has $5\\times 5$ zones.  For one\n   of the sub-domains, the perimeter of ghost cells is illustrated as\n   the red boundary.}\n \\end{figure}\n\n\\item {\\em AMR for structured grids}: adaptive mesh refinement uses a\n  hierarchy of grids to focus resolution in regions of complex flow.\n  For finite-volume codes, the standard reference for AMR is Berger \\&\n  Colella~\\cite{berger-colella}.  Each level is an even integer\n  multiple finer in resolution, and the grid cells line up with one\n  another (i.e.\\ in two-dimensions, four fine cells will be completely\n  enclosed by a single coarse cell, when using a jump in resolution of\n  2$\\times$.)  This provides a natural way to enforce conservation.\n  At coarse-fine interfaces, corrections are done to ensure\n  consistency.\n\n\\item {\\em Mapped grids}: mapped grids are (usually) logically\n  Cartesian grids that use a function (map) to transform from\n  a rectangular mesh (that corresponds to the array in memory)\n  to a general curvilinear grid.  These maintain the performance\n  characteristics of structured grids, since any zone can be\n  directly indexed with an integer for each spatial dimension. \\MarginPar{reference the Calhoun paper}\n\n\\item {\\em Lagrangian grids}: a hybrid of particle and grid methods is\n  provided by methods that move particles in a Lagrangian fashion and\n  use a Voronoi tessellation of these particles to define the grid\n  that finite-volume methods are applied to.  See, for example,\n  the Arepo code paper~\\cite{arepo}.\n\\end{itemize}\n"
  },
  {
    "path": "finite-volume/main.tex",
    "content": "\\documentclass[11pt]{article}\n\n% margins\n\\usepackage[margin=0.75in]{geometry}\n\n% figures\n\\usepackage{graphicx}\n\n% font\n\\usepackage{mathpazo}\n\n\\usepackage{helvet}\n\n% coloring\n\\usepackage{color}\n\\definecolor{mygray}{gray}{0.5}\n\n% footer\n\\usepackage{fancyhdr}\n\\pagestyle{fancy}\n\\fancyfoot[LO,LE]{\\footnotesize\\sffamily \\color{mygray} M.\\ Zingale---Notes on Grids}\n\\fancyfoot[RO,RE]{\\footnotesize \\sffamily \\color{mygray} (\\today)}\n\\fancyfoot[CO,CE]{\\thepage}\n\\fancyhead{}\n\\renewcommand{\\headrulewidth}{0.0pt}\n\\renewcommand{\\footrulewidth}{0.0pt}\n\n% captions\n\\usepackage{caption}\n\\renewcommand{\\captionfont}{\\footnotesize}\n\\renewcommand{\\captionlabelfont}{\\footnotesize}\n\\setlength{\\captionmargin}{5em}\n\n\n\\usepackage{sectsty}\n\\allsectionsfont{\\sffamily}\n\n\\newcounter{exercise}\n\\usepackage{changepage}\n\n\\newenvironment{exercise}% environment name \n{% begin code \n  \\begin{adjustwidth}{1.0cm}{0.5cm}\n  \\par\\vspace{\\baselineskip}\\noindent \n  \\refstepcounter{exercise}%\n  \\textbf{Exercise \\theexercise\\ }\\begin{itshape}% \n  %\\par\\vspace{\\baselineskip}\\noindent\\ignorespaces \n}% \n{% end code \n  \\end{itshape}\\end{adjustwidth}\\vspace{\\baselineskip}\\ignorespacesafterend \n}\n\n\\begin{document}\n\n\\begin{center}\n{\\LARGE \\textsf{\\textbf{\nNotes on Grids}}\n}\n\\end{center}\n\n\\input finite-volume\n\n\\bibliographystyle{plain}\n\\bibliography{../refs}\n\n\\end{document}\n"
  },
  {
    "path": "finite-volume/old-scripts/domain-1d.py",
    "content": "import pylab\nimport numpy\n\n\ndef drawBox(ll, uu, nx, ny, gridColor=\"0.5\", ng=0):\n\n    # draw the frame\n    pylab.plot([ll[0], ll[0], uu[0], uu[0], ll[0]],\n               [ll[1], uu[1], uu[1], ll[1], ll[1]], color=\"k\", lw=2)\n\n    # draw the x grid lines\n    dx = (uu[0] - ll[0])/nx\n    n = 1\n    while (n < nx):\n        pylab.plot([ll[0]+n*dx, ll[0]+n*dx],\n                   [ll[1], uu[1]], color=gridColor, ls=\":\")\n\n        n += 1\n\n    # draw the y grid lines\n    dy = (uu[1] - ll[1])/ny\n    n = 1\n    while (n < ny):\n        pylab.plot([ll[0], uu[0]],\n                   [ll[1]+n*dy, ll[1]+n*dy], color=gridColor, ls=\":\")\n\n        n += 1\n    \n    # ghostcells?  \n    if (ng > 0):\n        print \"here\"\n        xmin = ll[0]-ng*dx\n        xmax = uu[0]+ng*dx\n        ymin = ll[1]-ng*dy\n        ymax = uu[1]+ng*dy\n        pylab.plot([xmin, xmin, xmax, xmax, xmin], \n                   [ymin, ymax, ymax, ymin, ymin], \n                   ls=\"--\", color=\"r\")\n\n\npylab.clf()\n\ndrawBox([0., 0.], [1., 6.], 5, 30)\ndrawBox([1., 0.], [2., 6.], 5, 30)\ndrawBox([2., 0.], [3., 6.], 5, 30)\ndrawBox([3., 0.], [4., 6.], 5, 30, ng=1)\ndrawBox([4., 0.], [5., 6.], 5, 30)\ndrawBox([5., 0.], [6., 6.], 5, 30)\n\npylab.xlim(-0.25,6.25)\npylab.ylim(-0.25,6.25)\n\na = pylab.gca()\na.set_aspect(\"equal\", \"datalim\")\npylab.axis(\"off\")\n\nf = pylab.gcf()\nf.set_size_inches(7.0,7.0)\n\npylab.tight_layout()\n\npylab.savefig(\"domain_1d.png\")\npylab.savefig(\"domain_1d.eps\", bbox_inches=\"tight\")\n\n"
  },
  {
    "path": "finite-volume/old-scripts/simplegrid.py",
    "content": "import math\nimport numpy\nimport pylab\nimport grid_plot_util as gpu\n\ndef simplegrid():\n\n    # grid info\n    nzones = 7\n    ng = 1\n\n    gr = gpu.grid(nzones, ng, xmin=0.0, xmax=1.0)\n\n\n    #------------------------------------------------------------------------\n    gpu.drawGrid(gr, emphasizeEnd=1, edgeTicks=0)\n\n\n    # label a few\n    gpu.labelCenter(gr, ng+nzones/2, r\"$i$\")\n    gpu.labelCenter(gr, ng+nzones/2-1, r\"$i-1$\")\n    gpu.labelCenter(gr, ng+nzones/2+1, r\"$i+1$\")\n\n\n    pylab.xlim(gr.xl[0]-0.5*gr.dx,gr.xr[2*ng+nzones-1]+0.5*gr.dx)\n\n    # label dx\n    pylab.plot([gr.xl[ng+nzones/2-2], gr.xl[ng+nzones/2-2]], \n               [-0.35,-0.25], color=\"k\")\n\n    pylab.plot([gr.xr[ng+nzones/2-2], gr.xr[ng+nzones/2-2]], \n               [-0.35,-0.25], color=\"k\")\n\n    pylab.plot([gr.xl[ng+nzones/2-2], gr.xr[ng+nzones/2-2]], \n               [-0.3,-0.3], color=\"k\")\n\n    pylab.text(gr.xc[ng+nzones/2-2], -0.5, r\"$\\Delta x$\", \n               horizontalalignment=\"center\")\n\n    pylab.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95)\n\n    pylab.ylim(-0.5, 1.5)\n    pylab.axis(\"off\")\n\n\n    f = pylab.gcf()\n    f.set_size_inches(10.0,2.5)\n\n\n    pylab.savefig(\"simplegrid.png\")\n    pylab.savefig(\"simplegrid.eps\")\n               \nif __name__== \"__main__\":\n    simplegrid()\n"
  },
  {
    "path": "gravity/gravity.tex",
    "content": "\\section{Self-gravity and hydrodynamics}\n\n\\section{Monopole approximation}\n\n -- radial sampling\n -- time-centering\n -- predictor\n -- RK formalism (solve each stage)\n -- what to use as center?\n\n\\section{Multipole expansions}\n\n\n\\section{Full Poisson gravity}\n\n -- isolated boundaries\n\n\\subsection{Special case: triply periodic and FFTs}\n\n\n\\section{Conservative formulations}\n\n\n\n\n"
  },
  {
    "path": "higher-order/higher-order-burgers.tex",
    "content": "\n\\section{WENO methods, nonlinear equations, and flux-splitting}\n\\label{sec:weno-burgers-fvs}\n\n%\\todo{Also note that I haven't followed the notation of the rest of the notes as I was too lazy to work out what it was.}\n\nFor a nonlinear scalar conservation law\n\\begin{equation}\n  \\label{eq:scalar_conslaw}\n  u_t + f(u)_x = 0\n\\end{equation}\nthe WENO method introduced in subsection~\\ref{sec:WENO} above does not work,\nas we have assumed the characteristic\ninformation travels in one direction only. We have also reconstructed the\nvariable $q$ and from that constructed the ``flux'' which we feed into the\ndifferencing formula of equation~\\eqref{sec:ho-fd} to update the solution: this\nwill not work either.\n\nIn the nonlinear case, we instead reconstruct the \\emph{flux} directly. That\nis, given the state $u_i$ at location $x_i$, we compute the flux $f_i = f(u_i)$\nand reconstruct the flux $f_{i+\\myhalf}$ using the WENO reconstruction above.\n\nThis has the significant problem that we have no characteristic information for\nthe flux: does it propagate to the left or to the right? To get around this we\nintroduce \\emph{flux splitting}. We write\n\\begin{equation}\n  \\label{eq:flux-splitting}\n  f(u) = f^{(+)}(u) + f^{(-)}(u)\n\\end{equation}\nwhere we choose the functions such that the information contained in $f^{(+)}$\npropagates to the right and that in $f^{(-)}$ propagates to the left. There are\nmany ways of doing this, but the simplest is the \\emph{Lax-Friedrichs} flux\nsplitting\n\\begin{equation}\n  \\label{eq:lf-flux-split}\n  f^{(\\pm)}(u) = \\tfrac{1}{2} \\left( f(u) \\pm \\alpha u \\right)\n\\end{equation}\nwhere $\\alpha \\ge \\max | \\partial_u f |$. With $\\alpha$ being larger than the maximum propagation speed, this means that\n\\begin{equation}\n  \\label{eq:lf-flux-split-speeds}\n  \\partial_u f^{(+)} \\ge 0, \\quad \\partial_u f^{(-)} \\le 0,\n\\end{equation}\nand the characteristic information contained in $f^{(\\pm)}$ will propagate as required.\n\nThe simplest flux-split algorithm is then\n\\begin{enumerate}\n  \\item Compute the maximum characteristic speed $\\alpha$ over the entire computational grid;\n  \\item Compute the split fluxes $f^{(\\pm)}$ from equation~\\eqref{eq:lf-flux-split};\n  \\item Reconstruct $f^{(+)}$ to the right using, e.g., the WENO algorithm of section~\\ref{sec:WENO};\n  \\item Reconstruct $f^{(-)}$ to the left using the same method;\n  \\item Compute $f_{i+\\myhalf} = f^{(+)}_{i+\\myhalf} + f^{(-)}_{i+\\myhalf}$;\n  \\item Update the state using equation~\\eqref{eq:consup}.\n\\end{enumerate}\n\n\\begin{figure}[t]\n\\centering\n% figure generated by hydro_examples/advection/weno.py\n\\includegraphics[width=0.8\\linewidth]{weno-vs-plm-burger}\n\\caption[Comparing PLM and WENO methods for Burgers' equation]\n{\\label{fig:weno-vs-plm-burger} Numerical solutions to Burgers' equation using the piecewise linear method as in figure~\\ref{fig:burgers-shock} and two flux-split WENO methods as outlined in section~\\ref{sec:weno-burgers-fvs}. This uses the same sinusoidal data as figure~\\ref{fig:burgers-shock} shortly after shock formation. Note that the WENO methods are not obviously any better at resolving the shock than the simpler piecewise linear approach.\\\\\n\\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/advection/weno_burgers.py}{weno\\_burgers.py}}}\n\\end{figure}\n%\n\nAs a direct comparison between the piecewise linear method used at the start of\nthis chapter, and the more complex WENO methods introduced here, we start with\nlooking at a shock forming from smooth initial data. Figure~\\ref{fig:weno-vs-plm-burger}\nshows the evolution of the sinusoidal initial data as used in figure~\\ref{fig:burgers-shock},\nbut only showing the result at one time, shortly after characteristic crossing\nleads to a shock. We see that there is very little difference between the\ndifferent numerical methods. The characteristic focusing means that all the\nmethods have a similar number of points within the shock. The piecewise linear\nmethod appears to capture the edges of the shock slightly more accurate than the\n(higher order) WENO method with $r=3$. This is likely as the piecewise linear\nmethod is using the exact solution to the Riemann Problem, whilst the WENO method\nis using the more diffusive Lax-Friedrichs flux splitting. At the higher $r=5$\norder there is a marginal advantage to the WENO method.\n\n\\begin{figure}[t]\n\\centering\n% figure generated by hydro_examples/advection/weno.py\n\\includegraphics[width=0.8\\linewidth]{weno-converge-burgers}\n\\caption[WENO convergence rates for Burgers' equation]\n{\\label{fig:weno-converge-burgers} WENO solutions from evolving Burgers'\nequation for an initial Gaussian profile, stopping before the solution is\ndiscontinuous. A fourth order Runge-Kutta method is used for time evolution.\nFor $r=3$ we see the expected fifth ($2 r - 1$) order convergence.\nFor $r=5$ we see seventh order convergence briefly, before the error from the\ntime integrator dominates. \\\\\n\\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/advection/weno_burgers.py}{weno\\_burgers.py}}}\n\\end{figure}\n%\n\nThe advantages of the WENO method arise when the profile is still continuous, as\nin the rarefaction problem, or at times before shocks have formed. To confirm\nthat high-order convergence is still possible for a nonlinear problem, an\ninitially Gaussian profile is evolved for a short time and compared to the\nexact solution (computed from characteristic tracing). The resulting error\nconvergence is shown in figure~\\ref{fig:weno-converge-burgers}. We see the\nexpected fifth and seventh order convergence rates, at least for some of the\nresolutions. At too low a resolution the higher order terms in the error\nexpansion affect the solution. At too high a resolution, for the higher-order\nmethod, the time integration error starts to dominate.\n"
  },
  {
    "path": "higher-order/higher-order-euler.tex",
    "content": "\n\\section{WENO methods for the Euler equations}\n\\label{sec:weno-euler}\n\nWhen dealing with nonlinear systems the flux split method introduced in\nsection~\\ref{sec:weno-burgers-fvs} generalizes directly. The simplest, and most\ndiffusive flux splitting uses the maximum characteristic speed $\\alpha$. For a\nsystem, such as the Euler equations, the simplest method is to compute $\\alpha$\nby maximizing over all characteristics. The directional fluxes $\\Fb^{(\\pm)}(u)$\ncan then be reconstructed using a high order WENO scheme component by component.\n\nHowever, this approach can lead to significant oscillations, even in moderate\ntests. Rather than doing the reconstruction of the components of the directional\nfluxes, a safer approach is to work with characteristic variables. The\ndiscussion in chapter~\\ref{ch:compressible-theory} suggests that, by computing\nthe left and right eigenvectors of the appropriate Jacobian matrix\n$\\Ab = \\partial \\Fb/\\partial \\Uc$, we can project the directional fluxes into\nthe directional \\emph{characteristic} fluxes (using the left eigenvectors),\nreconstruct these characteristic fluxes using a high order method, and then\ncompute the reconstructed directional fluxes required (using the right\neigenvectors). As the characteristic fluxes should contain information about\nonly a single wave at a time, this \\emph{should} give similar results to the\nscalar case and reduce problems from close waves from different families.\n\nThe Jacobian matrix needs to be computed separately for each point at which the\nflux needs computing. The best choice of state from which the Jacobian is\ncomputed is not clear: often the arithmetic average of neighbouring states is\nused, but more complex choices can give better results.\n\nA direct comparison of component-wise and characteristic-wise flux-vector split\nWENO methods, applied to the Sod test, is given in\nfigure~\\ref{fig:weno-euler-r3} for $r=3$ and in figure~\\ref{fig:weno-euler-r5}\nfor $r=5$. The comparison to the PPM method in figure~\\ref{fig:Euler:sod:ppm} is\nparticularly instructive. The PPM method is specifically designed for\nhydrodynamic problems, and is much better at cleanly capturing the\ndiscontinuities, especially the contact. The WENO approach will have a higher\norder of accuracy, which will be clearest on tests with more smooth variation.\nThe oscillations introduced by the WENO methods are more pronounced as the\norder is increased, but are reduced by the use of characteristic-wise\nreconstruction. This is most clearly seen in the results for the internal energy\nin the $r=5$ case shown in figure~\\ref{fig:weno-euler-r5}.\n\n\\begin{figure}[t]\n\\centering\n% figure generated by hydro_examples/compressible/weno-euler.py\n\\includegraphics[width=0.8\\linewidth]{weno-euler-r3}\n\\caption[WENO $r=3$ for the Sod test]\n{\\label{fig:weno-euler-r3} The Sod test solved with WENO methods, $r=3$, using characteristic-wise and component-wise reconstruction, using 64 zones and a cfl of $0.5$. To compare with Godunov's method see figure~\\ref{fig:Euler:sod:god} and with PPM see figure~\\ref{fig:Euler:sod:ppm}. The WENO method does not capture discontinuities, especially the contact, as well as the PPM method which is specifically designed for the Euler equations. The small oscillations visible in the component-wise reconstruction are damped by using the more complex and expensive characteristic-wise approach. \\\\\n\\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/compressible/weno_euler.py}{weno\\_euler.py}}}\n\\end{figure}\n%\n\n\\begin{figure}[t]\n\\centering\n% figure generated by hydro_examples/compressible/weno-euler.py\n\\includegraphics[width=0.8\\linewidth]{weno-euler-r5}\n\\caption[WENO $r=5$ for the Sod test]\n{\\label{fig:weno-euler-r5} The Sod test solved with WENO methods, $r=5$, using characteristic-wise and component-wise reconstruction, using 64 zones and a cfl of $0.5$. Compare with figure~\\ref{fig:weno-euler-r3} to see the effect of the higher order of WENO scheme. The oscillations in the component-wise approach are much more pronounced as the order of the reconstruction is increased, and the characteristic-wise approach continues to help with this. \\\\\n\\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/compressible/weno_euler.py}{weno\\_euler.py}}}\n\\end{figure}\n%\n\nAnother illustration of the advantages of a numerical method, such as PPM, that\nis specifically designed for the Euler equations, is shown in the double\nrarefaction test in figure~\\ref{fig:weno-euler-rarefaction-r3}. The\nadvantages of the higher order methods are shown by how well the WENO schemes\ncapture the edges of the rarefaction waves, even with so few points and the\ndiffusive Lax-Friedrichs flux splitting. However, the artificial heating effect\nseen at the trivial contact at the center of the domain is about as bad as that\nfrom piecewise constant reconstruction, and nowhere near as good as PPM. Whilst\nincreasing the reconstruction order has a small impact, a less diffusive flux\nsplitting would be needed to approach PPM's performance.\n\n\\begin{figure}[t]\n\\centering\n% figure generated by hydro_examples/compressible/weno-euler.py\n\\includegraphics[width=0.8\\linewidth]{weno-euler-rarefaction-r3}\n\\caption[WENO $r=3$ for the double rarefaction test]\n{\\label{fig:weno-euler-rarefaction-r3} The double rarefaction test solved with WENO methods, $r=3$, using characteristic-wise and component-wise reconstruction, using 64 zones and a cfl of $0.5$. To compare with Godunov's method see figure~\\ref{fig:Euler:doublerare:god} and with PPM see figure~\\ref{fig:Euler:doublerare:ppm}. The WENO method captures the edges of the rarefactions almost as well as the PPM method, but is as almost as poor as Godunov's method at the trivial contact at the center. A less diffusive flux-splitting may help here. Increasing the reconstruction order has little effect.\\\\\n\\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/compressible/weno_euler.py}{weno\\_euler.py}}}\n\\end{figure}\n%\n\n\\subsection{Extensions}\n\nThe main advantage of the WENO method is that it retains its high order when\nextending to multiple dimensions using dimensional splitting. From the\nfinite-difference form there are no transverse Riemann problems to solve. From\nusing the Method of Lines there is no issue about ordering the dimensional\nsweeps: the updates in each direction are computed separately but applied\ntogether in the time integrator. This formally retains the high-order accuracy\nbut can lose significant absolute accuracy -- compare the advection of a\ntop-hat function with lower order methods using transverse Riemann problem\nsolutions.\n\nThe global Lax-Friedrichs flux splitting above can be excessively diffusive. A\nlocal Lax-Friedrichs flux splitting, where $\\alpha$ is computed at each point\nby maximising over the characteristic speed within the stencil at that point, is\nless diffusive but slightly more expensive. Roe-style flux splittings are\npossible but not always stable.\n"
  },
  {
    "path": "higher-order/weno-coefficients.ipynb",
    "content": "{\n \"cells\": [\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"# WENO coefficients\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"The starting point for a WENO code is a standard Lagrange interpolating polynomial.\\n\",\n    \"\\n\",\n    \"Assume we're trying to reconstruct the function $q(x)$, and that we're given the cell-centred information $q_i = \\\\langle q \\\\rangle_i$ and the cells $I_i$ centred at $x_i$, where all the cells have width $\\\\Delta x$. Then we want to use the stencil $I_{i-r+k+1}, \\\\dots I_{i+k}$ which uses $r$ cells to construct the Lagrange interpolation polynomial of order $r$. This polynomial is sticks out the right by $k$ cells.\\n\",\n    \"\\n\",\n    \"The Lagrange interpolation polynomial $p_{r, k}(x)$ is given by (equation (2.11) of Shu's review with notation and order altered)\\n\",\n    \"$$\\n\",\n    \"  p_{r, k}(x) = \\\\sum_{m=0}^r \\\\sum_{j=0}^{m-1} q_{i-r+k+1+j} \\\\frac{1}{\\\\Delta x^r} \\\\frac{\\\\sum_{l=0, l \\\\ne m}^r \\\\prod_{s=0, s \\\\ne l, m}^r \\\\left( x - x_{i-r+k+1+s+1/2} \\\\right)}{\\\\prod_{l=0, l \\\\ne m}^r \\\\left( m - l \\\\right)}.\\n\",\n    \"$$\\n\",\n    \"\\n\",\n    \"When explicitly evaluated at $x_{i+1/2}$ we get (equation (2.19) of Shu's review, again altered to match Gerolymos et al)\\n\",\n    \"$$\\n\",\n    \"  p_{r, k, i+1/2} = \\\\sum_{l=0}^{r-1} \\\\left( \\\\sum_{j=r-l}^r \\\\frac{\\\\sum_{m=0, m \\\\ne j}^r \\\\prod_{s=0, s \\\\ne j, m}^r \\\\left( r - k - s \\\\right)}{\\\\prod_{m=0, m \\\\ne j}^r \\\\left( j - m \\\\right)} \\\\right) q_{i+k-l} .\\n\",\n    \"$$\\n\",\n    \"\\n\",\n    \"Thus we write\\n\",\n    \"$$\\n\",\n    \"  q_{r, k, i+1/2} = \\\\sum_{l=0}^{r-1} a_{r, k, l} q_{i+k-l}\\n\",\n    \"$$\\n\",\n    \"where the constants $a_{r, k, l}$ are given by\\n\",\n    \"$$\\n\",\n    \"  a_{r, k, l} = \\\\sum_{j=r-l}^{r} \\\\frac{\\\\sum_{m=0, m \\\\ne j}^r \\\\prod_{s=0, s \\\\ne j, m}^r \\\\left( r - k - s \\\\right)}{\\\\prod_{m=0, m \\\\ne j}^r \\\\left( j - m \\\\right)}.\\n\",\n    \"$$\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 1,\n   \"metadata\": {\n    \"collapsed\": true\n   },\n   \"outputs\": [],\n   \"source\": [\n    \"import sympy\\n\",\n    \"sympy.init_printing()\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 2,\n   \"metadata\": {\n    \"collapsed\": true\n   },\n   \"outputs\": [],\n   \"source\": [\n    \"def a_coefficient(order_r, shift_k, index_l):\\n\",\n    \"    coefficient = sympy.Rational(0, 1)\\n\",\n    \"    for j in range(order_r - index_l, order_r+1):\\n\",\n    \"        numerator = 0\\n\",\n    \"        denominator = 1\\n\",\n    \"        for m in range(order_r+1):\\n\",\n    \"            if m == j:\\n\",\n    \"                pass\\n\",\n    \"            else:\\n\",\n    \"                product = 1\\n\",\n    \"                for s in range(order_r+1):\\n\",\n    \"                    if s == j or s == m:\\n\",\n    \"                        pass\\n\",\n    \"                    else:\\n\",\n    \"                        product *= order_r - shift_k - s\\n\",\n    \"                numerator += product\\n\",\n    \"                denominator *= j - m\\n\",\n    \"        coefficient += sympy.Rational(numerator, denominator)\\n\",\n    \"    return coefficient\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 3,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"0 0 11/6\\n\",\n      \"0 1 -7/6\\n\",\n      \"0 2 1/3\\n\",\n      \"1 0 1/3\\n\",\n      \"1 1 5/6\\n\",\n      \"1 2 -1/6\\n\",\n      \"2 0 -1/6\\n\",\n      \"2 1 5/6\\n\",\n      \"2 2 1/3\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"order_r = 3\\n\",\n    \"for k in range(order_r):\\n\",\n    \"    for l in range(order_r):\\n\",\n    \"        print(k, l, a_coefficient(order_r, k, l))\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"So an individual stencil, of order $r$, shifted by $k$, will have the form\\n\",\n    \"$$\\n\",\n    \"  q_{r, k, i+1/2} = \\\\sum_{l=0}^{r-1} a_{r, k, l} q_{i+k-l}\\n\",\n    \"$$\\n\",\n    \"which we can show symbolically:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 4,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [],\n   \"source\": [\n    \"q = sympy.symbols(r'q:{}'.format(2*order_r-1))\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 5,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [],\n   \"source\": [\n    \"def stencil(order_r, shift_k, q, offset=0):\\n\",\n    \"    stencil_sum = sympy.Rational(0, 1)\\n\",\n    \"    for l in range(order_r):\\n\",\n    \"        stencil_sum += a_coefficient(order_r, shift_k, l) * q[order_r+shift_k-l-1-offset]\\n\",\n    \"    return stencil_sum\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 6,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"0 q0/3 - 7*q1/6 + 11*q2/6\\n\",\n      \"1 -q1/6 + 5*q2/6 + q3/3\\n\",\n      \"2 q2/3 + 5*q3/6 - q4/6\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"for k in range(order_r):\\n\",\n    \"    print(k, stencil(order_r, k, q))\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"We can, at the same time, compute the *single* stencil of order $2 r - 1$ that uses the same data as all of the stencils of order $r$, and compare the two:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 7,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"q0/30 - 13*q1/60 + 47*q2/60 + 9*q3/20 - q4/20\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"print(stencil(2*order_r-1, order_r-1, q, offset=order_r-1))\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"Now, we can form a convex combination of each of the stencils of order $r$, so that \\n\",\n    \"$$\\n\",\n    \"  q_{r, \\\\text{Optimal}, i+1/2} = \\\\sum_{k=0}^{r-1} C_{r, k} q_{r, k, i+1/2}.\\n\",\n    \"$$\\n\",\n    \"We can match the stencils and so compute the $C_{r, k}$ coefficients:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 8,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"[Eq(1/30, C0/3), Eq(-13/60, -7*C0/6 - C1/6), Eq(47/60, 11*C0/6 + 5*C1/6 + C2/3)]\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"C = sympy.symbols(r'C:{}'.format(order_r))\\n\",\n    \"sum_stencil = 0\\n\",\n    \"for k in range(order_r):\\n\",\n    \"    sum_stencil += C[k] * stencil(order_r, k, q)\\n\",\n    \"eqns=[]\\n\",\n    \"optimal_stencil = stencil(2*order_r-1, order_r-1, q, offset=order_r-1)\\n\",\n    \"for k in range(order_r):\\n\",\n    \"    eqns.append(sympy.Eq(optimal_stencil.expand().coeff(q[k]),\\n\",\n    \"                         sum_stencil.expand().coeff(q[k])))\\n\",\n    \"print(eqns)\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 9,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"0 1/10\\n\",\n      \"1 3/5\\n\",\n      \"2 3/10\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"sol=sympy.solve(eqns, C)\\n\",\n    \"for k in range(order_r):\\n\",\n    \"    print(k, sol[C[k]])\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"We convert that into a function:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 10,\n   \"metadata\": {\n    \"collapsed\": true\n   },\n   \"outputs\": [],\n   \"source\": [\n    \"def C_coefficients(order_r):\\n\",\n    \"    coefficients = []\\n\",\n    \"    q = sympy.symbols(r'q:{}'.format(2*order_r-1))\\n\",\n    \"    C = sympy.symbols(r'C:{}'.format(order_r))\\n\",\n    \"    sum_stencil = 0\\n\",\n    \"    for k in range(order_r):\\n\",\n    \"        sum_stencil += C[k] * stencil(order_r, k, q)\\n\",\n    \"    eqns=[]\\n\",\n    \"    optimal_stencil = stencil(2*order_r-1, order_r-1, q, offset=order_r-1)\\n\",\n    \"    for k in range(order_r):\\n\",\n    \"        eqns.append(sympy.Eq(optimal_stencil.expand().coeff(q[k]),\\n\",\n    \"                             sum_stencil.expand().coeff(q[k])))\\n\",\n    \"    solution = sympy.solve(eqns, C)\\n\",\n    \"    for k in range(order_r):\\n\",\n    \"        coefficients.append(solution[C[k]])\\n\",\n    \"    return coefficients\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 11,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"1/24310\\n\",\n      \"36/12155\\n\",\n      \"504/12155\\n\",\n      \"2352/12155\\n\",\n      \"882/2431\\n\",\n      \"3528/12155\\n\",\n      \"1176/12155\\n\",\n      \"144/12155\\n\",\n      \"9/24310\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"check_r = 9\\n\",\n    \"C = C_coefficients(check_r)\\n\",\n    \"for k in range(check_r):\\n\",\n    \"    print(C[k])\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"Now, using the optimal coefficients is ideal when the interpolating polynomial $p(x)$ is nice and smooth. When there are discontinuities it will give large oscillations. So we want some indication of when these oscillations will occur.\\n\",\n    \"\\n\",\n    \"The Jiang-Shu smoothness indicators are given by the sum of the integrals of the square of the derivatives of the interpolating polynomial:\\n\",\n    \"$$\\n\",\n    \"  \\\\beta_{r, k, i+1/2} = \\\\sum_{m=1}^{r-1} \\\\Delta x^{2 m -1} \\\\int_{-\\\\Delta x / 2}^{+\\\\Delta x / 2} \\\\text{d} (\\\\xi - x_i) \\\\, \\\\frac{\\\\text{d}^m}{\\\\text{d} \\\\xi^m} p_{r, k}(\\\\xi-x_i).\\n\",\n    \"$$\\n\",\n    \"Using the square guarantees that it is positive. The factors of $\\\\Delta x$ should cancel out once the definition of $p_{r, k}$ is included, so (without loss of generality) we set $\\\\Delta x = 1$.\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 12,\n   \"metadata\": {\n    \"collapsed\": true\n   },\n   \"outputs\": [],\n   \"source\": [\n    \"def interpolating_p(order_r, shift_k, xi_m_x, q):\\n\",\n    \"    shift_r = order_r - shift_k - 1\\n\",\n    \"    p = sympy.Rational(0, 1)\\n\",\n    \"    for m in range(0, order_r+1):\\n\",\n    \"        for j in range(0, m):\\n\",\n    \"            numerator = 0\\n\",\n    \"            denominator = 1\\n\",\n    \"            for l in range(order_r+1):\\n\",\n    \"                if l == m:\\n\",\n    \"                    pass\\n\",\n    \"                else:\\n\",\n    \"                    product = 1\\n\",\n    \"                    for s in range(order_r+1):\\n\",\n    \"                        if s == l or s == m:\\n\",\n    \"                            pass\\n\",\n    \"                        else:\\n\",\n    \"                            product *= xi_m_x + shift_r - s + sympy.Rational(1, 2)\\n\",\n    \"                    numerator += product\\n\",\n    \"                    denominator *= m - l\\n\",\n    \"            p += q[order_r-shift_r+j-1] * numerator / denominator\\n\",\n    \"    return p\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 13,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [],\n   \"source\": [\n    \"def beta_integral(order_r, shift_k, q):\\n\",\n    \"    xi_m_x = sympy.symbols(r'\\\\xi')\\n\",\n    \"    half = sympy.Rational(1, 2)\\n\",\n    \"    sum_m = 0\\n\",\n    \"    for m in range(1, order_r):\\n\",\n    \"        sum_m += sympy.integrate(sympy.diff(interpolating_p(order_r, shift_k, xi_m_x, q).simplify(), xi_m_x, m)**2, \\n\",\n    \"                                 (xi_m_x, -half, half))\\n\",\n    \"    return sum_m\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 14,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"0 4*q0**2/3 - 19*q0*q1/3 + 11*q0*q2/3 + 25*q1**2/3 - 31*q1*q2/3 + 10*q2**2/3\\n\",\n      \"1 4*q1**2/3 - 13*q1*q2/3 + 5*q1*q3/3 + 13*q2**2/3 - 13*q2*q3/3 + 4*q3**2/3\\n\",\n      \"2 10*q2**2/3 - 31*q2*q3/3 + 11*q2*q4/3 + 25*q3**2/3 - 19*q3*q4/3 + 4*q4**2/3\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"for k in range(order_r):\\n\",\n    \"    print(k, beta_integral(order_r, k, q).expand())\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"In order to pre-calculate the smoothness indicators in a useful form, we want to match this against the expression that Gerolymos et al use, where the smoothness indicator is given as\\n\",\n    \"$$\\n\",\n    \"  \\\\beta_{r, k, i+1/2} = \\\\sum_{l=0}^{r-1} \\\\sum_{m=0}^{l} \\\\sigma_{r, k, l, m} q_{i+k-l} q_{i+k-m}.\\n\",\n    \"$$\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 15,\n   \"metadata\": {\n    \"collapsed\": true\n   },\n   \"outputs\": [],\n   \"source\": [\n    \"def beta_sigma(order_r, shift_k, q, sigma):\\n\",\n    \"    sum_beta = 0\\n\",\n    \"    for l in range(order_r):\\n\",\n    \"        for m in range(l+1):\\n\",\n    \"            sum_beta += sigma[l, m] * q[order_r+shift_k-l-1] * q[order_r+shift_k-m-1]\\n\",\n    \"    return sum_beta.simplify()\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 16,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"0 q0**2*\\\\sigma[2, 2] + q0*q1*\\\\sigma[2, 1] + q0*q2*\\\\sigma[2, 0] + q1**2*\\\\sigma[1, 1] + q1*q2*\\\\sigma[1, 0] + q2**2*\\\\sigma[0, 0]\\n\",\n      \"1 q1**2*\\\\sigma[2, 2] + q1*q2*\\\\sigma[2, 1] + q1*q3*\\\\sigma[2, 0] + q2**2*\\\\sigma[1, 1] + q2*q3*\\\\sigma[1, 0] + q3**2*\\\\sigma[0, 0]\\n\",\n      \"2 q2**2*\\\\sigma[2, 2] + q2*q3*\\\\sigma[2, 1] + q2*q4*\\\\sigma[2, 0] + q3**2*\\\\sigma[1, 1] + q3*q4*\\\\sigma[1, 0] + q4**2*\\\\sigma[0, 0]\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"sigma = sympy.MatrixSymbol('\\\\sigma', order_r, order_r)\\n\",\n    \"for k in range(order_r):\\n\",\n    \"    print(k, beta_sigma(order_r, k, q, sigma))\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"We can now read off the sigma coefficients:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 17,\n   \"metadata\": {\n    \"collapsed\": true\n   },\n   \"outputs\": [],\n   \"source\": [\n    \"def sigma_coefficients(order_r, shift_k):\\n\",\n    \"    coefficients = []\\n\",\n    \"    q = sympy.symbols(r'q:{}'.format(2*order_r-1))\\n\",\n    \"    integral_beta = beta_integral(order_r, shift_k, q).expand()\\n\",\n    \"    for l in range(order_r):\\n\",\n    \"        l_coeffs = []\\n\",\n    \"        for m in range(l+1):\\n\",\n    \"            if l == m:\\n\",\n    \"                l_coeffs.append(integral_beta.coeff(q[order_r+shift_k-l-1]**2))\\n\",\n    \"            else:\\n\",\n    \"                l_coeffs.append(integral_beta.coeff(q[order_r+shift_k-l-1]).coeff(q[order_r+shift_k-m-1]))\\n\",\n    \"        coefficients.append(l_coeffs)\\n\",\n    \"    return coefficients\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 18,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"0 [[10/3], [-31/3, 25/3], [11/3, -19/3, 4/3]]\\n\",\n      \"1 [[4/3], [-13/3, 13/3], [5/3, -13/3, 4/3]]\\n\",\n      \"2 [[4/3], [-19/3, 25/3], [11/3, -31/3, 10/3]]\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"for k in range(order_r):\\n\",\n    \"    print(k, sigma_coefficients(order_r, k))\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"We can now do the higher order terms, which are all tabulated in Gerolymos et al:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 19,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"a coefficients\\n\",\n      \"0 0 25/12\\n\",\n      \"0 1 -23/12\\n\",\n      \"0 2 13/12\\n\",\n      \"0 3 -1/4\\n\",\n      \"1 0 1/4\\n\",\n      \"1 1 13/12\\n\",\n      \"1 2 -5/12\\n\",\n      \"1 3 1/12\\n\",\n      \"2 0 -1/12\\n\",\n      \"2 1 7/12\\n\",\n      \"2 2 7/12\\n\",\n      \"2 3 -1/12\\n\",\n      \"3 0 1/12\\n\",\n      \"3 1 -5/12\\n\",\n      \"3 2 13/12\\n\",\n      \"3 3 1/4\\n\",\n      \"C coefficients\\n\",\n      \"0 1/35\\n\",\n      \"1 12/35\\n\",\n      \"2 18/35\\n\",\n      \"3 4/35\\n\",\n      \"sigma coefficients\\n\",\n      \"0 [[2107/240], [-1567/40, 11003/240], [3521/120, -8623/120, 7043/240], [-309/40, 2321/120, -647/40, 547/240]]\\n\",\n      \"1 [[547/240], [-1261/120, 3443/240], [961/120, -2983/120, 2843/240], [-247/120, 267/40, -821/120, 89/80]]\\n\",\n      \"2 [[89/80], [-821/120, 2843/240], [267/40, -2983/120, 3443/240], [-247/120, 961/120, -1261/120, 547/240]]\\n\",\n      \"3 [[547/240], [-647/40, 7043/240], [2321/120, -8623/120, 11003/240], [-309/40, 3521/120, -1567/40, 2107/240]]\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"order_r=4\\n\",\n    \"print(\\\"a coefficients\\\")\\n\",\n    \"for k in range(order_r):\\n\",\n    \"    for l in range(order_r):\\n\",\n    \"        print(k, l, a_coefficient(order_r, k, l))\\n\",\n    \"print(\\\"C coefficients\\\")\\n\",\n    \"C = C_coefficients(order_r)\\n\",\n    \"for k in range(order_r):\\n\",\n    \"    print(k, C[k])\\n\",\n    \"print(\\\"sigma coefficients\\\")\\n\",\n    \"for k in range(order_r):\\n\",\n    \"    print(k, sigma_coefficients(order_r, k))\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"We'll now write functions to generate a string representation of the coefficients:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 22,\n   \"metadata\": {\n    \"collapsed\": true\n   },\n   \"outputs\": [],\n   \"source\": [\n    \"import numpy\\n\",\n    \"\\n\",\n    \"def print_a(order_r):\\n\",\n    \"    a_numerators = []\\n\",\n    \"    numers = []\\n\",\n    \"    denoms = []\\n\",\n    \"    lcms = []\\n\",\n    \"    for k in range(order_r):\\n\",\n    \"        k_numers = []\\n\",\n    \"        k_denoms = []\\n\",\n    \"        for l in range(order_r):\\n\",\n    \"            coeff = a_coefficient(order_r, k, l)\\n\",\n    \"            k_numers.append(sympy.numer(coeff))\\n\",\n    \"            k_denoms.append(sympy.denom(coeff))\\n\",\n    \"        numers.append(k_numers)\\n\",\n    \"        denoms.append(k_denoms)\\n\",\n    \"        lcms.append(sympy.lcm_list(k_denoms))\\n\",\n    \"    lcm = sympy.lcm_list(lcms)\\n\",\n    \"    for k in range(order_r):\\n\",\n    \"        common_numers = []\\n\",\n    \"        for l in range(order_r):\\n\",\n    \"            common_numers.append(numers[k][l] * sympy.Rational(lcm, denoms[k][l]))\\n\",\n    \"        a_numerators.append(common_numers)\\n\",\n    \"    max_len = len(str(numpy.max(numpy.abs(numpy.array(a_numerators)))))\\n\",\n    \"    string1 = \\\"{{:>{}}}, \\\".format(max_len+1)\\n\",\n    \"    string2 = \\\"{{:>{}}}],\\\\n\\\".format(max_len+1)\\n\",\n    \"    spacing_string = \\\"     \\\" + len(str(order_r))*\\\" \\\"\\n\",\n    \"    a_string = \\\"a_{} = numpy.array([\\\\n\\\".format(order_r)\\n\",\n    \"    for k in range(order_r):\\n\",\n    \"        a_string += spacing_string\\n\",\n    \"        a_string += \\\"             [\\\"\\n\",\n    \"        for l in range(order_r-1):\\n\",\n    \"            a_string += string1.format(int(a_numerators[k][l]))\\n\",\n    \"        a_string += string2.format(int(a_numerators[k][-1]))\\n\",\n    \"    a_string += spacing_string\\n\",\n    \"    a_string += \\\"            ]) / {}\\\".format(int(lcm))\\n\",\n    \"    return a_string\\n\",\n    \"\\n\",\n    \"\\n\",\n    \"def print_C(order_r):\\n\",\n    \"    C_coeffs = C_coefficients(order_r)\\n\",\n    \"    denoms = [sympy.denom(i) for i in C_coeffs]\\n\",\n    \"    lcm = sympy.lcm_list(denoms)\\n\",\n    \"    C_numerators = [sympy.numer(i) * sympy.Rational(lcm, sympy.denom(i)) for i in C_coeffs]\\n\",\n    \"    max_len = len(str(numpy.max(numpy.abs(numpy.array(C_numerators)))))\\n\",\n    \"    string1 = \\\"{{:>{}}}, \\\".format(max_len+1)\\n\",\n    \"    string2 = \\\"{{:>{}}}\\\".format(max_len+1)\\n\",\n    \"    C_string = \\\"C_{} = numpy.array([\\\".format(order_r)\\n\",\n    \"    for k in range(order_r-1):\\n\",\n    \"        C_string += string1.format(int(C_numerators[k]))\\n\",\n    \"    C_string += string2.format(int(C_numerators[-1]))\\n\",\n    \"    C_string += \\\" ]) / {}\\\".format(int(lcm))\\n\",\n    \"    return C_string\\n\",\n    \"\\n\",\n    \"\\n\",\n    \"def print_sigma(order_r):\\n\",\n    \"    sigma_numerators = []\\n\",\n    \"    numers = []\\n\",\n    \"    denoms = []\\n\",\n    \"    lcms = []\\n\",\n    \"    for k in range(order_r):\\n\",\n    \"        k_numers = []\\n\",\n    \"        k_denoms = []\\n\",\n    \"        coeffs = sigma_coefficients(order_r, k)\\n\",\n    \"        for l in range(order_r):\\n\",\n    \"            l_numers = []\\n\",\n    \"            l_denoms = []\\n\",\n    \"            for m in range(l+1):\\n\",\n    \"                l_numers.append(sympy.numer(coeffs[l][m]))\\n\",\n    \"                l_denoms.append(sympy.denom(coeffs[l][m]))\\n\",\n    \"            for m in range(l+1, order_r):\\n\",\n    \"                l_numers.append(0)\\n\",\n    \"                l_denoms.append(1)\\n\",\n    \"            lcms.append(sympy.lcm_list(l_denoms))\\n\",\n    \"            k_numers.append(l_numers)\\n\",\n    \"            k_denoms.append(l_denoms)\\n\",\n    \"        numers.append(k_numers)\\n\",\n    \"        denoms.append(k_denoms)\\n\",\n    \"    lcm = sympy.lcm_list(lcms)\\n\",\n    \"    for k in range(order_r):\\n\",\n    \"        k_common_numers = []\\n\",\n    \"        for l in range(order_r):\\n\",\n    \"            l_common_numers = []\\n\",\n    \"            for m in range(order_r):\\n\",\n    \"                l_common_numers.append(numers[k][l][m] * sympy.Rational(lcm, denoms[k][l][m]))\\n\",\n    \"            k_common_numers.append(l_common_numers)\\n\",\n    \"        sigma_numerators.append(k_common_numers)\\n\",\n    \"    max_len = len(str(numpy.max(numpy.abs(numpy.array(sigma_numerators)))))\\n\",\n    \"    string1 = \\\"{{:>{}}}, \\\".format(max_len+1)\\n\",\n    \"    string2 = \\\"{{:>{}}}],\\\".format(max_len+1)\\n\",\n    \"    string3 = \\\"{{:>{}}}]\\\".format(max_len+1)\\n\",\n    \"    string4 = \\\"{{:>{}}}] ]\\\\n\\\".format(max_len+1)\\n\",\n    \"    spacing_string = \\\"     \\\" + len(str(order_r))*\\\" \\\"\\n\",\n    \"    sigma_string = \\\"sigma_{} = numpy.array([\\\".format(order_r)\\n\",\n    \"    start_k_string = \\\"\\\\n\\\" + spacing_string + \\\"                  [\\\"\\n\",\n    \"    start_l_string = \\\"\\\\n\\\" + spacing_string + \\\"                    [\\\"\\n\",\n    \"    end_k_string   = \\\"\\\\n\\\" + spacing_string + \\\"                  ],\\\"\\n\",\n    \"    end_l_string   = \\\"\\\\n\\\" + spacing_string + \\\"                    ],\\\"\\n\",\n    \"    for k in range(order_r-1):\\n\",\n    \"        sigma_string += start_k_string\\n\",\n    \"        for l in range(order_r-1):\\n\",\n    \"            sigma_string += start_l_string\\n\",\n    \"            for m in range(order_r-1):\\n\",\n    \"                sigma_string += string1.format(int(sigma_numerators[k][l][m]))\\n\",\n    \"            sigma_string += string2.format(int(sigma_numerators[k][l][-1]))\\n\",\n    \"        sigma_string += start_l_string\\n\",\n    \"        for m in range(order_r-1):\\n\",\n    \"            sigma_string += string1.format(int(sigma_numerators[k][-1][m]))\\n\",\n    \"        sigma_string += string3.format(int(sigma_numerators[k][-1][-1]))\\n\",\n    \"        sigma_string += end_k_string\\n\",\n    \"    sigma_string += start_k_string\\n\",\n    \"    for l in range(order_r-1):\\n\",\n    \"        sigma_string += start_l_string\\n\",\n    \"        for m in range(order_r-1):\\n\",\n    \"            sigma_string += string1.format(int(sigma_numerators[-1][l][m]))\\n\",\n    \"        sigma_string += string2.format(int(sigma_numerators[-1][l][-1]))\\n\",\n    \"    sigma_string += start_l_string\\n\",\n    \"    for m in range(order_r-1):\\n\",\n    \"        sigma_string += string1.format(int(sigma_numerators[-1][-1][m]))\\n\",\n    \"    sigma_string += string3.format(int(sigma_numerators[-1][-1][-1]))\\n\",\n    \"    sigma_string += \\\"\\\\n\\\" + spacing_string + \\\"                  ]\\\" + \\\"\\\\n\\\"\\n\",\n    \"    if lcm == 1:\\n\",\n    \"        sigma_string += \\\"                      ])\\\"\\n\",\n    \"    else:\\n\",\n    \"        sigma_string += \\\"                      ]) / {}\\\".format(int(lcm))\\n\",\n    \"    return sigma_string\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 21,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [],\n   \"source\": [\n    \"# You really don't want to do this - it's only there to\\n\",\n    \"# check that the output from the functions is executable\\n\",\n    \"# Python code\\n\",\n    \"\\n\",\n    \"#exec(print_sigma(4))\\n\",\n    \"#sigma_4\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"From the string representations we can create a \\\"human readable\\\" (hmm...) Python file that could be used with the WENO reconstruction routines in eg `pyro`.\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 23,\n   \"metadata\": {\n    \"collapsed\": true\n   },\n   \"outputs\": [],\n   \"source\": [\n    \"# Output the first \\\"few\\\" sets of coefficients to a file\\n\",\n    \"# This takes around 5 minutes on my machine...\\n\",\n    \"\\n\",\n    \"order_lo = 2\\n\",\n    \"order_hi = 7\\n\",\n    \"\\n\",\n    \"with open('./weno_coefficients.py', 'tw') as f:\\n\",\n    \"    f.write(\\\"import numpy\\\\n\\\")\\n\",\n    \"    f.write(\\\"\\\\n\\\\n\\\")\\n\",\n    \"    for order in range(order_lo, order_hi+1):\\n\",\n    \"        f.write(\\\"# Coefficients of order r={}\\\\n\\\".format(order))\\n\",\n    \"        f.write(\\\"# On smooth solutions this should converge with order r={}\\\\n\\\".format(2*order-1))\\n\",\n    \"        f.write(print_C(order))\\n\",\n    \"        f.write(\\\"\\\\n\\\")\\n\",\n    \"        f.write(print_a(order))\\n\",\n    \"        f.write(\\\"\\\\n\\\")\\n\",\n    \"        f.write(print_sigma(order))\\n\",\n    \"        f.write(\\\"\\\\n\\\\n\\\")\\n\",\n    \"    for var in \\\"C\\\", \\\"a\\\", \\\"sigma\\\":\\n\",\n    \"        f.write(var + \\\"_all = {\\\\n\\\")\\n\",\n    \"        for order in range(order_lo, order_hi):\\n\",\n    \"            f.write((len(var)+9)*\\\" \\\" + \\\"{0} : {1}_{0},\\\\n\\\".format(order, var))\\n\",\n    \"        f.write((len(var)+9)*\\\" \\\" + \\\"{0} : {1}_{0}\\\\n\\\".format(order_hi, var))\\n\",\n    \"        f.write((len(var)+7)*\\\" \\\" + \\\"}\\\\n\\\\n\\\")\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"Now we can use the coefficients, and repeat the convergence plot from the figures file.\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 24,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [],\n   \"source\": [\n    \"import weno_coefficients\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 72,\n   \"metadata\": {\n    \"collapsed\": true\n   },\n   \"outputs\": [],\n   \"source\": [\n    \"def weno(order, q):\\n\",\n    \"    \\\"\\\"\\\"\\n\",\n    \"    Do WENO reconstruction\\n\",\n    \"    \\n\",\n    \"    Parameters\\n\",\n    \"    ----------\\n\",\n    \"    \\n\",\n    \"    order : int\\n\",\n    \"        The stencil width\\n\",\n    \"    q : numpy array\\n\",\n    \"        Scalar data to reconstruct\\n\",\n    \"        \\n\",\n    \"    Returns\\n\",\n    \"    -------\\n\",\n    \"    \\n\",\n    \"    qL : numpy array\\n\",\n    \"        Reconstructed data - boundary points are zero\\n\",\n    \"    \\\"\\\"\\\"\\n\",\n    \"    C = weno_coefficients.C_all[order]\\n\",\n    \"    a = weno_coefficients.a_all[order]\\n\",\n    \"    sigma = weno_coefficients.sigma_all[order]\\n\",\n    \"\\n\",\n    \"    qL = numpy.zeros_like(q)\\n\",\n    \"    beta = numpy.zeros((order, len(q)))\\n\",\n    \"    w = numpy.zeros_like(beta)\\n\",\n    \"    np = len(q) - 2 * order\\n\",\n    \"    epsilon = 1e-16\\n\",\n    \"    for i in range(order, np+order):\\n\",\n    \"        q_stencils = numpy.zeros(order)\\n\",\n    \"        alpha = numpy.zeros(order)\\n\",\n    \"        for k in range(order):\\n\",\n    \"            for l in range(order):\\n\",\n    \"                for m in range(l+1):\\n\",\n    \"                    beta[k, i] += sigma[k, l, m] * q[i+k-l] * q[i+k-m]\\n\",\n    \"            alpha[k] = C[k] / (epsilon + beta[k, i]**2)\\n\",\n    \"            for l in range(order):\\n\",\n    \"                q_stencils[k] += a[k, l] * q[i+k-l]\\n\",\n    \"        w[:, i] = alpha / numpy.sum(alpha)\\n\",\n    \"        qL[i] = numpy.dot(w[:, i], q_stencils)\\n\",\n    \"    \\n\",\n    \"    return qL\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"We use a smooth $\\\\sin$ function as the exact solution. The smaples to be reconstructed are the cell integral averages which, of course, come from a $\\\\cos$ function appropriately evaluated at the cell edges.\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 82,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [],\n   \"source\": [\n    \"f_smooth = lambda x: numpy.sin(2 * numpy.pi * x)\\n\",\n    \"f_antiderivative = lambda x: -numpy.cos(2 * numpy.pi * x) / (2 * numpy.pi)\\n\",\n    \"\\n\",\n    \"# Convergence check, using the lazy fix to get error norms.\\n\",\n    \"orders = numpy.arange(min(weno_coefficients.C_all.keys()), max(weno_coefficients.C_all.keys())+1)\\n\",\n    \"points = 4 * 2**numpy.arange(1, 7)\\n\",\n    \"error_1norm = numpy.zeros((len(points), len(orders)))\\n\",\n    \"error_2norm = numpy.zeros((len(points), len(orders)))\\n\",\n    \"for i, pts in enumerate(points):\\n\",\n    \"    dx = 1 / pts\\n\",\n    \"    x_recon_interior = numpy.arange(dx/2, 1, dx)\\n\",\n    \"    recon_exact = f_smooth(x_recon_interior + dx/2)\\n\",\n    \"    for j, order in enumerate(orders):\\n\",\n    \"        x_recon = numpy.arange(dx * (1/2 - order), 1 + dx * order, dx)\\n\",\n    \"        s_recon = (f_antiderivative(x_recon + dx/2) -\\n\",\n    \"                   f_antiderivative(x_recon - dx/2)) / dx\\n\",\n    \"        s_weno = weno(order, s_recon)\\n\",\n    \"        error_1norm[i, j] = numpy.linalg.norm(s_weno[order:-order] - recon_exact, 1) / pts\\n\",\n    \"        error_2norm[i, j] = numpy.linalg.norm(s_weno[order:-order] - recon_exact, 2) / numpy.sqrt(pts)\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 79,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [],\n   \"source\": [\n    \"%matplotlib inline\\n\",\n    \"from matplotlib import pyplot\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 87,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"data\": {\n      \"image/png\": 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ndZ/7EK0lfz5s3V7eLi4u44lt8VxL+zcD7grl/3cvz4cfXmm29ef1+Z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\\nAABJRU5ErkJggg==\\n\",\n      \"text/plain\": [\n       \"<matplotlib.figure.Figure at 0x112eb8f98>\"\n      ]\n     },\n     \"metadata\": {},\n     \"output_type\": \"display_data\"\n    }\n   ],\n   \"source\": [\n    \"dxs = 1 / points\\n\",\n    \"fig, axes = pyplot.subplots(nrows=1, ncols=2, figsize=(10, 8))\\n\",\n    \"markers = '+x^osd<'\\n\",\n    \"colours = 'kbrgcm'\\n\",\n    \"for i, order in enumerate(orders):\\n\",\n    \"    axes[0].loglog(dxs, error_1norm[:, i], label=r\\\"$r={}$\\\".format(order),\\n\",\n    \"                   linestyle=None, linewidth=0,\\n\",\n    \"                   marker=markers[i], mec=colours[i], markersize=6, mew=2)\\n\",\n    \"    axes[0].loglog(dxs, error_1norm[0, i] * (dxs/dxs[0])**(2*order-1),\\n\",\n    \"                   linewidth=2, color=colours[i])\\n\",\n    \"    axes[1].loglog(dxs, error_2norm[:, i], \\n\",\n    \"                   linestyle=None, linewidth=0,\\n\",\n    \"                   marker=markers[i], mec=colours[i], markersize=6, mew=2)\\n\",\n    \"    axes[1].loglog(dxs, error_1norm[0, i] * (dxs/dxs[0])**(2*order-1),\\n\",\n    \"                   linewidth=2, color=colours[i],\\n\",\n    \"                   label=r\\\"$\\\\propto \\\\Delta x^{{{}}}$\\\".format(2*order-1))\\n\",\n    \"axes[0].set_xlabel(r\\\"$\\\\Delta x$\\\")\\n\",\n    \"axes[1].set_xlabel(r\\\"$\\\\Delta x$\\\")\\n\",\n    \"axes[0].set_ylabel(r\\\"$|$Error$|_1$\\\")\\n\",\n    \"axes[1].set_ylabel(r\\\"$|$Error$|_2$\\\")\\n\",\n    \"axes[0].set_ylim(1e-16, 1)\\n\",\n    \"axes[1].set_ylim(1e-16, 1)\\n\",\n    \"axes[0].legend()\\n\",\n    \"axes[1].legend()\\n\",\n    \"fig.tight_layout()\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"Note here that floating point error kicks in very quickly for the high order schemes. Equally for the lowest order scheme we probably need higher resolution to get clean convergence.\"\n   ]\n  }\n ],\n \"metadata\": {\n  \"anaconda-cloud\": {},\n  \"kernelspec\": {\n   \"display_name\": \"Python [default]\",\n   \"language\": \"python\",\n   \"name\": \"python3\"\n  },\n  \"language_info\": {\n   \"codemirror_mode\": {\n    \"name\": \"ipython\",\n    \"version\": 3\n   },\n   \"file_extension\": \".py\",\n   \"mimetype\": \"text/x-python\",\n   \"name\": \"python\",\n   \"nbconvert_exporter\": \"python\",\n   \"pygments_lexer\": \"ipython3\",\n   \"version\": \"3.4.5\"\n  }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 0\n}\n"
  },
  {
    "path": "hydro-test-problems/hydro-test-problems.tex",
    "content": "\\section{Testing}\n\nThere are a large number of standard hydrodynamics test problems that\nshould be run on any implementation of an algorithm.  These problems\nuncover strengths and weaknesses in your choice of algorithm and also\nsimple coding bugs.\n\nAn ideal test problem has an analytic solution that can be compared to\ndirectly.\n\nConvergence of your results is also an important test.\n\nRecall that for a finite-volume discretization, to second-order, the\nvalue of a function evaluated at the cell-center is the same as the\naverage of that function over the zone.  Usually this means that you\ncan simply initialize your test problem using cell-center coordinates.\nBut for problems that are not well aligned with your grid, e.g., a\nspherical initial function mapped onto your Cartesian grid, it is\nbeneficial to try to initialize the average value in the zone.  A\ncommon way to do this is to sub-divide a zone into a number of\nsub-zones, initialize each of the sub-zones, and then average the\nsub-zones back to the original zone.  \\MarginPar{show figure}\n\n\n\\section{Shock tube problems}\n\n\nThese tests are one-dimensional, and as such, they can provide good\ntests of symmetry in your code.  Run the shock tube in the $x$, $y$,\nand $z$ directions separately and then compare the profiles---they\nshould be identical (atleast to machine precision).  If not, there may\nbe a simple indexing bug, or something like that in your code.\n\nSince these tests start out with a discontinuity, they are not the\nbest tests to use for convergence testing.  Wherever there is an\ninitiali discontinuity, the limiters will kick in and drop your\nmethod to first-order accurate.\n\n\n\\subsection{Slow moving shock}\n\n\n\\section{Advection}\n\n\n\\section{Sedov}\n\n\n\n\\section{Gresho vortex}\n\nThe Gresho vortex is a vortex in with a stabilizing pressure gradient\ndesigned such that the overall structure is unchanging in time.  It\nhas a nice feature in that it allows you to set the Mach number of\nthe flow as one of the parameters.\n\n\n\\section{Odd-even decompiling}\n\n\n\\section{Hydrostatic equilibrium}\n\n"
  },
  {
    "path": "hydro1d/hydro1d.tex",
    "content": "\\label{app:hydro1d}\n\n%% \\begin{center}\n%% \\includegraphics[width=0.125\\linewidth]{pyro-sm}\n%% \\end{center}\n\n\\section{Introduction}\n\n\\hydrooned\\ is a one-dimensional compressible hydrodynamics code written\nin modern Fortran.  In particular, it implements the piecewise parabolic\nmethod described in \\S~\\ref{sec:hydro:ppm}, in both Cartesian and spherical\ngeometries.  It assumes a gamma-law equation of state and supports gravity.\n\n\n\\section{Getting \\hydrooned}\n\n\\hydrooned\\ can be downloaded from its github repository, \\url{https://github.com/zingale/hydro1d} as:\n\\begin{verbatim}\ngit clone https://github.com/zingale/hydro1d\n\\end{verbatim}\n\nSome details on the code can be found on its webpage:\n\\url{http://zingale.github.io/hydro1d/}\\, .  As with the other codes detailed here,\nif you wish to contribute, fix bugs, etc., you can create issues or pull requests\nthrough the code's github page.\n\n\n\n\\section{\\hydrooned 's structure}\n\nThere a few major data structures in \\hydrooned, defined in the {\\tt\n  grid.f90} file:\n\n\\begin{itemize}\n\\item {\\tt grid\\_t} : this holds the grid information, with fields for\n  the low and high indicies of the valid domain, and coordinate\n  information (including interface areas and volumes, to support\n  spherical geometry).\n\n\\item {\\tt gridvar\\_t} : data that lives on a grid.  This holds both\n  a {\\tt data} array and the {\\tt grid\\_t} that it is defined on.\n  \n\\item {\\tt gridedgevar\\_t} : like {\\tt gridvar\\_t}, but for data\n  defined at the interfaces between zones.\n\n\\end{itemize}\n\nThe integer keys defined in the {\\tt variables\\_module} allow you\nto index the difference state fields in the data arrays of the \ngrid variables.\n\n\n\\section{Running \\hydrooned}\n\nEach problem is in it's own directory, and the code is built and run\nthere.  For example, to run the Sod shock tube problem, do:\n\\begin{verbatim}\ncd hydro1d/sod\nmake\n./hydro1d inputs-sod-xp\n\\end{verbatim}\n\nAs the code is built, object files and modules will be output into the\n{\\tt \\_build} subdirectory.  {\\tt make realclean} will clean up the\nobjects.  Things are setup for {\\tt gfortran} by default---you will\nneed to edit the {\\tt Ghydro.mak} with different options for different\ncompilers. Some bits of Fortran 2003 and 2008 are used, so an\nup-to-date compiler is needed.\n\nA number of runtime options can be set---these are listed in {\\tt\n  params.f90} and {\\tt probparams.f90} (the latter is problem-specific\nparameters).\n\n\n\\section{Problem setups}\n\nThe following problem setups are provided:\n\\begin{itemize}\n\\item {\\tt advect} : a simple advection test where a density profile\n  is advected through periodic boundaries, in a uniform pressure\n  medium (to suppress dynamics)\n\n\\item {\\tt hse} : a simple 1-d atmosphere in hydrostatic equilibrium.\n  This test works to see if the atmosphere stays in HSE.\n\n\\item {\\tt sedov} : a 1-d spherical Sedov explosion.\n\n\\item {\\tt sod} : a shock tube setup for testing against exact Riemann\n  solvers.\n\\end{itemize}\n\n\n"
  },
  {
    "path": "hydro_examples/hydro_examples.tex",
    "content": "\\label{app:hydroex}\n\n\\section{Introduction}\n\nThe \\hydroex\\ codes are simple 1-d solvers written in python that\nillustrate many of the ideas in these nodes.  They are used for making\nmany of the figures found throughout (wherever the ``\\hydroexdoit{}''\nnote is found).  Most are written for python 3.x.\n\n\\section{Getting \\hydroex}\n\nThe \\hydroex\\ codes are hosted on github:\\\\\n\\url{https://github.com/zingale/hydro_examples}\n\nThere are a few ways to get them.  The simplest way is to just clone\nfrom github on the commandline:\n\n\\begin{verbatim}\ngit clone https://github.com/zingale/hydro_examples\n\\end{verbatim}\n\nThis will create a local git repo on your machine with all of the\ncode.  You can then run the scripts, ``as is'' without needing to go\nto github anymore.  Periodically, there may be updates to the scripts,\nwhich you can obtain by issuing ``{\\tt git pull}'' in the {\\tt\nhydro\\_examples/} directory.  Note that if you have made any local\nchanges to the scripts, \\git\\ may complain.  The process of merging\nchanges is descibed online in various places,\ne.g. ``\\href{https://help.github.com/articles/resolving-a-merge-conflict-from-the-command-line}{Resolving\na merge conflict from the command line}'' from github.\n\nAlternately, you can use github's web interface to fork it.  Logon to\ngithub (or create an account) and click on the ``Fork'' icon on the\n\\hydroex\\ page.  You can then interact with your version of the repo\nas needed.  This method will allow you to push changes back to github,\nand, if you think they should be included in the main \\hydroex\\ repo,\nissue a pull-request.\n\n\\section{\\hydroex\\ codes}\n\nThe codes in \\hydroex\\ are organized into directories named after the\nchapters in these notes.  Each directory is self-contained.  The\nfollowing scripts are available:\n\n\\begin{itemize}\n\n\\item {\\tt advection/}\n  \n  \\begin{itemize}\n  \\item {\\tt advection.py}: a 1-d second-order linear advection solver\n    with a wide range of limiters.\n\n  \\item {\\tt fdadvect\\_implicit.py}: a 1-d first-order implicit\n    finite-difference advection solver using periodic boundary\n    conditions.\n\n  \\item {\\tt fdadvect.py}: a 1-d first-order explicit\n    finite-difference linear advection solver using upwinded\n    differencing.\n\n  \\end{itemize}\n\n\\item {\\tt burgers/}\n\n  \\begin{itemize}\n  \\item {\\tt burgers.py}: a 1-d second-order solver for the inviscid\n    Burgers' equation, with initial conditions corresponding to a\n    shock and a rarefaction.\n\n  \\end{itemize}\n\n\\item {\\tt compressible/}\n\n  \\begin{itemize}\n  \\item {\\tt cfl.py}: a simple brute force script used to find the stability \n    limit of a multidimensional advection discretization.\n\n  \\item {\\tt euler.ipynb}: an {\\sf Jupyter} notebook using {\\sf SymPy}\n      that derives the eigensystem for the primitive-variable form\n      of the Euler equations.\n\n  \\item {\\tt euler-generaleos.ipynb}: an {\\sf Jupyter} notebook using\n    {\\sf SymPy} that derives the eigensystem for the\n    primitive-variable form of various forms of Euler equations with a\n    general EOS (with either $(\\rho e)$ or $\\gamma_e$ augmenting the\n    system) and for gray radiation hydrodynamics.  Additionally, this \n    notebook computes the full form of the interface states.\n      \n  \\item {\\tt riemann.py}: the main Riemann class for gamma-law Euler\n    equations.  The {\\tt RiemannProblem} class has methods to find\n    the star state, plot the Hugoniot curves, and sample the solution.\n\n  \\item {\\tt riemann-2shock.py}: draw the Hugoniot curves in the $u$-$p$\n    plane for a pair of states that comprise a Riemann problem, {\\em under\n      the 2-shock approximation}.\n\n  \\item {\\tt riemann-phase.py}: draw the Hugoniot curves in the\n    $u$-$p$ plane for a pair of states that comprise a Riemann\n    problem.\n\n  \\item {\\tt riemann-sod.py}: solve the Riemann problem for the Sod\n    initial conditions and plot the exact solution.\n\n  \\end{itemize}\n\n\\item {\\tt diffusion/}\n\n  \\begin{itemize}\n    \\item {\\tt diff\\_converge.py}: make a convergence plot (error\n      vs.\\ resolution) of different explicit and implicit diffusion\n      methods.\n\n    \\item {\\tt diffusion\\_explicit.py}: solve the constant-diffusivity\n      diffusion equation explicitly.  The method is first-order\n      accurate in time, but second-order in space.  A Gaussian profile\n      is diffused---the analytic solution is also a Gaussian.\n\n    \\item {\\tt diffusion\\_fo\\_implicit.py}: solve the\n      constant-diffusivity diffusion equation implicitly.\n      Backwards-difference time-discretization is used, resulting in a\n      first-order-in-time method.  Spatial discretization is second\n      order.  A Gaussian profile is diffused.\n\n    \\item {\\tt diffusion\\_implicit.py}: solve the constant-diffusivity\n      diffusion equation implicitly.  Crank-Nicolson time-discretization\n      is used, resulting in a second-order method.  A Gaussian profile\n      is diffused.\n\n  \\end{itemize}\n  \n\\item {\\tt elliptic/}\n\n  \\begin{itemize}\n    \\item {\\tt poisson\\_fft.py}: an FFT solver for a 2-d Poisson problem\n    with periodic boundaries.\n  \\end{itemize}\n\n\\item {\\tt finite-volume/}\n\n  \\begin{itemize}\n  \\item {\\tt conservative-interpolation.ipynb}: an {\\sf Jupyter} notebook\n    using {\\sf SymPy} that derives conservative interpolants.\n  \\end{itemize}\n\n\\item {\\tt incompressible/}\n\n  \\begin{itemize}\n  \\item {\\tt project.py}: a simple example of using a projection \n    to recover a divergence-free velocity field.\n  \\end{itemize}\n\n\\item {\\tt multigrid/}\n\n  \\begin{itemize}\n  \\item {\\tt mg\\_converge.py}: a convergence test of the multigrid\n    solver.  A Poisson problem is solved at various resolutions and\n    compared to the exact solution.  This demonstrates second-order\n    accuracy.\n\n  \\item {\\tt mg\\_test.py}: a simple driver for the multigrid solver.\n    This sets up and solves a Poisson problem and plots the behavior \n    of the solution as a function of V-cycle number.\n\n  \\item{\\tt multigrid.py}: a multigrid class for cell-centered data.\n    This implements pure V-cycles.  A square domain with $2^N$ zones\n    ($N$ a positive integer) is required.\n\n  \\item {\\tt patch1d.py}: a class for 1-d cell-centered data that\n    lives on a grid.  This manages the data, handles boundary\n    conditions, and provides routines for prolongation and restriction\n    to other grids.\n\n  \\item {\\tt smooth.py}: solve a Poisson equation with smoothing\n    only (but we still use the interface through the multigrid object).\n\n  \\end{itemize}\n\n\\item {\\tt multiphysics/}\n\n  \\begin{itemize}\n  \\item {\\tt burgersvisc.py}: solve the viscous Burgers equation.\n     The advective terms are treated explicitly with a second-order\n     accurate method.  The diffusive term is solved using an \n     implicit Crank-Nicolson discretization.  The overall coupling\n     is second-order accurate.\n\n  \\item {\\tt diffusion-reaction.py}: solve a diffusion-reaction\n    equation that propagates a diffusive reacting front (flame).  A\n    simple reaction term is modeled.  The diffusion is solved using\n    a second-order Crank-Nicolson discretization.  The reactions\n    are evolved using the VODE ODE solver (via SciPy).  The two\n    processes are coupled together using Strang-splitting to be\n    second-order accurate in time.\n     \n\n  \\end{itemize}\n\n\\end{itemize}\n\n\n"
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    "path": "incompressible/GNUmakefile",
    "content": "EPStoPDF = epstopdf\n\nALL: incompressible.pdf\n\n\nincompressible.pdf: main.tex incompressible.tex\n\tpdflatex -jobname=incompressible main.tex < /dev/null\n\tbibtex incompressible.aux\n\tpdflatex -jobname=incompressible main.tex < /dev/null\n\tpdflatex -jobname=incompressible main.tex < /dev/null\n\n\nclean:\n\t$(RM) *.aux *.log *.dvi *.bbl *.blg \n\t$(RM) *~\n\n.PHONY: clean\n"
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    "content": "%!PS-Adobe-3.0 EPSF-3.0\n%%Title: MAC.eps\n%%Creator: matplotlib version 1.0.1, http://matplotlib.sourceforge.net/\n%%CreationDate: Sun Nov 25 18:20:15 2012\n%%Orientation: portrait\n%%BoundingBox: 18 108 594 684\n%%EndComments\n%%BeginProlog\n/mpldict 10 dict def\nmpldict begin\n/m { moveto } bind def\n/l { lineto } bind def\n/r { rlineto } bind def\n/c { curveto } bind def\n/cl { closepath } bind def\n/box {\nm\n1 index 0 r\n0 exch r\nneg 0 r\ncl\n} bind def\n/clipbox {\nbox\nclip\nnewpath\n} bind def\n%!PS-Adobe-3.0 Resource-Font\n%%Title: cmsy10\n%%Copyright: Copyright (C) 1994, Basil K. Malyshev. 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_c\n362 340 361 331 359 322 _c\n264 -57 _l\n257 -84 244 -109 225 -131 _c\n205 -153 182 -171 156 -184 _c\n129 -197 102 -204 74 -204 _c\n52 -204 32 -198 14 -188 _c\n-3 -177 -12 -161 -12 -141 _c\n303 606 _m\n303 620 308 633 320 644 _c\n331 655 344 661 359 661 _c\n370 661 379 657 387 650 _c\n394 643 398 634 398 624 _c\n398 609 392 596 380 585 _c\n368 573 355 568 341 568 _c\n330 568 321 571 314 579 _c\n306 586 303 595 303 606 _c\n_cl}_e}_d\nend readonly def\n\n/BuildGlyph\n {exch begin\n CharStrings exch\n 2 copy known not{pop /.notdef}if\n true 3 1 roll get exec\n end}_d\n\n/BuildChar {\n 1 index /Encoding get exch get\n 1 index /BuildGlyph get exec\n}_d\n\nFontName currentdict end definefont pop\n%!PS-Adobe-3.0 Resource-Font\n%%Title: cmr10\n%%Copyright: Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B.\n%%Creator: Converted from TrueType to type 3 by PPR\n25 dict begin\n/_d{bind def}bind def\n/_m{moveto}_d\n/_l{lineto}_d\n/_cl{closepath eofill}_d\n/_c{curveto}_d\n/_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d\n/_e{exec}_d\n/FontName /Cmr10 def\n/PaintType 0 def\n/FontMatrix[.001 0 0 .001 0 0]def\n/FontBBox[-43 -249 1009 750]def\n/FontType 3 def\n/Encoding [ /v /two /d /plus /one /a ] def\n/FontInfo 10 dict dup begin\n/FamilyName (cmr10) def\n/FullName (cmr10) def\n/Notice (Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. ) def\n/Weight (Regular) def\n/Version (1.1/12-Nov-94) def\n/ItalicAngle 0.0 def\n/isFixedPitch false def\n/UnderlinePosition -133 def\n/UnderlineThickness 20 def\nend readonly def\n/CharStrings 6 dict dup begin\n/v{{527 0 19 -10 508 431 _sc\n244 0 _m\n98 367 _l\n92 379 82 387 69 391 _c\n56 394 39 396 19 396 _c\n19 431 _l\n229 431 _l\n229 396 _l\n191 396 173 388 173 372 _c\n173 369 173 367 174 366 _c\n286 84 _l\n387 339 _l\n389 344 390 349 390 355 _c\n390 367 385 377 375 385 _c\n365 392 354 396 342 396 _c\n342 431 _l\n508 431 _l\n508 396 _l\n487 396 469 391 454 382 _c\n438 372 426 358 418 340 _c\n283 0 _l\n280 -6 275 -10 267 -10 _c\n260 -10 _l\n252 -10 246 -6 244 0 _c\n}_e{_cl}_e}_d\n/two{{500 0 50 0 449 666 _sc\n50 0 _m\n50 27 _l\n50 28 50 30 52 32 _c\n207 204 _l\n230 229 249 250 264 268 _c\n278 285 293 305 307 327 _c\n321 349 333 372 341 396 _c\n349 419 354 444 354 470 _c\n354 497 349 523 339 548 _c\n329 573 314 593 294 608 _c\n274 623 249 631 221 631 _c\n192 631 166 622 143 605 _c\n119 587 103 565 94 537 _c\n96 537 100 538 105 538 _c\n119 538 132 533 143 523 _c\n153 513 159 500 159 484 _c\n159 468 153 455 143 445 _c\n132 434 119 429 105 429 _c\n89 429 76 434 66 445 _c\n55 456 50 469 50 484 _c\n50 508 54 532 64 554 _c\n73 576 86 595 104 613 _c\n122 630 142 643 164 652 _c\n186 661 210 666 236 666 _c\n274 666 309 658 342 642 _c\n375 626 401 603 420 573 _c\n439 543 449 509 449 470 _c\n449 441 442 414 430 388 _c\n417 362 401 338 381 317 _c\n361 295 336 271 305 244 _c\n274 217 254 199 244 190 _c\n131 81 _l\n227 81 _l\n}_e{274 81 313 81 345 82 _c\n377 82 394 84 396 86 _c\n404 94 412 125 420 178 _c\n449 178 _l\n421 0 _l\n50 0 _l\n_cl}_e}_d\n/d{{555 0 33 -10 529 694 _sc\n244 -10 _m\n204 -10 168 0 136 21 _c\n103 42 78 70 60 105 _c\n42 139 33 176 33 216 _c\n33 256 42 293 62 328 _c\n82 362 109 390 143 411 _c\n177 431 214 442 255 442 _c\n279 442 302 436 324 426 _c\n346 416 365 402 381 384 _c\n381 592 _l\n381 610 378 623 373 631 _c\n367 639 360 643 350 645 _c\n340 647 324 648 303 648 _c\n303 683 _l\n451 694 _l\n451 91 _l\n451 73 453 60 459 52 _c\n464 44 472 40 482 38 _c\n492 36 507 35 529 35 _c\n529 0 _l\n378 -10 _l\n378 52 _l\n360 32 340 16 316 6 _c\n292 -4 268 -10 244 -10 _c\n142 87 _m\n153 65 168 47 187 34 _c\n206 21 227 15 250 15 _c\n278 15 303 23 326 39 _c\n349 55 366 75 378 101 _c\n378 341 _l\n}_e{370 355 360 368 348 380 _c\n336 391 323 400 308 406 _c\n293 412 277 416 261 416 _c\n226 416 198 406 177 386 _c\n156 366 141 341 133 311 _c\n124 280 120 248 120 215 _c\n120 188 121 165 124 145 _c\n126 125 132 105 142 87 _c\n_cl}_e}_d\n/plus{{777 0 56 -82 721 583 _sc\n75 230 _m\n69 230 65 232 61 236 _c\n57 240 56 245 56 250 _c\n56 254 57 259 61 263 _c\n65 267 69 270 75 270 _c\n369 270 _l\n369 565 _l\n369 570 371 574 375 578 _c\n379 581 383 583 389 583 _c\n393 583 398 581 402 578 _c\n406 574 409 570 409 565 _c\n409 270 _l\n703 270 _l\n707 270 711 267 715 263 _c\n719 259 721 254 721 250 _c\n721 245 719 240 715 236 _c\n711 232 707 230 703 230 _c\n409 230 _l\n409 -64 _l\n409 -69 406 -73 402 -77 _c\n398 -80 393 -82 389 -82 _c\n383 -82 379 -80 375 -77 _c\n371 -73 369 -69 369 -64 _c\n369 230 _l\n75 230 _l\n_cl}_e}_d\n/one{500 0 87 0 421 666 _sc\n93 0 _m\n93 35 _l\n176 35 218 45 218 67 _c\n218 592 _l\n183 575 139 567 87 567 _c\n87 602 _l\n168 602 230 623 272 666 _c\n286 666 _l\n288 666 291 665 293 663 _c\n295 661 296 659 296 657 _c\n296 67 _l\n296 45 337 35 421 35 _c\n421 0 _l\n93 0 _l\n_cl}_d\n/a{{500 0 40 -10 493 448 _sc\n40 98 _m\n40 138 55 170 87 195 _c\n118 219 155 237 199 247 _c\n243 257 284 263 324 263 _c\n324 304 _l\n324 323 319 342 311 360 _c\n303 378 291 392 275 404 _c\n259 416 241 422 222 422 _c\n178 422 144 412 121 392 _c\n133 392 144 387 152 377 _c\n160 367 165 356 165 344 _c\n165 330 160 319 151 310 _c\n141 300 130 296 117 296 _c\n103 296 92 300 83 310 _c\n73 319 69 330 69 344 _c\n69 379 85 405 117 422 _c\n149 439 184 448 222 448 _c\n249 448 276 442 304 431 _c\n331 419 353 403 371 381 _c\n388 359 397 334 397 306 _c\n397 81 _l\n397 68 399 56 405 45 _c\n411 34 419 29 431 29 _c\n442 29 450 34 456 45 _c\n461 56 464 68 464 81 _c\n464 145 _l\n493 145 _l\n493 81 _l\n}_e{493 66 489 52 481 38 _c\n473 24 463 14 450 6 _c\n436 -1 422 -5 407 -5 _c\n387 -5 370 2 356 17 _c\n342 32 334 50 333 71 _c\n320 45 302 25 278 11 _c\n254 -3 228 -10 201 -10 _c\n175 -10 150 -6 126 0 _c\n101 7 80 19 64 35 _c\n48 51 40 72 40 98 _c\n121 98 _m\n121 74 129 55 147 39 _c\n164 23 184 15 208 15 _c\n229 15 248 20 266 31 _c\n284 41 298 56 308 75 _c\n318 93 324 113 324 134 _c\n324 238 _l\n294 238 263 233 231 223 _c\n199 213 173 197 152 176 _c\n131 155 121 129 121 98 _c\n_cl}_e}_d\nend readonly def\n\n/BuildGlyph\n {exch begin\n CharStrings exch\n 2 copy known not{pop /.notdef}if\n true 3 1 roll get exec\n end}_d\n\n/BuildChar {\n 1 index /Encoding get exch get\n 1 index /BuildGlyph get exec\n}_d\n\nFontName currentdict end definefont pop\nend\n%%EndProlog\nmpldict begin\n18 108 translate\n576 576 0 0 clipbox\n1.000 setlinewidth\n1 setlinejoin\n2 setlinecap\n[] 0 setdash\n1.000 setgray\ngsave\n0 0 m\n576 0 l\n576 576 l\n0 576 l\ncl\ngsave\nfill\ngrestore\nstroke\ngrestore\n/p0_0 {\nnewpath\ntranslate\n2.82095 2.82095 m\n0 0 l\n-2.82095 2.82095 l\n0 0 l\n-2.82095 -2.82095 l\n0 0 l\n2.82095 -2.82095 l\n0 0 l\ncl\n} bind def\n0 setlinecap\n0.000 setgray\ngsave\n553 553 11.52 11.52 clipbox\n380.16 306.432 p0_0\ngsave\n0.000 0.000 1.000 setrgbcolor\nfill\ngrestore\nstroke\ngrestore\n/p1_0 {\nnewpath\ntranslate\n2.82095 2.82095 m\n0 0 l\n-2.82095 2.82095 l\n0 0 l\n-2.82095 -2.82095 l\n0 0 l\n2.82095 -2.82095 l\n0 0 l\ncl\n} bind def\ngsave\n553 553 11.52 11.52 clipbox\n232.704 306.432 p1_0\ngsave\n0.000 0.000 1.000 setrgbcolor\nfill\ngrestore\nstroke\ngrestore\n/p2_0 {\nnewpath\ntranslate\n2.82095 2.82095 m\n0 0 l\n-2.82095 2.82095 l\n0 0 l\n-2.82095 -2.82095 l\n0 0 l\n2.82095 -2.82095 l\n0 0 l\ncl\n} bind def\ngsave\n553 553 11.52 11.52 clipbox\n306.432 380.16 p2_0\ngsave\n0.000 0.000 1.000 setrgbcolor\nfill\ngrestore\nstroke\ngrestore\n/p3_0 {\nnewpath\ntranslate\n2.82095 2.82095 m\n0 0 l\n-2.82095 2.82095 l\n0 0 l\n-2.82095 -2.82095 l\n0 0 l\n2.82095 -2.82095 l\n0 0 l\ncl\n} bind def\ngsave\n553 553 11.52 11.52 clipbox\n306.432 232.704 p3_0\ngsave\n0.000 0.000 1.000 setrgbcolor\nfill\ngrestore\nstroke\ngrestore\n2.000 setlinewidth\n2 setlinecap\ngsave\n553 553 11.52 11.52 clipbox\n48.384 85.248 m\n564.48 85.248 l\nstroke\ngrestore\ngsave\n553 553 11.52 11.52 clipbox\n48.384 232.704 m\n564.48 232.704 l\nstroke\ngrestore\ngsave\n553 553 11.52 11.52 clipbox\n48.384 380.16 m\n564.48 380.16 l\nstroke\ngrestore\ngsave\n553 553 11.52 11.52 clipbox\n48.384 527.616 m\n564.48 527.616 l\nstroke\ngrestore\ngsave\n553 553 11.52 11.52 clipbox\n85.248 48.384 m\n85.248 564.48 l\nstroke\ngrestore\ngsave\n553 553 11.52 11.52 clipbox\n232.704 48.384 m\n232.704 564.48 l\nstroke\ngrestore\ngsave\n553 553 11.52 11.52 clipbox\n380.16 48.384 m\n380.16 564.48 l\nstroke\ngrestore\ngsave\n553 553 11.52 11.52 clipbox\n527.616 48.384 m\n527.616 564.48 l\nstroke\ngrestore\ngsave\n296.432000 307.304800 translate\n0.000000 rotate\n/Cmmi10 findfont\n15.0 scalefont\nsetfont\n0.000000 6.375000 moveto\n/u glyphshow\n\n/Cmmi10 findfont\n10.5 scalefont\nsetfont\n8.576660 2.548594 moveto\n/i glyphshow\n\n12.191162 2.548594 moveto\n/comma glyphshow\n\n15.303564 2.548594 moveto\n/j glyphshow\n\n\ngrestore\ngsave\n296.932000 292.559200 translate\n0.000000 rotate\n/Cmmi10 findfont\n15.0 scalefont\nsetfont\n0.000000 6.359375 moveto\n/v glyphshow\n\n/Cmmi10 findfont\n10.5 scalefont\nsetfont\n7.258301 2.532969 moveto\n/i glyphshow\n\n10.872803 2.532969 moveto\n/comma glyphshow\n\n13.985205 2.532969 moveto\n/j glyphshow\n\n\ngrestore\ngsave\n435.888000 307.304800 translate\n0.000000 rotate\n/Cmmi10 findfont\n15.0 scalefont\nsetfont\n0.000000 6.375000 moveto\n/u glyphshow\n\n/Cmmi10 findfont\n10.5 scalefont\nsetfont\n8.576660 2.548594 moveto\n/i glyphshow\n\n/Cmr10 findfont\n10.5 scalefont\nsetfont\n14.034814 2.548594 moveto\n/plus glyphshow\n\n22.856592 2.548594 moveto\n/one glyphshow\n\n/Cmmi10 findfont\n10.5 scalefont\nsetfont\n28.106592 2.548594 moveto\n/comma glyphshow\n\n31.218994 2.548594 moveto\n/j glyphshow\n\n\ngrestore\ngsave\n436.388000 292.559200 translate\n0.000000 rotate\n/Cmmi10 findfont\n15.0 scalefont\nsetfont\n0.000000 6.359375 moveto\n/v glyphshow\n\n/Cmmi10 findfont\n10.5 scalefont\nsetfont\n7.258301 2.532969 moveto\n/i glyphshow\n\n/Cmr10 findfont\n10.5 scalefont\nsetfont\n12.716455 2.532969 moveto\n/plus glyphshow\n\n21.538232 2.532969 moveto\n/one glyphshow\n\n/Cmmi10 findfont\n10.5 scalefont\nsetfont\n26.788232 2.532969 moveto\n/comma glyphshow\n\n29.900635 2.532969 moveto\n/j glyphshow\n\n\ngrestore\ngsave\n141.976000 307.304800 translate\n0.000000 rotate\n/Cmmi10 findfont\n15.0 scalefont\nsetfont\n0.000000 6.375000 moveto\n/u glyphshow\n\n/Cmmi10 findfont\n10.5 scalefont\nsetfont\n8.576660 2.548594 moveto\n/i glyphshow\n\n/Cmsy10 findfont\n10.5 scalefont\nsetfont\n12.191162 2.548594 moveto\n/minus glyphshow\n\n/Cmr10 findfont\n10.5 scalefont\nsetfont\n20.348145 2.548594 moveto\n/one glyphshow\n\n/Cmmi10 findfont\n10.5 scalefont\nsetfont\n25.598145 2.548594 moveto\n/comma glyphshow\n\n28.710547 2.548594 moveto\n/j glyphshow\n\n\ngrestore\ngsave\n142.976000 292.559200 translate\n0.000000 rotate\n/Cmmi10 findfont\n15.0 scalefont\nsetfont\n0.000000 6.359375 moveto\n/v glyphshow\n\n/Cmmi10 findfont\n10.5 scalefont\nsetfont\n7.258301 2.532969 moveto\n/i glyphshow\n\n/Cmsy10 findfont\n10.5 scalefont\nsetfont\n10.872803 2.532969 moveto\n/minus glyphshow\n\n/Cmr10 findfont\n10.5 scalefont\nsetfont\n19.029785 2.532969 moveto\n/one glyphshow\n\n/Cmmi10 findfont\n10.5 scalefont\nsetfont\n24.279785 2.532969 moveto\n/comma glyphshow\n\n27.392187 2.532969 moveto\n/j glyphshow\n\n\ngrestore\ngsave\n288.432000 454.760800 translate\n0.000000 rotate\n/Cmmi10 findfont\n15.0 scalefont\nsetfont\n0.000000 6.375000 moveto\n/u glyphshow\n\n/Cmmi10 findfont\n10.5 scalefont\nsetfont\n8.576660 2.548594 moveto\n/i glyphshow\n\n12.191162 2.548594 moveto\n/comma glyphshow\n\n15.303564 2.548594 moveto\n/j glyphshow\n\n/Cmr10 findfont\n10.5 scalefont\nsetfont\n21.464111 2.548594 moveto\n/plus glyphshow\n\n30.285889 2.548594 moveto\n/one glyphshow\n\n\ngrestore\ngsave\n288.932000 440.015200 translate\n0.000000 rotate\n/Cmmi10 findfont\n15.0 scalefont\nsetfont\n0.000000 6.359375 moveto\n/v glyphshow\n\n/Cmmi10 findfont\n10.5 scalefont\nsetfont\n7.258301 2.532969 moveto\n/i glyphshow\n\n10.872803 2.532969 moveto\n/comma glyphshow\n\n13.985205 2.532969 moveto\n/j glyphshow\n\n/Cmr10 findfont\n10.5 scalefont\nsetfont\n20.145752 2.532969 moveto\n/plus glyphshow\n\n28.967529 2.532969 moveto\n/one glyphshow\n\n\ngrestore\ngsave\n289.432000 159.848800 translate\n0.000000 rotate\n/Cmmi10 findfont\n15.0 scalefont\nsetfont\n0.000000 6.375000 moveto\n/u glyphshow\n\n/Cmmi10 findfont\n10.5 scalefont\nsetfont\n8.576660 2.548594 moveto\n/i glyphshow\n\n12.191162 2.548594 moveto\n/comma glyphshow\n\n15.303564 2.548594 moveto\n/j glyphshow\n\n/Cmsy10 findfont\n10.5 scalefont\nsetfont\n19.620459 2.548594 moveto\n/minus glyphshow\n\n/Cmr10 findfont\n10.5 scalefont\nsetfont\n27.777441 2.548594 moveto\n/one glyphshow\n\n\ngrestore\ngsave\n290.432000 145.103200 translate\n0.000000 rotate\n/Cmmi10 findfont\n15.0 scalefont\nsetfont\n0.000000 6.359375 moveto\n/v glyphshow\n\n/Cmmi10 findfont\n10.5 scalefont\nsetfont\n7.258301 2.532969 moveto\n/i glyphshow\n\n10.872803 2.532969 moveto\n/comma glyphshow\n\n13.985205 2.532969 moveto\n/j glyphshow\n\n/Cmsy10 findfont\n10.5 scalefont\nsetfont\n18.302100 2.532969 moveto\n/minus glyphshow\n\n/Cmr10 findfont\n10.5 scalefont\nsetfont\n26.459082 2.532969 moveto\n/one glyphshow\n\n\ngrestore\ngsave\n387.532800 329.932000 translate\n270.000000 rotate\n/Cmmi10 findfont\n15.0 scalefont\nsetfont\n0.000000 7.338281 moveto\n/u glyphshow\n\n/Cmr10 findfont\n10.5 scalefont\nsetfont\n8.576660 13.715625 moveto\n/a glyphshow\n\n13.826660 13.715625 moveto\n/d glyphshow\n\n19.656006 13.715625 moveto\n/v glyphshow\n\n/Cmmi10 findfont\n10.5 scalefont\nsetfont\n8.576660 3.511875 moveto\n/i glyphshow\n\n/Cmr10 findfont\n10.5 scalefont\nsetfont\n14.034814 3.511875 moveto\n/plus glyphshow\n\n22.856592 3.511875 moveto\n/one glyphshow\n\n/Cmmi10 findfont\n10.5 scalefont\nsetfont\n28.106592 3.511875 moveto\n/slash glyphshow\n\n/Cmr10 findfont\n10.5 scalefont\nsetfont\n33.356592 3.511875 moveto\n/two glyphshow\n\n/Cmmi10 findfont\n10.5 scalefont\nsetfont\n38.606592 3.511875 moveto\n/comma glyphshow\n\n41.718994 3.511875 moveto\n/j glyphshow\n\n\ngrestore\ngsave\n204.331200 328.432000 translate\n270.000000 rotate\n/Cmmi10 findfont\n15.0 scalefont\nsetfont\n0.000000 7.338281 moveto\n/u glyphshow\n\n/Cmr10 findfont\n10.5 scalefont\nsetfont\n8.576660 13.715625 moveto\n/a glyphshow\n\n13.826660 13.715625 moveto\n/d glyphshow\n\n19.656006 13.715625 moveto\n/v glyphshow\n\n/Cmmi10 findfont\n10.5 scalefont\nsetfont\n8.576660 3.511875 moveto\n/i glyphshow\n\n/Cmsy10 findfont\n10.5 scalefont\nsetfont\n12.191162 3.511875 moveto\n/minus glyphshow\n\n/Cmr10 findfont\n10.5 scalefont\nsetfont\n20.348145 3.511875 moveto\n/one glyphshow\n\n/Cmmi10 findfont\n10.5 scalefont\nsetfont\n25.598145 3.511875 moveto\n/slash glyphshow\n\n/Cmr10 findfont\n10.5 scalefont\nsetfont\n30.848145 3.511875 moveto\n/two glyphshow\n\n/Cmmi10 findfont\n10.5 scalefont\nsetfont\n36.098145 3.511875 moveto\n/comma glyphshow\n\n39.210547 3.511875 moveto\n/j glyphshow\n\n\ngrestore\ngsave\n283.932000 387.532800 translate\n0.000000 rotate\n/Cmmi10 findfont\n15.0 scalefont\nsetfont\n0.000000 7.338281 moveto\n/v glyphshow\n\n/Cmr10 findfont\n10.5 scalefont\nsetfont\n7.258301 13.715625 moveto\n/a glyphshow\n\n12.508301 13.715625 moveto\n/d glyphshow\n\n18.337646 13.715625 moveto\n/v glyphshow\n\n/Cmmi10 findfont\n10.5 scalefont\nsetfont\n7.258301 3.511875 moveto\n/i glyphshow\n\n10.872803 3.511875 moveto\n/comma glyphshow\n\n13.985205 3.511875 moveto\n/j glyphshow\n\n/Cmr10 findfont\n10.5 scalefont\nsetfont\n20.145752 3.511875 moveto\n/plus glyphshow\n\n28.967529 3.511875 moveto\n/one glyphshow\n\n/Cmmi10 findfont\n10.5 scalefont\nsetfont\n34.217529 3.511875 moveto\n/slash glyphshow\n\n/Cmr10 findfont\n10.5 scalefont\nsetfont\n39.467529 3.511875 moveto\n/two glyphshow\n\n\ngrestore\ngsave\n284.932000 204.331200 translate\n0.000000 rotate\n/Cmmi10 findfont\n15.0 scalefont\nsetfont\n0.000000 7.338281 moveto\n/v glyphshow\n\n/Cmr10 findfont\n10.5 scalefont\nsetfont\n7.258301 13.715625 moveto\n/a glyphshow\n\n12.508301 13.715625 moveto\n/d glyphshow\n\n18.337646 13.715625 moveto\n/v glyphshow\n\n/Cmmi10 findfont\n10.5 scalefont\nsetfont\n7.258301 3.511875 moveto\n/i glyphshow\n\n10.872803 3.511875 moveto\n/comma glyphshow\n\n13.985205 3.511875 moveto\n/j glyphshow\n\n/Cmsy10 findfont\n10.5 scalefont\nsetfont\n18.302100 3.511875 moveto\n/minus glyphshow\n\n/Cmr10 findfont\n10.5 scalefont\nsetfont\n26.459082 3.511875 moveto\n/one glyphshow\n\n/Cmmi10 findfont\n10.5 scalefont\nsetfont\n31.709082 3.511875 moveto\n/slash glyphshow\n\n/Cmr10 findfont\n10.5 scalefont\nsetfont\n36.959082 3.511875 moveto\n/two glyphshow\n\n\ngrestore\ngsave\n145.976000 32.638400 translate\n0.000000 rotate\n/Cmmi10 findfont\n16.0 scalefont\nsetfont\n0.000000 1.343750 moveto\n/i glyphshow\n\n/Cmsy10 findfont\n16.0 scalefont\nsetfont\n5.507812 1.343750 moveto\n/minus glyphshow\n\n/Cmr10 findfont\n16.0 scalefont\nsetfont\n17.937500 1.343750 moveto\n/one glyphshow\n\n\ngrestore\ngsave\n303.432000 32.638400 translate\n0.000000 rotate\n/Cmmi10 findfont\n16.0 scalefont\nsetfont\n0.000000 1.421875 moveto\n/i glyphshow\n\n\ngrestore\ngsave\n438.888000 31.638400 translate\n0.000000 rotate\n/Cmmi10 findfont\n16.0 scalefont\nsetfont\n0.000000 2.343750 moveto\n/i glyphshow\n\n/Cmr10 findfont\n16.0 scalefont\nsetfont\n8.317187 2.343750 moveto\n/plus glyphshow\n\n21.767187 2.343750 moveto\n/one glyphshow\n\n\ngrestore\ngsave\n33.638400 151.476000 translate\n0.000000 rotate\n/Cmmi10 findfont\n16.0 scalefont\nsetfont\n0.000000 4.343750 moveto\n/j glyphshow\n\n/Cmsy10 findfont\n16.0 scalefont\nsetfont\n6.578125 4.343750 moveto\n/minus glyphshow\n\n/Cmr10 findfont\n16.0 scalefont\nsetfont\n19.007812 4.343750 moveto\n/one glyphshow\n\n\ngrestore\ngsave\n33.638400 298.932000 translate\n0.000000 rotate\n/Cmmi10 findfont\n16.0 scalefont\nsetfont\n0.000000 4.421875 moveto\n/j glyphshow\n\n\ngrestore\ngsave\n33.638400 446.388000 translate\n0.000000 rotate\n/Cmmi10 findfont\n16.0 scalefont\nsetfont\n0.000000 4.343750 moveto\n/j glyphshow\n\n/Cmr10 findfont\n16.0 scalefont\nsetfont\n9.387500 4.343750 moveto\n/plus glyphshow\n\n22.837500 4.343750 moveto\n/one glyphshow\n\n\ngrestore\n\nend\nshowpage\n"
  },
  {
    "path": "incompressible/MAC_solve.eps",
    "content": "%!PS-Adobe-3.0 EPSF-3.0\n%%Title: MAC_solve.eps\n%%Creator: matplotlib version 1.3.1, http://matplotlib.org/\n%%CreationDate: Tue Nov 18 20:16:08 2014\n%%Orientation: portrait\n%%BoundingBox: -115 185 727 606\n%%EndComments\n%%BeginProlog\n/mpldict 10 dict def\nmpldict begin\n/m { moveto } bind def\n/l { lineto } bind def\n/r { rlineto } bind def\n/c { curveto } bind def\n/cl { closepath } bind def\n/box {\nm\n1 index 0 r\n0 exch r\nneg 0 r\ncl\n} bind def\n/clipbox {\nbox\nclip\nnewpath\n} bind def\n%!PS-Adobe-3.0 Resource-Font\n%%Title: cmmi10\n%%Copyright: Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B.\n%%Creator: Converted from TrueType to type 3 by PPR\n25 dict begin\n/_d{bind def}bind def\n/_m{moveto}_d\n/_l{lineto}_d\n/_cl{closepath eofill}_d\n/_c{curveto}_d\n/_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d\n/_e{exec}_d\n/FontName /Cmmi10 def\n/PaintType 0 def\n/FontMatrix[.001 0 0 .001 0 0]def\n/FontBBox[-34 -250 1048 750]def\n/FontType 3 def\n/Encoding [ /phi /comma /i /v /D /U /u /slash /j /G ] def\n/FontInfo 10 dict dup begin\n/FamilyName (cmmi10) def\n/FullName (cmmi10) def\n/Notice (Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. ) def\n/Weight (Regular) def\n/Version (1.1/12-Nov-94) def\n/ItalicAngle 0.0 def\n/isFixedPitch false def\n/UnderlinePosition -133 def\n/UnderlineThickness 20 def\nend readonly def\n/CharStrings 10 dict dup begin\n/phi{{595 0 47 -204 575 694 _sc\n194 -194 _m\n239 -13 _l\n205 -13 173 -5 144 10 _c\n114 25 91 47 73 75 _c\n55 103 47 134 47 168 _c\n47 204 56 238 74 271 _c\n92 304 116 334 147 361 _c\n178 387 211 408 247 423 _c\n283 437 318 445 354 445 _c\n414 688 _l\n415 692 418 694 422 694 _c\n434 694 _l\n436 694 438 692 440 690 _c\n442 688 443 686 443 684 _c\n383 445 _l\n407 445 431 440 455 431 _c\n}_e{479 421 500 408 518 392 _c\n536 376 550 356 560 334 _c\n570 312 575 288 575 263 _c\n575 227 566 192 548 159 _c\n530 126 506 96 476 70 _c\n446 44 412 23 375 9 _c\n337 -5 302 -13 268 -13 _c\n222 -198 _l\n220 -202 218 -204 214 -204 _c\n202 -204 _l\n199 -204 197 -202 196 -200 _c\n194 -198 194 -196 194 -194 _c\n245 12 _m\n347 419 _l\n330 419 312 415 294 408 _c\n275 401 257 392 241 382 _c\n}_e{225 372 210 360 197 347 _c\n180 331 165 311 153 289 _c\n141 267 131 243 125 219 _c\n119 195 116 171 116 147 _c\n116 121 121 98 132 78 _c\n142 58 157 42 177 30 _c\n197 18 219 12 245 12 _c\n275 12 _m\n300 12 327 19 355 33 _c\n383 47 406 64 425 84 _c\n449 108 469 139 484 175 _c\n498 211 506 247 506 284 _c\n506 308 500 331 490 352 _c\n479 372 464 389 444 401 _c\n}_e{424 413 402 419 377 419 _c\n275 12 _l\n_cl}_e}_d\n/comma{277 0 84 -193 205 110 _sc\n99 -179 _m\n99 -176 100 -173 103 -171 _c\n127 -147 145 -120 159 -90 _c\n172 -60 179 -28 179 4 _c\n179 16 _l\n168 5 155 0 139 0 _c\n123 0 110 5 100 16 _c\n89 26 84 39 84 55 _c\n84 71 89 84 100 94 _c\n110 104 123 110 139 110 _c\n163 110 180 99 190 77 _c\n200 55 205 30 205 4 _c\n205 -32 197 -67 183 -101 _c\n168 -134 147 -164 120 -191 _c\n117 -192 115 -193 114 -193 _c\n110 -193 107 -191 104 -188 _c\n100 -185 99 -182 99 -179 _c\n_cl}_d\n/i{{344 0 27 -10 296 661 _sc\n78 72 _m\n78 84 80 97 86 111 _c\n167 326 _l\n175 350 180 370 180 386 _c\n180 406 172 416 158 416 _c\n132 416 110 402 93 375 _c\n75 347 63 317 55 284 _c\n53 280 51 278 47 278 _c\n35 278 _l\n29 278 27 281 27 287 _c\n27 290 _l\n37 330 54 365 76 396 _c\n98 426 126 442 160 442 _c\n184 442 204 434 220 418 _c\n236 402 245 383 245 359 _c\n}_e{245 346 242 333 237 320 _c\n156 105 _l\n146 83 142 63 142 45 _c\n142 25 149 15 165 15 _c\n191 15 212 29 230 57 _c\n247 85 259 115 267 147 _c\n268 151 271 153 275 153 _c\n287 153 _l\n289 153 291 152 293 150 _c\n295 148 296 146 296 144 _c\n296 143 295 142 295 141 _c\n285 103 269 68 247 37 _c\n224 5 196 -10 163 -10 _c\n139 -10 119 -2 103 13 _c\n86 29 78 48 78 72 _c\n}_e{191 606 _m\n191 620 196 632 208 644 _c\n220 655 233 661 247 661 _c\n258 661 267 657 275 650 _c\n282 643 286 634 286 624 _c\n286 609 280 596 268 585 _c\n256 573 243 568 229 568 _c\n218 568 209 571 202 579 _c\n194 586 191 595 191 606 _c\n_cl}_e}_d\n/v{{484 0 27 -10 467 443 _sc\n107 113 _m\n107 127 108 144 112 162 _c\n116 180 119 194 123 206 _c\n126 218 134 240 147 274 _c\n160 307 167 324 167 326 _c\n175 350 180 370 180 386 _c\n180 406 172 416 158 416 _c\n132 416 110 402 93 375 _c\n75 347 63 317 55 284 _c\n53 280 51 278 47 278 _c\n35 278 _l\n29 278 27 281 27 287 _c\n27 290 _l\n37 330 54 365 76 396 _c\n}_e{98 426 126 442 160 442 _c\n184 442 204 434 220 418 _c\n236 402 245 383 245 359 _c\n245 346 242 333 237 320 _c\n223 286 213 258 206 238 _c\n198 218 191 195 185 170 _c\n179 144 176 121 176 100 _c\n176 75 181 55 193 39 _c\n204 23 222 15 247 15 _c\n293 15 334 50 372 122 _c\n384 145 395 172 405 203 _c\n415 234 421 260 421 280 _c\n421 300 417 315 411 327 _c\n}_e{404 338 395 350 385 362 _c\n375 374 370 383 370 391 _c\n370 404 375 416 386 427 _c\n396 437 408 443 422 443 _c\n438 443 449 435 456 421 _c\n463 406 467 389 467 371 _c\n467 349 463 322 457 292 _c\n451 261 442 229 431 196 _c\n419 163 409 137 400 119 _c\n356 33 304 -10 246 -10 _c\n204 -10 170 0 145 21 _c\n119 41 107 72 107 113 _c\n_cl}_e}_d\n/D{{827 0 39 0 803 683 _sc\n49 0 _m\n42 0 39 4 39 13 _c\n39 15 40 18 41 22 _c\n41 26 43 29 45 31 _c\n47 33 49 35 52 35 _c\n92 35 121 37 137 42 _c\n145 44 152 53 156 69 _c\n293 618 _l\n294 624 295 629 295 632 _c\n295 639 291 643 283 644 _c\n270 646 246 648 211 648 _c\n204 648 201 652 201 661 _c\n201 663 202 666 203 670 _c\n204 674 205 677 207 679 _c\n}_e{209 681 211 683 214 683 _c\n573 683 _l\n619 683 660 671 695 649 _c\n730 626 757 595 775 557 _c\n793 519 803 477 803 431 _c\n803 379 792 328 771 277 _c\n750 226 720 179 682 137 _c\n644 95 600 61 552 37 _c\n503 12 453 0 402 0 _c\n49 0 _l\n236 43 _m\n236 37 247 35 270 35 _c\n386 35 _l\n423 35 459 42 493 58 _c\n527 73 558 94 586 121 _c\n611 146 633 179 652 220 _c\n}_e{671 260 686 303 697 347 _c\n707 391 713 432 713 469 _c\n713 496 709 520 701 542 _c\n693 564 681 582 667 598 _c\n653 614 635 626 613 635 _c\n591 643 566 648 539 648 _c\n429 648 _l\n411 648 399 646 393 643 _c\n387 640 381 630 377 614 _c\n240 65 _l\n237 57 236 49 236 43 _c\n_cl}_e}_d\n/U{{682 0 66 -21 762 683 _sc\n147 139 _m\n147 115 151 93 161 74 _c\n171 54 185 39 203 29 _c\n221 18 242 13 267 13 _c\n295 13 323 18 351 30 _c\n378 42 402 58 424 79 _c\n446 100 464 124 480 151 _c\n495 177 506 204 513 230 _c\n603 591 _l\n604 598 605 603 605 606 _c\n605 634 579 648 528 648 _c\n521 648 518 652 518 661 _c\n520 669 521 675 523 678 _c\n524 681 528 683 535 683 _c\n}_e{752 683 _l\n754 683 757 681 759 678 _c\n761 675 762 672 762 670 _c\n762 669 761 666 760 662 _c\n758 657 757 653 755 651 _c\n753 649 751 648 749 648 _c\n685 648 648 627 638 587 _c\n548 226 _l\n540 194 528 164 510 135 _c\n492 105 470 79 444 55 _c\n418 31 390 12 359 -1 _c\n328 -14 296 -21 264 -21 _c\n227 -21 193 -13 163 2 _c\n133 18 109 40 92 69 _c\n74 98 66 132 66 170 _c\n}_e{66 192 68 212 73 230 _c\n170 618 _l\n171 624 172 629 172 632 _c\n172 639 168 643 160 644 _c\n147 646 123 648 88 648 _c\n81 648 78 652 78 661 _c\n80 669 82 675 83 678 _c\n84 681 88 683 95 683 _c\n364 683 _l\n370 683 374 678 374 670 _c\n374 668 373 665 372 661 _c\n371 657 370 654 368 652 _c\n366 649 364 648 361 648 _c\n320 648 292 645 276 641 _c\n267 637 261 628 257 614 _c\n}_e{160 226 _l\n151 190 147 161 147 139 _c\n_cl}_e}_d\n/u{{572 0 27 -10 546 442 _sc\n106 109 _m\n106 131 109 153 115 176 _c\n121 198 129 224 139 252 _c\n149 280 159 305 167 326 _c\n175 350 180 370 180 386 _c\n180 406 172 416 158 416 _c\n132 416 110 402 93 375 _c\n75 347 63 317 55 284 _c\n53 280 51 278 47 278 _c\n35 278 _l\n29 278 27 281 27 287 _c\n27 290 _l\n37 330 54 365 76 396 _c\n98 426 126 442 160 442 _c\n}_e{184 442 204 434 220 418 _c\n236 402 245 383 245 359 _c\n245 346 242 333 237 320 _c\n233 312 226 294 216 267 _c\n206 239 198 217 193 199 _c\n187 181 183 163 179 146 _c\n175 128 174 111 174 94 _c\n174 72 178 53 188 38 _c\n197 22 212 15 234 15 _c\n276 15 314 41 346 94 _c\n346 96 347 99 348 101 _c\n348 103 349 106 349 109 _c\n421 399 _l\n423 407 428 415 436 421 _c\n}_e{444 427 453 431 463 431 _c\n471 431 478 428 484 423 _c\n490 418 493 411 493 403 _c\n493 399 492 396 492 395 _c\n420 107 _l\n415 87 413 70 413 58 _c\n413 29 422 15 442 15 _c\n463 15 479 28 490 55 _c\n500 81 509 112 517 147 _c\n518 151 521 153 525 153 _c\n537 153 _l\n539 153 541 151 543 149 _c\n545 147 546 145 546 143 _c\n538 114 531 89 525 69 _c\n518 49 507 30 493 14 _c\n}_e{479 -2 461 -10 440 -10 _c\n418 -10 399 -4 383 6 _c\n366 17 355 33 349 53 _c\n333 33 316 18 296 7 _c\n276 -4 254 -10 232 -10 _c\n193 -10 162 0 140 21 _c\n117 41 106 71 106 109 _c\n_cl}_e}_d\n/slash{500 0 56 -249 443 750 _sc\n56 -229 _m\n56 -227 56 -225 57 -225 _c\n405 738 _l\n406 742 408 745 412 747 _c\n415 749 419 750 423 750 _c\n429 750 433 748 437 744 _c\n441 740 443 736 443 730 _c\n443 726 _l\n95 -237 _l\n91 -245 84 -249 76 -249 _c\n70 -249 66 -247 62 -243 _c\n58 -239 56 -234 56 -229 _c\n_cl}_d\n/j{{411 0 -12 -204 398 661 _sc\n-12 -141 _m\n-12 -125 -6 -111 4 -99 _c\n15 -87 29 -81 45 -81 _c\n55 -81 64 -84 72 -91 _c\n79 -97 83 -106 83 -117 _c\n83 -129 79 -141 71 -151 _c\n63 -161 53 -168 41 -172 _c\n53 -176 64 -178 76 -178 _c\n104 -178 129 -165 150 -139 _c\n170 -113 185 -83 194 -51 _c\n288 324 _l\n292 343 295 359 295 373 _c\n295 401 285 416 265 416 _c\n235 416 210 402 188 376 _c\n}_e{166 349 148 318 135 284 _c\n133 280 131 278 127 278 _c\n115 278 _l\n109 278 107 281 107 287 _c\n107 290 _l\n121 330 142 365 170 396 _c\n198 426 230 442 267 442 _c\n295 442 317 433 335 416 _c\n353 398 362 376 362 350 _c\n362 340 361 331 359 322 _c\n264 -57 _l\n257 -84 244 -109 225 -131 _c\n205 -153 182 -171 156 -184 _c\n129 -197 102 -204 74 -204 _c\n52 -204 32 -198 14 -188 _c\n}_e{-3 -177 -12 -161 -12 -141 _c\n303 606 _m\n303 620 308 633 320 644 _c\n331 655 344 661 359 661 _c\n370 661 379 657 387 650 _c\n394 643 398 634 398 624 _c\n398 609 392 596 380 585 _c\n368 573 355 568 341 568 _c\n330 568 321 571 314 579 _c\n306 586 303 595 303 606 _c\n_cl}_e}_d\n/G{{786 0 51 -21 761 705 _sc\n146 219 _m\n146 177 154 141 170 110 _c\n186 79 210 55 241 38 _c\n272 21 308 13 349 13 _c\n389 13 426 22 462 41 _c\n498 59 520 86 530 122 _c\n552 208 _l\n552 210 552 212 552 214 _c\n552 216 553 217 553 219 _c\n553 225 551 229 547 231 _c\n530 235 497 238 447 238 _c\n440 238 437 242 437 251 _c\n439 259 440 264 442 268 _c\n444 271 448 273 454 273 _c\n}_e{712 273 _l\n718 273 722 268 722 260 _c\n722 258 721 255 720 251 _c\n719 247 718 244 716 242 _c\n714 239 712 238 709 238 _c\n687 238 670 236 660 232 _c\n650 228 643 219 639 204 _c\n617 118 _l\n589 7 _l\n587 2 584 0 580 0 _c\n574 0 566 7 556 22 _c\n546 37 538 50 534 60 _c\n511 32 480 12 441 -1 _c\n402 -14 362 -21 322 -21 _c\n270 -21 223 -9 182 13 _c\n}_e{140 36 108 68 85 110 _c\n62 152 51 199 51 252 _c\n51 308 63 363 88 418 _c\n112 472 146 521 189 564 _c\n231 607 280 641 334 667 _c\n388 692 443 705 499 705 _c\n521 705 542 701 562 695 _c\n582 688 601 678 619 664 _c\n637 650 651 635 662 617 _c\n741 703 _l\n741 704 742 705 746 705 _c\n752 705 _l\n754 705 756 704 758 702 _c\n760 700 761 697 761 695 _c\n693 425 _l\n}_e{693 420 690 418 685 418 _c\n667 418 _l\n661 418 659 421 659 429 _c\n661 443 663 461 663 481 _c\n663 513 657 544 646 573 _c\n635 602 618 625 595 643 _c\n571 661 543 670 509 670 _c\n455 670 405 656 360 629 _c\n314 601 276 565 244 521 _c\n212 477 188 428 171 374 _c\n154 320 146 269 146 219 _c\n_cl}_e}_d\nend readonly def\n\n/BuildGlyph\n {exch begin\n CharStrings exch\n 2 copy known not{pop /.notdef}if\n true 3 1 roll get exec\n end}_d\n\n/BuildChar {\n 1 index /Encoding get exch get\n 1 index /BuildGlyph get exec\n}_d\n\nFontName currentdict end definefont pop\n%!PS-Adobe-3.0 Resource-Font\n%%Title: cmsy10\n%%Copyright: Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B.\n%%Creator: Converted from TrueType to type 3 by PPR\n25 dict begin\n/_d{bind def}bind def\n/_m{moveto}_d\n/_l{lineto}_d\n/_cl{closepath eofill}_d\n/_c{curveto}_d\n/_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d\n/_e{exec}_d\n/FontName /Cmsy10 def\n/PaintType 0 def\n/FontMatrix[.001 0 0 .001 0 0]def\n/FontBBox[-29 -960 1123 779]def\n/FontType 3 def\n/Encoding [ /minus ] def\n/FontInfo 10 dict dup begin\n/FamilyName (cmsy10) def\n/FullName (cmsy10) def\n/Notice (Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. ) def\n/Weight (Regular) def\n/Version (1.1/12-Nov-94) def\n/ItalicAngle 0.0 def\n/isFixedPitch false def\n/UnderlinePosition -133 def\n/UnderlineThickness 20 def\nend readonly def\n/CharStrings 1 dict dup begin\n/minus{777 0 83 230 694 270 _sc\n102 230 _m\n96 230 92 232 88 236 _c\n84 240 83 245 83 250 _c\n83 254 84 259 88 263 _c\n92 267 96 270 102 270 _c\n676 270 _l\n681 270 685 267 689 263 _c\n692 259 694 254 694 250 _c\n694 245 692 240 689 236 _c\n685 232 681 230 676 230 _c\n102 230 _l\n_cl}_d\nend readonly def\n\n/BuildGlyph\n {exch begin\n CharStrings exch\n 2 copy known not{pop /.notdef}if\n true 3 1 roll get exec\n end}_d\n\n/BuildChar {\n 1 index /Encoding get exch get\n 1 index /BuildGlyph get exec\n}_d\n\nFontName currentdict end definefont pop\n%!PS-Adobe-3.0 Resource-Font\n%%Title: cmr10\n%%Copyright: Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B.\n%%Creator: Converted from TrueType to type 3 by PPR\n25 dict begin\n/_d{bind def}bind def\n/_m{moveto}_d\n/_l{lineto}_d\n/_cl{closepath eofill}_d\n/_c{curveto}_d\n/_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d\n/_e{exec}_d\n/FontName /Cmr10 def\n/PaintType 0 def\n/FontMatrix[.001 0 0 .001 0 0]def\n/FontBBox[-44 -250 1009 750]def\n/FontType 3 def\n/Encoding [ /v /two /parenright /d /plus /one /a /parenleft ] def\n/FontInfo 10 dict dup begin\n/FamilyName (cmr10) def\n/FullName (cmr10) def\n/Notice (Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. ) def\n/Weight (Regular) def\n/Version (1.1/12-Nov-94) def\n/ItalicAngle 0.0 def\n/isFixedPitch false def\n/UnderlinePosition -133 def\n/UnderlineThickness 20 def\nend readonly def\n/CharStrings 8 dict dup begin\n/v{{527 0 19 -10 508 431 _sc\n244 0 _m\n98 367 _l\n92 379 82 387 69 391 _c\n56 394 39 396 19 396 _c\n19 431 _l\n229 431 _l\n229 396 _l\n191 396 173 388 173 372 _c\n173 369 173 367 174 366 _c\n286 84 _l\n387 339 _l\n389 344 390 349 390 355 _c\n390 367 385 377 375 385 _c\n365 392 354 396 342 396 _c\n342 431 _l\n508 431 _l\n508 396 _l\n487 396 469 391 454 382 _c\n}_e{438 372 426 358 418 340 _c\n283 0 _l\n280 -6 275 -10 267 -10 _c\n260 -10 _l\n252 -10 246 -6 244 0 _c\n_cl}_e}_d\n/two{{500 0 50 0 449 666 _sc\n50 0 _m\n50 27 _l\n50 28 50 30 52 32 _c\n207 204 _l\n230 229 249 250 264 268 _c\n278 285 293 305 307 327 _c\n321 349 333 372 341 396 _c\n349 419 354 444 354 470 _c\n354 497 349 523 339 548 _c\n329 573 314 593 294 608 _c\n274 623 249 631 221 631 _c\n192 631 166 622 143 605 _c\n119 587 103 565 94 537 _c\n96 537 100 538 105 538 _c\n}_e{119 538 132 533 143 523 _c\n153 513 159 500 159 484 _c\n159 468 153 455 143 445 _c\n132 434 119 429 105 429 _c\n89 429 76 434 66 445 _c\n55 456 50 469 50 484 _c\n50 508 54 532 64 554 _c\n73 576 86 595 104 613 _c\n122 630 142 643 164 652 _c\n186 661 210 666 236 666 _c\n274 666 309 658 342 642 _c\n375 626 401 603 420 573 _c\n439 543 449 509 449 470 _c\n}_e{449 441 442 414 430 388 _c\n417 362 401 338 381 317 _c\n361 295 336 271 305 244 _c\n274 217 254 199 244 190 _c\n131 81 _l\n227 81 _l\n274 81 313 81 345 82 _c\n377 82 394 84 396 86 _c\n404 94 412 125 420 178 _c\n449 178 _l\n421 0 _l\n50 0 _l\n_cl}_e}_d\n/parenright{{388 0 56 -249 291 750 _sc\n65 -249 _m\n59 -249 56 -246 56 -240 _c\n56 -237 56 -235 58 -234 _c\n172 -122 229 38 229 250 _c\n229 462 172 623 60 733 _c\n57 734 56 737 56 741 _c\n56 743 57 745 59 747 _c\n61 749 63 750 65 750 _c\n74 750 _l\n76 750 77 749 79 748 _c\n127 710 167 664 199 610 _c\n231 556 254 498 269 437 _c\n283 376 291 314 291 250 _c\n291 202 287 156 279 110 _c\n}_e{271 64 258 20 241 -24 _c\n223 -68 201 -108 175 -146 _c\n148 -184 116 -217 79 -247 _c\n77 -248 76 -249 74 -249 _c\n65 -249 _l\n_cl}_e}_d\n/d{{555 0 33 -10 529 694 _sc\n244 -10 _m\n204 -10 168 0 136 21 _c\n103 42 78 70 60 105 _c\n42 139 33 176 33 216 _c\n33 256 42 293 62 328 _c\n82 362 109 390 143 411 _c\n177 431 214 442 255 442 _c\n279 442 302 436 324 426 _c\n346 416 365 402 381 384 _c\n381 592 _l\n381 610 378 623 373 631 _c\n367 639 360 643 350 645 _c\n340 647 324 648 303 648 _c\n303 683 _l\n451 694 _l\n451 91 _l\n}_e{451 73 453 60 459 52 _c\n464 44 472 40 482 38 _c\n492 36 507 35 529 35 _c\n529 0 _l\n378 -10 _l\n378 52 _l\n360 32 340 16 316 6 _c\n292 -4 268 -10 244 -10 _c\n142 87 _m\n153 65 168 47 187 34 _c\n206 21 227 15 250 15 _c\n278 15 303 23 326 39 _c\n349 55 366 75 378 101 _c\n378 341 _l\n370 355 360 368 348 380 _c\n336 391 323 400 308 406 _c\n}_e{293 412 277 416 261 416 _c\n226 416 198 406 177 386 _c\n156 366 141 341 133 311 _c\n124 280 120 248 120 215 _c\n120 188 121 165 124 145 _c\n126 125 132 105 142 87 _c\n_cl}_e}_d\n/plus{{777 0 56 -82 721 583 _sc\n75 230 _m\n69 230 65 232 61 236 _c\n57 240 56 245 56 250 _c\n56 254 57 259 61 263 _c\n65 267 69 270 75 270 _c\n369 270 _l\n369 565 _l\n369 570 371 574 375 578 _c\n379 581 383 583 389 583 _c\n393 583 398 581 402 578 _c\n406 574 409 570 409 565 _c\n409 270 _l\n703 270 _l\n707 270 711 267 715 263 _c\n719 259 721 254 721 250 _c\n721 245 719 240 715 236 _c\n}_e{711 232 707 230 703 230 _c\n409 230 _l\n409 -64 _l\n409 -69 406 -73 402 -77 _c\n398 -80 393 -82 389 -82 _c\n383 -82 379 -80 375 -77 _c\n371 -73 369 -69 369 -64 _c\n369 230 _l\n75 230 _l\n_cl}_e}_d\n/one{500 0 87 0 421 666 _sc\n93 0 _m\n93 35 _l\n176 35 218 45 218 67 _c\n218 592 _l\n183 575 139 567 87 567 _c\n87 602 _l\n168 602 230 623 272 666 _c\n286 666 _l\n288 666 291 665 293 663 _c\n295 661 296 659 296 657 _c\n296 67 _l\n296 45 337 35 421 35 _c\n421 0 _l\n93 0 _l\n_cl}_d\n/a{{500 0 40 -10 493 448 _sc\n40 98 _m\n40 138 55 170 87 195 _c\n118 219 155 237 199 247 _c\n243 257 284 263 324 263 _c\n324 304 _l\n324 323 319 342 311 360 _c\n303 378 291 392 275 404 _c\n259 416 241 422 222 422 _c\n178 422 144 412 121 392 _c\n133 392 144 387 152 377 _c\n160 367 165 356 165 344 _c\n165 330 160 319 151 310 _c\n141 300 130 296 117 296 _c\n103 296 92 300 83 310 _c\n}_e{73 319 69 330 69 344 _c\n69 379 85 405 117 422 _c\n149 439 184 448 222 448 _c\n249 448 276 442 304 431 _c\n331 419 353 403 371 381 _c\n388 359 397 334 397 306 _c\n397 81 _l\n397 68 399 56 405 45 _c\n411 34 419 29 431 29 _c\n442 29 450 34 456 45 _c\n461 56 464 68 464 81 _c\n464 145 _l\n493 145 _l\n493 81 _l\n493 66 489 52 481 38 _c\n473 24 463 14 450 6 _c\n}_e{436 -1 422 -5 407 -5 _c\n387 -5 370 2 356 17 _c\n342 32 334 50 333 71 _c\n320 45 302 25 278 11 _c\n254 -3 228 -10 201 -10 _c\n175 -10 150 -6 126 0 _c\n101 7 80 19 64 35 _c\n48 51 40 72 40 98 _c\n121 98 _m\n121 74 129 55 147 39 _c\n164 23 184 15 208 15 _c\n229 15 248 20 266 31 _c\n284 41 298 56 308 75 _c\n318 93 324 113 324 134 _c\n324 238 _l\n}_e{294 238 263 233 231 223 _c\n199 213 173 197 152 176 _c\n131 155 121 129 121 98 _c\n_cl}_e}_d\n/parenleft{{388 0 97 -249 332 750 _sc\n310 -247 _m\n272 -217 240 -184 214 -146 _c\n187 -108 165 -67 148 -25 _c\n131 17 118 62 110 109 _c\n101 155 97 202 97 250 _c\n97 298 101 345 110 392 _c\n118 438 131 483 149 527 _c\n166 570 188 611 215 649 _c\n242 687 274 720 310 748 _c\n310 749 311 750 315 750 _c\n324 750 _l\n326 750 327 749 329 747 _c\n331 745 332 743 332 741 _c\n332 738 331 736 330 735 _c\n}_e{297 703 270 668 248 632 _c\n226 596 209 557 196 516 _c\n183 474 174 432 168 388 _c\n162 344 159 298 159 250 _c\n159 38 215 -122 329 -232 _c\n331 -234 332 -236 332 -240 _c\n332 -242 331 -244 329 -246 _c\n327 -248 326 -249 324 -249 _c\n315 -249 _l\n311 -249 310 -248 310 -247 _c\n_cl}_e}_d\nend readonly def\n\n/BuildGlyph\n {exch begin\n CharStrings exch\n 2 copy 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    "content": "\\label{ch:incompressible}\n\n\\section{Incompressible flow}\n\nAs a fluid parcel advects through a domain, it compresses and expands\ndue to a variety of effects (stratification, local heat release,\nacoustic/shock waves).  The Lagrangian derivative of the density\ncaptures the changes in the fluid, and is a measure of its compressibility.\nFrom the continuity equation, we see:\n\\begin{equation}\n-\\frac{1}{\\rho}\\frac{D\\rho}{Dt} = \\nabla \\cdot \\Ub\n\\end{equation}\nNote that for $\\nabla \\cdot \\Ub > 0$, we have $-(1/\\rho) (D\\rho/Dt) > 0$,\nwhich means that $\\rho$ gets smaller---this is expansion of the Lagrangian\nfluid element.\n\nA fluid in which the density (and therefore volume) of a fluid element is not\nallowed to change is called {\\em incompressible}.  An incompressible\nfluid obeys the velocity constraint:\n\\begin{equation}\n\\nabla \\cdot \\Ub = 0\n\\end{equation}\n(since $D\\rho / Dt = 0$).  The incompressible fluid approximation is \na reasonable approximation when the Mach number of the fluid is small\n($\\ll 1$).  To complete the system, we add the momentum equation.  If \nwe take the density to be constant everywhere in the domain (not just\nin our fluid element), then we have:\n\\begin{equation}\n\\frac{\\partial \\Ub}{\\partial t} + \\Ub \\cdot \\nabla \\Ub + \\nabla p = 0\n\\end{equation}\nNote that $p$ here looks like a pressure, but it is not subject to\nany equation of state.  This system is closed as written.  The value\nof $p$ is determined such that the velocity constraint is satisfied.\n\nWe can gain more insight into the applicability of the incompressible \nequations by doing an asymptotic expansion.  Starting with the momentum\nequation from the Euler equations:\n\\begin{equation}\n\\frac{\\partial (\\rho {\\bf U})}{\\partial t} + {\\nabla}\\cdot (\\rho {\\bf U}{\\bf U}) + {\\nabla}p = 0\n\\end{equation}\nwe can introduce dimensionless variables:\n\\begin{equation}\n\\bar{\\bf U} = \\frac{{\\bf U}}{|{\\bf U}_0|} ; \\quad \\bar{t} = \\frac{t}{t_0} ; \\quad \\bar{\\rho} = \\frac{\\rho}{\\rho_0} ; \\quad \\bar{x} = \\frac{x}{L_0} ; \\quad \\bar{p} = \\frac{p}{\\rho_0 c_0^2}\n\\end{equation}\nand we find the dimensionless evolution equation:\n\\begin{equation}\n\\frac{\\partial (\\bar{\\rho} \\bar{\\bf U})}{\\partial \\bar{t}} + \\bar{\\nabla} \\cdot (\\bar{\\rho} \\bar{\\bf U}\\bar{\\bf U}) + \\frac{1}{M^2} \\bar{\\nabla} \\bar{p} = 0\n\\end{equation}\nNow we introduce an expansion of pressure in terms of Mach number:\n\\begin{equation}\np = p_0 + p_1 M + p_2 M^2\n\\end{equation}\nGrouping terms by power of Mach number, we find:\n\\begin{equation}\n\\nabla p_0 = \\nabla p_1 = 0\n\\end{equation}\nand\n\\begin{equation}\n\\frac{\\partial (\\rho {\\bf U})}{\\partial t} + {\\nabla}\\cdot (\\rho {\\bf U}{\\bf U}) + {\\nabla}p_2 = 0\n\\end{equation}\nOnly $p_2$, which is sometimes called the {\\em dynamical pressure} appears in the dynamics.\nAll of the thermodynamic content is in $p_0$, consistent with our argument previously\nthat no equation of state is needed.\n\nRecalling that pressure evolves according to\n\\begin{equation}\n\\frac{\\partial p}{\\partial t} + \\gamma p {\\nabla}\\cdot {\\bf U} + {\\bf U}\\cdot {\\nabla}p = 0\n\\end{equation}\nThe general expression for the velocity divergence is:\n\\begin{equation}\n{\\nabla}\\cdot {U} = -\\frac{1}{\\gamma p} \\frac{Dp}{Dt}\n\\end{equation}\nfrom our pressure expansion, we see that\n\\begin{equation}\n\\nabla \\cdot {\\bf U} \\sim  \\mathcal{O}(M^2)\n\\end{equation}\nThis justifies our argument that the incompressible equations apply at low Mach number.\n\n\n\\section{Projection methods}\n\nThe basic idea behind a projection method is that any vector field\ncan be decomposed into a divergence free part and the gradient of \na scalar (this is sometimes called a {\\em Hodge decomposition}).  \nGiven a velocity field $\\Ub^\\star$, we can express it in terms of the \ndivergence free part $\\Ub^d$ and a scalar, $\\phi$ as:\n\\begin{equation}\n\\Ub^\\star = \\Ub^d + \\nabla \\phi\n\\end{equation}\nTaking the divergence of each side, and noting that $\\nabla \\cdot \\Ub^d\n= 0$, we have\n\\begin{equation}\n\\nabla \\cdot \\Ub^\\star = \\nabla^2 \\phi\n\\end{equation}\nThis is an elliptic equation.  Given suitable boundary conditions, we\ncan solve for $\\phi$ (for instance, using multigrid) and recover the\ndivergence free part of $\\Ub^\\star$ as:\n\\begin{equation}\n\\Ub^d = \\Ub^\\star - \\nabla \\phi\n\\end{equation}\nWe call this operation of extracting the divergence free part of a\nvelocity field a {\\em projection}.  This can be expressed by defining\nan operator $P$, such that $P\\Ub^\\star = \\Ub^d$, and $(I - P)\\Ub^\\star =\n\\nabla \\phi$.  From the momentum equation, we see that $\\partial\n\\Ub/\\partial t + \\nabla p$ is in the form of a divergence free term\n$+$ the gradient of a scalar.  This means that advancing the velocity\nfield subject to the constraint involves solving:\n\\begin{equation}\n\\Ub_t = P(\\Ub_t + \\nabla p) = P(-\\Ub\\cdot \\nabla \\Ub)\n\\label{eq:projevolve}\n\\end{equation}\nSee Bell, Colella, and Howell~\\cite{BCH} for a nice discussion of this.\n\nThe original projection method for incompressible flows goes back to\nChorin~\\cite{chorin:1968}.  Instead of evolving\nEq.~\\ref{eq:projevolve} directly, we break the update into pieces.\nThe basic idea is to evolve the velocity advection equation without\nregard to the constraint, yielding a {\\em provisional velocity} field\nwhich is then subjected to a projection to enforce the divergence-free\nconstraint.  Bell, Colella, and Glaz (BCG)~\\cite{BCG} introduced a projection\nmethod that uses standard Godunov methods for the advection terms\n(much like is done with compressible flow) and then solves an elliptic\nequation to enforce the constraint.  This division of the operations\nin the algorithm (advect then project) is a type of {\\em fractional\n  step} method.\n\nThere are different ways to discretize the operators that make up the\nprojection.  We denote the discretized divergence as $D$ and the\ndiscretized gradient operation as $G$.  For an exact projection, the\ndiscretized Laplacian, $L$, would be the same as applying $G$ and $D$\nin succession (i.e. $L = DG$).  Depending on our data centerings, we\nmay prefer to discretize the Laplacian independently to $D$ and $G$,\nsuch that $L \\ne DG$.  This is called an {\\em approximate projection}.\nNote that for an approximate projection, $P$ is not\nidempotent: $P^2 \\Ub \\ne P\\Ub$.\n\nMany variations on this basic idea exist, using alternate forms of the\nprojection, different grid centerings of the $\\phi$ variable, and\nadditional physics.\n\nFigure~\\ref{fig:incomp:project} shows an example of a projection.  The\ninitial velocity field is specified as:\n\\begin{align}\n\\label{eq:incomp:proju}\n  u &= -\\sin(\\pi x)^2 \\sin(2\\pi y) \\\\\n\\label{eq:incomp:projv}\n  v &= \\phantom{+} \\sin(\\pi y)^2 \\sin(2\\pi x)\n\\end{align}\nwhich is divergence free and doubly-periodic.  This is then modified\nby adding the gradient of a scalar, $\\phi$, of the form:\n\\begin{equation}\n\\label{eq:incomp:projphi}\n\\phi = \\frac{1}{10} \\cos(2\\pi y) \\cos(2\\pi x) \n\\end{equation}\nyielding a new velocity, $\\Ub_p = \\Ub + \\nabla \\phi$.\nThe result is the middle panel in the figure, and is not divergence free.\nA projection is then done, to recover $\\phi$ by solving\n\\begin{equation}\n\\nabla^2 \\phi = \\nabla \\cdot \\Ub_p\n\\end{equation}\nFor this figure, we use the standard 5-point Laplacian and a divergence\nbuilt via centered-differences:\n\\begin{equation}\n\\nabla \\cdot \\Ub = \\frac{u_{i+1,j} - u_{i-1,j}}{2\\dx} + \\frac{v_{i,j+1} - v_{i,j-1}}{2\\dy}\n\\end{equation}\nTogether, these represent an approximation projection. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{project-u}\n\\caption[Example of a projection]{\\label{fig:incomp:project} An\n  example of an approximate projection, on a $128^2$ grid.  The top\n  shows the original velocity field, Eqs.~\\ref{eq:incomp:proju} and\n  \\ref{eq:incomp:projv}.  This was poluted by adding the gradient of a\n  scalar, Eq.~\\ref{eq:incomp:projphi} (middle) and then a projection\n  was done to recover the original velocity field (bottom).  In this\n  case, the L$_2$ norm of the error on the velocity (comparing before\n  and after projection) is $0.000189$. \\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/incompressible/project.py}{project.py}}}\n\\end{figure}\n\n\\begin{exercise}[An approximate projection]\nImplement an approximate projection, using pure smoothing for solving\nthe elliptic equation.  Start with the velocity field described\nby Eqs.~\\ref{eq:incomp:proju} and \\ref{eq:incomp:projv} and the scalar\nfrom Eq.~\\ref{eq:incomp:projphi}.  Compute a poluted velocity field,\n$\\Ub_p = \\Ub + \\nabla \\phi$ and then project it to recover the \noriginal velocity field.\n\\end{exercise}\n\n\\section{Cell-centered approximate projection solver}\n\nHere we describe an incompressible algorithm that uses cell-centered\ndata throughout---$\\Ub$ and $p$ are both cell-centered.  The\nprojection at the end is an approximate projection.  The basic\nalgorithm flow is \n\\begin{itemize}\n\\item Create the time-centered advective velocities through the faces\n  of the zones.\n\\item Project the advective velocities such that they obey the\n  velocity constraint\n\\item Construct the time-centered interface states of all quantities\n  on the faces of the zones using the advective velocity.\n\\item Update the velocity to the new time.  This is defines the\n  provisional velocity field---it does not yet satisfy the constraint.\n\\item Enforce the velocity constraint by projecting the velocity.\n\\end{itemize}\n\nThe description below is pieced together from a variety of sources.\nBCH describes a cell-centered method, but with an exact projection\n(with a larger, decoupled stencil).  Almgren, Bell, and Szymczak\n(ABS)~\\cite{ABS} describes an approximate projection method, but with\na node-centered final projection.  We follow this paper closely up\nuntil the projection.  Martin and Colella~\\cite{MartinColella} (and\nMartin's PhD thesis) method uses a cell-centered projection, as is\ndescribed here.  They go into additional effort to describe this for a\nrefined grid.  All of these method are largely alike, aside from how\nthe discretization of the final projection is handled.\n\n\\subsection{Advective velocity}\n\nIn predicting the interface states, we first seek to construct the\nvelocities through the interfaces.  A key concept throughout the\nadvection step is that once we have the normal velocities on the\ninterfaces, we can use these to upwind left and right states of any\nquantity to get their interface value.  The advective velocities we\nconstruct here, ${u}^\\mathrm{adv}$ and ${v}^\\mathrm{adv}$,\nwill later be used to upwind the various states we predict to the\ninterfaces.  We only need the velocity through the interfaces, as\nshown in the figure \\ref{fig:MAC}. This staggered grid arrangement is\nsometimes called a MAC grid.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{MAC}\n\\caption[MAC grid for velocity]\n  {\\label{fig:MAC} The staggered `MAC' grid for the advective\n  velocities.}\n\\end{figure}\n\n\nWe follow ABS.  Our velocity evolution system (writing out the\nindividual components of $\\Ub$: $u$ and $v$) is\n\\begin{eqnarray}\n\\frac{\\partial u}{\\partial t} &=& -u \\frac{\\partial u}{\\partial x} \n                                  -v \\frac{\\partial u}{\\partial y} \n                                  -\\frac{\\partial p}{\\partial x} = 0 \\\\\n\\frac{\\partial v}{\\partial t} &=& -u \\frac{\\partial v}{\\partial x} \n                                  -v \\frac{\\partial v}{\\partial y} \n                                  -\\frac{\\partial p}{\\partial y} = 0 \n\\end{eqnarray}\nOur goal in this step is to predict time-centered interface values of\nthe normal velocity ($u$ on x-edges and $v$ on y-edges).  The\nprediction follows from Taylor expanding the state to the interface\n(through $\\Delta x/2$ or $\\Delta y/2$) and to the half-time (through\n$\\Delta t/2$).  As with the regular advection, we can have left and\nright states which we will resolve by solving a Riemann problem.  The\nleft interface state of $u$ at $i+\\myhalf,j$ is found as:\n\\begin{eqnarray}\nu_{i+\\myhalf,j,L}^{n+\\myhalf} \n  &=& u_{i,j} \n    + \\frac{\\Delta x}{2} \\left . \\frac{\\partial u}{\\partial x} \\right |_{i,j}\n    + \\frac{\\Delta t}{2} \\left . \\frac{\\partial u}{\\partial t} \\right |_{i,j}\\\\\n  &=& u_{i,j} \n    + \\frac{\\Delta x}{2} \\left . \\frac{\\partial u}{\\partial x} \\right |_{i,j}\n    + \\frac{\\Delta t}{2} \\left . \\left (-u \\frac{\\partial u}{\\partial x}\n                                -v \\frac{\\partial u}{\\partial y}\n                                -\\frac{\\partial p}{\\partial x} \\right ) \\right |_{i,j}\\\\\n  &=& u_{i,j} \n    + \\frac{\\Delta x}{2} \\left ( 1 - \\frac{\\Delta t}{\\Delta x} u_{i,j} \\right )\n                         \\left .  \\frac{\\partial u}{\\partial x}\\right |_{i,j}\n    - \\frac{\\Delta t}{2} \\left ( v \\frac{\\partial u}{\\partial y}\\right )_{i,j}\n    - \\frac{\\Delta t}{2} \\left . \\frac{\\partial p}{\\partial x} \\right |_{i,j}\\\\\n\\end{eqnarray}\nWe express ${\\partial u}/{\\partial x} |_{i,j}$ as $\\Dux_{i,j} / \\Delta\nx$, where $\\Dux_{i,j}$ is the limited slope of $u$ in the\n$x$-direction in zone $i,j$.  Our interface state is then:\n\\begin{equation}\nu_{i+\\myhalf,j,L}^{n+\\myhalf} \n    = \\underbrace{u_{i,j} + \\frac{1}{2} \\left ( 1 - \\frac{\\Delta t}{\\Delta x} u \\right ) \\Dux_{i,j}}_{\\equiv \\hat{u}_{i+\\myhalf,j,L}^{n+\\myhalf}}\n    - \\underbrace{\\frac{\\Delta t}{2} \\left ( v \\frac{\\partial u}{\\partial y} \\right )_{i,j}}_{\\mathrm{transverse~term}}\n    - \\frac{\\Delta t}{2} \\left . \\frac{\\partial p}{\\partial x} \\right |_{i,j}\n\\label{eq:uMAC_l}\n\\end{equation} \n\n\nSimilarly, for $v$ through the $y$ faces, we find:\n\\begin{equation}\nv_{i,j+\\myhalf,L}^{n+\\myhalf} \n    = \\underbrace{v_{i,j} + \\frac{1}{2} \\left ( 1 - \\frac{\\Delta t}{\\Delta x} v \\right ) \\Dvy_{i,j}}_{\\equiv \\hat{v}_{i,j+\\myhalf,L}^{n+\\myhalf}}\n    - \\underbrace{\\frac{\\Delta t}{2} \\left ( u \\frac{\\partial v}{\\partial x} \\right )_{i,j}}_{\\mathrm{transverse~term}}\n    - \\frac{\\Delta t}{2} \\left . \\frac{\\partial p}{\\partial y} \\right |_{i,j}\n\\label{eq:vMAC_l}\n\\end{equation} \n\nAs indicated above (and following ABS and the similar notation used by\nColella~\\cite{colella:1990}), we denote the quantities that will be used to evaluate\nthe transverse states (consisting only of the normal predictor) with a\n`$\\hat{~}$'.  These will be used to evaluate the transverse terms\nlabeled above.\n\nWe predict $u$ and $v$ to both the $x$ and $y$ interfaces, using only\nthe normal part of the predictor.  This gives us the left and right `hat'\nstates on each interface.\n\n\\noindent $u$ on $x$-interfaces:\n\\begin{eqnarray}\n\\hat{u}_{i+\\myhalf,j,L}^{n+\\myhalf} &=&\n   u_{i,j} + \\frac{1}{2} \\left ( 1 - \\frac{\\Delta t}{\\Delta x} u_{i,j} \\right )\n       \\Dux_{i,j} \\\\ \n%\n\\hat{u}_{i+\\myhalf,j,R}^{n+\\myhalf} &=&\n   u_{i+1,j} - \\frac{1}{2} \\left ( 1 + \\frac{\\Delta t}{\\Delta x} u_{i+1,j} \\right )\n       \\Dux_{i+1,j} \n\\end{eqnarray}\n\n\\noindent $v$ on $x$-interfaces:\n\\begin{eqnarray}\n\\hat{v}_{i+\\myhalf,j,L}^{n+\\myhalf} &=&\n   v_{i,j} + \\frac{1}{2} \\left ( 1 - \\frac{\\Delta t}{\\Delta x} u_{i,j} \\right )\n       \\Dvx_{i,j} \\\\ \n%\n\\hat{v}_{i,j+\\myhalf,R}^{n+\\myhalf} &=&\n   v_{i+1,j} - \\frac{1}{2} \\left ( 1 + \\frac{\\Delta t}{\\Delta x} u_{i+1,j} \\right )\n       \\Dvx_{i+1,j} \n\\end{eqnarray}\n\n\\noindent $u$ on $y$-interfaces:\n\\begin{eqnarray}\n\\hat{u}_{i,j+\\myhalf,L}^{n+\\myhalf} &=&\n   u_{i,j} + \\frac{1}{2} \\left ( 1 - \\frac{\\Delta t}{\\Delta y} v_{i,j} \\right )\n       \\Duy_{i,j} \\\\ \n%\n\\hat{u}_{i,j+\\myhalf,R}^{n+\\myhalf} &=&\n   u_{i,j+1} - \\frac{1}{2} \\left ( 1 + \\frac{\\Delta t}{\\Delta y} v_{i,j+1} \\right )\n       \\Duy_{i,j+1} \n\\end{eqnarray}\n\n\\noindent $v$ on $y$-interfaces:\n\\begin{eqnarray}\n\\hat{v}_{i,j+\\myhalf,L}^{n+\\myhalf} &=&\n   v_{i,j} + \\frac{1}{2} \\left ( 1 - \\frac{\\Delta t}{\\Delta y} v_{i,j} \\right )\n       \\Dvy_{i,j} \\\\ \n%\n\\hat{v}_{i,j+\\myhalf,R}^{n+\\myhalf} &=&\n   v_{i,j+1} - \\frac{1}{2} \\left ( 1 + \\frac{\\Delta t}{\\Delta y} v_{i,j+1} \\right )\n       \\Dvy_{i,j+1} \n\\end{eqnarray}\n\nNote that the `right' state is constructed using the data to the right\nof the interface.  Also note that in constructing these transverse\nvelocities, we do not include the $p$ term.\n\nNext we find the advective velocity through each interface.  \nThe incompressible\nvelocity equation looks like the inviscid Burger's equation, and the\nRiemann solver follows that construction.  BCG provide the implementation\nused here (and throughout the incompressible literature).  Also see Toro~\\cite{toro:1997}.\nWe denote the resulting velocities with the `adv' superscript, as these\nare the normal velocities used to advect the hat states.  The Riemann\nproblem solution is:\n\\begin{equation}\n\\mathcal{R}(q_L,q_R) = \\left \\{ \\begin{array}{cl}\n   q_L  & \\mathrm{~if~} q_L > 0, \\qquad q_L + q_R > 0 \\\\\n   0          & \\mathrm{~if~} q_L \\le 0, \\qquad q_R \\ge 0 \\\\\n   q_R  & \\mathrm{~otherwise}\n  \\end{array}\n  \\right .\n\\end{equation}\nWe solve this for each of the normal velocities, giving:\n\\begin{eqnarray}\n\\hat{u}^\\mathrm{adv}_{i+\\myhalf,j} &=& \n    \\mathcal{R}(\\hat{u}_{i+\\myhalf,j,L}^{n+\\myhalf}, \\hat{u}_{i+\\myhalf,j,R}^{n+\\myhalf}) \\\\\n%\n\\hat{v}^\\mathrm{adv}_{i,j+\\myhalf} &=& \n    \\mathcal{R}(\\hat{v}_{i,j+\\myhalf,L}^{n+\\myhalf}, \\hat{v}_{i,j+\\myhalf,R}^{n+\\myhalf})\n\\end{eqnarray}\nThese advective velocities (sometimes called the {\\em transverse\n  velocities}) are used to resolve the left and right states of all the\nhat quantities by simple upwinding.  For a $u$ or $v$ state on the\n$x$-interface, we upwind based on $\\hat{u}^\\mathrm{adv}$; and for a\n$u$ or $v$ state on the $y$-interface, we upwind based on\n$\\hat{v}^\\mathrm{adv}$.  If we write the upwinding as:\n\\begin{equation}\n\\mathcal{U}[s^\\mathrm{adv}](q_L, q_R) =\n  \\left \\{\n  \\begin{array}{cl}\n  q_L                    & \\mathrm{~if~} s^\\mathrm{adv} > 0 \\\\\n  \\frac{1}{2}(q_L + q_R) & \\mathrm{~if~} s^\\mathrm{adv} = 0 \\\\\n  q_R                    & \\mathrm{~if~} s^\\mathrm{adv} < 0 \\\\\n  \\end{array}\n  \\right .\n\\end{equation}\nThen the interface states are:\n\\begin{eqnarray}\n\\hat{u}_{i+\\myhalf,j} &=& \\mathcal{U}[\\hat{u}_{i+\\myhalf,j}^\\mathrm{adv}]\n             (\\hat{u}_{i+\\myhalf,j,L}^{n+\\myhalf}, \\hat{u}_{i+\\myhalf,j,R}^{n+\\myhalf} ) \\\\\n\\hat{v}_{i+\\myhalf,j} &=& \\mathcal{U}[\\hat{u}_{i+\\myhalf,j}^\\mathrm{adv}]\n             (\\hat{v}_{i+\\myhalf,j,L}^{n+\\myhalf}, \\hat{v}_{i+\\myhalf,j,R}^{n+\\myhalf} ) \\\\\n\\hat{u}_{i,j+\\myhalf} &=& \\mathcal{U}[\\hat{v}_{i,j+\\myhalf}^\\mathrm{adv}]\n             (\\hat{u}_{i,j+\\myhalf,L}^{n+\\myhalf}, \\hat{u}_{i,j+\\myhalf,R}^{n+\\myhalf} ) \\\\\n\\hat{v}_{i,j+\\myhalf} &=& \\mathcal{U}[\\hat{v}_{i,j+\\myhalf}^\\mathrm{adv}]\n             (\\hat{v}_{i,j+\\myhalf,L}^{n+\\myhalf}, \\hat{v}_{i,j+\\myhalf,R}^{n+\\myhalf} ) \n\\end{eqnarray}\n\nNow we can construct the full left and right predictions for the\nnormal velocities on each interface (Eqs.~\\ref{eq:uMAC_l} and\n\\ref{eq:vMAC_l}).  This involves simply adding the transverse term to\nthe hat quantities and adding the pressure gradient.\n\\begin{equation}\nu_{i+\\myhalf,j,L}^{n+\\myhalf} = \\hat{u}_{i+\\myhalf,j,L}^{n+\\myhalf} \n    - \\frac{\\Delta t}{2} \n       \\left [ \\frac{1}{2} \\left (\\hat{v}^\\mathrm{adv}_{i,j-\\myhalf} +\n                                  \\hat{v}^\\mathrm{adv}_{i,j+\\myhalf} \\right )\n      \\right ]\n      \\left ( \\frac{\\hat{u}_{i,j+\\myhalf}^{n+\\myhalf} - \n                    \\hat{u}_{i,j-\\myhalf}^{n+\\myhalf}}{\\Delta y} \\right )\n    - \\frac{\\Delta t}{2} (Gp)^{(x),n-\\myhalf}_{i,j}\n\\end{equation}\nand\n\\begin{equation}\nv_{i,j+\\myhalf,L}^{n+\\myhalf} = \\hat{v}_{i,j+\\myhalf,L}^{n+\\myhalf} \n    - \\frac{\\Delta t}{2} \n       \\left [ \\frac{1}{2} \\left (\\hat{u}^\\mathrm{adv}_{i-\\myhalf,j} +\n                                  \\hat{u}^\\mathrm{adv}_{i+\\myhalf,j} \\right )\n      \\right ]\n      \\left ( \\frac{\\hat{v}_{i+\\myhalf,j}^{n+\\myhalf} - \n                    \\hat{v}_{i-\\myhalf,j}^{n+\\myhalf}}{\\Delta x} \\right )\n    - \\frac{\\Delta t}{2} (Gp)^{(y),n-\\myhalf}_{i,j}\n\\end{equation}\nHere $ (Gp)^{(x),n-\\myhalf}_{i,j}$ and $ (Gp)^{(y),n-\\myhalf}_{i,j}$ are\ndifference-approximations to $\\nabla p$ in the $x$ and $y$\ndirections respectively.  Note that they are lagged---these come from\nthe projection at the end of the previous timestep.  See BCG for\na discussion.  A similar\nconstruction is done for the right states at the interface.\n\nFinally, we do a Riemann solve (again, using the Burger's form of the\nRiemann problem) followed by upwinding to get the normal advective\nvelocities.  This is basically the $\\mathcal{R}(.,.)$ operation followed \nby $\\mathcal{U}(.,.)$.  Together, it is:\n\\begin{equation}\nu^\\mathrm{adv}_{i+\\myhalf,j} = \\left \\{ \\begin{array}{cl}\n   u_{i+\\myhalf,j,L}^{n+\\myhalf}  & \n         \\mathrm{~if~} u_{i+\\myhalf,j,L}^{n+\\myhalf} > 0, \\qquad \n                       u_{i+\\myhalf,j,L}^{n+\\myhalf} + u_{i+\\myhalf,j,R}^{n+\\myhalf} > 0 \\\\\n   \\frac{1}{2} \\left ( u_{i+\\myhalf,j,L}^{n+\\myhalf} +\n                       u_{i+\\myhalf,j,R}^{n+\\myhalf} \\right )  &\n          \\mathrm{~if~} u_{i+\\myhalf,j,L}^{n+\\myhalf} \\le 0, \\qquad \n                        u_{i+\\myhalf,j,R}^{n+\\myhalf} \\ge 0 \\\\\n   u_{i+\\myhalf,j,R}^{n+\\myhalf}  & \\mathrm{~otherwise}\n  \\end{array}\n  \\right .\n\\end{equation}\nand similar for $v^\\mathrm{adv}_{i,j+\\myhalf}$.  \nThese velocities are sometimes referred to as the MAC velocities.\n\n\n\\subsection{MAC projection}\n\nWe could simply use these time-centered advective velocities to\nconstruct the fluxes through the interfaces and update to the new time\nlevel.  However BCH showed that such a method is unstable for CFL $>\n0.5$.  The fix is to enforce the velocity constraint on these advective\nvelocities.  This involves projecting the velocity field onto the\nspace that is divergence free.  This projection is usually called the\nMAC projection.  Once the MAC-projected advective velocities are\ncomputed, we can reconstruct the interface states using this\ndivergence-free velocity field.  Figure~\\ref{fig:incomp:MACsolve}\nshows the location of the various quantities that participate\nit the MAC projection.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{MAC_solve}\n\\caption[MAC grid data centerings] {\\label{fig:incomp:MACsolve} The\n  centerings of the various components that make up the MAC\n  projection.}\n\\end{figure}\n\nThe divergence of the MAC velocities is cell-centered and constructed as:\n\\begin{equation}\n(D \\Ub)_{i,j} = \\frac{u^\\mathrm{adv}_{i+\\myhalf,j} - u^\\mathrm{adv}_{i-\\myhalf,j}}\n                   {\\Delta x} +\n              \\frac{v^\\mathrm{adv}_{i,j+\\myhalf} - v^\\mathrm{adv}_{i,j-\\myhalf}}\n                   {\\Delta y}\n\\end{equation}\nWe define a cell-centered $\\phi$.  $G\\phi$ will then be edge-centered\non a MAC grid, and $L\\phi = DG\\phi$ is again cell-centered.  Since $L =\nDG$, this makes the MAC projection an exact projection.\n\nWe solve\n\\begin{equation}\nL\\phi = D\\Ub\n\\end{equation}\nusing multigrid V-cycles and then update the MAC velocities as:\n\\begin{eqnarray}\nu^\\mathrm{adv}_{i+\\myhalf,j} &=& u^\\mathrm{adv}_{i+\\myhalf,j} - \n    \\frac{\\phi_{i+1,j} - \\phi_{i,j}}{\\Delta x} \\\\\n%\nv^\\mathrm{adv}_{i,j+\\myhalf} &=& v^\\mathrm{adv}_{i,j+\\myhalf} - \n    \\frac{\\phi_{i,j+1} - \\phi_{i,j}}{\\Delta y}\n\\end{eqnarray}\n\n\n\\subsection{Reconstruct interface states}\n\nNext we redo the construction of the interface states.  This procedure\nis identical to that above---construct the interface states\n$\\hat{u}_{L,R}$, $\\hat{v}_{L,R}$ on all edges, upwind based on\n$\\hat{u}^\\mathrm{adv}$, $\\hat{v}^\\mathrm{adv}$, and use these to\nconstruct the full states (including transverse terms).  Now however,\nwe construct the interface states of $u$ and $v$ on both the $x$ and\n$y$-interfaces (not just the normal component at each interface).\nFinally, instead of solving a Riemann problem to resolve the left and\nright states, we simply upwind using the MAC-projected\n$u^\\mathrm{adv}$ and $v^\\mathrm{adv}$.  This results in the interface\nstate $u^{n+\\myhalf}_{i+\\myhalf,j}$, $v^{n+\\myhalf}_{i+\\myhalf,j}$, $u^{n+\\myhalf}_{i,j+\\myhalf}$,\n$v^{n+\\myhalf}_{i,j+\\myhalf}$.\n\nThe only reason we need to do this step over, instead of using the\ninterface states that we predicted previously is we want to ensure\nthat they are consistent with the MAC-projected advective velocities\n(and therefore, consistent with the constraint).\n\n\\subsection{Provisional update}\n\nOnce we have the time-centered interface states that are consistent with\nthe MAC-projected advective velocities, we can update the velocities to\nthe new time by discretizing the advective terms ($\\Ub\\cdot \\nabla \\Ub$).\nWe express the advective terms for $u$ as $A^{(u),n+\\myhalf}_{i,j}$ and\nthose for $v$ as $A^{(v),n+\\myhalf}_{i,j}$.  These have the form:\n\\begin{eqnarray}\nA^{(u),n+\\myhalf}_{i,j} =   \n   && \\frac{1}{2}\n     \\left ( u^\\mathrm{adv}_{i-\\myhalf,j} + u^\\mathrm{adv}_{i+\\myhalf,j} \\right )\n     \\frac{u^{n+\\myhalf}_{i+\\myhalf,j} - u^{n+\\myhalf}_{i-\\myhalf,j}}{\\Delta x} + \\nonumber \\\\\n   && \\frac{1}{2}\n     \\left ( v^\\mathrm{adv}_{i,j-\\myhalf} + v^\\mathrm{adv}_{i,j+\\myhalf} \\right )\n     \\frac{u^{n+\\myhalf}_{i,j+\\myhalf} - u^{n+\\myhalf}_{i,j-\\myhalf}}{\\Delta y} \\\\\n%\nA^{(v),n+\\myhalf}_{i,j} = \n   && \\frac{1}{2}\n     \\left ( u^\\mathrm{adv}_{i-\\myhalf,j} + u^\\mathrm{adv}_{i+\\myhalf,j} \\right )\n     \\frac{v^{n+\\myhalf}_{i+\\myhalf,j} - v^{n+\\myhalf}_{i-\\myhalf,j}}{\\Delta x} \\nonumber +\\\\\n   && \\frac{1}{2}\n     \\left ( v^\\mathrm{adv}_{i,j-\\myhalf} + v^\\mathrm{adv}_{i,j+\\myhalf} \\right )\n     \\frac{v^{n+\\myhalf}_{i,j+\\myhalf} - v^{n+\\myhalf}_{i,j-\\myhalf}}{\\Delta y}\n\\end{eqnarray}\n\nThe normal update for $u$, $v$ would include the $Gp$ term and appear\nas:\n\\begin{eqnarray}\nu^{\\star}_{i,j} &=& u^n_{i,j} - \\Delta t A^{(u),n+\\myhalf}_{i,j} - \\Delta t (G p)^{(x),n-\\myhalf}_{i,j} \\\\\nv^{\\star}_{i,j} &=& v^n_{i,j} - \\Delta t A^{(v),n+\\myhalf}_{i,j} - \\Delta t (G p)^{(y),n-\\myhalf}_{i,j} \n\\end{eqnarray}\nNote that at this point, we don't have an updated $p$, so we use a lagged\nvalue from the previous step's projection.\n\nAlternately, we can note that for an exact projection, $Gp$, is the gradient\nof a scalar and would be removed by the projection, so we can omit it in\nthis update, giving an alternate provisional update:\n\\begin{eqnarray}\nu^{\\star\\star}_{i,j} &=& u^n_{i,j} - \\Delta t A^{(u),n+\\myhalf}_{i,j}  \\\\\nv^{\\star\\star}_{i,j} &=& v^n_{i,j} - \\Delta t A^{(v),n+\\myhalf}_{i,j}  \n\\end{eqnarray}\n\n\nFollowing the notation in Martin, we distinguish between these with\nan `$\\star$' vs.\\ `$\\star\\star$' \\footnote{Note that these are identical to\nABC~\\cite{ABC} approximation projections (1) and (2) (a quick look at ABC might\nsuggest the opposite, but note that their definition of $\\Ub^\\star$\nalready includes a $-Gp^{n-\\myhalf}$ term, so by explicitly adding it back\nin, you are dealing with the case where $\\Ub^\\star$ was updated without\nany $Gp^{n-\\myhalf}$ term, like the `$\\star\\star$' case above.)}.\n\n\n\\subsection{Approximate projection}\n\nThis provisional velocity field does not yet obey the constraint.  To\nenforce the constraint, we need to do a projection.  Here is where we\nhave the flexibility on whether to include the $G p^{n-\\myhalf}$ term.  If\nwe were doing an exact projection, then adding the gradient of a\nscalar would not change the divergence-free velocity field, so there\nwould be no need to add it.  \n\nBCH did an exact projection on a cell-centered grid.  There, the\ndivergence operator is:\n\\begin{equation}\n(D \\Ub)_{i,j} = \\frac{u_{i+1,j} - u_{i-1,j}}{2\\Delta x} +\n              \\frac{v_{i,j+1} - v_{i,j-1}}{2\\Delta y} \\label{eq:ccdiv}\n\\end{equation}\nThis gives a cell-centered $D\\Ub$.  If we want $\\phi$ cell-centered, then\nthe gradient, $G\\phi$ must also be cell centered so $L\\phi = DG\\phi$ is \ncell-centered.  This means that we must have\n\\begin{eqnarray}\n(G\\phi)^{(x)}_{i,j} &=& \\frac{\\phi_{i+1,j} - \\phi_{i-1,j}}{2\\Delta x} \\\\\n(G\\phi)^{(y)}_{i,j} &=& \\frac{\\phi_{i,j+1} - \\phi_{i,j-1}}{2\\Delta y}\n\\end{eqnarray}\nThe resulting Laplacian would then be a 5-point stencil that skips over\nthe immediate neighbors:\n\\begin{equation}\n(L\\phi)_{i,j} = \\frac{\\phi_{i+2,j} -2\\phi_{i,j} + \\phi_{i-2,j}}{4 \\Delta x^2} +\n    \\frac{\\phi_{i,j+2} -2\\phi_{i,j} + \\phi_{i,j-2}}{4 \\Delta y^2}\n\\end{equation}\nThis decomposes the domain into 4 distinct grids that are only linked \ntogether at the boundaries.  While an exact projection, this decoupling\ncan be undesirable.\n\nApproximate projections relax the idea that $L = DG$.  In an exact\nprojection, when you apply the projection operator, $P$, in\nsuccession, the result is unchanged ($P^2 = P$).  This is not the case\nfor an approximate projection.  As a result, exactly what form you\nproject matters.  For an approximate projection, we can use the \nstandard 5-point stencil for the Laplacian,\n\\begin{equation}\n(L\\phi)_{i,j} = \\frac{\\phi_{i+1,j} -2\\phi_{i,j} + \\phi_{i-1,j}}{\\Delta x^2} +\n    \\frac{\\phi_{i,j+1} -2\\phi_{i,j} + \\phi_{i,j-1}}{\\Delta y^2}\n\\end{equation}\ntogether with the cell-centered divergence above (Eq.~\\ref{eq:ccdiv}).\n\n\nRider~\\cite{rider} and Almgren, Bell, and Crutchfield (ABC)~\\cite{ABC}\nexplore various forms of what to project when doing the approximate\nprojection.  For instance, do we include the $Gp^{n-\\myhalf}$ term in the\nprovisional velocity or not?  Chorin noted that if viscosity is being\nmodeled, then it is necessary to include it here to get second-order\naccuracy.  Also, one can project the update to the velocity, $(\\Ub^\\star\n- \\Ub^n)/\\Delta t$ instead of just the new velocity, since $\\Ub^n$ should\nalready be divergence free.  Rider argues that projecting the update\nis not desirable with approximate projections, since any error in\n$\\Ub^n$ being divergence-free is carried forward to the new $\\Ub^{n+1}$.\nOne issue is that a cell-centered approximate projection cannot remove\nall sources of divergence (see Rider and Martin's PhD thesis).\n\nWhen projecting the new velocity, we scale by $\\Delta t$ to get a quantity\nthat has dimensions of pressure.  The procedure for the projection\ndiffers slightly depending on whether we project $\\Ub^\\star$ or $\\Ub^{\\star\\star}$:\n\\begin{itemize}\n\\item case I: projecting $\\Ub^{\\star}/\\Delta t$.  \n\nFrom the expression above,\nthis looks like:\n\\begin{equation}\n\\frac{\\Ub^\\star}{\\Delta t} = \\frac{\\Ub^n}{\\Delta t} - A^{(u),n+\\myhalf} - (G p)^{(x),n-\\myhalf}\n\\end{equation}\nIdeally, $\\Ub^n$ is already divergence free, and $Gp^{n-\\myhalf}$ is the\ngradient of a scalar, which will be removed, so the projection should\npick out the divergence free portion of $A^{(u)}$.  We solve:\n\\begin{equation}\nL\\phi = D (\\Ub^\\star/\\Delta t)\n\\end{equation}\nusing multigrid V-cycles.\nWe then find the new, divergence free velocity field as:\n\\begin{equation}\n\\Ub^{n+1} = \\Ub^\\star - \\Delta t G\\phi\n\\end{equation}\n\nSince we already included $Gp^{n-\\myhalf}$ in what we projected, $G\\phi$\nwill be the correction,\n\\begin{equation}\nG\\phi = Gp^{n+\\myhalf} - Gp^{n-\\myhalf}\n\\end{equation}\nor\n\\begin{equation}\nGp^{n+\\myhalf} = Gp^{n-\\myhalf} + G\\phi\n\\end{equation}\n(see Martin 2.5 or ABC).  We store this for the next timestep.\n\n\n\\item case II: projecting $\\Ub^{\\star\\star}/\\Delta t$.  \n\nFrom the expression above,\nthis looks like:\n\\begin{equation}\n\\frac{\\Ub^{\\star\\star}}{\\Delta t} = \\frac{\\Ub^n}{\\Delta t} - A^{(u),n+\\myhalf} \n\\end{equation}\nThere is no explicit $Gp^{n-\\myhalf}$ term.  We solve:\n\\begin{equation}\nL\\phi = D (\\Ub^{\\star\\star}/\\Delta t)\n\\end{equation}\nusing multigrid V-cycles.\nWe then find the new, divergence free velocity field as:\n\\begin{equation}\n\\Ub^{n+1} = \\Ub^{\\star\\star} - \\Delta t G\\phi\n\\end{equation}\n\nSince there was no $Gp^{n-\\myhalf}$ in what we projected, $p^{n+\\myhalf} = \\phi$,\nand \n\\begin{equation}\nGp^{n+\\myhalf} = G\\phi\n\\end{equation}\nWe store this for the next timestep.\n\\end{itemize}\n\nOne pathology of this form of the projection is that $(D\\Ub)_{i,j}$ does\nnot actually make use of the velocity field in zone $(i,j)$.  This\ndecoupling from the local zone can result in a checkerboarding pattern\nin the projected velocity field.\n\n\\section{Boundary conditions}\n\nFor the advection portion of the algorithm, the boundary conditions\non $u$ and $v$ are implemented in the usual way, using ghost cells.\nFor the projection, \n\nFor a periodic domain, the boundary conditions on $\\phi$ are likewise\nperiodic.  At a solid wall or inflow boundary, we already predicted\nthe velocity that we want at the wall (in the advection step), and we\ndo not want this value to change in the projection step.  Since the\ncorrection is:\n\\begin{equation}\n\\Ub^{n+1} = \\Ub^\\star - \\nabla \\phi\n\\end{equation}\nwe want $\\nabla \\phi \\cdot n = 0$.  \n\nAt outflow boundaries, we do not want to introduce any shear as we go\nthrough the boundary.  This means that we do not want any tangential\nacceleration.  Setting $\\phi = 0$ on the boundary enforces $\\nabla\n\\phi \\cdot t = 0$, where $t$ is the unit vector tangential to the\nboundary.\n\nSee ABS for a discussion of the boundary conditions.\n\n\\section{Bootstrapping}\n\nAt step 0, we do not have a value of $Gp^{-\\myhalf}$.  To get an initial value\nfor $Gp$, we run through the entire evolution algorithm starting with the\ninitial data.  At the end of a step, we reset $u$ and $v$ to the initial\nvalues and store the $Gp$ at the end of this step as $Gp^{-\\myhalf}$.\n\nIt is also common to precede this initialization by first projecting\nthe velocity field to ensure it is divergence free.  This way, we do\nnot have to rely on the initial conditions to always set a divergence\nfree velocity field.\n\n\n\\section{Test problems}\n\n\\subsection{Convergence test}\n\nMinion~\\cite{minion:1996} introduced a simple test problem with an analytic solution.\nThe velocity field is initialized as:\n\\begin{eqnarray}\nu(x,y) &=& 1 - 2 \\cos(2 \\pi x) \\sin(2 \\pi y) \\\\\nv(x,y) &=& 1 + 2 \\sin(2 \\pi x) \\cos(2 \\pi y)\n\\end{eqnarray}\nThe exact solution at some time $t$ is:\n\\begin{eqnarray}\nu(x,y,t) &=& 1 - 2 \\cos(2 \\pi (x - t)) \\sin(2 \\pi (y - t)) \\\\\nv(x,y,t) &=& 1 + 2 \\sin(2 \\pi (x - t)) \\cos(2 \\pi (y - t))\n\\end{eqnarray}\nMinion also gives the pressure, but this is not needed for the\nsolution.  This is run on a doubly-periodic unit square domain.  The\nmain utility of this set of initial conditions is that we can use the\nanalytic solution to measure the convergence behavior of the\nalgorithm. \n\n\n\\section{Extensions}\n\n\\begin{itemize}\n\n\\item {\\em Variable density incompressible}: Bell \\& Marcus\n  \\cite{BellMarcus} describe how to extend these methods to variable\n  density flows.  This means that the density in the domain may not be\n  constant, but within a fluid element, the density does not change.\n  This can arise, for instance, in modeling the Rayleigh-Taylor\n  instability.\n\n  The basic idea follows the method described above.  Now the \n  mass continuity equation is also evolved:\n  \\begin{equation}\n  \\frac{\\partial \\rho}{\\partial t} + \\nabla \\cdot (\\rho \\Ub) = 0\n  \\end{equation}\n  The density is predicted to the interfaces follow the same procedure\n  above and upwinded using the MAC velocities.\n  %\n  For the projection, the decomposition is written as:\n  \\begin{equation}\n  \\Ub = \\Ub^d + \\frac{1}{\\rho}\\nabla \\phi\n  \\end{equation}\n  and the elliptic equation is now a variable-coefficient equation:\n  \\begin{equation}\n  \\nabla \\cdot \\frac{1}{\\rho} \\nabla \\phi = \\nabla \\cdot \\Ub\n  \\end{equation}\n\n\\item {\\em Viscosity}: When viscosity is included in the system, our\n  momentum equation becomes:\n  \\begin{equation}\n  \\Ub_t + \\Ub\\cdot \\nabla \\Ub + \\nabla p = \\epsilon \\nabla^2 \\Ub\n  \\end{equation}\n  The solution process for this equation is largely the same.\n  Following BCH, first the advective term is computed by predicting\n  the velocities to the interfaces, doing the MAC projection, and then\n  forming $A^{n+\\myhalf}$.  Now there is an explicit viscosity term (at\n  time-level $n$) present in the prediction, as a source term.  The\n  provision velocity update is no longer a simple flux update, but\n  instead requires solving two decoupled diffusion-like equations (one\n  for $u$ and one for $v$).  These are differenced using\n  Crank-Nicolson centering:\n  \\begin{eqnarray}\n  \\frac{u^\\star - u^n}{\\Delta t} &=& - A^{(u),n+\\myhalf} - \\nabla p^{(x),n-\\myhalf} \n       + \\frac{\\epsilon }{2} \\nabla^2 (u^n + u^\\star) \\\\\n  \\frac{v^\\star - v^n}{\\Delta t} &=& - A^{(v),n+\\myhalf} - \\nabla p^{(y),n-\\myhalf} \n       + \\frac{\\epsilon }{2} \\nabla^2 (v^n + v^\\star) \n  \\end{eqnarray}\n  This involves two separate multigrid solves.  Once $\\Ub^\\star$ is found,\n  the final projection is done as usual.\n\n\\item {\\em Low Mach number combustion}: In low-Mach number combustion\n  flows, the fluid is allowed to respond to local heat release \n  by expanding.  The velocity constraint is derived by differentiating\n  the equation of state along particle paths, leading to the appearance\n  of a source term:\n  \\begin{equation}\n  \\nabla \\cdot \\Ub = S\n  \\end{equation}\n  Here, $S$, incorporates the compressibility effects due to the \n  heat release and diffusion.  This system is used when modeling\n  burning fronts (flames).  This type of flow can be thought\n  of as linking two incompressible states (the fuel and the ash)\n  by the expansion across the interface.\n\n  The solution technique largely follows that of the incompressible\n  flow.  One caveat, because the constraint now has a local heat\n  source, $S$, doing the cell-centered divergence described above\n  leads to a decoupling of $D\\Ub$ from the local source, since the\n  stencil of $D\\Ub$ does not include the zone upon which it is centered.\n\n  This system is described in detail in \\cite{pember-flame,\n    DayBell:2000,SNpaper}.\n\n\\item {\\em Stratified flows}: When the background is stratified, a\n  different velocity constraint can arise, capturing the expansion of\n  the fluid element due to the pressure change with altitude.  For example,\n  with an ideal gas, the equation of state can be recast as:\n  \\begin{equation}\n  \\nabla \\cdot (p_0^{1/\\gamma}) = 0\n  \\end{equation}\n  where $p_0(z)$ is the pressure as a function of height,\n  representing the hydrostatic background, and $\\gamma$ is the ratio\n  of specific heats.  For a general equation of state,\n  $p_0^(1/\\gamma)$ is replaced with something more complicated (see\n  \\cite{ABRZ:I,ABRZ:II,ABNZ:III} for the development of a low Mach\n  hydrodynamics method for stratified astrophysical flows).\n\n\\item {\\em Nodal projection}: instead of discretizing the final projection\n  using a cell-centered $\\phi$, ABS use a node-centered $\\phi$.  While\n  this is still an approximate projection, this discretization couples\n  in the zone we are centered on, and is said to be able to do a better\n  job removing pathological divergent velocity fields that cell-centered\n  projections stumble on.\n\n\\end{itemize}\n\n\n"
  },
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    "path": "incompressible/main.tex",
    "content": "\\documentclass[11pt]{article}\n\n% margins\n\\usepackage[margin=0.75in]{geometry}\n\n% figures\n\\usepackage{graphicx}\n\n% font\n\\usepackage{mathpazo}\n\n\\usepackage{helvet}\n\n% coloring\n\\usepackage{color}\n\\definecolor{mygray}{gray}{0.5}\n\n% footer\n\\usepackage{fancyhdr}\n\\pagestyle{fancy}\n\\fancyfoot[LO,LE]{\\footnotesize\\sffamily \\color{mygray} M.\\ Zingale---Notes on incompressible solvers}\n\\fancyfoot[RO,RE]{\\footnotesize \\sffamily  \\color{mygray} (\\today)}\n\\fancyfoot[CO,CE]{\\thepage}\n\\fancyhead{}\n\\renewcommand{\\headrulewidth}{0.0pt}\n\\renewcommand{\\footrulewidth}{0.0pt}\n\n% captions\n\\usepackage{caption}\n\\renewcommand{\\captionfont}{\\footnotesize}\n\\renewcommand{\\captionlabelfont}{\\footnotesize}\n\\setlength{\\captionmargin}{5em}\n\n\\usepackage{sectsty}\n\\allsectionsfont{\\sffamily}\n\n% shortcuts\n\\newcommand{\\Dux}{\\overline{\\Delta u}^{(x)}}\n\\newcommand{\\Duy}{\\overline{\\Delta u}^{(y)}}\n\n\\newcommand{\\Dvx}{\\overline{\\Delta v}^{(x)}}\n\\newcommand{\\Dvy}{\\overline{\\Delta v}^{(y)}}\n\n\\setlength{\\marginparwidth}{0.75in}\n\\newcommand{\\MarginPar}[1]{\\marginpar{\\vskip-\\baselineskip\\raggedright\\tiny\\sffamily\\hrule\\smallskip{\\color{red}#1}\\par\\smallskip\\hrule}}\n\n\n\n\n\\begin{document}\n\n\\begin{center}\n{\\LARGE \\textsf{\\textbf{\nNotes on incompressible solvers}}\n}\n\\end{center}\n\n\\begin{quote}\n\\noindent {\\em These summarize methods for solving the incompressible\n  hydrodynamics equations using an cell-centered approximate\n  projection method.}\n\\end{quote}\n\n\\input incompressible\n\n\\bibliographystyle{plain}\n\\bibliography{../refs}\n\n\n\\end{document}\n"
  },
  {
    "path": "instabilities/instabilities.tex",
    "content": "\\section{Dimensionless numbers}\n\n\nRe\n\nRa\n\nM\n\nPr\n\nRossby\n\nTaylor\n\nAtwood\n\n\nTurbulence and instabilities are ubitiquous in astrophysical flows.  \nA proper treatment of them requires resolving the important lengthscales\nFurthermore, it is three-dimensional\n\n\n\\section{Turbulence}\n\nAstrophysical flows are characterized by large Reynolds numbers.  \n\nTurbulent eddies cascade between scales, owing to the nonlinear advective \nterm in the momentum equation: $\\mathbf{U} \\cdot \\nabla \\mathbf{U}$.  \n\nBurgers' turbulence?\n\nintegral scale and intensity\n\ndifference between 2- and 3-d\n\nsubgrid scale modeling / bottleneck\nILES / LES / DNS\n\n\n\n\n\n\n\n\n\\section{Convection}\n\nMLT\n\nLedoux\n\ninternally driven vs. plates\n\n\n\n\n\n\\section{Rayleigh-Taylor}\n\nlinear growth eate\n\nalpha\n\nincompressible vs. compressible\n\n\n\n\\section{Kelvin-Helmholtz}\n\n\nGalilean invariance\n\n\n\n\\section{Reacting flow}\n\nLD\n\nfire-polishing length\n\n\n"
  },
  {
    "path": "intro/intro.tex",
    "content": "\\section{What is simulation?}\n\nAstronomy is an observational science.  Unlike in terrestrial physics,\nwe do not have the luxury of being able to build a model system and do\nphysical experimentation on it to understand the core physics.  We\nhave to take what nature gives us.  Simulation enables us to build a\nmodel of a system and allows us to do virtual experiments to\nunderstand how this system reacts to a range of conditions and\nassumptions.\n\nIt's tempting to think that one can download a simulation code, set a\nfew parameters, maybe edit some initial conditions, run, and then have\na virtual realization of some astrophysical system that you are\ninterested in.  Just like that.  In practice, it is not this simple.\nAll simulation codes make approximations---these start even before one\nturns to the computer, simply by making a choice of what equations are\nto be solved.  The main approximation that we will follow here, is the\n{\\em fluid approximation} (see Figure~\\ref{fig:intro:fluid_scale}).\nWe don't want to focus on the motions of the individual atoms, nuclei,\nelectrons, and photons in our system, so we work on a scale that is\nmuch larger than the mean free path of the system.  This allows us to\ndescribe the bulk properties of a fluid element, which in turn is\nsmall compared to the system of interest.\n\nWithin the fluid approximation, additional approximations are made,\nboth in terms of the physics included and how we represent a continuous\nfluid in the finite-memory of a computer (the {\\em discretization} process).\n\nTypically, we have a system of PDEs, and we need to\nconvert the continuous functional form of our system into a discrete\nform that can be represented in the finite memory of a computer.  This\nintroduces yet more approximation.\n\n%V\\&V\n\nBlindly trusting the numbers that come out of the code is a recipe\nfor disaster.  You don't stop being a physicist the moment you execute\nthe code---your job as a computational scientist is to make sure that\nthe code is producing reasonable results, by testing it against known\nproblems and your physical intuition.\n\nSimulations should be used to gain insight and provide a physical\nunderstanding.  Because the systems we solve are so nonlinear, small\nchanges in the code or the programming environment (compilers,\noptimization, etc.)  can produce large differences in the numbers\ncoming out of the code.  That's not a reason to panic.  As such it is\nbest not to obsess about precise numbers, but rather the trends our\nsimulations reveal.  To really understand the limits of your\nsimulations, you should do parameter and convergence studies.\n\nThere is no ``\\\"uber-code''.  Every algorithm begins with\napproximations and has limitations.  \nComparisons between different codes are important and common in our\nfield (see, for example,\n\\cite{frenk:1999,dimonte:2004,devalborro:2006}), and build confidence\nin the results that we are on the right track.\n\n%\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{fluid_scale}\n\\caption[The fluid scale.]{\\label{fig:intro:fluid_scale} The fluid\n  scale sits in an intermediate range---much smaller than the system\n  of interest (a star in this case), but much larger than the mean\n  free path.}\n\\end{figure}\n%\n\nTo really understand your simulations, you need to know what the code\nyour are using is doing under the hood.  This means understanding the\ncore methods used in our field.  These notes are designed to provide a\nbasic tour of some of the more popular methods, referring to the key\npapers for full derivations and details.  The best way to learn is to\ncode up these methods for yourself.  A companion python code, {\\sf\n  pyro} is available to help, and most of the exercises (or\ncorresponding figures) have links to simple codes that are part of the \\hydroex\\\nrepository\\footnote{look for the \\hydroexsymb\\ symbol.}. Descriptions\nand links to these codes are found in the appendices.\n\n\n\\section{Numerical basics}\n\nWe assume a familiarity with basic numerical methods, which we\nsummarize below.  Any book on numerical methods can provide a\ndeeper discussion of these methods.  Some good choices are the\ntexts by Garcia~\\cite{garcia}, Newman~\\cite{newman}, and Pang~\\cite{pang}.\n\n\\subsection{Sources of error}\n\nWith any algorithm, there are two sources of error we are concerned\nwith: {\\em roundoff error} and {\\em truncation error}.\n\nRoundoff arises from the error inherent in representing a floating\npoint number with a finite number of bits in the computer memory.  An\nexcellent introduction to the details of how computers represent\nnumbers is provided in \\cite{goldberg:1991}.\n\n\\begin{exercise}[Floating point]\nIn your choice of programming language, create a floating point\nvariable and initialize it to 0.1.  Now, print it out in full\nprecision (you may need to use format specifiers in your language to\nget all significant digits the computer tracks).  \n\nYou should see that it is not exactly 0.1 to the computer---this is\nthe floating point error.  The number 0.1 is not exactly representable\nin the binary format used for floating point.\n\\end{exercise}\n\n\n\\begin{exercise}[Machine epsilon]\nTo see roundoff error in action, write a program to find the value of\n$\\epsilon$ for which $1 + \\epsilon = 1$.  Start with $\\epsilon = 1$\nand iterate, halving $\\epsilon$ each iteration until $1 + \\epsilon =\n1$. This last value of $\\epsilon$ for which this was not true is\nthe {\\em machine epsilon}.  You will get a different value for\nsingle- vs.\\ double-precision floating point arithmetic.\n\\end{exercise}\n\nSome reorganization of algorithms can help minimize roundoff,\ne.g.\\ avoiding the subtraction of two very large numbers by factoring as:\n\\begin{equation}\nx^3 - y^3 = (x - y)(x^2 + xy + y^2) \\enskip ,\n\\end{equation}\nbut roundoff error will always be present at some level.\n\nTruncation error is a feature of an algorithm---we typically\napproximate an operator or function by expanding about some small\nquantity.  When we throw away higher-order terms, we are truncating\nour expression, and introducing an error in the representation.  If\nthe quantity we expand about truly is small, then the error is small.\nA simple example is to consider the Taylor series representation of\n$\\sin(x)$:\n\\begin{equation}\n\\sin(x) = \\sum_{n=1}^\\infty (-1)^{n-1} \\frac{x^{2n-1}}{(2n-1)!}\n\\end{equation}\nFor $|x| \\ll 1$, we can approximate this as:\n\\begin{equation}\n\\sin(x) \\approx x - \\frac{x^3}{6}\n\\end{equation}\nin this case, our truncation error has the leading term $\\propto x^5$,\nand we say that our approximation is $\\mathcal{O}(x^5)$, or\n$5^\\mathrm{th}$-order accurate.\n\n\\begin{exercise}[Convergence and order-of-accuracy]\nWe will be concerned with the order-of-accuracy of our methods, and a\ngood way to test whether our method is behaving properly is to perform\na convergence test.  Consider our $5^\\mathrm{th}$-order accurate\napproximation to $\\sin(x)$ above.  Pick a range of $x$'s ($< 1$), and\ncompute the error in our approximation as\n\\begin{equation*}\n\\epsilon \\equiv \\sin(x) - [  x - x^3/6 ]\n\\end{equation*}\nand show that as you cut $x$ in half, $|\\epsilon|$\nreduces by $2^5$---demonstrating $5^\\mathrm{th}$-order accuracy.\n\\end{exercise}\n\nThis demonstration of measuring the error as we vary the size\nof our small parameter is an example of a {\\em convergence test}.\n\n\\subsection{Differentiation and integration}\n\nFor both differentiation and integration, there are two cases we might\nencounter:\n\\begin{enumerate}\n\\item We have function values, $f_0$, $f_1$, $\\ldots$, at a discrete\n  number of points, $x_0$, $x_1$, $\\ldots$, and we want to compute the\n  derivative at a point or integration over a range of points.\n\\item We know a function analytically and we want to construct a\n  derivative or integral of this function numerically. \n\\end{enumerate}\n\n In these notes, we will mainly be concerned with the first case.\n\n\\subsubsection{Differentiation of discretely-sampled function}\n\\label{ch:intro:diff}\nConsider a collection of equally spaced points, labeled with an index\n$i$, with the physical spacing between them denoted $\\Delta\nx$.  \\MarginPar{show a figure} We can express the first derivative of\na quantity $a$ at $i$ as:\n\\begin{equation}\n\\left . \\frac{\\partial a}{\\partial x} \\right |_i \\approx \\frac{a_i - a_{i-1}}{\\Delta x}\n\\end{equation}\nor\n\\begin{equation}\n\\left . \\frac{\\partial a}{\\partial x} \\right |_i \\approx \\frac{a_{i+1} - a_i}{\\Delta x}\n\\end{equation}\n%\n(Indeed, as $\\Delta x \\rightarrow 0$, this is the definition of a\nderivative we learned in calculus.)  Both of these are {\\em one-sided\n  differences}.  By Taylor expanding the data about $x_i$, we see\n\\begin{equation}\na_{i+1} = a_i + \\Delta x \\left . \\frac{\\partial a}{\\partial x} \\right |_i + \\frac{1}{2} \\Delta x^2 \\left . \\frac{\\partial^2 a}{\\partial x^2} \\right |_i + \\ldots\n\\end{equation}\nSolving for ${\\partial a}/{\\partial x} |_i$, we see\n\\begin{align}\n\\left . \\frac{\\partial a}{\\partial x} \\right |_i &= \n    \\frac{a_i - a_{i-1}}{\\Delta x} - \\frac{1}{2}\\Delta x \\left . \\frac{\\partial^2 a}{\\partial x^2} \\right |_i + \\ldots \\\\ \n%\n &= \\frac{a_i - a_{i-1}}{\\Delta x} + \\mathcal{O}(\\Delta x)\n\\end{align}\nwhere $\\mathcal{O}(\\Delta x)$ indicates that the leading term in the\nerror for this approximation is $\\sim \\Delta x$\\footnote{in some\n  texts, you see this $\\mathcal{O}(\\Delta x^n)$ referred to as ``big O\n  notation''}.  We say that this is {\\em first order accurate}.  This\nmeans that we are neglecting terms that scale as $\\Delta x$ or to\nhigher powers.  This is fine if $\\Delta x$ is small.  This error is our\ntruncation error---just as discussed above, it arises because our\nnumerical approximation throws away higher order terms.  The\napproximation ${\\partial a}/{\\partial x} |_i = ({a_{i+1} -\n  a_i})/{\\Delta x}$ has the same order of accuracy.\n\n\\begin{exercise}[Truncation error]\n{Show that a centered difference,\n \\begin{equation*}\n\\left .\\frac{\\partial a}{\\partial x} \\right |_i = \\frac{a_{i+1} - a_{i-1}}{2 \\Delta x}\n\\end{equation*}\nis second order\naccurate, i.e.\\ its truncation error is $\\mathcal{O}(\\Delta x^2)$.}\n\\end{exercise}\n\n% this figure was created by hydro_examples/basics/derivatives/derivatives.py\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{derivs}\n\\caption[Difference approximations to the derivative of $\\sin(x)$]\n        {\\label{fig:derivs} A comparison of one-sided and centered\n          difference approximations to the derivative of $\\sin(x)$.\n\\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/basic_numerics/derivatives/derivatives.py}{derivatives.py}}}\n\\end{figure}\n\nFigure~\\ref{fig:derivs} shows the left- and right-sided first-order\ndifferences and the central difference as approximations to\n$\\sin(x)$. Generally speaking, higher-order methods have lower\nnumerical error associated with them, and also involve a wider range\nof data points.\n\nSecond- and higher-order derivatives can be constructed in the same fashion.\n\n\\begin{exercise}[Second derivative]\n{Using the Taylor expansions for $a_{i+1}$ and $a_{i-1}$, find a difference\napproximation to the second derivative at $i$.}\n\\end{exercise}\n\n\\subsubsection{Differentiation of an analytic function}\n\nAn alternate scenario is when you know the analytic form of the\nfunction, $f(x)$, and are free to choose the points where you evaluate\nit.  Here you can pick a $\\delta x$ and evaluate the derivative as\n\\begin{equation}\n\\left . \\frac{df}{dx} \\right |_{x=x_0} =  \\frac{f(x_0+\\delta x) - f(x_0)}{\\delta x}\n\\end{equation}\nAn optimal value for $\\delta x$ requires a balance of truncation error\n(which wants a small $\\delta x$) and roundoff error (which becomes\nlarge when $\\delta x$ is close to machine $\\epsilon$).\nFigure~\\ref{fig:deriv_error} shows the error for the numerical\nderivative of $f(x) = \\sin(x)$ at the point $x_0 = 1$, as a function\nof $\\delta x$.  A nice discussion of this is given in \\cite{yak}.  A\ngood rule-of-thumb is to pick $\\delta x \\approx \\sqrt{\\epsilon}$,\nwhere $\\epsilon$ is machine epsilon, to balance roundoff and\ntruncation error.\n\nComparing the result with different choices of $\\delta x$ allows for\nerror estimation and an improvement of the result by combining the\nestimates using $\\delta x$ and $\\delta x/2$ (this is the basis for a\nmethod called {\\em Richardson extrapolation}).\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.8\\linewidth]{deriv_error}\n  \\caption[Error in numerical derivatives] {\\label{fig:deriv_error}\n    Error in the numerical approximation of the derivative of $f(x) =\n    \\sin(x)$ at $x_0 = 1$ as a function of the spacing $\\delta x$.  For\n    small $\\delta x$, roundoff error dominates the error in the\n    approximation.  For large $\\delta x$, truncation error dominates. \\\\\n    \\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/basic_numerics/derivatives/deriv_error.py}{deriv\\_error.py}}}\n\\end{figure}\n\n\n\n\n\\subsubsection{Integration}\n\nIn numerical analysis, any integration method that is composed as a\nweighted sum of the function evaluated at discrete points is called a\n{\\em quadrature rule}.\n\nIf we have a function sampled at a number of\nequally-spaced points, $x_0 \\equiv a, x_1, \\ldots, x_N \\equiv\nb$\\footnote{Note that this is $N$ intervals and $N+1$ points}, we can\nconstruct a discrete approximation to an integral as:\n\\begin{equation}\nI \\equiv \\int_a^b f(x) dx \\approx \\Delta x \\sum_{i = 0}^{N-1} f(x_i)\n\\end{equation}\nwhere $\\Delta x \\equiv (b-a)/N$ is the width of the intervals (see the\ntop-left panel in Figure~\\ref{fig:integration}).  This is a very crude\nmethod, but in the limit that $\\Delta x \\rightarrow 0$ (or $N\n\\rightarrow \\infty$), this will converge to the true integral.  This\nmethod is called the {\\em rectangle rule}.  Note that here we\nexpressing the integral over the $N$ intervals using a simple\nquadrature rule in each interval.  Summing together the results of the\nintegral over each interval to get the result in our domain is called\n{\\em compound} integration.\n\nWe can get a more accurate answer for $I$ by interpolating between the\npoints.  The simplest case is to connect the sampled function values,\n$f(x_0), f(x_1), \\ldots, f(x_N)$ with a line, creating a trapezoid in\neach interval, and then simply add up the area of all of the\ntrapezoids:\n\\begin{equation}\nI \\equiv \\int_a^b f(x) dx \\approx\n  \\Delta x \\sum_{i = 0}^{N-1} \\frac{f(x_i) + f(x_{i+1})}{2}\n\\end{equation}\nThis is called the {\\em trapezoid rule} (see the top-right panel in\nFigure~\\ref{fig:integration}).  Note here we assume that the points\nare equally spaced.\n\nOne can keep going, but practically speaking, a quadratic interpolation\nis as high as one usually encounters.  Fitting a quadratic polynomial\nrequires three points.\n\n\\begin{exercise}[Simpson's rule for integration]\n\\label{ex:simpsons}\nConsider a function, $f(x)$, sampled at three equally-spaced points,\n$\\alpha, \\beta, \\gamma$, with corresponding function values $f_\\alpha,\nf_\\beta, f_\\gamma$.  Derive the expression for Simpson's rule by\nfitting a quadratic $\\hat{f}(x) = A(x - \\alpha)^2 + B(x - \\alpha) + C$\nto the three points (this gives you $A$, $B$, and $C$), and then\nanalytically integrating $\\hat{f}(x)$ in the interval\n$[\\alpha,\\gamma]$.  You should find\n\\begin{equation}\nI = \\frac{\\gamma-\\alpha}{6} (f_\\alpha + 4f_\\beta + f_\\gamma)\n\\end{equation}\nNote that $(\\gamma-\\alpha)/6 = \\Delta x/3$\n\\end{exercise}\n\nFor a number of samples, $N$, in $[a,b]$, we will\nconsider every two intervals together.  The resulting expression is:\n\\begin{align}\nI &\\equiv \\int_a^b f(x) dx\n\\approx  \\frac{\\Delta x}{3} \\sum_{i = 0}^{(N-2)/2} \\left [f(x_{2i}) + 4 f(x_{2i+1}) + f(x_{2i+2}) \\right ] \\\\\n&= f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + 2f(x_4) + \\ldots + \\nonumber 2f(x_{N-2}) + 4f(x_{N-1}) + f(x_N)\n\\end{align}\nThis method is called {\\em Simpson's rule}.  Note that for 2 intervals\n/ 3 sample points ($N=2$), we only have 1 term in the sum, $(N-2)/2 =\n0$, and we get the result derived in Exercise~\\ref{ex:simpsons}.\n\n\nFigure~\\ref{fig:integration} shows\nthese different approximations for the case of two intervals (three points).\n\n% these figures were created with figures/intro/integrals.py\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\linewidth]{rectangle}\n\\includegraphics[width=0.49\\linewidth]{trapezoid} \\\\\n\\includegraphics[width=0.49\\linewidth]{simpsons}\n\\begin{minipage}[b]{0.49\\linewidth}\n\\caption[Integration rules]{\\label{fig:integration} The rectangle rule\n  (top left), trapezoid rule (top right) and Simpson's rule (left) for\n  integration.}\n\\end{minipage}\n\\end{figure}\n\n\nAnalogous expressions exist for the case of unequally-spaced points.\n\nThe compound trapezoid rule converges as second-order over the\ninterval $[a,b]$, while Simpson's rule converges as fourth-order.\n\nAs with differentiation, if you are free to pick the points where you\nevaluate $f(x)$, you can get a much higher-order accurate result.\n{\\em Gaussian quadrature} is a very powerful technique that uses the\nzeros of different polynomials as the evaluation points for the\nfunction to give extremely accurate results.  See the text by\nGarcia~\\cite{garcia} for a nice introduction to these methods.\n\n\n\\subsection{Root finding}\n\nOften we want to find the root of a function (or the vector that zeros\na vector of functions).  The most popular method for root finding is\nthe {\\em Newton-Raphson method}.  We want to find $x$, such that $f(x)\n= 0$.  Start with an initial guess $x_0$ that you believe is close to\nthe root, then you can improve the guess to the root by an amount\n$\\delta x$ by looking at the Taylor expansion about $x_0$:\n\\begin{equation}\nf(x_0 + \\delta x) \\sim f(x_0) + f^\\prime(x_0) \\delta x + \\ldots = 0\n\\end{equation}\nKeeping only to $\\mathcal{O}(\\delta x)$, we can solve for the correction, $\\delta x$:\n\\begin{equation}\n\\label{eq:intro:newtonsmethod}\n  \\delta x = -\\frac{f(x_0)}{f^\\prime(x_0)}\n\\end{equation}\nThis can be used to correct out guess as $x_0 \\leftarrow x_0 + \\delta\nx$, and we can iterate on this procedure until $\\delta x$ falls below\nsome tolerance.  Figure~\\ref{fig:newtonsmethod} illustrates this\niteration.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\linewidth]{newton_00}\n\\includegraphics[width=0.49\\linewidth]{newton_01} \\\\[1em]\n\\includegraphics[width=0.49\\linewidth]{newton_02}\n\\includegraphics[width=0.49\\linewidth]{newton_03}\n\\caption[Convergence of Newton's method for root\n  finding]{\\label{fig:newtonsmethod} The convergence of Newton's\n  method for finding the root.  In each pane, the red point is the\n  current guess for the root.  The solid gray line is the\n  extrapolation of the slope at the guess to the $x$-axis, which\n  defines the next approximation to the root.  The vertical dotted\n  line to the function shows the new slope that will be used for\n  extrapolation in the next iteration. \\\\\n\\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/basic_numerics/roots/roots_plot.py}{roots\\_plot.py}}}\n\\end{figure}\n\n\nThe main ``got-ya'' with this technique is that you need a good initial\nguess.  In the Taylor expansion, we threw away the $\\delta x^2$ term,\nbut if our guess was far from the root, this (and higher-orders) term\nmay not be small.  Obviously, the derivative must be non-zero in the\nregion around the root that you search as well.\n\n\\begin{exercise}[Newton's method]\n{Code up Newton's method for finding the root of a function $f(x)$\nand test it on several different test functions.}\n\\end{exercise}\nThe {\\em secant method} for root finding is essentially\nNewton-Raphson, but instead of using an analytic derivative,\n$f^\\prime$ is estimated as a numerical difference.\n\nNewton's method has pathologies---it is possible to get into a cycle\nwhere you don't converge but simply pass through the same set of root\napproximations.  Many other root finding methods exist, including {\\em\n  bisection}, which iteratively halves an interval known to contain a\nroot by looking for the change in sign of the function $f(x)$.  {\\em\n  Brent's method} combines several different methods to produce a\nrobust procedure for root-finding.  Most numerical analysis texts will\ngive a description of these.\n\n\\subsection{Norms}\n\n\\label{intro:sec:norm}\n\nOften we will need to measure the ``size'' of an error to a discrete\napproximation.  For example, imagine that we know the exact function,\n$f(x)$ and we have have an approximation, $f_i$ defined at $N$ points,\n$i = 0, \\ldots, N-1$.  The error at each point $i$ is $\\epsilon_i =\n|f_i - f(x_i)|$.  But this is $N$ separate errors---we want a single\nnumber that represents the error of our approximation.  This is the\njob of a vector norm.  \n\nThere are many different norms we can define.  For a vector ${\\bf q}$,\nwe write the norm as $\\|q\\|$.  Often, we'll put a subscript after the\nnorm brackets to indicate which norm was used.  Some popular norms\nare:\n\\begin{itemize}\n\\item {\\em inf norm}: \n  \\begin{equation} \n    \\| q \\|_\\infty = \\max_i |q_i|\n  \\end{equation}\n\n\\item {\\em L1 norm}:\n  \\begin{equation}\n    \\| q \\|_1 = \\frac{1}{N} \\sum_{i=0}^{N-1} |q_i|\n  \\end{equation}\n\n\\item {\\em L2 norm}:\n  \\begin{equation}\n    \\| q \\|_2 = \\left [ \\frac{1}{N} \\sum_{i=0}^{N-1} |q_i|^2 \\right ]^{1/2}\n  \\end{equation}\n\n\\item {\\em general p-norm}\n  \\begin{equation}\n    \\| q \\|_p = \\left [ \\frac{1}{N} \\sum_{i=0}^{N-1} |q_i|^p \\right ]^{1/p}\n  \\end{equation}\n\n\\end{itemize}\n\nNote that these norms are defined such that they are normalized---if\nyou double the number of elements ($N$), the normalization gives you a\nnumber that can still be meaningfully compared to the smaller set.\nFor this reason, we will use these norms when we look at the\nconvergence with resolution of our numerical methods.\n\nWe'll look into how the choice of norm influences you convergence\ncriterion, but inspection shows that the inf-norm is local---a element\nin your vector is given the entire weight, whereas the other norms are\nmore global.\n\n\\subsection{ODEs}\n\nConsider a system of first-order ordinary differential equations,\n\\begin{equation}\n\\dot{\\bf y} = {\\bf f}(t, {\\bf y}(t))\n\\end{equation}\nIf $k$ represents an index into the vector ${\\bf y}$, then the\n$k^\\mathrm{th}$ ODE is\n\\begin{equation}\n\\dot{y}_k = f_k(t, {\\bf y}(t))\n\\end{equation}\nWe want to solve for the vector ${\\bf y}$ as a function of time.  Note\nhigher-order ODEs can always be written as a system of first-order\nODEs by introducing new variables\\footnote{As an example, consider\n  Newton's second law, $\\ddot{x} = F/m$.  We can write this as a system\n  of two ODEs by introducing the velocity, $v$, giving us: $\\dot{v} =\n  F/m$, $\\dot{x} = v$.}.\n\nBroadly speaking, methods for integrating ODEs can be broken down into\nexplicit and implicit methods.  Explicit methods difference the system\nto form an update that uses only the current (and perhaps previous)\nvalues of the dependent variables in evaluating $f$.  For example, a\nfirst-order explicit update ({\\em Euler's method}) appears as:\n\\begin{equation}\ny^{n+1}_k = y^n_k + \\Delta t f_k(t^n, {\\bf y}^n)\n\\end{equation}\nwhere $\\Delta t$ is the stepsize.  Implicit methods instead evaluate\nthe righthand side, ${\\bf f}$ using the new-time value of ${\\bf y}$\nwe are solving for.\n\nPerhaps the most popular explicit method for ODEs is the 4th-order\nRunge-Kutta method (RK4).  This is a multistage method where several\nextrapolations to the midpoint and endpoint of the interval are\nmade to estimate slopes and then a weighted average of these slopes\nare used to advance the solution.  The various slopes\nare illustrated in Figure~\\ref{fig:rk} and the overall procedure looks like:\n\\begin{equation}\ny_k^{n+1} = y_k^n + \\frac{\\Delta t}{6} (k_1 + 2 k_2 + 2 k_3 + k_4)\n\\end{equation}\nwhere the slopes are:\n\\begin{align}\nk_1 &= f(t^n, y_k^n) \\\\\nk_2 &= f(t^n + \\tfrac{\\Delta t}{2}, y_k^n + \\tfrac{\\Delta t}{2} k_1) \\\\\nk_3 &= f(t^n + \\tfrac{\\Delta t}{2}, y_k^n + \\tfrac{\\Delta t}{2} k_2) \\\\\nk_4 &= f(t^n + \\Delta t, y_k^n + \\Delta t k_3)\n\\end{align}\nNote the similarity to Simpson's method for integration.  This is\nfourth-order accurate overall.\n\n\\afterpage{\\clearpage}\n% rk4_plot.py creates rk4_k[1-4].pdf, rk4_final.pdf\n\\begin{figure}[p]\n\\centering\n\\includegraphics[width=0.48\\linewidth]{rk4_k1} \n\\includegraphics[width=0.48\\linewidth]{rk4_k2} \\\\\n\\includegraphics[width=0.48\\linewidth]{rk4_k3} \n\\includegraphics[width=0.48\\linewidth]{rk4_k4} \\\\\n\\includegraphics[width=0.6\\linewidth]{rk4_final}\n%\n\\caption[The $4^\\mathrm{th}$-order Runge-Kutta method] {\\label{fig:rk}\n  A graphical illustration of the four steps in the\n  $4^\\mathrm{th}$-order Runge-Kutta method.  This example\n  is integrating $dy/dt = -y$.}\n\\end{figure}\n\n\n\\begin{exercise}[ODE accuracy]\nConsider the orbit of Earth around the Sun.  If we work in the units\nof astronomical units, years, and solar masses, then Newton's gravitational\nconstant and the solar mass together are simply $G M = 4\\pi^2$ (this\nshould look familiar as Kepler's third law).  We\ncan write the ODE system describing the motion of Earth as:\n\\begin{align}\n\\dot{\\bf x} &= {\\bf v} \\\\\n\\dot{\\bf v} &= -\\frac{G M {\\bf r}}{r^3}\n\\end{align}\nIf we take the coordinate system such that the Sun is at the origin,\nthen, ${\\bf x} = (x, y)^\\intercal$ is the position of the Earth\nand ${\\bf r} = x {\\bf \\hat{x}} + y {\\bf \\hat{y}}$ is the radius\nvector pointing from the Sun to the Earth.\n\nTake as initial conditions the planet at perihelion:\n\\begin{align*}\nx_0 &= 0 \\\\\ny_0 &= a (1-e) \\\\\n({\\bf v}\\cdot {\\bf \\hat{x}})_0 &= -\\sqrt{\\frac{GM}{a} \\frac{1+e}{1-e}} \\\\\n({\\bf v}\\cdot {\\bf \\hat{y}})_0 &= 0\n\\end{align*}\nwhere $a$ is the semi-major axis and $e$ is the eccentricity of the orbit\n(these expressions can be found in any introductory astronomy text that\ndiscusses Kepler's laws).\n\nIntegrate this system for a single orbital period with the first-order\nEuler and the RK4 method and measure the convergence by integrating at\na number of different $\\Delta t$'s.  Note: you'll need to define some\nmeasure of error, you can consider a number of different metrics, e.g.,\nthe change in radius after a single orbit.\n\\end{exercise}\n\\MarginPar{show figure}\n\nThe choice of the stepsize, $\\Delta t$, in the method greatly affects\nthe accuracy.  In practice you want to balance the desire for accuracy\nwith the expense of taking lots of small steps.  A powerful technique\nfor doing this is to use error estimation and an adaptive stepsize with\nyour ODE integrator.  This monitors the size of the truncation error\nand adjusts the stepsize, as needed, to achieve a desired accuracy.  A\nnice introduction to how this works for RK4 is given in \\cite{garcia}.\n\n\\subsubsection{Implicit methods}\n\nImplicit methods difference the system in a way that includes the new\nvalue of the dependent variables in the evaluation of $f_k(t,{\\bf\n  y}(t))$---the resulting implicit system is usually solved using, for\nexample, Newton-Raphson iteration.\n\nA first-order implicit update\n(called {\\em backward Euler}) is:\n\\begin{equation}\n  \\label{eq:backwardeuler}\n{\\bf y}^{n+1} = {\\bf y}^n + \\Delta t {\\bf f}(t^{n+1}, {\\bf y}^{n+1})\n\\end{equation}\nThis is more complicated to solve than the explicit methods above, and\ngenerally will require some linear algebra.  If we take $\\Delta t$ to\nbe small, then the change in the solution, $\\Delta {\\bf y}$ will be\nsmall as well, and we can Taylor-expand the system.\n\nTo solve this, we pick a guess, ${\\bf y}^{n+1}_0$, that we think is\nclose, to the solution and we will solve for a correction, $\\Delta\n{\\bf y}_0$ such that\n\\begin{equation}\n\\label{eq:impl_correction}\n  {\\bf y}^{n+1} = {\\bf y}^{n+1}_0 + \\Delta {\\bf y}_0\n\\end{equation}\nUsing this approximation, we can expand the righthand side vector,\n\\begin{equation}\n\\label{eq:impl_rhs_expansion}\n{\\bf f}(t^{n+1},{\\bf y}^{n+1}) = {\\bf f}(t^{n+1}, {\\bf y}^{n+1}_0) +\n     \\left . \\frac{\\partial {\\bf f}}{\\partial {\\bf y}} \\right |_0 \\Delta {\\bf y}_0 + \\ldots\n\\end{equation}\nHere we recognize the Jacobian matrix, ${\\bf J} \\equiv \\partial {\\bf\n  f}/\\partial {\\bf y}$,\n\\begin{equation}\n\\renewcommand\\arraystretch{1.5}\n{\\bf J} = \\left (\n\\begin{array}{ccccc}\n\\partial f_1/\\partial y_1 & \\partial f_1/\\partial y_2 &\n\\partial f_1/\\partial y_3 & \\ldots & \\partial f_1/\\partial y_n \\\\\n%\n\\partial f_2/\\partial y_1 & \\partial f_2/\\partial y_2 &\n\\partial f_2/\\partial y_3 & \\ldots & \\partial f_2/\\partial y_n \\\\\n%\n\\vdots  & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n%\n\\partial f_n/\\partial y_1 & \\partial f_n/\\partial y_2 &\n\\partial f_n/\\partial y_3 & \\ldots & \\partial f_n/\\partial y_n \\\\\n\\end{array}\n\\right )\n\\end{equation}\n%\nPutting Eqs.~\\ref{eq:impl_correction} and \\ref{eq:impl_rhs_expansion}\ninto Eq.~\\ref{eq:backwardeuler}, we have:\n\\begin{equation}\n{\\bf y}^{n+1}_0 + \\Delta {\\bf y}_0 = {\\bf y}^n + \\Delta t \\left [\n  {\\bf f}(t^{n+1}, {\\bf y}^{n+1}_0) +\n     \\left .{\\bf J}\\right |_0\\, \\Delta {\\bf y}_0 \\right ]\n\\end{equation}\nWriting this as a system for the unknown correction, $\\Delta {\\bf\n  y}_0$, we have\n\\begin{equation}\n  \\left ({\\bf I} - \\Delta t \\left . {\\bf J} \\right |_0 \\right ) \\Delta {\\bf y}_0 =\n   {\\bf y}^n - {\\bf y}_0^{n+1} + \\Delta t {\\bf f}(t^{n+1}, {\\bf y}^{n+1}_0)\n\\end{equation}\nThis is a linear system (a matrix $\\times$ vector $=$ vector) that can\nbe solved using standard matrix techniques (numerical methods for\nlinear algebra can be found in most numerical analysis texts).  After\nsolving, we can correct our initial guess:\n\\begin{equation}\n  {\\bf y}^{n+1}_1 = {\\bf y}^{n+1}_0 + \\Delta {\\bf y}_0\n\\end{equation}\n\nWritten this way, we see that we can iterate.  To kick things off, we\nneed a suitable guess---an obvious choice is ${\\bf y}^{n+1}_0 = {\\bf\n  y}^n$.  Then we correct this guess by iterating, with the $k$-th\niteration looking like:\n\\begin{align}\n  \\left ({\\bf I} - \\Delta t \\left . {\\bf J} \\right |_{k-1} \\right ) \\Delta {\\bf y}_{k-1} &=\n        {\\bf y}^n - {\\bf y}_{k-1}^{n+1} + \\Delta t {\\bf f}(t^{n+1}, {\\bf y}^{n+1}_{k-1}) \\\\\n  {\\bf y}^{n+1}_k &= {\\bf y}^{n+1}_{k-1} + \\Delta {\\bf y}_{k-1}\n\\end{align}\nWe will iterate until we find $\\| \\Delta {\\bf y}_k\\| < \\epsilon \\|\n{\\bf y}^n \\|$.  Here $\\epsilon$ is a small tolerance, and we use ${\\bf\n  y}^n$ to produce a reference scale for meaningful comparison.  Note\nthat here we use a vector norm to give a single number for comparison.\n\nNote that the role of the Jacobian here is the same as the first\nderivative in the scalar Newton's method for root finding\n(Eq.~\\ref{eq:intro:newtonsmethod})---it points from the current guess\nto the solution.  Sometimes an approximation to the Jacobian, which is\ncheaper to evaluate, may work well enough for the method to converge.\n\nExplicit methods are easier to program and run faster (for a given $\n\\Delta t$), but implicit methods work better for {\\em stiff}\nproblems---those characterized by widely disparate timescales over\nwhich the solution changes~\\cite{byrnehindmarsh}\\footnote{Defining\n  whether a problem is stiff can be tricky (see \\cite{byrnehindmarsh}\n  for some definitions).  For a system of ODEs, a large range in the\n  eigenvalues of the Jacobian usually means it is stiff.}.  A good\nexample of this issue in astrophysical problems is with nuclear\nreaction networks (see, e.g., \\cite{timmes_networks}).  As with the\nexplicit case, higher-order methods exist that can provide better\naccuracy at reduced cost.\n\n\n\n\\subsection{FFTs}\n\n\\label{sec:intro:ffts}\n\nThe discrete Fourier transform converts a discretely-sampled function\nfrom real space to frequency space, and identifies the amount of power\nassociate with discrete wavenumbers.  This is useful for both\nanalysis, as well as for solving certain linear problems (see, e.g.,\n\\S~\\ref{elliptic:sec:fft}).\n\nFor a function, $f(x)$ sampled at $N$ equally-spaced points (such that\n$f_n = f(x_n)$), the discrete Fourier transform, $\\mathcal{F}_k$ is\nwritten as:\n\\begin{equation}\n\\mathcal{F}_k = \\sum_{n=0}^{N-1} f_n e^{-2\\pi i n k /N} \\qquad k \\in [0, N-1]\n\\end{equation}\nThe exponential in the sum brings in a real (cosine terms, symmetric\nfunctions) and imaginary (sine terms, antisymmetric functions) part.\n\\begin{align}\n\\mathrm{Re}(\\mathcal{F}_k) &= \\sum_{n=0}^{N-1} f_n \\cos\\left(\\frac{2\\pi n k}{N}\\right) \\\\\n\\mathrm{Im}(\\mathcal{F}_k) &= \\sum_{n=0}^{N-1} f_n \\sin\\left(\\frac{2\\pi n k}{N}\\right)\n\\end{align}\nAlternately, it is sometimes useful to combine the real and imaginary\nparts into an amplitude and phase.\n\nThe inverse transform is:\n\\begin{equation}\nf_n = \\frac{1}{N} \\sum_{k=0}^{N-1} \\mathcal{F}_k e^{2\\pi i n k /N} \\qquad n \\in [0, N-1]\n\\end{equation}\nThe $1/N$ normalization is a consequence of Parseval's theorem---the\ntotal power in real space must equal the total power in frequency\nspace.  One way to see that this must be the case is to consider the\ndiscrete transform of $f(x) = 1$, which should be a delta function.\n\nThe FFT of a discrete function has the same amount of information as\nthe original discrete function.  Note that if $f(x)$ is real-valued,\nthen the transform $\\mathcal{F}_k$ has 2 numbers (the real and\nimaginary parts) for each of our original $N$ real values.  This would\nmean that we have $2N$ pieces of information in frequency space where\nwe only had $N$ pieces of information in real space.  Since we cannot\ncreate information in frequency space where there was no corresponding\nreal-space information, half of the $\\mathcal{F}_k$'s are redundant\n(and $\\mathcal{F}_{-k} = \\mathcal{F}^\\star_k$).\n\nDirectly computing $\\mathcal{F}_k$ for each $k$ takes $\\mathcal{O}(N^2)$\noperations.  The fast Fourier transform (FFT) is a reordering of the\nsums in the discrete Fourier transform to reuse partial sums and compute\nthe same result in $\\mathcal{O}(N\\log N)$ work.  Many numerical methods\nbooks can give a good introduction to how to design an FFT algorithm.\n\n\\begin{exercise}[FFTs]\n{Learn how to use a FFT library or the built-in FFT method in your\n  programming language of choice.  There are various ways to define\n  the normalization in the FFT, that can vary from one library to the\n  next.  To ensure that you are doing things correctly, compute the\n  following transforms:\n  \\begin{itemize}\n  \\item $\\sin(2\\pi k_0 x)$ with $k_0 = 0.2$.  The transform should\n    have all of the power in the imaginary component only at the \n    frequency $0.2$.\n\n  \\item $\\cos(2\\pi k_0 x)$ with $k_0 = 0.2$.  The transform should\n    have all of the power in the real component only at the \n    frequency $0.2$.\n\n  \\item $\\sin(2\\pi k_0 x + \\pi/4)$ with $k_0 = 0.2$.  The transform should\n    have equal power in the real and imaginary components, only at the \n    frequency $0.2$.  Since the power is 1, the amplitude of the real and\n    imaginary parts will be $1/\\sqrt{2}$.\n  \\end{itemize}\n\nAn example of the transform of a single-frequency sine wave with a\nphase shift is shown in Figure~\\ref{fig:intro:fft}.\n\n}\n\\end{exercise}\n\nThe FFT assumes that a function is periodic and that the points are\nevenly spaced.  If the function is not periodic, then a signal will\nshow up in the FFT at a wavenumber corresponding to the size of the\ndomain.  Related to this, you need to ensure that your points are\nevenly spaced even at the boundary, e.g., if you have a function on\n$[0, 1]$ represented by 5 points, you want the points to be $\\{0, 0.2,\n0.4, 0.6, 0.8\\}$ and not $\\{0, 0.25, 0.5, 0.75, 1\\}$.  In the latter\ncase, $0$ and $1$ are the same function value (due to periodicity),\nbut the FFT assumes that all points are evenly spaced (even across the\nboundary), so it will think that there is a spacing of $0.25$ between\nthese end points.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{fft-sine-phase}\n\\caption[Fourier transform of $f(x) = \\sin(2\\pi k_0 x +\n  \\pi/4)$]{\\label{fig:intro:fft} The Fourier transform of a sine wave\n  with a phase of $\\pi/4$, $f(x) = \\sin(2\\pi k_0 x + \\pi/4)$ with $k_0\n  = 0.2$.  The top shows the original function.  The second panel\n  shows the real and imaginary components---we see all of the power is\n  at our input wavenumber, split equally between the real and\n  imaginary parts.  The third pane shows the power (the absolute value\n  of the transform).  Finally, the bottom panel shows the inverse\n  transform of our transform, giving us back our original function. \\\\\n  \\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/basic_numerics/FFT/fft_simple_examples.py}{fft\\_simple\\_examples.py}}}\n\\end{figure}\n\n\n\n\n%% \\section{A Sample of Public Codes}\n\n%% There are a large number of freely-available codes for modeling astrophysical\n%% flows.\n"
  },
  {
    "path": "low_mach/low_mach.tex",
    "content": "\\newcommand{\\teta}{\\tilde{\\eta}}\n\nIncompressible flow represents the zero-Mach number limit of fluid\nflow---no compressibility effects are modeled.  We can extend the\nideas of incompressibe flow to allow us to model some compressibility\neffects, giving rise to low Mach number methods.\n\n\\section{Low Mach divergence constraints}\n\\label{ch:lm:constraints}\n\nThe key idea in solvers for low Mach number flows is that, as a fluid\nelement advects, its pressure remains the same as the background\nstate.  For an atmosphere, this background state is a hydrostatic\nprofile.  For a smallscale combustion flow, this background state is a\nspatially constant pressure (and constant-in-time if the domain is\nopen).  We'll denote this background pressure as $p_0(r,t)$, where \n$r$ is a radial coordinate.\n\nPreviously, we derived the pressure evolution with reactive sources\n(see Eq.~\\ref{eq:reacting:p}) by following the evolution of pressure\nas a fluid element moves (e.g., the Lagrangian derivative).  For the \nlow Mach flows, we want to see the evolution $p_0$ in the fluid\nelement, which is expressed as:\n\\begin{equation}\n\\ddt{p_0} + \\Ub \\cdot \\nabla p_0 + \\Gamma_1 p_0 \\nabla \\cdot \\Ub = \\Gamma_1 p_0 \\sigma \\Hnuc\n\\end{equation}\nWe can rewrite this as a constraint on the velocity field by solving for $\\nabla \\cdot \\Ub$:\n\\begin{equation}\n  \\nabla \\cdot \\Ub + \\frac{1}{\\Gamma_1 p_0}\\frac{Dp_0}{Dt} = \\sigma \\Hnuc \\equiv S\n\\end{equation}\nwith\n\\begin{equation}\n\\sigma = \\frac{\\partial p_0/\\partial T |_\\rho}{\\rho c_p \\partial p_0/\\partial \\rho |_T}\n\\end{equation}\n\nThis is the general constraint equation for low Mach flow.  Note that the\nonly approximation we made is $p \\rightarrow p_0$.  This form of the constraint,\nfor a general equation of state, was originally derived in \\cite{ABRZ:I}.  \n\n\\subsubsection{Combustion limit}\n\nA useful limit is smallscale combustion.  In an open domain, we can take\n$p_0$ as constant, so $Dp_0/Dt = 0$, and we are left with\n\\begin{equation}\n\\nabla \\cdot \\Ub = S\n\\end{equation}\nThis looks like the constraint for incompressible flow, but with a source\nto the divergence.  This source captures the compressible effects due\nto local heat release---as a fluid parcel moves, the only changes to\nits density will come through local heat release.  Methods for this\ntype of constraint are discussed in \\cite{pember-flame,DayBell:2000,SNpaper}.\n\n\\subsubsection{Atmospheric case}\n\nAnother interesting case is that of an atmosphere.  If we consider an\nideal gas, then $\\Gamma_1 = \\gamma = \\mathrm{constant}$.  A second\nappromation we take is that $p_0 = p_0(r)$---i.e., no time\ndependence to the atmosphere.  Finally, we'll consider the case\nwith no local heat sources ($S = 0$).  Then we have\n\\begin{equation}\n\\nabla \\cdot \\Ub + \\frac{1}{\\gamma p_0} \\Ub \\cdot \\nabla p_0 = 0\n\\end{equation}\nwhich is equivalent to\n\\begin{equation}\n\\nabla \\cdot \\left ( p_0^{1/\\gamma} \\Ub \\right ) = 0\n\\end{equation}\nThis constraint captures the changes in compressibilty due to the \nbackground stratification of an atmosphere.  This form was originally\nderived for atmospheric flows by \\cite{durran:1989} and generalized to\nstellar flows in \\cite{ABRZ:I}.  If the structure of the\natmosphere is isentropic, then we know that $d\\log p_0 = \\gamma d\\log \\rho_0$,\nwhere we use $\\rho_0$ to represent the density corresponding to $p_0$, and\nwe can write this constraint as:\n\\begin{equation}\n\\nabla \\cdot (\\rho_0 \\Ub) = 0\n\\end{equation}\nThis is the traditional anelastic constraint.\n\nExtensions involving external heating sources or reactions are\ndiscussed in \\cite{ABRZ:II,ABNZ:III}, and dealing with the\nnon-constant $\\Gamma_1$ is discussed in \\cite{ABRZ:I,KP:2012}.  The\ntime-dependence of the background atmosphere is explored in\n\\cite{ABRZ:II} for plane-parallel atmospheres, following the ideas\nfrom \\cite{almgren:2000}, and in \\cite{multilevel} for spherical,\nself-gravitating bodies.\n\n\\section{Multigrid for Variable-Density Flows}\n\nThe solution methodology for these low Mach number systems follows \nthat of the incompressible flow, but with two additions.  First,\nwe need to incorporate a density (mass continuity) evolution equation---this\nwill follow the same techniques we already saw for advection, as we'll\nsee later.\nNext, we need to be able to enforce more general forms of the \ndivergence constraint, which as we'll see in a moment, require\nus to solve a variable-coefficient elliptic equation.  Our\nmultigrid technique will need to be suitably modified.\n\n\\label{sec:lm:vcelliptic}\n\nWe now need to solve an elliptic equation of the form:\n\\begin{equation}\n\\nabla \\cdot (\\eta \\nabla \\phi) = f\n\\end{equation}\n\nIf we denote the discrete divergence and gradient operators as $D$ and $G$,\nthen our operator will be $L_\\eta \\equiv D \\eta G$.  If we wish to\nuse a cell-centered discretization for $\\phi$, then using a standard \ncentered-difference for $D$ and $G$ will result in a stencil that reaches\ntwo zones on either side of the current zone.  This can lead to an\nodd-even decoupling. \\MarginPar{cite the appropriate Bell paper}\n\n\\MarginPar{there is a Pao and Colella (or Pao's thesis?) that also \ndiscusses issues with cell-centered}\n\nInstead, we again use an approximate projection.  We discretize the\nvariable-coefficient Laplacian as:\n\\begin{align}\n(L_\\eta \\phi)_{i,j} = \n & \\frac{\\eta_{i+\\myhalf,j} (\\phi_{i+1,j} - \\phi_{i,j}) -\n        \\eta_{i-\\myhalf,j} (\\phi_{i,j} - \\phi_{i-1,j})}{\\Delta x^2} + \\nonumber \\\\\n & \\frac{\\eta_{i,j+\\myhalf} (\\phi_{i,j+1} - \\phi_{i,j}) -\n        \\eta_{i,j-\\myhalf} (\\phi_{i,j} - \\phi_{i,j-1})}{\\Delta y^2}\n\\label{lm:eq:lap}\n\\end{align}\n\nWe can define the interface values of $\\eta$ as the averages of the\ncell-centered values, e.g.,\n\\begin{equation}\n  \\eta_{i,j+\\myhalf} = \\frac{1}{2}(\\eta_{i,j} + \\eta_{i,j+1})\n\\end{equation}\nOur elliptic equation is then \n\\begin{equation}\n(L_\\eta \\phi)_{i,j} = f_{i,j}\n\\end{equation}\n\nThe relaxation method for this operator again relies on isolating\n$\\phi_{i,j}$, yielding:\n\\begin{equation}\n\\phi_{i,j} = \\frac{\\teta_{i+\\myhalf,j}\\phi_{i+1,j} + \\teta_{i-\\myhalf,j}\\phi_{i-1,j} +\n                   \\teta_{i,j+\\myhalf}\\phi_{i,j+1} + \\teta_{i,j-\\myhalf}\\phi_{i,j-1} -\n                   f_{i,j} } \n                  {\\teta_{i+\\myhalf,j} + \\teta_{i-\\myhalf,j} + \n                   \\teta_{i,j+\\myhalf} + \\teta_{i,j-\\myhalf}}\n\\label{lm:eq:smooth}\n\\end{equation}\nwith the shorthand that $\\teta_{i\\pm\\myhalf,j} = \\eta_{i\\pm\\myhalf,j}/\\Delta x^2$\nand $\\teta_{i,j\\pm\\myhalf} = \\eta_{i,j\\pm\\myhalf}/\\Delta y^2$.\n\nTo put this into our multigrid framework, there are three changes we\nneed to make:\n\\begin{itemize}\n\\item The smoothing function needs to implement the more general smoothing\ndescribed by Eq.~\\ref{lm:eq:smooth}.\n\n\\item The residual function needs to compute $(L_\\eta \\phi)_{i,j}$ according\nto Eq.~\\ref{lm:eq:lap}, and then $r_{i,j} = f_{i,j} - (L_\\eta \\phi)_{i,j}$.\n\n\\item The coefficients, $\\eta$ should be averaged to the edges on the fine\ngrid and then restricted down the multigrid hierarchy as edge-based \nquantities.\n\\end{itemize}\n\n\\subsection{Test problem}\n\n\\subsubsection{Periodic}\n\n\\label{sec:lm:periodicbcs}\n\nTo test the solver, we need to devise a problem with a known analytic\nsolution.  The easiest way to do this is to pick an $\\eta(x)$ and\n$\\phi$ and then do the divergence and gradients to find the required\nrighthand side, $f$.  We'll use periodic BCs, and for our\nequation $\\nabla \\cdot ( \\eta \\nabla \\phi ) = f$, the following\nprovide a well-posed test problem:\n\\begin{align}\n\\eta &= 2 + \\cos(2\\pi x) \\cos(2\\pi y)  \\label{eq:vc:lap}\n\\\\\nf &= -16.0 \\pi^2 \\left [ \\cos(2\\pi x)\\cos(2\\pi y) + 1 \\right ] \\sin(2\\pi x)\\sin(2 \\pi y) \\nonumber\n\\end{align}\nwith the solution:\n\\begin{equation}\n\\phi^\\mathrm{true} = \\sin(2 \\pi x) \\sin(2\\pi y) \\nonumber \\\\\n\\end{equation}\n\nThere is an important caveat when dealing with a purely-periodic\nproblem.  Since there is no boundary values to ``anchor'' the solution,\nit is free to float.  Solving the elliptic problem will give use the\ncorrect $\\nabla \\phi$, but the average of $\\phi$ over the domain is \nunconstrained.  For our algorithms, it is $\\nabla \\phi$ that matters\n(that is the forcing term that enters into the momentum equation).\n\nWhen for checking convergence, we want to compare to the exact solution.\nWe therefore normalize $\\phi$ by subtracting off its average value before\ncomputing the norm of the error with respect to the exact solution:\n\\begin{equation}\n\\epsilon = \\| \\phi_{i,j} - \\bar{\\phi} - \\phi^\\mathrm{true}_{i,j} \\| \\enskip ,\n\\end{equation}\nwhere\n\\begin{equation}\n\\bar{\\phi} = \\frac{1}{N_x N_y} \\sum_{i,j} \\phi_{i,j}\n\\end{equation}\nAs discussed in \\S~\\ref{sec:multigrid:other}, this can arise if\nthe discrete form the righthand side, $f_{i,j}$ does not sum exactly\nto zero.  Figure~\\ref{fig:mg_vc} shows the solution to this problem\nwith a $512^2$ grid. and the convergence of the solver described\nhere is shown in Figure~\\ref{fig:mg_vc_converge}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{mg_vc_periodic_test}\n\\caption[Solution and error of a variable-coefficient Poisson problem]{\\label{fig:mg_vc} Solution and error to the variable-coefficient \nPoisson problem defined in Eq.~\\ref{eq:vc:lap}.  This test can be\nrun with \\pyro\\ {\\tt multigrid/test\\_mg\\_vc\\_periodic.py}.}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{mg_vc_periodic_converge}\n\\caption[Convergence of the variable-coefficient Poisson solver]{\\label{fig:mg_vc_converge} Convergence of the variable-coefficient\nmultigrid solver for the test problem defined in Eq.~\\ref{eq:vc:lap}.  This test can be\nrun with \\pyro\\ {\\tt multigrid/test\\_mg\\_vc\\_periodic.py}.}\n\\end{figure}\n\n\\subsubsection{Dirichlet}\n\nWe can run the same problem with Dirichlet boundary conditions on $\\phi$,\nand we are free to pick different boundary conditions for $\\eta$, since\nit represents a different physical quantity.  Since we only have homogeneous\nDirichlet or Neumann BCs implemented, we'll run with Neumann BCs on $\\eta$.\n\n\n\n\\section{Atmospheric flows}\n\\label{sec:lm:atm} \n\n\\subsection{Equation Set}\n\nFor atmospheric flows, we define a one-dimensional base state that is\nin hydrostatic equilibrium.  In general, this base state can be\ntime-dependent, expanding in response to heating in the atmosphere\n(see e.g. \\cite{almgren:2000,ABRZ:II}.  Here we'll consider only a\ntime-independent state.\n\nWe'll follow the procedure defined in \\cite{multilevel}: we define\n$\\rho_0$ as the lateral average of $\\rho$:\n\\begin{equation}\n{\\rho_0}_j = \\frac{1}{N_x} \\sum_i \\rho_{i,j}\n\\end{equation}\nand then we define the base state pressure, $p_0$, by integrating the\nequation of hydrostatic equilibrium, $dp_0/dy = \\rho g$, as:\n\\begin{equation}\n{p_0}_{j+1} = {p_0}_j + \\frac{\\Delta y}{2} ({\\rho_0}_j + {\\rho_0}_{j+1}) g\n\\end{equation}\nwith an initial condition of \n\\begin{equation}\n{p_0}_\\mathrm{jlo} = \\frac{1}{N_x} \\sum_{i} p^\\mathrm{initial}_{i,\\mathrm{jlo}}\n\\end{equation}\n\nThe compressible momentum equation (written in terms of velocity is):\n\\begin{equation}\n\\rho \\frac{\\partial \\Ub}{\\partial t} + \\rho \\Ub \\cdot \\nabla \\Ub + \\nabla p = \\rho \\gb\n\\end{equation}\nSubtracting off the base state, and defining the perturbational\npressure (sometimes called the dynamic pressure) as $p^\\prime = p -\np_0$, and perturbational density as $\\rho' = \\rho - \\rho_0$, we have:\n\\begin{equation}\n\\rho \\frac{\\partial \\Ub}{\\partial t} + \\rho \\Ub \\cdot \\nabla \\Ub + \\nabla p' = \\rho' \\gb\n\\end{equation}\nor \n\\begin{equation}\n\\frac{\\partial \\Ub}{\\partial t} + \\Ub \\cdot \\nabla \\Ub + \\frac{1}{\\rho} \\nabla p^\\prime = \n   \\frac{\\rho^\\prime}{\\rho} \\gb\n\\end{equation}\n\nSeveral authors \\cite{KP:2012,VLBWZ:2013} explored the idea of energy\nconservation in a low Mach number system and found that an additional term (which can\nlook like a buoyancy) is needed in the low Mach number formulation, yielding:\n\\begin{equation}\n\\frac{\\partial \\Ub}{\\partial t} + \\Ub \\cdot \\nabla \\Ub + \n   \\frac{\\beta_0}{\\rho} \\nabla \\left (\\frac{p^\\prime}{\\beta_0} \\right ) = \n   \\frac{\\rho^\\prime}{\\rho} \\gb\n\\end{equation}\n\nCompleting the system are the continuity equation,\n\\begin{equation}\n\\frac{\\partial \\rho}{\\partial t} + \\nabla \\cdot (\\rho \\Ub) = 0\n\\end{equation}\nand the constraint,\n\\begin{equation}\n\\nabla \\cdot (\\beta_0 \\Ub) = 0\n\\end{equation}\nwith $\\beta_0 = p_0^{1/\\gamma}$.\n\n  \n\\subsection{Solution Procedure}\n\n\\label{sec:lm:density}\n\nThe general solution procedure is for a single step is:\n\\begin{enumerate}\n  \\renewcommand{\\labelenumi}{\\Roman{enumi}.}\n  \n\\item {\\em Predict $\\Ub$ to the interfaces}\n\n\\item {\\em Enforce the divergence constraint on the interface $\\Ub$'s (the\n  MAC projection) to get $\\Ub^\\mathrm{adv}$.}\n\n  Decomposing the velocity field as\n  \\begin{equation}\n    \\Ub^\\star = \\Ub^d + \\frac{\\beta_0}{\\rho} \\nabla \\phi\n  \\end{equation}\n  as suggested from the form of the momentum equation, our Poisson\n  equation can be defined by multiplying by $\\beta_0$ and taking the\n  divergence, and using $\\nabla \\cdot (\\beta_0 \\Ub^d) = 0$, giving\n  \\begin{equation}\n    \\nabla \\cdot \\frac{\\beta_0^2}{\\rho} \\nabla \\phi =\n    \\nabla \\cdot (\\beta_0 \\Ub^\\star)\n  \\end{equation}\n  Note that when written like this, $\\phi$ has units of\n  $p^\\prime/\\beta_0$. \\MarginPar{there is a $\\Delta t$ here too}\n  \n  For the MAC projection, we have edge-centered velocities (in their\n  respective coordinate direction).  We define an edge-centered $\\beta_0$ as\n  \\begin{equation}\n    {\\beta_0}_{j+\\myhalf} = \\frac{1}{2} ( {\\beta_0}_j + {\\beta_0}_{j+1} )\n  \\end{equation}\n  (note that, since $\\beta_0$ is one-dimensional, we only average in\n  the vertical direction).  The divergence term is then:\n  \\begin{equation}\n    \\left [ \\nabla \\cdot (\\beta_0 \\Ub) \\right ]_{i,j}^\\mathrm{adv} =\n          {\\beta_0}_j \\frac{u^\\mathrm{adv}_{i+\\myhalf,j} - \n            u^\\mathrm{adv}_{i-\\myhalf,j}}{\\Delta x} +\n          \\frac{{\\beta_0}_{j+\\myhalf} v^\\mathrm{adv}_{i,j+\\myhalf} - \n            {\\beta_0}_{j-\\myhalf} v^\\mathrm{adv}_{i,j-\\myhalf} }{\\Delta y}\n  \\end{equation}\n  \n  We define the coefficient, $\\eta$, in the Poisson equation as:\n  \\begin{equation}\n    \\eta_{i,j} = \\frac{{\\beta_0}_j^2}{\\rho_{i,j}}\n  \\end{equation}\n  and bring it to edges simply through averaging.  Multigrid is\n  used to solve for $\\phi_{i,j}$.\n\n  We then solve\n  \\begin{equation}\n    (L_\\eta\\phi)_{i,j} =\n    \\left [ \\nabla \\cdot (\\beta_0 \\Ub) \\right ]_{i,j}^\\mathrm{adv}\n  \\end{equation}\n  Once we solve for $\\phi$, we correct the velocity as:\n  \\begin{equation}\n    \\Ub^\\mathrm{new} = \\Ub^\\star - \\frac{\\beta_0}{\\rho} \\nabla \\phi\n  \\end{equation}\n  Since the MAC velocities are edge-centered, our correction appears as:\n  \\begin{align}\n    u^\\mathrm{adv}_{i+\\myhalf,j} &= u^\\mathrm{adv}_{i+\\myhalf,j} -\n    \\left (\\frac{\\beta_0}{\\rho}\\right )_{i+\\myhalf,j}\n    \\frac{\\phi_{i+1,j} - \\phi_{i,j}}{\\Delta x} \\\\\n    %\n    v^\\mathrm{adv}_{i,j+\\myhalf} &= v^\\mathrm{adv}_{i,j+\\myhalf} -\n    \\left (\\frac{\\beta_0}{\\rho}\\right )_{i,j+\\myhalf}\n    \\frac{\\phi_{i,j+1} - \\phi_{i,j}}{\\Delta y} \n  \\end{align}\n  \n\\item {\\em Predict $\\rho$ to the interfaces}\n\nWe need to solve the continuity equation.  We use the same techniques\nthat were used for advection.  Our equation is:\n\\begin{equation}\n\\rho_t + (\\rho u)_x + (\\rho v)_y = 0\n\\end{equation}\nThe $x$-interface left state would be:\n\\begin{align}\n\\rho_{i+\\myhalf,j,L}^{n+\\myhalf} &= \\rho_{i,j}^n + \n   \\frac{\\Delta x}{2} \\frac{\\partial \\rho}{\\partial x} +\n   \\frac{\\Delta t}{2} \\frac{\\partial \\rho}{\\partial t} + \\ldots \\nonumber \\\\\n%\n &= \\rho_{i,j}^n + \n    \\frac{\\Delta x}{2} \\frac{\\partial \\rho}{\\partial x} +\n    \\frac{\\Delta t}{2} \\left [ -(\\rho u)_x -(\\rho v)_y \\right ]_{i,j} \\nonumber\\\\\n%\n &= \\underbrace{\\rho_{i,j}^n + \n   \\frac{\\Delta x}{2} \\left ( 1 - \\frac{\\Delta t}{\\Delta x} u_{i,j} \\right )\n        \\frac{\\partial \\rho}{\\partial x}}_{\\hat{\\rho}_{i+\\myhalf,j,L}^{n+\\myhalf}}\n   - \\frac{\\Delta t}{2} \\left [\\rho u_x \\right ]_{i,j} \n   - \\frac{\\Delta t}{2} \\left [ (\\rho v)_y \\right ]_{i,j}\n\\end{align}\nA similar construction would yield the right state at that interface, and the\n$y$-interface states.\n\nSince we already have the MAC-projected advected velocities at this\npoint, we can use them in the construction of the interface states and\nthe upwinding.  As before, we split this into a normal part (the\n$\\hat{\\rho}$ term) and ``transverse'' part (which here includes the\nnon-advective part of the divergence).  We first construct the normal\nparts as\n\\begin{equation}\n\\hat{\\rho}_{i+\\myhalf,j,L}^{n+\\myhalf} = \\rho_{i,j}^n + \n   \\frac{1}{2} \\left ( 1 - \\frac{\\Delta t}{\\Delta x} u^\\mathrm{adv}_{i+\\myhalf,j} \\right )\n   \\overline{\\Delta \\rho}^{(x)}_{i,j}\n\\end{equation}\nand then solve the Riemann problem (upwinding) for these:\n\\begin{equation}\n\\hat{\\rho}_{i+\\myhalf,j}^{n+\\myhalf} = \n   \\mathcal{U}[u^\\mathrm{adv}_{i+\\myhalf,j}](\\hat{\\rho}_{i+\\myhalf,j,L}^{n+\\myhalf},\n                                     \\hat{\\rho}_{i+\\myhalf,j,R}^{n+\\myhalf}) =\n  \\left \\{\n  \\begin{array}{cl}\n     \\hat{\\rho}_{i+\\myhalf,j,L}^{n+\\myhalf} & u^\\mathrm{adv}_{i+\\myhalf,j} > 0 \\\\\n     \\hat{\\rho}_{i+\\myhalf,j,R}^{n+\\myhalf} & u^\\mathrm{adv}_{i+\\myhalf,j} < 0 \n  \\end{array} \\right .\n\\end{equation}\nThe same procedure is done for the $y$-interfaces.\n\nThe full states are then constructed using these ``hat'' states.  For example,\n\\begin{align}\n\\rho_{i+\\myhalf,j,L}^{n+\\myhalf} = \\hat{\\rho}_{i+\\myhalf,j,L}^{n+\\myhalf} \n   &- \\frac{\\Delta t}{2} \\rho_{i,j} \n     \\frac{u^\\mathrm{adv}_{i+\\myhalf,j} - u^\\mathrm{adv}_{i-\\myhalf,j}}{\\Delta x} \\nonumber \\\\\n   &- \\frac{\\Delta t}{2} \n        \\frac{\\hat{\\rho}_{i,j+\\myhalf}^{n+\\myhalf} v^\\mathrm{adv}_{i,j+\\myhalf} -\n              \\hat{\\rho}_{i,j-\\myhalf}^{n+\\myhalf} v^\\mathrm{adv}_{i,j-\\myhalf}}{\\Delta y}\n\\end{align}\nOnce the new states on both the $x$- and $y$-interfaces are constructed, we\nagain upwind to find the final $\\rho$ state on each interface:\n\\begin{equation}\n\\rho_{i+\\myhalf,j}^{n+\\myhalf} = \n   \\mathcal{U}[u^\\mathrm{adv}_{i+\\myhalf,j}](\\rho_{i+\\myhalf,j,L}^{n+\\myhalf},\n                                     \\rho_{i+\\myhalf,j,R}^{n+\\myhalf}) =\n  \\left \\{\n  \\begin{array}{cl}\n     \\rho_{i+\\myhalf,j,L}^{n+\\myhalf} & u^\\mathrm{adv}_{i+\\myhalf,j} > 0 \\\\\n     \\rho_{i+\\myhalf,j,R}^{n+\\myhalf} & u^\\mathrm{adv}_{i+\\myhalf,j} < 0 \n  \\end{array} \\right .\n\\end{equation}\n\n\n  \n\\item {\\em Do the conservative update of $\\rho$}\n\n  Once the interface states are found, we can conservatively-difference\nthe continuity equation:\n\\begin{align}\n\\rho^{n+1}_{i,j} = \\rho^n_{i,j} \n   &- \\frac{\\Delta t}{\\Delta x} \n      \\left (\\rho_{i+\\myhalf,j}^{n+\\myhalf} u^\\mathrm{adv}_{i+\\myhalf,j} -\n             \\rho_{i-\\myhalf,j}^{n+\\myhalf} u^\\mathrm{adv}_{i-\\myhalf,j} \\right ) \\nonumber \\\\\n   &- \\frac{\\Delta t}{\\Delta y}\n      \\left (\\rho_{i,j+\\myhalf}^{n+\\myhalf} v^\\mathrm{adv}_{i,j+\\myhalf} -\n             \\rho_{i,j-\\myhalf}^{n+\\myhalf} v^\\mathrm{adv}_{i,j-\\myhalf} \\right )\n\\end{align}\n\n\n\\item {\\em Update $\\Ub^n$ to $\\Ub^{n+1,\\star}$}\n  \n\\item {\\em Enforce the divergence constraint on $\\Ub^{n+1}$}\n\n  For the final projection, we have cell-centered velocties.  We \n  define the divergence term as:\n  \\begin{equation}\n    \\left [ \\nabla \\cdot (\\beta_0 \\Ub) \\right ]_{i,j}^\\star =\n          {\\beta_0}_j \\frac{u^\\star_{i+1,j} - \n            u^\\star_{i-1,j}}{2\\Delta x} +\n          \\frac{{\\beta_0}_{j+1} v^\\star_{i,j+1} - \n            {\\beta_0}_{j-1} v^\\star_{i,j-1} }{2\\Delta y}\n  \\end{equation}\n\n  \n\\end{enumerate}\n\n\n\\subsection{Timestep constraint}\n\nIn addition to the advective timestep constraint, an additional constraint\nis needed if the velocity field is initialially zero.  In \\cite{ABNZ:III},\na constraint based on the buoyancy forcing is used.  First, we compute\nthe maxiumum buoyancy acceleration,\n\\begin{equation}\na_\\mathrm{max} = \\max \\left \\{ \\left |\\frac{\\rho'\\gb}{\\rho} \\right | \\right \\}\n\\end{equation}\nand then \n\\begin{equation}\n\\Delta t_\\mathrm{force} = \\left ( \\frac{2 \\Delta x}{a_\\mathrm{max}} \\right )^{\\myhalf} \n\\end{equation}\nThis is based simply on $\\Delta x = (1/2) a t^2$---the\ntime it takes for the buoyant force to accelerate a fluid element\nacross a zone width.\n\n\n\\subsection{Bootstrapping}\n\nFirst we need to make sure that the initial velocity field satisfies our constraint\n\nNext we need to find $\\nabla {p^\\prime}^{n-\\myhalf}$ for the first step.\n\n\n\n\\section{Combustion}\n\n\nTaking $p = p_0 + p^\\prime$, with $p_0 = \\mathrm{constant}$, the system \nbecomes:\n\\begin{align}\n\\frac{\\partial \\rho}{\\partial t} + \\nabla \\cdot (\\rho \\Ub) &= 0 \\\\\n\\frac{\\partial \\rho \\Ub}{\\partial t} + \\nabla \\cdot (\\rho \\Ub \\Ub) + \\nabla p^\\prime &= 0 \\\\\n\\nabla \\cdot \\Ub &= S\n\\end{align}\n\n\\subsection{Species}\n\n\n\\subsection{Constraint}\n\nOur constraint equation is $\\nabla \\cdot \\Ub = S$.  Decomposing the\nvelocity field as\n\\begin{equation}\n\\Ub^\\star = \\Ub^d + \\frac{1}{\\rho} \\nabla \\phi\n\\end{equation}\nour Poisson equation can be defined by taking the divergence, and\nusing $\\nabla \\cdot \\Ub^d = S$, giving\n\\begin{equation}\n\\nabla \\cdot \\frac{1}{\\rho} \\nabla \\phi = \\nabla \\cdot \\Ub^\\star - S\n\\end{equation}\n\n\n\\subsection{Solution Procedure}\n\nThe general solution procedure is for a single step is:\n\\begin{itemize}\n\n  \\item React for $\\Delta t/2$\n\n  \\item Do the hydrodynamcis for $\\Delta t$\n\n    \\begin{itemize}\n    \\item Predict $\\Ub$ to the interfaces \n    \\item Enforce the divergence constraint on the interface $\\Ub$'s (the\n      MAC projection) to get $\\Ub^\\mathrm{adv}$.\n    \\item Predict $\\rho$ to the interfaces\n    \\item Do the conservative update of $\\rho$\n    \\item Update $\\Ub^n$ to $\\Ub^{n+1}$\n    \\item Enforce the divergence constraint on $\\Ub^{n+1}$\n    \\end{itemize}\n\n  \\item React for $\\Delta t/2$\n\n\\end{itemize}\n"
  },
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    "path": "low_mach/mg_vc_periodic_converge.eps",
    "content": "%!PS-Adobe-3.0 EPSF-3.0\n%%Title: mg_vc_periodic_converge.eps\n%%Creator: matplotlib version 1.3.1, http://matplotlib.org/\n%%CreationDate: Fri Mar 20 21:03:27 2015\n%%Orientation: portrait\n%%BoundingBox: 54 180 558 612\n%%EndComments\n%%BeginProlog\n/mpldict 8 dict def\nmpldict begin\n/m { moveto } bind def\n/l { lineto } bind def\n/r { rlineto } bind def\n/c { curveto } bind def\n/cl { closepath } bind def\n/box {\nm\n1 index 0 r\n0 exch r\nneg 0 r\ncl\n} bind def\n/clipbox {\nbox\nclip\nnewpath\n} bind def\n%!PS-Adobe-3.0 Resource-Font\n%%Title: Bitstream Vera Sans\n%%Copyright: Copyright (c) 2003 by Bitstream, Inc. All Rights Reserved.\n%%Creator: Converted from TrueType to type 3 by PPR\n25 dict begin\n/_d{bind def}bind def\n/_m{moveto}_d\n/_l{lineto}_d\n/_cl{closepath eofill}_d\n/_c{curveto}_d\n/_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d\n/_e{exec}_d\n/FontName /BitstreamVeraSans-Roman def\n/PaintType 0 def\n/FontMatrix[.001 0 0 .001 0 0]def\n/FontBBox[-183 -236 1287 928]def\n/FontType 3 def\n/Encoding [ /hyphen /zero /one /two /three /four /five /six /N /e /o /r ] def\n/FontInfo 10 dict dup begin\n/FamilyName (Bitstream Vera Sans) def\n/FullName (Bitstream Vera Sans) def\n/Notice (Copyright (c) 2003 by Bitstream, Inc. All Rights Reserved. Bitstream Vera is a trademark of Bitstream, Inc.) def\n/Weight (Roman) def\n/Version (Release 1.10) def\n/ItalicAngle 0.0 def\n/isFixedPitch false def\n/UnderlinePosition -213 def\n/UnderlineThickness 143 def\nend readonly def\n/CharStrings 12 dict dup begin\n/hyphen{361 0 49 234 312 314 _sc\n49 314 _m\n312 314 _l\n312 234 _l\n49 234 _l\n49 314 _l\n_cl}_d\n/zero{636 0 66 -13 570 742 _sc\n318 664 _m\n267 664 229 639 203 589 _c\n177 539 165 464 165 364 _c\n165 264 177 189 203 139 _c\n229 89 267 64 318 64 _c\n369 64 407 89 433 139 _c\n458 189 471 264 471 364 _c\n471 464 458 539 433 589 _c\n407 639 369 664 318 664 _c\n318 742 _m\n399 742 461 709 505 645 _c\n548 580 570 486 570 364 _c\n570 241 548 147 505 83 _c\n461 19 399 -13 318 -13 _c\n236 -13 173 19 130 83 _c\n87 147 66 241 66 364 _c\n66 486 87 580 130 645 _c\n173 709 236 742 318 742 _c\n_cl}_d\n/one{636 0 110 0 544 729 _sc\n124 83 _m\n285 83 _l\n285 639 _l\n110 604 _l\n110 694 _l\n284 729 _l\n383 729 _l\n383 83 _l\n544 83 _l\n544 0 _l\n124 0 _l\n124 83 _l\n_cl}_d\n/two{{636 0 73 0 536 742 _sc\n192 83 _m\n536 83 _l\n536 0 _l\n73 0 _l\n73 83 _l\n110 121 161 173 226 239 _c\n290 304 331 346 348 365 _c\n380 400 402 430 414 455 _c\n426 479 433 504 433 528 _c\n433 566 419 598 392 622 _c\n365 646 330 659 286 659 _c\n255 659 222 653 188 643 _c\n154 632 117 616 78 594 _c\n78 694 _l\n118 710 155 722 189 730 _c\n223 738 255 742 284 742 _c\n}_e{359 742 419 723 464 685 _c\n509 647 532 597 532 534 _c\n532 504 526 475 515 449 _c\n504 422 484 390 454 354 _c\n446 344 420 317 376 272 _c\n332 227 271 164 192 83 _c\n_cl}_e}_d\n/three{{636 0 76 -13 556 742 _sc\n406 393 _m\n453 383 490 362 516 330 _c\n542 298 556 258 556 212 _c\n556 140 531 84 482 45 _c\n432 6 362 -13 271 -13 _c\n240 -13 208 -10 176 -4 _c\n144 1 110 10 76 22 _c\n76 117 _l\n103 101 133 89 166 81 _c\n198 73 232 69 268 69 _c\n330 69 377 81 409 105 _c\n441 129 458 165 458 212 _c\n458 254 443 288 413 312 _c\n383 336 341 349 287 349 _c\n}_e{202 349 _l\n202 430 _l\n291 430 _l\n339 430 376 439 402 459 _c\n428 478 441 506 441 543 _c\n441 580 427 609 401 629 _c\n374 649 336 659 287 659 _c\n260 659 231 656 200 650 _c\n169 644 135 635 98 623 _c\n98 711 _l\n135 721 170 729 203 734 _c\n235 739 266 742 296 742 _c\n370 742 429 725 473 691 _c\n517 657 539 611 539 553 _c\n539 513 527 479 504 451 _c\n481 423 448 403 406 393 _c\n_cl}_e}_d\n/four{636 0 49 0 580 729 _sc\n378 643 _m\n129 254 _l\n378 254 _l\n378 643 _l\n352 729 _m\n476 729 _l\n476 254 _l\n580 254 _l\n580 172 _l\n476 172 _l\n476 0 _l\n378 0 _l\n378 172 _l\n49 172 _l\n49 267 _l\n352 729 _l\n_cl}_d\n/five{{636 0 77 -13 549 729 _sc\n108 729 _m\n495 729 _l\n495 646 _l\n198 646 _l\n198 467 _l\n212 472 227 476 241 478 _c\n255 480 270 482 284 482 _c\n365 482 429 459 477 415 _c\n525 370 549 310 549 234 _c\n549 155 524 94 475 51 _c\n426 8 357 -13 269 -13 _c\n238 -13 207 -10 175 -6 _c\n143 -1 111 6 77 17 _c\n77 116 _l\n106 100 136 88 168 80 _c\n199 72 232 69 267 69 _c\n}_e{323 69 368 83 401 113 _c\n433 143 450 183 450 234 _c\n450 284 433 324 401 354 _c\n368 384 323 399 267 399 _c\n241 399 214 396 188 390 _c\n162 384 135 375 108 363 _c\n108 729 _l\n_cl}_e}_d\n/six{{636 0 70 -13 573 742 _sc\n330 404 _m\n286 404 251 388 225 358 _c\n199 328 186 286 186 234 _c\n186 181 199 139 225 109 _c\n251 79 286 64 330 64 _c\n374 64 409 79 435 109 _c\n461 139 474 181 474 234 _c\n474 286 461 328 435 358 _c\n409 388 374 404 330 404 _c\n526 713 _m\n526 623 _l\n501 635 476 644 451 650 _c\n425 656 400 659 376 659 _c\n310 659 260 637 226 593 _c\n}_e{192 549 172 482 168 394 _c\n187 422 211 444 240 459 _c\n269 474 301 482 336 482 _c\n409 482 467 459 509 415 _c\n551 371 573 310 573 234 _c\n573 159 550 99 506 54 _c\n462 9 403 -13 330 -13 _c\n246 -13 181 19 137 83 _c\n92 147 70 241 70 364 _c\n70 479 97 571 152 639 _c\n206 707 280 742 372 742 _c\n396 742 421 739 447 735 _c\n472 730 498 723 526 713 _c\n_cl}_e}_d\n/N{748 0 98 0 650 729 _sc\n98 729 _m\n231 729 _l\n554 119 _l\n554 729 _l\n650 729 _l\n650 0 _l\n517 0 _l\n194 610 _l\n194 0 _l\n98 0 _l\n98 729 _l\n_cl}_d\n/e{{615 0 55 -13 562 560 _sc\n562 296 _m\n562 252 _l\n149 252 _l\n153 190 171 142 205 110 _c\n238 78 284 62 344 62 _c\n378 62 412 66 444 74 _c\n476 82 509 95 541 113 _c\n541 28 _l\n509 14 476 3 442 -3 _c\n408 -9 373 -13 339 -13 _c\n251 -13 182 12 131 62 _c\n80 112 55 181 55 268 _c\n55 357 79 428 127 481 _c\n175 533 241 560 323 560 _c\n397 560 455 536 498 489 _c\n}_e{540 441 562 377 562 296 _c\n472 322 _m\n471 371 457 410 431 440 _c\n404 469 368 484 324 484 _c\n274 484 234 469 204 441 _c\n174 413 156 373 152 322 _c\n472 322 _l\n_cl}_e}_d\n/o{612 0 55 -13 557 560 _sc\n306 484 _m\n258 484 220 465 192 427 _c\n164 389 150 338 150 273 _c\n150 207 163 156 191 118 _c\n219 80 257 62 306 62 _c\n354 62 392 80 420 118 _c\n448 156 462 207 462 273 _c\n462 337 448 389 420 427 _c\n392 465 354 484 306 484 _c\n306 560 _m\n384 560 445 534 490 484 _c\n534 433 557 363 557 273 _c\n557 183 534 113 490 63 _c\n445 12 384 -13 306 -13 _c\n227 -13 165 12 121 63 _c\n77 113 55 183 55 273 _c\n55 363 77 433 121 484 _c\n165 534 227 560 306 560 _c\n_cl}_d\n/r{411 0 91 0 411 560 _sc\n411 463 _m\n401 469 390 473 378 476 _c\n366 478 353 480 339 480 _c\n288 480 249 463 222 430 _c\n194 397 181 350 181 288 _c\n181 0 _l\n91 0 _l\n91 547 _l\n181 547 _l\n181 462 _l\n199 495 224 520 254 536 _c\n284 552 321 560 365 560 _c\n371 560 378 559 386 559 _c\n393 558 401 557 411 555 _c\n411 463 _l\n_cl}_d\nend readonly def\n\n/BuildGlyph\n {exch begin\n CharStrings exch\n 2 copy known not{pop /.notdef}if\n true 3 1 roll get exec\n end}_d\n\n/BuildChar {\n 1 index /Encoding get exch get\n 1 index /BuildGlyph get exec\n}_d\n\nFontName currentdict end definefont pop\nend\n%%EndProlog\nmpldict begin\n54 180 translate\n504 432 0 0 clipbox\ngsave\n0 0 m\n504 0 l\n504 432 l\n0 432 l\ncl\n1.000 setgray\nfill\ngrestore\ngsave\n59.4298 50.652 m\n480.37 50.652 l\n480.37 411.174 l\n59.4298 411.174 l\ncl\n1.000 setgray\nfill\ngrestore\n0.500 setlinewidth\n1 setlinejoin\n0 setlinecap\n[] 0 setdash\n1.000 0.000 0.000 setrgbcolor\ngsave\n420.9 360.5 59.43 50.65 clipbox\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n-3 -3 m\n3 3 l\n-3 3 m\n3 -3 l\ngsave\n1.000 0.000 0.000 setrgbcolor\nfill\ngrestore\nstroke\ngrestore\n} bind def\n102.391 393.101 o\n165.749 338.557 o\n229.107 284.223 o\n292.464 229.942 o\n355.822 175.674 o\n419.18 121.409 o\ngrestore\n1.000 setlinewidth\n[6 6] 0 setdash\n0.000 setgray\ngsave\n420.9 360.5 59.43 50.65 clipbox\n102.391 393.101 m\n165.749 338.837 l\n229.107 284.573 l\n292.464 230.309 l\n355.822 176.045 l\n419.18 121.781 l\nstroke\ngrestore\n0.500 setlinewidth\n[] 0 setdash\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 4 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n59.4298 50.652 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -4 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n59.4298 411.174 o\ngrestore\ngsave\n47.429834 35.152000 translate\n0.000000 rotate\n/BitstreamVeraSans-Roman findfont\n12.0 scalefont\nsetfont\n0.000000 0.575000 moveto\n/one glyphshow\n\n7.634766 0.575000 moveto\n/zero glyphshow\n\n/BitstreamVeraSans-Roman findfont\n8.4 scalefont\nsetfont\n15.269531 6.875000 moveto\n/one glyphshow\n\n\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 4 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n269.9 50.652 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -4 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n269.9 411.174 o\ngrestore\ngsave\n257.899890 35.152000 translate\n0.000000 rotate\n/BitstreamVeraSans-Roman findfont\n12.0 scalefont\nsetfont\n0.000000 0.465625 moveto\n/one glyphshow\n\n7.634766 0.465625 moveto\n/zero 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setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n186.145 411.174 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n206.542 50.652 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n206.542 411.174 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n223.207 50.652 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n223.207 411.174 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n237.298 50.652 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n237.298 411.174 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n249.503 50.652 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n249.503 411.174 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n260.269 50.652 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n260.269 411.174 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n333.258 50.652 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n333.258 411.174 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n370.32 50.652 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n370.32 411.174 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n396.615 50.652 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n396.615 411.174 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n417.012 50.652 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n417.012 411.174 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n433.677 50.652 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n433.677 411.174 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n447.768 50.652 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n447.768 411.174 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n459.973 50.652 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n459.973 411.174 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 2 l\ngsave\n0.000 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    "path": "low_mach/mg_vc_periodic_test.eps",
    "content": "%!PS-Adobe-3.0 EPSF-3.0\n%%Title: mg_vc_periodic_test.eps\n%%Creator: matplotlib version 1.3.1, http://matplotlib.org/\n%%CreationDate: Tue Jul 15 19:12:24 2014\n%%Orientation: portrait\n%%BoundingBox: -46 256 658 535\n%%EndComments\n%%BeginProlog\n/mpldict 8 dict def\nmpldict begin\n/m { moveto } bind def\n/l { lineto } bind def\n/r { rlineto } bind def\n/c { curveto } bind def\n/cl { closepath } bind def\n/box {\nm\n1 index 0 r\n0 exch r\nneg 0 r\ncl\n} bind def\n/clipbox {\nbox\nclip\nnewpath\n} bind def\n%!PS-Adobe-3.0 Resource-Font\n%%Title: Bitstream Vera Sans\n%%Copyright: Copyright (c) 2003 by Bitstream, Inc. All Rights Reserved.\n%%Creator: Converted from TrueType to type 3 by PPR\n25 dict begin\n/_d{bind def}bind def\n/_m{moveto}_d\n/_l{lineto}_d\n/_cl{closepath eofill}_d\n/_c{curveto}_d\n/_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d\n/_e{exec}_d\n/FontName /BitstreamVeraSans-Roman def\n/PaintType 0 def\n/FontMatrix[.001 0 0 .001 0 0]def\n/FontBBox[-183 -236 1287 928]def\n/FontType 3 def\n/Encoding [ /space /period /zero /one /two /four /five /six /seven /eight /equal /e /n /o /r /x /y /minus ] def\n/FontInfo 10 dict dup begin\n/FamilyName (Bitstream Vera Sans) def\n/FullName (Bitstream Vera Sans) def\n/Notice (Copyright (c) 2003 by Bitstream, Inc. All Rights Reserved. Bitstream Vera is a trademark of Bitstream, Inc.) def\n/Weight (Roman) def\n/Version (Release 1.10) def\n/ItalicAngle 0.0 def\n/isFixedPitch false def\n/UnderlinePosition -213 def\n/UnderlineThickness 143 def\nend readonly def\n/CharStrings 18 dict dup begin\n/space{318 0 0 0 0 0 _sc\n}_d\n/period{318 0 107 0 210 124 _sc\n107 124 _m\n210 124 _l\n210 0 _l\n107 0 _l\n107 124 _l\n_cl}_d\n/zero{636 0 66 -13 570 742 _sc\n318 664 _m\n267 664 229 639 203 589 _c\n177 539 165 464 165 364 _c\n165 264 177 189 203 139 _c\n229 89 267 64 318 64 _c\n369 64 407 89 433 139 _c\n458 189 471 264 471 364 _c\n471 464 458 539 433 589 _c\n407 639 369 664 318 664 _c\n318 742 _m\n399 742 461 709 505 645 _c\n548 580 570 486 570 364 _c\n570 241 548 147 505 83 _c\n461 19 399 -13 318 -13 _c\n236 -13 173 19 130 83 _c\n87 147 66 241 66 364 _c\n66 486 87 580 130 645 _c\n173 709 236 742 318 742 _c\n_cl}_d\n/one{636 0 110 0 544 729 _sc\n124 83 _m\n285 83 _l\n285 639 _l\n110 604 _l\n110 694 _l\n284 729 _l\n383 729 _l\n383 83 _l\n544 83 _l\n544 0 _l\n124 0 _l\n124 83 _l\n_cl}_d\n/two{{636 0 73 0 536 742 _sc\n192 83 _m\n536 83 _l\n536 0 _l\n73 0 _l\n73 83 _l\n110 121 161 173 226 239 _c\n290 304 331 346 348 365 _c\n380 400 402 430 414 455 _c\n426 479 433 504 433 528 _c\n433 566 419 598 392 622 _c\n365 646 330 659 286 659 _c\n255 659 222 653 188 643 _c\n154 632 117 616 78 594 _c\n78 694 _l\n118 710 155 722 189 730 _c\n223 738 255 742 284 742 _c\n}_e{359 742 419 723 464 685 _c\n509 647 532 597 532 534 _c\n532 504 526 475 515 449 _c\n504 422 484 390 454 354 _c\n446 344 420 317 376 272 _c\n332 227 271 164 192 83 _c\n_cl}_e}_d\n/four{636 0 49 0 580 729 _sc\n378 643 _m\n129 254 _l\n378 254 _l\n378 643 _l\n352 729 _m\n476 729 _l\n476 254 _l\n580 254 _l\n580 172 _l\n476 172 _l\n476 0 _l\n378 0 _l\n378 172 _l\n49 172 _l\n49 267 _l\n352 729 _l\n_cl}_d\n/five{{636 0 77 -13 549 729 _sc\n108 729 _m\n495 729 _l\n495 646 _l\n198 646 _l\n198 467 _l\n212 472 227 476 241 478 _c\n255 480 270 482 284 482 _c\n365 482 429 459 477 415 _c\n525 370 549 310 549 234 _c\n549 155 524 94 475 51 _c\n426 8 357 -13 269 -13 _c\n238 -13 207 -10 175 -6 _c\n143 -1 111 6 77 17 _c\n77 116 _l\n106 100 136 88 168 80 _c\n199 72 232 69 267 69 _c\n}_e{323 69 368 83 401 113 _c\n433 143 450 183 450 234 _c\n450 284 433 324 401 354 _c\n368 384 323 399 267 399 _c\n241 399 214 396 188 390 _c\n162 384 135 375 108 363 _c\n108 729 _l\n_cl}_e}_d\n/six{{636 0 70 -13 573 742 _sc\n330 404 _m\n286 404 251 388 225 358 _c\n199 328 186 286 186 234 _c\n186 181 199 139 225 109 _c\n251 79 286 64 330 64 _c\n374 64 409 79 435 109 _c\n461 139 474 181 474 234 _c\n474 286 461 328 435 358 _c\n409 388 374 404 330 404 _c\n526 713 _m\n526 623 _l\n501 635 476 644 451 650 _c\n425 656 400 659 376 659 _c\n310 659 260 637 226 593 _c\n}_e{192 549 172 482 168 394 _c\n187 422 211 444 240 459 _c\n269 474 301 482 336 482 _c\n409 482 467 459 509 415 _c\n551 371 573 310 573 234 _c\n573 159 550 99 506 54 _c\n462 9 403 -13 330 -13 _c\n246 -13 181 19 137 83 _c\n92 147 70 241 70 364 _c\n70 479 97 571 152 639 _c\n206 707 280 742 372 742 _c\n396 742 421 739 447 735 _c\n472 730 498 723 526 713 _c\n_cl}_e}_d\n/seven{636 0 82 0 551 729 _sc\n82 729 _m\n551 729 _l\n551 687 _l\n286 0 _l\n183 0 _l\n432 646 _l\n82 646 _l\n82 729 _l\n_cl}_d\n/eight{{636 0 68 -13 568 742 _sc\n318 346 _m\n271 346 234 333 207 308 _c\n180 283 167 249 167 205 _c\n167 161 180 126 207 101 _c\n234 76 271 64 318 64 _c\n364 64 401 76 428 102 _c\n455 127 469 161 469 205 _c\n469 249 455 283 429 308 _c\n402 333 365 346 318 346 _c\n219 388 _m\n177 398 144 418 120 447 _c\n96 476 85 511 85 553 _c\n85 611 105 657 147 691 _c\n188 725 245 742 318 742 _c\n}_e{390 742 447 725 489 691 _c\n530 657 551 611 551 553 _c\n551 511 539 476 515 447 _c\n491 418 459 398 417 388 _c\n464 377 501 355 528 323 _c\n554 291 568 251 568 205 _c\n568 134 546 80 503 43 _c\n459 5 398 -13 318 -13 _c\n237 -13 175 5 132 43 _c\n89 80 68 134 68 205 _c\n68 251 81 291 108 323 _c\n134 355 171 377 219 388 _c\n183 544 _m\n183 506 194 476 218 455 _c\n}_e{242 434 275 424 318 424 _c\n360 424 393 434 417 455 _c\n441 476 453 506 453 544 _c\n453 582 441 611 417 632 _c\n393 653 360 664 318 664 _c\n275 664 242 653 218 632 _c\n194 611 183 582 183 544 _c\n_cl}_e}_d\n/equal{838 0 106 172 732 454 _sc\n106 454 _m\n732 454 _l\n732 372 _l\n106 372 _l\n106 454 _l\n106 255 _m\n732 255 _l\n732 172 _l\n106 172 _l\n106 255 _l\n_cl}_d\n/e{{615 0 55 -13 562 560 _sc\n562 296 _m\n562 252 _l\n149 252 _l\n153 190 171 142 205 110 _c\n238 78 284 62 344 62 _c\n378 62 412 66 444 74 _c\n476 82 509 95 541 113 _c\n541 28 _l\n509 14 476 3 442 -3 _c\n408 -9 373 -13 339 -13 _c\n251 -13 182 12 131 62 _c\n80 112 55 181 55 268 _c\n55 357 79 428 127 481 _c\n175 533 241 560 323 560 _c\n397 560 455 536 498 489 _c\n}_e{540 441 562 377 562 296 _c\n472 322 _m\n471 371 457 410 431 440 _c\n404 469 368 484 324 484 _c\n274 484 234 469 204 441 _c\n174 413 156 373 152 322 _c\n472 322 _l\n_cl}_e}_d\n/n{634 0 91 0 549 560 _sc\n549 330 _m\n549 0 _l\n459 0 _l\n459 327 _l\n459 379 448 417 428 443 _c\n408 469 378 482 338 482 _c\n289 482 251 466 223 435 _c\n195 404 181 362 181 309 _c\n181 0 _l\n91 0 _l\n91 547 _l\n181 547 _l\n181 462 _l\n202 494 227 519 257 535 _c\n286 551 320 560 358 560 _c\n420 560 468 540 500 501 _c\n532 462 549 405 549 330 _c\n_cl}_d\n/o{612 0 55 -13 557 560 _sc\n306 484 _m\n258 484 220 465 192 427 _c\n164 389 150 338 150 273 _c\n150 207 163 156 191 118 _c\n219 80 257 62 306 62 _c\n354 62 392 80 420 118 _c\n448 156 462 207 462 273 _c\n462 337 448 389 420 427 _c\n392 465 354 484 306 484 _c\n306 560 _m\n384 560 445 534 490 484 _c\n534 433 557 363 557 273 _c\n557 183 534 113 490 63 _c\n445 12 384 -13 306 -13 _c\n227 -13 165 12 121 63 _c\n77 113 55 183 55 273 _c\n55 363 77 433 121 484 _c\n165 534 227 560 306 560 _c\n_cl}_d\n/r{411 0 91 0 411 560 _sc\n411 463 _m\n401 469 390 473 378 476 _c\n366 478 353 480 339 480 _c\n288 480 249 463 222 430 _c\n194 397 181 350 181 288 _c\n181 0 _l\n91 0 _l\n91 547 _l\n181 547 _l\n181 462 _l\n199 495 224 520 254 536 _c\n284 552 321 560 365 560 _c\n371 560 378 559 386 559 _c\n393 558 401 557 411 555 _c\n411 463 _l\n_cl}_d\n/x{592 0 29 0 559 547 _sc\n549 547 _m\n351 281 _l\n559 0 _l\n453 0 _l\n294 215 _l\n135 0 _l\n29 0 _l\n241 286 _l\n47 547 _l\n153 547 _l\n298 352 _l\n443 547 _l\n549 547 _l\n_cl}_d\n/y{592 0 30 -207 562 547 _sc\n322 -50 _m\n296 -114 271 -157 247 -177 _c\n223 -197 191 -207 151 -207 _c\n79 -207 _l\n79 -132 _l\n132 -132 _l\n156 -132 175 -126 189 -114 _c\n203 -102 218 -75 235 -31 _c\n251 9 _l\n30 547 _l\n125 547 _l\n296 119 _l\n467 547 _l\n562 547 _l\n322 -50 _l\n_cl}_d\n/minus{838 0 106 272 732 355 _sc\n106 355 _m\n732 355 _l\n732 272 _l\n106 272 _l\n106 355 _l\n_cl}_d\nend readonly def\n\n/BuildGlyph\n {exch begin\n CharStrings exch\n 2 copy known not{pop /.notdef}if\n true 3 1 roll get exec\n end}_d\n\n/BuildChar {\n 1 index /Encoding get exch get\n 1 index /BuildGlyph get exec\n}_d\n\nFontName currentdict end definefont pop\nend\n%%EndProlog\nmpldict 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l\n274.851 82.4838 l\n274.851 81.5681 l\n} bind def\n/p0_39 {\nnewpath\ntranslate\n274.851 82.4838 m\n286.572 82.4838 l\n286.572 83.3995 l\n274.851 83.3995 l\n274.851 82.4838 l\n} bind def\n/p0_3a {\nnewpath\ntranslate\n274.851 83.3995 m\n286.572 83.3995 l\n286.572 84.3152 l\n274.851 84.3152 l\n274.851 83.3995 l\n} bind def\n/p0_3b {\nnewpath\ntranslate\n274.851 84.3152 m\n286.572 84.3152 l\n286.572 85.2309 l\n274.851 85.2309 l\n274.851 84.3152 l\n} bind def\n/p0_3c {\nnewpath\ntranslate\n274.851 85.2309 m\n286.572 85.2309 l\n286.572 86.1466 l\n274.851 86.1466 l\n274.851 85.2309 l\n} bind def\n/p0_3d {\nnewpath\ntranslate\n274.851 86.1466 m\n286.572 86.1466 l\n286.572 87.0623 l\n274.851 87.0623 l\n274.851 86.1466 l\n} bind def\n/p0_3e {\nnewpath\ntranslate\n274.851 87.0623 m\n286.572 87.0623 l\n286.572 87.978 l\n274.851 87.978 l\n274.851 87.0623 l\n} bind def\n/p0_3f {\nnewpath\ntranslate\n274.851 87.978 m\n286.572 87.978 l\n286.572 88.8937 l\n274.851 88.8937 l\n274.851 87.978 l\n} bind def\n/p0_40 {\nnewpath\ntranslate\n274.851 88.8937 m\n286.572 88.8937 l\n286.572 89.8093 l\n274.851 89.8093 l\n274.851 88.8937 l\n} bind def\n/p0_41 {\nnewpath\ntranslate\n274.851 89.8093 m\n286.572 89.8093 l\n286.572 90.725 l\n274.851 90.725 l\n274.851 89.8093 l\n} bind def\n/p0_42 {\nnewpath\ntranslate\n274.851 90.725 m\n286.572 90.725 l\n286.572 91.6407 l\n274.851 91.6407 l\n274.851 90.725 l\n} bind def\n/p0_43 {\nnewpath\ntranslate\n274.851 91.6407 m\n286.572 91.6407 l\n286.572 92.5564 l\n274.851 92.5564 l\n274.851 91.6407 l\n} bind def\n/p0_44 {\nnewpath\ntranslate\n274.851 92.5564 m\n286.572 92.5564 l\n286.572 93.4721 l\n274.851 93.4721 l\n274.851 92.5564 l\n} bind def\n/p0_45 {\nnewpath\ntranslate\n274.851 93.4721 m\n286.572 93.4721 l\n286.572 94.3878 l\n274.851 94.3878 l\n274.851 93.4721 l\n} bind def\n/p0_46 {\nnewpath\ntranslate\n274.851 94.3878 m\n286.572 94.3878 l\n286.572 95.3035 l\n274.851 95.3035 l\n274.851 94.3878 l\n} bind def\n/p0_47 {\nnewpath\ntranslate\n274.851 95.3035 m\n286.572 95.3035 l\n286.572 96.2192 l\n274.851 96.2192 l\n274.851 95.3035 l\n} bind def\n/p0_48 {\nnewpath\ntranslate\n274.851 96.2192 m\n286.572 96.2192 l\n286.572 97.1348 l\n274.851 97.1348 l\n274.851 96.2192 l\n} bind def\n/p0_49 {\nnewpath\ntranslate\n274.851 97.1348 m\n286.572 97.1348 l\n286.572 98.0505 l\n274.851 98.0505 l\n274.851 97.1348 l\n} bind def\n/p0_4a {\nnewpath\ntranslate\n274.851 98.0505 m\n286.572 98.0505 l\n286.572 98.9662 l\n274.851 98.9662 l\n274.851 98.0505 l\n} bind def\n/p0_4b {\nnewpath\ntranslate\n274.851 98.9662 m\n286.572 98.9662 l\n286.572 99.8819 l\n274.851 99.8819 l\n274.851 98.9662 l\n} bind def\n/p0_4c {\nnewpath\ntranslate\n274.851 99.8819 m\n286.572 99.8819 l\n286.572 100.798 l\n274.851 100.798 l\n274.851 99.8819 l\n} bind def\n/p0_4d {\nnewpath\ntranslate\n274.851 100.798 m\n286.572 100.798 l\n286.572 101.713 l\n274.851 101.713 l\n274.851 100.798 l\n} bind def\n/p0_4e {\nnewpath\ntranslate\n274.851 101.713 m\n286.572 101.713 l\n286.572 102.629 l\n274.851 102.629 l\n274.851 101.713 l\n} bind def\n/p0_4f {\nnewpath\ntranslate\n274.851 102.629 m\n286.572 102.629 l\n286.572 103.545 l\n274.851 103.545 l\n274.851 102.629 l\n} bind def\n/p0_50 {\nnewpath\ntranslate\n274.851 103.545 m\n286.572 103.545 l\n286.572 104.46 l\n274.851 104.46 l\n274.851 103.545 l\n} bind def\n/p0_51 {\nnewpath\ntranslate\n274.851 104.46 m\n286.572 104.46 l\n286.572 105.376 l\n274.851 105.376 l\n274.851 104.46 l\n} bind def\n/p0_52 {\nnewpath\ntranslate\n274.851 105.376 m\n286.572 105.376 l\n286.572 106.292 l\n274.851 106.292 l\n274.851 105.376 l\n} bind def\n/p0_53 {\nnewpath\ntranslate\n274.851 106.292 m\n286.572 106.292 l\n286.572 107.207 l\n274.851 107.207 l\n274.851 106.292 l\n} bind def\n/p0_54 {\nnewpath\ntranslate\n274.851 107.207 m\n286.572 107.207 l\n286.572 108.123 l\n274.851 108.123 l\n274.851 107.207 l\n} bind def\n/p0_55 {\nnewpath\ntranslate\n274.851 108.123 m\n286.572 108.123 l\n286.572 109.039 l\n274.851 109.039 l\n274.851 108.123 l\n} bind def\n/p0_56 {\nnewpath\ntranslate\n274.851 109.039 m\n286.572 109.039 l\n286.572 109.954 l\n274.851 109.954 l\n274.851 109.039 l\n} bind def\n/p0_57 {\nnewpath\ntranslate\n274.851 109.954 m\n286.572 109.954 l\n286.572 110.87 l\n274.851 110.87 l\n274.851 109.954 l\n} bind def\n/p0_58 {\nnewpath\ntranslate\n274.851 110.87 m\n286.572 110.87 l\n286.572 111.786 l\n274.851 111.786 l\n274.851 110.87 l\n} bind def\n/p0_59 {\nnewpath\ntranslate\n274.851 111.786 m\n286.572 111.786 l\n286.572 112.702 l\n274.851 112.702 l\n274.851 111.786 l\n} bind def\n/p0_5a {\nnewpath\ntranslate\n274.851 112.702 m\n286.572 112.702 l\n286.572 113.617 l\n274.851 113.617 l\n274.851 112.702 l\n} bind def\n/p0_5b {\nnewpath\ntranslate\n274.851 113.617 m\n286.572 113.617 l\n286.572 114.533 l\n274.851 114.533 l\n274.851 113.617 l\n} bind def\n/p0_5c {\nnewpath\ntranslate\n274.851 114.533 m\n286.572 114.533 l\n286.572 115.449 l\n274.851 115.449 l\n274.851 114.533 l\n} bind def\n/p0_5d {\nnewpath\ntranslate\n274.851 115.449 m\n286.572 115.449 l\n286.572 116.364 l\n274.851 116.364 l\n274.851 115.449 l\n} bind def\n/p0_5e {\nnewpath\ntranslate\n274.851 116.364 m\n286.572 116.364 l\n286.572 117.28 l\n274.851 117.28 l\n274.851 116.364 l\n} bind def\n/p0_5f {\nnewpath\ntranslate\n274.851 117.28 m\n286.572 117.28 l\n286.572 118.196 l\n274.851 118.196 l\n274.851 117.28 l\n} bind def\n/p0_60 {\nnewpath\ntranslate\n274.851 118.196 m\n286.572 118.196 l\n286.572 119.111 l\n274.851 119.111 l\n274.851 118.196 l\n} bind def\n/p0_61 {\nnewpath\ntranslate\n274.851 119.111 m\n286.572 119.111 l\n286.572 120.027 l\n274.851 120.027 l\n274.851 119.111 l\n} bind def\n/p0_62 {\nnewpath\ntranslate\n274.851 120.027 m\n286.572 120.027 l\n286.572 120.943 l\n274.851 120.943 l\n274.851 120.027 l\n} bind def\n/p0_63 {\nnewpath\ntranslate\n274.851 120.943 m\n286.572 120.943 l\n286.572 121.858 l\n274.851 121.858 l\n274.851 120.943 l\n} bind def\n/p0_64 {\nnewpath\ntranslate\n274.851 121.858 m\n286.572 121.858 l\n286.572 122.774 l\n274.851 122.774 l\n274.851 121.858 l\n} bind def\n/p0_65 {\nnewpath\ntranslate\n274.851 122.774 m\n286.572 122.774 l\n286.572 123.69 l\n274.851 123.69 l\n274.851 122.774 l\n} bind def\n/p0_66 {\nnewpath\ntranslate\n274.851 123.69 m\n286.572 123.69 l\n286.572 124.606 l\n274.851 124.606 l\n274.851 123.69 l\n} bind def\n/p0_67 {\nnewpath\ntranslate\n274.851 124.606 m\n286.572 124.606 l\n286.572 125.521 l\n274.851 125.521 l\n274.851 124.606 l\n} bind def\n/p0_68 {\nnewpath\ntranslate\n274.851 125.521 m\n286.572 125.521 l\n286.572 126.437 l\n274.851 126.437 l\n274.851 125.521 l\n} bind def\n/p0_69 {\nnewpath\ntranslate\n274.851 126.437 m\n286.572 126.437 l\n286.572 127.353 l\n274.851 127.353 l\n274.851 126.437 l\n} bind def\n/p0_6a {\nnewpath\ntranslate\n274.851 127.353 m\n286.572 127.353 l\n286.572 128.268 l\n274.851 128.268 l\n274.851 127.353 l\n} bind def\n/p0_6b {\nnewpath\ntranslate\n274.851 128.268 m\n286.572 128.268 l\n286.572 129.184 l\n274.851 129.184 l\n274.851 128.268 l\n} bind def\n/p0_6c {\nnewpath\ntranslate\n274.851 129.184 m\n286.572 129.184 l\n286.572 130.1 l\n274.851 130.1 l\n274.851 129.184 l\n} bind def\n/p0_6d {\nnewpath\ntranslate\n274.851 130.1 m\n286.572 130.1 l\n286.572 131.015 l\n274.851 131.015 l\n274.851 130.1 l\n} bind def\n/p0_6e {\nnewpath\ntranslate\n274.851 131.015 m\n286.572 131.015 l\n286.572 131.931 l\n274.851 131.931 l\n274.851 131.015 l\n} bind def\n/p0_6f {\nnewpath\ntranslate\n274.851 131.931 m\n286.572 131.931 l\n286.572 132.847 l\n274.851 132.847 l\n274.851 131.931 l\n} bind def\n/p0_70 {\nnewpath\ntranslate\n274.851 132.847 m\n286.572 132.847 l\n286.572 133.762 l\n274.851 133.762 l\n274.851 132.847 l\n} bind def\n/p0_71 {\nnewpath\ntranslate\n274.851 133.762 m\n286.572 133.762 l\n286.572 134.678 l\n274.851 134.678 l\n274.851 133.762 l\n} bind def\n/p0_72 {\nnewpath\ntranslate\n274.851 134.678 m\n286.572 134.678 l\n286.572 135.594 l\n274.851 135.594 l\n274.851 134.678 l\n} bind def\n/p0_73 {\nnewpath\ntranslate\n274.851 135.594 m\n286.572 135.594 l\n286.572 136.509 l\n274.851 136.509 l\n274.851 135.594 l\n} bind def\n/p0_74 {\nnewpath\ntranslate\n274.851 136.509 m\n286.572 136.509 l\n286.572 137.425 l\n274.851 137.425 l\n274.851 136.509 l\n} bind def\n/p0_75 {\nnewpath\ntranslate\n274.851 137.425 m\n286.572 137.425 l\n286.572 138.341 l\n274.851 138.341 l\n274.851 137.425 l\n} bind def\n/p0_76 {\nnewpath\ntranslate\n274.851 138.341 m\n286.572 138.341 l\n286.572 139.257 l\n274.851 139.257 l\n274.851 138.341 l\n} bind def\n/p0_77 {\nnewpath\ntranslate\n274.851 139.257 m\n286.572 139.257 l\n286.572 140.172 l\n274.851 140.172 l\n274.851 139.257 l\n} bind def\n/p0_78 {\nnewpath\ntranslate\n274.851 140.172 m\n286.572 140.172 l\n286.572 141.088 l\n274.851 141.088 l\n274.851 140.172 l\n} bind def\n/p0_79 {\nnewpath\ntranslate\n274.851 141.088 m\n286.572 141.088 l\n286.572 142.004 l\n274.851 142.004 l\n274.851 141.088 l\n} bind def\n/p0_7a {\nnewpath\ntranslate\n274.851 142.004 m\n286.572 142.004 l\n286.572 142.919 l\n274.851 142.919 l\n274.851 142.004 l\n} bind def\n/p0_7b {\nnewpath\ntranslate\n274.851 142.919 m\n286.572 142.919 l\n286.572 143.835 l\n274.851 143.835 l\n274.851 142.919 l\n} bind def\n/p0_7c {\nnewpath\ntranslate\n274.851 143.835 m\n286.572 143.835 l\n286.572 144.751 l\n274.851 144.751 l\n274.851 143.835 l\n} bind def\n/p0_7d {\nnewpath\ntranslate\n274.851 144.751 m\n286.572 144.751 l\n286.572 145.666 l\n274.851 145.666 l\n274.851 144.751 l\n} bind def\n/p0_7e {\nnewpath\ntranslate\n274.851 145.666 m\n286.572 145.666 l\n286.572 146.582 l\n274.851 146.582 l\n274.851 145.666 l\n} bind def\n/p0_7f {\nnewpath\ntranslate\n274.851 146.582 m\n286.572 146.582 l\n286.572 147.498 l\n274.851 147.498 l\n274.851 146.582 l\n} bind def\n/p0_80 {\nnewpath\ntranslate\n274.851 147.498 m\n286.572 147.498 l\n286.572 148.413 l\n274.851 148.413 l\n274.851 147.498 l\n} bind def\n/p0_81 {\nnewpath\ntranslate\n274.851 148.413 m\n286.572 148.413 l\n286.572 149.329 l\n274.851 149.329 l\n274.851 148.413 l\n} bind def\n/p0_82 {\nnewpath\ntranslate\n274.851 149.329 m\n286.572 149.329 l\n286.572 150.245 l\n274.851 150.245 l\n274.851 149.329 l\n} bind def\n/p0_83 {\nnewpath\ntranslate\n274.851 150.245 m\n286.572 150.245 l\n286.572 151.16 l\n274.851 151.16 l\n274.851 150.245 l\n} bind def\n/p0_84 {\nnewpath\ntranslate\n274.851 151.16 m\n286.572 151.16 l\n286.572 152.076 l\n274.851 152.076 l\n274.851 151.16 l\n} bind def\n/p0_85 {\nnewpath\ntranslate\n274.851 152.076 m\n286.572 152.076 l\n286.572 152.992 l\n274.851 152.992 l\n274.851 152.076 l\n} bind def\n/p0_86 {\nnewpath\ntranslate\n274.851 152.992 m\n286.572 152.992 l\n286.572 153.908 l\n274.851 153.908 l\n274.851 152.992 l\n} bind def\n/p0_87 {\nnewpath\ntranslate\n274.851 153.908 m\n286.572 153.908 l\n286.572 154.823 l\n274.851 154.823 l\n274.851 153.908 l\n} bind def\n/p0_88 {\nnewpath\ntranslate\n274.851 154.823 m\n286.572 154.823 l\n286.572 155.739 l\n274.851 155.739 l\n274.851 154.823 l\n} bind def\n/p0_89 {\nnewpath\ntranslate\n274.851 155.739 m\n286.572 155.739 l\n286.572 156.655 l\n274.851 156.655 l\n274.851 155.739 l\n} bind def\n/p0_8a {\nnewpath\ntranslate\n274.851 156.655 m\n286.572 156.655 l\n286.572 157.57 l\n274.851 157.57 l\n274.851 156.655 l\n} bind def\n/p0_8b {\nnewpath\ntranslate\n274.851 157.57 m\n286.572 157.57 l\n286.572 158.486 l\n274.851 158.486 l\n274.851 157.57 l\n} bind def\n/p0_8c {\nnewpath\ntranslate\n274.851 158.486 m\n286.572 158.486 l\n286.572 159.402 l\n274.851 159.402 l\n274.851 158.486 l\n} bind def\n/p0_8d {\nnewpath\ntranslate\n274.851 159.402 m\n286.572 159.402 l\n286.572 160.317 l\n274.851 160.317 l\n274.851 159.402 l\n} bind def\n/p0_8e {\nnewpath\ntranslate\n274.851 160.317 m\n286.572 160.317 l\n286.572 161.233 l\n274.851 161.233 l\n274.851 160.317 l\n} bind def\n/p0_8f {\nnewpath\ntranslate\n274.851 161.233 m\n286.572 161.233 l\n286.572 162.149 l\n274.851 162.149 l\n274.851 161.233 l\n} bind def\n/p0_90 {\nnewpath\ntranslate\n274.851 162.149 m\n286.572 162.149 l\n286.572 163.064 l\n274.851 163.064 l\n274.851 162.149 l\n} bind def\n/p0_91 {\nnewpath\ntranslate\n274.851 163.064 m\n286.572 163.064 l\n286.572 163.98 l\n274.851 163.98 l\n274.851 163.064 l\n} bind def\n/p0_92 {\nnewpath\ntranslate\n274.851 163.98 m\n286.572 163.98 l\n286.572 164.896 l\n274.851 164.896 l\n274.851 163.98 l\n} bind def\n/p0_93 {\nnewpath\ntranslate\n274.851 164.896 m\n286.572 164.896 l\n286.572 165.811 l\n274.851 165.811 l\n274.851 164.896 l\n} bind def\n/p0_94 {\nnewpath\ntranslate\n274.851 165.811 m\n286.572 165.811 l\n286.572 166.727 l\n274.851 166.727 l\n274.851 165.811 l\n} bind def\n/p0_95 {\nnewpath\ntranslate\n274.851 166.727 m\n286.572 166.727 l\n286.572 167.643 l\n274.851 167.643 l\n274.851 166.727 l\n} bind def\n/p0_96 {\nnewpath\ntranslate\n274.851 167.643 m\n286.572 167.643 l\n286.572 168.559 l\n274.851 168.559 l\n274.851 167.643 l\n} bind def\n/p0_97 {\nnewpath\ntranslate\n274.851 168.559 m\n286.572 168.559 l\n286.572 169.474 l\n274.851 169.474 l\n274.851 168.559 l\n} bind def\n/p0_98 {\nnewpath\ntranslate\n274.851 169.474 m\n286.572 169.474 l\n286.572 170.39 l\n274.851 170.39 l\n274.851 169.474 l\n} bind def\n/p0_99 {\nnewpath\ntranslate\n274.851 170.39 m\n286.572 170.39 l\n286.572 171.306 l\n274.851 171.306 l\n274.851 170.39 l\n} bind def\n/p0_9a {\nnewpath\ntranslate\n274.851 171.306 m\n286.572 171.306 l\n286.572 172.221 l\n274.851 172.221 l\n274.851 171.306 l\n} bind def\n/p0_9b {\nnewpath\ntranslate\n274.851 172.221 m\n286.572 172.221 l\n286.572 173.137 l\n274.851 173.137 l\n274.851 172.221 l\n} bind def\n/p0_9c {\nnewpath\ntranslate\n274.851 173.137 m\n286.572 173.137 l\n286.572 174.053 l\n274.851 174.053 l\n274.851 173.137 l\n} bind def\n/p0_9d {\nnewpath\ntranslate\n274.851 174.053 m\n286.572 174.053 l\n286.572 174.968 l\n274.851 174.968 l\n274.851 174.053 l\n} bind def\n/p0_9e {\nnewpath\ntranslate\n274.851 174.968 m\n286.572 174.968 l\n286.572 175.884 l\n274.851 175.884 l\n274.851 174.968 l\n} bind def\n/p0_9f {\nnewpath\ntranslate\n274.851 175.884 m\n286.572 175.884 l\n286.572 176.8 l\n274.851 176.8 l\n274.851 175.884 l\n} bind def\n/p0_a0 {\nnewpath\ntranslate\n274.851 176.8 m\n286.572 176.8 l\n286.572 177.715 l\n274.851 177.715 l\n274.851 176.8 l\n} bind def\n/p0_a1 {\nnewpath\ntranslate\n274.851 177.715 m\n286.572 177.715 l\n286.572 178.631 l\n274.851 178.631 l\n274.851 177.715 l\n} bind def\n/p0_a2 {\nnewpath\ntranslate\n274.851 178.631 m\n286.572 178.631 l\n286.572 179.547 l\n274.851 179.547 l\n274.851 178.631 l\n} bind def\n/p0_a3 {\nnewpath\ntranslate\n274.851 179.547 m\n286.572 179.547 l\n286.572 180.462 l\n274.851 180.462 l\n274.851 179.547 l\n} bind def\n/p0_a4 {\nnewpath\ntranslate\n274.851 180.462 m\n286.572 180.462 l\n286.572 181.378 l\n274.851 181.378 l\n274.851 180.462 l\n} bind def\n/p0_a5 {\nnewpath\ntranslate\n274.851 181.378 m\n286.572 181.378 l\n286.572 182.294 l\n274.851 182.294 l\n274.851 181.378 l\n} bind def\n/p0_a6 {\nnewpath\ntranslate\n274.851 182.294 m\n286.572 182.294 l\n286.572 183.21 l\n274.851 183.21 l\n274.851 182.294 l\n} bind def\n/p0_a7 {\nnewpath\ntranslate\n274.851 183.21 m\n286.572 183.21 l\n286.572 184.125 l\n274.851 184.125 l\n274.851 183.21 l\n} bind def\n/p0_a8 {\nnewpath\ntranslate\n274.851 184.125 m\n286.572 184.125 l\n286.572 185.041 l\n274.851 185.041 l\n274.851 184.125 l\n} bind def\n/p0_a9 {\nnewpath\ntranslate\n274.851 185.041 m\n286.572 185.041 l\n286.572 185.957 l\n274.851 185.957 l\n274.851 185.041 l\n} bind def\n/p0_aa {\nnewpath\ntranslate\n274.851 185.957 m\n286.572 185.957 l\n286.572 186.872 l\n274.851 186.872 l\n274.851 185.957 l\n} bind def\n/p0_ab {\nnewpath\ntranslate\n274.851 186.872 m\n286.572 186.872 l\n286.572 187.788 l\n274.851 187.788 l\n274.851 186.872 l\n} bind def\n/p0_ac {\nnewpath\ntranslate\n274.851 187.788 m\n286.572 187.788 l\n286.572 188.704 l\n274.851 188.704 l\n274.851 187.788 l\n} bind def\n/p0_ad {\nnewpath\ntranslate\n274.851 188.704 m\n286.572 188.704 l\n286.572 189.619 l\n274.851 189.619 l\n274.851 188.704 l\n} bind def\n/p0_ae {\nnewpath\ntranslate\n274.851 189.619 m\n286.572 189.619 l\n286.572 190.535 l\n274.851 190.535 l\n274.851 189.619 l\n} bind def\n/p0_af {\nnewpath\ntranslate\n274.851 190.535 m\n286.572 190.535 l\n286.572 191.451 l\n274.851 191.451 l\n274.851 190.535 l\n} bind def\n/p0_b0 {\nnewpath\ntranslate\n274.851 191.451 m\n286.572 191.451 l\n286.572 192.366 l\n274.851 192.366 l\n274.851 191.451 l\n} bind def\n/p0_b1 {\nnewpath\ntranslate\n274.851 192.366 m\n286.572 192.366 l\n286.572 193.282 l\n274.851 193.282 l\n274.851 192.366 l\n} bind def\n/p0_b2 {\nnewpath\ntranslate\n274.851 193.282 m\n286.572 193.282 l\n286.572 194.198 l\n274.851 194.198 l\n274.851 193.282 l\n} bind def\n/p0_b3 {\nnewpath\ntranslate\n274.851 194.198 m\n286.572 194.198 l\n286.572 195.114 l\n274.851 195.114 l\n274.851 194.198 l\n} bind def\n/p0_b4 {\nnewpath\ntranslate\n274.851 195.114 m\n286.572 195.114 l\n286.572 196.029 l\n274.851 196.029 l\n274.851 195.114 l\n} bind def\n/p0_b5 {\nnewpath\ntranslate\n274.851 196.029 m\n286.572 196.029 l\n286.572 196.945 l\n274.851 196.945 l\n274.851 196.029 l\n} bind def\n/p0_b6 {\nnewpath\ntranslate\n274.851 196.945 m\n286.572 196.945 l\n286.572 197.861 l\n274.851 197.861 l\n274.851 196.945 l\n} bind def\n/p0_b7 {\nnewpath\ntranslate\n274.851 197.861 m\n286.572 197.861 l\n286.572 198.776 l\n274.851 198.776 l\n274.851 197.861 l\n} bind def\n/p0_b8 {\nnewpath\ntranslate\n274.851 198.776 m\n286.572 198.776 l\n286.572 199.692 l\n274.851 199.692 l\n274.851 198.776 l\n} bind def\n/p0_b9 {\nnewpath\ntranslate\n274.851 199.692 m\n286.572 199.692 l\n286.572 200.608 l\n274.851 200.608 l\n274.851 199.692 l\n} bind def\n/p0_ba {\nnewpath\ntranslate\n274.851 200.608 m\n286.572 200.608 l\n286.572 201.523 l\n274.851 201.523 l\n274.851 200.608 l\n} bind def\n/p0_bb {\nnewpath\ntranslate\n274.851 201.523 m\n286.572 201.523 l\n286.572 202.439 l\n274.851 202.439 l\n274.851 201.523 l\n} bind def\n/p0_bc {\nnewpath\ntranslate\n274.851 202.439 m\n286.572 202.439 l\n286.572 203.355 l\n274.851 203.355 l\n274.851 202.439 l\n} bind def\n/p0_bd {\nnewpath\ntranslate\n274.851 203.355 m\n286.572 203.355 l\n286.572 204.27 l\n274.851 204.27 l\n274.851 203.355 l\n} bind def\n/p0_be {\nnewpath\ntranslate\n274.851 204.27 m\n286.572 204.27 l\n286.572 205.186 l\n274.851 205.186 l\n274.851 204.27 l\n} bind def\n/p0_bf {\nnewpath\ntranslate\n274.851 205.186 m\n286.572 205.186 l\n286.572 206.102 l\n274.851 206.102 l\n274.851 205.186 l\n} bind def\n/p0_c0 {\nnewpath\ntranslate\n274.851 206.102 m\n286.572 206.102 l\n286.572 207.017 l\n274.851 207.017 l\n274.851 206.102 l\n} bind def\n/p0_c1 {\nnewpath\ntranslate\n274.851 207.017 m\n286.572 207.017 l\n286.572 207.933 l\n274.851 207.933 l\n274.851 207.017 l\n} bind def\n/p0_c2 {\nnewpath\ntranslate\n274.851 207.933 m\n286.572 207.933 l\n286.572 208.849 l\n274.851 208.849 l\n274.851 207.933 l\n} bind def\n/p0_c3 {\nnewpath\ntranslate\n274.851 208.849 m\n286.572 208.849 l\n286.572 209.765 l\n274.851 209.765 l\n274.851 208.849 l\n} bind def\n/p0_c4 {\nnewpath\ntranslate\n274.851 209.765 m\n286.572 209.765 l\n286.572 210.68 l\n274.851 210.68 l\n274.851 209.765 l\n} bind def\n/p0_c5 {\nnewpath\ntranslate\n274.851 210.68 m\n286.572 210.68 l\n286.572 211.596 l\n274.851 211.596 l\n274.851 210.68 l\n} bind def\n/p0_c6 {\nnewpath\ntranslate\n274.851 211.596 m\n286.572 211.596 l\n286.572 212.512 l\n274.851 212.512 l\n274.851 211.596 l\n} bind def\n/p0_c7 {\nnewpath\ntranslate\n274.851 212.512 m\n286.572 212.512 l\n286.572 213.427 l\n274.851 213.427 l\n274.851 212.512 l\n} bind def\n/p0_c8 {\nnewpath\ntranslate\n274.851 213.427 m\n286.572 213.427 l\n286.572 214.343 l\n274.851 214.343 l\n274.851 213.427 l\n} bind def\n/p0_c9 {\nnewpath\ntranslate\n274.851 214.343 m\n286.572 214.343 l\n286.572 215.259 l\n274.851 215.259 l\n274.851 214.343 l\n} bind def\n/p0_ca {\nnewpath\ntranslate\n274.851 215.259 m\n286.572 215.259 l\n286.572 216.174 l\n274.851 216.174 l\n274.851 215.259 l\n} bind def\n/p0_cb {\nnewpath\ntranslate\n274.851 216.174 m\n286.572 216.174 l\n286.572 217.09 l\n274.851 217.09 l\n274.851 216.174 l\n} bind def\n/p0_cc {\nnewpath\ntranslate\n274.851 217.09 m\n286.572 217.09 l\n286.572 218.006 l\n274.851 218.006 l\n274.851 217.09 l\n} bind def\n/p0_cd {\nnewpath\ntranslate\n274.851 218.006 m\n286.572 218.006 l\n286.572 218.921 l\n274.851 218.921 l\n274.851 218.006 l\n} bind def\n/p0_ce {\nnewpath\ntranslate\n274.851 218.921 m\n286.572 218.921 l\n286.572 219.837 l\n274.851 219.837 l\n274.851 218.921 l\n} bind def\n/p0_cf {\nnewpath\ntranslate\n274.851 219.837 m\n286.572 219.837 l\n286.572 220.753 l\n274.851 220.753 l\n274.851 219.837 l\n} bind def\n/p0_d0 {\nnewpath\ntranslate\n274.851 220.753 m\n286.572 220.753 l\n286.572 221.668 l\n274.851 221.668 l\n274.851 220.753 l\n} bind def\n/p0_d1 {\nnewpath\ntranslate\n274.851 221.668 m\n286.572 221.668 l\n286.572 222.584 l\n274.851 222.584 l\n274.851 221.668 l\n} bind def\n/p0_d2 {\nnewpath\ntranslate\n274.851 222.584 m\n286.572 222.584 l\n286.572 223.5 l\n274.851 223.5 l\n274.851 222.584 l\n} bind def\n/p0_d3 {\nnewpath\ntranslate\n274.851 223.5 m\n286.572 223.5 l\n286.572 224.416 l\n274.851 224.416 l\n274.851 223.5 l\n} bind def\n/p0_d4 {\nnewpath\ntranslate\n274.851 224.416 m\n286.572 224.416 l\n286.572 225.331 l\n274.851 225.331 l\n274.851 224.416 l\n} bind def\n/p0_d5 {\nnewpath\ntranslate\n274.851 225.331 m\n286.572 225.331 l\n286.572 226.247 l\n274.851 226.247 l\n274.851 225.331 l\n} bind def\n/p0_d6 {\nnewpath\ntranslate\n274.851 226.247 m\n286.572 226.247 l\n286.572 227.163 l\n274.851 227.163 l\n274.851 226.247 l\n} bind def\n/p0_d7 {\nnewpath\ntranslate\n274.851 227.163 m\n286.572 227.163 l\n286.572 228.078 l\n274.851 228.078 l\n274.851 227.163 l\n} bind def\n/p0_d8 {\nnewpath\ntranslate\n274.851 228.078 m\n286.572 228.078 l\n286.572 228.994 l\n274.851 228.994 l\n274.851 228.078 l\n} bind def\n/p0_d9 {\nnewpath\ntranslate\n274.851 228.994 m\n286.572 228.994 l\n286.572 229.91 l\n274.851 229.91 l\n274.851 228.994 l\n} bind def\n/p0_da {\nnewpath\ntranslate\n274.851 229.91 m\n286.572 229.91 l\n286.572 230.825 l\n274.851 230.825 l\n274.851 229.91 l\n} bind def\n/p0_db {\nnewpath\ntranslate\n274.851 230.825 m\n286.572 230.825 l\n286.572 231.741 l\n274.851 231.741 l\n274.851 230.825 l\n} bind def\n/p0_dc {\nnewpath\ntranslate\n274.851 231.741 m\n286.572 231.741 l\n286.572 232.657 l\n274.851 232.657 l\n274.851 231.741 l\n} bind def\n/p0_dd {\nnewpath\ntranslate\n274.851 232.657 m\n286.572 232.657 l\n286.572 233.572 l\n274.851 233.572 l\n274.851 232.657 l\n} bind def\n/p0_de {\nnewpath\ntranslate\n274.851 233.572 m\n286.572 233.572 l\n286.572 234.488 l\n274.851 234.488 l\n274.851 233.572 l\n} bind def\n/p0_df {\nnewpath\ntranslate\n274.851 234.488 m\n286.572 234.488 l\n286.572 235.404 l\n274.851 235.404 l\n274.851 234.488 l\n} bind def\n/p0_e0 {\nnewpath\ntranslate\n274.851 235.404 m\n286.572 235.404 l\n286.572 236.319 l\n274.851 236.319 l\n274.851 235.404 l\n} bind def\n/p0_e1 {\nnewpath\ntranslate\n274.851 236.319 m\n286.572 236.319 l\n286.572 237.235 l\n274.851 237.235 l\n274.851 236.319 l\n} bind def\n/p0_e2 {\nnewpath\ntranslate\n274.851 237.235 m\n286.572 237.235 l\n286.572 238.151 l\n274.851 238.151 l\n274.851 237.235 l\n} bind def\n/p0_e3 {\nnewpath\ntranslate\n274.851 238.151 m\n286.572 238.151 l\n286.572 239.067 l\n274.851 239.067 l\n274.851 238.151 l\n} bind def\n/p0_e4 {\nnewpath\ntranslate\n274.851 239.067 m\n286.572 239.067 l\n286.572 239.982 l\n274.851 239.982 l\n274.851 239.067 l\n} bind def\n/p0_e5 {\nnewpath\ntranslate\n274.851 239.982 m\n286.572 239.982 l\n286.572 240.898 l\n274.851 240.898 l\n274.851 239.982 l\n} bind def\n/p0_e6 {\nnewpath\ntranslate\n274.851 240.898 m\n286.572 240.898 l\n286.572 241.814 l\n274.851 241.814 l\n274.851 240.898 l\n} bind def\n/p0_e7 {\nnewpath\ntranslate\n274.851 241.814 m\n286.572 241.814 l\n286.572 242.729 l\n274.851 242.729 l\n274.851 241.814 l\n} bind def\n/p0_e8 {\nnewpath\ntranslate\n274.851 242.729 m\n286.572 242.729 l\n286.572 243.645 l\n274.851 243.645 l\n274.851 242.729 l\n} bind def\n/p0_e9 {\nnewpath\ntranslate\n274.851 243.645 m\n286.572 243.645 l\n286.572 244.561 l\n274.851 244.561 l\n274.851 243.645 l\n} bind def\n/p0_ea {\nnewpath\ntranslate\n274.851 244.561 m\n286.572 244.561 l\n286.572 245.476 l\n274.851 245.476 l\n274.851 244.561 l\n} bind def\n/p0_eb {\nnewpath\ntranslate\n274.851 245.476 m\n286.572 245.476 l\n286.572 246.392 l\n274.851 246.392 l\n274.851 245.476 l\n} bind def\n/p0_ec {\nnewpath\ntranslate\n274.851 246.392 m\n286.572 246.392 l\n286.572 247.308 l\n274.851 247.308 l\n274.851 246.392 l\n} bind def\n/p0_ed {\nnewpath\ntranslate\n274.851 247.308 m\n286.572 247.308 l\n286.572 248.223 l\n274.851 248.223 l\n274.851 247.308 l\n} bind def\n/p0_ee {\nnewpath\ntranslate\n274.851 248.223 m\n286.572 248.223 l\n286.572 249.139 l\n274.851 249.139 l\n274.851 248.223 l\n} bind def\n/p0_ef {\nnewpath\ntranslate\n274.851 249.139 m\n286.572 249.139 l\n286.572 250.055 l\n274.851 250.055 l\n274.851 249.139 l\n} bind def\n/p0_f0 {\nnewpath\ntranslate\n274.851 250.055 m\n286.572 250.055 l\n286.572 250.97 l\n274.851 250.97 l\n274.851 250.055 l\n} bind def\n/p0_f1 {\nnewpath\ntranslate\n274.851 250.97 m\n286.572 250.97 l\n286.572 251.886 l\n274.851 251.886 l\n274.851 250.97 l\n} bind def\n/p0_f2 {\nnewpath\ntranslate\n274.851 251.886 m\n286.572 251.886 l\n286.572 252.802 l\n274.851 252.802 l\n274.851 251.886 l\n} bind def\n/p0_f3 {\nnewpath\ntranslate\n274.851 252.802 m\n286.572 252.802 l\n286.572 253.718 l\n274.851 253.718 l\n274.851 252.802 l\n} bind def\n/p0_f4 {\nnewpath\ntranslate\n274.851 253.718 m\n286.572 253.718 l\n286.572 254.633 l\n274.851 254.633 l\n274.851 253.718 l\n} bind def\n/p0_f5 {\nnewpath\ntranslate\n274.851 254.633 m\n286.572 254.633 l\n286.572 255.549 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def\n/p1_16 {\nnewpath\ntranslate\n609.242 50.2626 m\n620.972 50.2626 l\n620.972 51.179 l\n609.242 51.179 l\n609.242 50.2626 l\n} bind def\n/p1_17 {\nnewpath\ntranslate\n609.242 51.179 m\n620.972 51.179 l\n620.972 52.0954 l\n609.242 52.0954 l\n609.242 51.179 l\n} bind def\n/p1_18 {\nnewpath\ntranslate\n609.242 52.0954 m\n620.972 52.0954 l\n620.972 53.0118 l\n609.242 53.0118 l\n609.242 52.0954 l\n} bind def\n/p1_19 {\nnewpath\ntranslate\n609.242 53.0118 m\n620.972 53.0118 l\n620.972 53.9283 l\n609.242 53.9283 l\n609.242 53.0118 l\n} bind def\n/p1_1a {\nnewpath\ntranslate\n609.242 53.9283 m\n620.972 53.9283 l\n620.972 54.8447 l\n609.242 54.8447 l\n609.242 53.9283 l\n} bind def\n/p1_1b {\nnewpath\ntranslate\n609.242 54.8447 m\n620.972 54.8447 l\n620.972 55.7611 l\n609.242 55.7611 l\n609.242 54.8447 l\n} bind def\n/p1_1c {\nnewpath\ntranslate\n609.242 55.7611 m\n620.972 55.7611 l\n620.972 56.6775 l\n609.242 56.6775 l\n609.242 55.7611 l\n} bind def\n/p1_1d {\nnewpath\ntranslate\n609.242 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{\nnewpath\ntranslate\n609.242 76.8389 m\n620.972 76.8389 l\n620.972 77.7553 l\n609.242 77.7553 l\n609.242 76.8389 l\n} bind def\n/p1_34 {\nnewpath\ntranslate\n609.242 77.7553 m\n620.972 77.7553 l\n620.972 78.6717 l\n609.242 78.6717 l\n609.242 77.7553 l\n} bind def\n/p1_35 {\nnewpath\ntranslate\n609.242 78.6717 m\n620.972 78.6717 l\n620.972 79.5881 l\n609.242 79.5881 l\n609.242 78.6717 l\n} bind def\n/p1_36 {\nnewpath\ntranslate\n609.242 79.5881 m\n620.972 79.5881 l\n620.972 80.5046 l\n609.242 80.5046 l\n609.242 79.5881 l\n} bind def\n/p1_37 {\nnewpath\ntranslate\n609.242 80.5046 m\n620.972 80.5046 l\n620.972 81.421 l\n609.242 81.421 l\n609.242 80.5046 l\n} bind def\n/p1_38 {\nnewpath\ntranslate\n609.242 81.421 m\n620.972 81.421 l\n620.972 82.3374 l\n609.242 82.3374 l\n609.242 81.421 l\n} bind def\n/p1_39 {\nnewpath\ntranslate\n609.242 82.3374 m\n620.972 82.3374 l\n620.972 83.2538 l\n609.242 83.2538 l\n609.242 82.3374 l\n} bind def\n/p1_3a {\nnewpath\ntranslate\n609.242 83.2538 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l\n609.242 90.5852 l\n609.242 89.6688 l\n} bind def\n/p1_42 {\nnewpath\ntranslate\n609.242 90.5852 m\n620.972 90.5852 l\n620.972 91.5017 l\n609.242 91.5017 l\n609.242 90.5852 l\n} bind def\n/p1_43 {\nnewpath\ntranslate\n609.242 91.5017 m\n620.972 91.5017 l\n620.972 92.4181 l\n609.242 92.4181 l\n609.242 91.5017 l\n} bind def\n/p1_44 {\nnewpath\ntranslate\n609.242 92.4181 m\n620.972 92.4181 l\n620.972 93.3345 l\n609.242 93.3345 l\n609.242 92.4181 l\n} bind def\n/p1_45 {\nnewpath\ntranslate\n609.242 93.3345 m\n620.972 93.3345 l\n620.972 94.2509 l\n609.242 94.2509 l\n609.242 93.3345 l\n} bind def\n/p1_46 {\nnewpath\ntranslate\n609.242 94.2509 m\n620.972 94.2509 l\n620.972 95.1674 l\n609.242 95.1674 l\n609.242 94.2509 l\n} bind def\n/p1_47 {\nnewpath\ntranslate\n609.242 95.1674 m\n620.972 95.1674 l\n620.972 96.0838 l\n609.242 96.0838 l\n609.242 95.1674 l\n} bind def\n/p1_48 {\nnewpath\ntranslate\n609.242 96.0838 m\n620.972 96.0838 l\n620.972 97.0002 l\n609.242 97.0002 l\n609.242 96.0838 l\n} bind def\n/p1_49 {\nnewpath\ntranslate\n609.242 97.0002 m\n620.972 97.0002 l\n620.972 97.9166 l\n609.242 97.9166 l\n609.242 97.0002 l\n} bind def\n/p1_4a {\nnewpath\ntranslate\n609.242 97.9166 m\n620.972 97.9166 l\n620.972 98.8331 l\n609.242 98.8331 l\n609.242 97.9166 l\n} bind def\n/p1_4b {\nnewpath\ntranslate\n609.242 98.8331 m\n620.972 98.8331 l\n620.972 99.7495 l\n609.242 99.7495 l\n609.242 98.8331 l\n} bind def\n/p1_4c {\nnewpath\ntranslate\n609.242 99.7495 m\n620.972 99.7495 l\n620.972 100.666 l\n609.242 100.666 l\n609.242 99.7495 l\n} bind def\n/p1_4d {\nnewpath\ntranslate\n609.242 100.666 m\n620.972 100.666 l\n620.972 101.582 l\n609.242 101.582 l\n609.242 100.666 l\n} bind def\n/p1_4e {\nnewpath\ntranslate\n609.242 101.582 m\n620.972 101.582 l\n620.972 102.499 l\n609.242 102.499 l\n609.242 101.582 l\n} bind def\n/p1_4f {\nnewpath\ntranslate\n609.242 102.499 m\n620.972 102.499 l\n620.972 103.415 l\n609.242 103.415 l\n609.242 102.499 l\n} bind def\n/p1_50 {\nnewpath\ntranslate\n609.242 103.415 m\n620.972 103.415 l\n620.972 104.332 l\n609.242 104.332 l\n609.242 103.415 l\n} bind def\n/p1_51 {\nnewpath\ntranslate\n609.242 104.332 m\n620.972 104.332 l\n620.972 105.248 l\n609.242 105.248 l\n609.242 104.332 l\n} bind def\n/p1_52 {\nnewpath\ntranslate\n609.242 105.248 m\n620.972 105.248 l\n620.972 106.164 l\n609.242 106.164 l\n609.242 105.248 l\n} bind def\n/p1_53 {\nnewpath\ntranslate\n609.242 106.164 m\n620.972 106.164 l\n620.972 107.081 l\n609.242 107.081 l\n609.242 106.164 l\n} bind def\n/p1_54 {\nnewpath\ntranslate\n609.242 107.081 m\n620.972 107.081 l\n620.972 107.997 l\n609.242 107.997 l\n609.242 107.081 l\n} bind def\n/p1_55 {\nnewpath\ntranslate\n609.242 107.997 m\n620.972 107.997 l\n620.972 108.914 l\n609.242 108.914 l\n609.242 107.997 l\n} bind def\n/p1_56 {\nnewpath\ntranslate\n609.242 108.914 m\n620.972 108.914 l\n620.972 109.83 l\n609.242 109.83 l\n609.242 108.914 l\n} bind def\n/p1_57 {\nnewpath\ntranslate\n609.242 109.83 m\n620.972 109.83 l\n620.972 110.747 l\n609.242 110.747 l\n609.242 109.83 l\n} bind def\n/p1_58 {\nnewpath\ntranslate\n609.242 110.747 m\n620.972 110.747 l\n620.972 111.663 l\n609.242 111.663 l\n609.242 110.747 l\n} bind def\n/p1_59 {\nnewpath\ntranslate\n609.242 111.663 m\n620.972 111.663 l\n620.972 112.579 l\n609.242 112.579 l\n609.242 111.663 l\n} bind def\n/p1_5a {\nnewpath\ntranslate\n609.242 112.579 m\n620.972 112.579 l\n620.972 113.496 l\n609.242 113.496 l\n609.242 112.579 l\n} bind def\n/p1_5b {\nnewpath\ntranslate\n609.242 113.496 m\n620.972 113.496 l\n620.972 114.412 l\n609.242 114.412 l\n609.242 113.496 l\n} bind def\n/p1_5c {\nnewpath\ntranslate\n609.242 114.412 m\n620.972 114.412 l\n620.972 115.329 l\n609.242 115.329 l\n609.242 114.412 l\n} bind def\n/p1_5d {\nnewpath\ntranslate\n609.242 115.329 m\n620.972 115.329 l\n620.972 116.245 l\n609.242 116.245 l\n609.242 115.329 l\n} bind def\n/p1_5e {\nnewpath\ntranslate\n609.242 116.245 m\n620.972 116.245 l\n620.972 117.162 l\n609.242 117.162 l\n609.242 116.245 l\n} bind def\n/p1_5f {\nnewpath\ntranslate\n609.242 117.162 m\n620.972 117.162 l\n620.972 118.078 l\n609.242 118.078 l\n609.242 117.162 l\n} bind def\n/p1_60 {\nnewpath\ntranslate\n609.242 118.078 m\n620.972 118.078 l\n620.972 118.994 l\n609.242 118.994 l\n609.242 118.078 l\n} bind def\n/p1_61 {\nnewpath\ntranslate\n609.242 118.994 m\n620.972 118.994 l\n620.972 119.911 l\n609.242 119.911 l\n609.242 118.994 l\n} bind def\n/p1_62 {\nnewpath\ntranslate\n609.242 119.911 m\n620.972 119.911 l\n620.972 120.827 l\n609.242 120.827 l\n609.242 119.911 l\n} bind def\n/p1_63 {\nnewpath\ntranslate\n609.242 120.827 m\n620.972 120.827 l\n620.972 121.744 l\n609.242 121.744 l\n609.242 120.827 l\n} bind def\n/p1_64 {\nnewpath\ntranslate\n609.242 121.744 m\n620.972 121.744 l\n620.972 122.66 l\n609.242 122.66 l\n609.242 121.744 l\n} bind def\n/p1_65 {\nnewpath\ntranslate\n609.242 122.66 m\n620.972 122.66 l\n620.972 123.577 l\n609.242 123.577 l\n609.242 122.66 l\n} bind def\n/p1_66 {\nnewpath\ntranslate\n609.242 123.577 m\n620.972 123.577 l\n620.972 124.493 l\n609.242 124.493 l\n609.242 123.577 l\n} bind def\n/p1_67 {\nnewpath\ntranslate\n609.242 124.493 m\n620.972 124.493 l\n620.972 125.409 l\n609.242 125.409 l\n609.242 124.493 l\n} bind def\n/p1_68 {\nnewpath\ntranslate\n609.242 125.409 m\n620.972 125.409 l\n620.972 126.326 l\n609.242 126.326 l\n609.242 125.409 l\n} bind def\n/p1_69 {\nnewpath\ntranslate\n609.242 126.326 m\n620.972 126.326 l\n620.972 127.242 l\n609.242 127.242 l\n609.242 126.326 l\n} bind def\n/p1_6a {\nnewpath\ntranslate\n609.242 127.242 m\n620.972 127.242 l\n620.972 128.159 l\n609.242 128.159 l\n609.242 127.242 l\n} bind def\n/p1_6b {\nnewpath\ntranslate\n609.242 128.159 m\n620.972 128.159 l\n620.972 129.075 l\n609.242 129.075 l\n609.242 128.159 l\n} bind def\n/p1_6c {\nnewpath\ntranslate\n609.242 129.075 m\n620.972 129.075 l\n620.972 129.991 l\n609.242 129.991 l\n609.242 129.075 l\n} bind def\n/p1_6d {\nnewpath\ntranslate\n609.242 129.991 m\n620.972 129.991 l\n620.972 130.908 l\n609.242 130.908 l\n609.242 129.991 l\n} bind def\n/p1_6e {\nnewpath\ntranslate\n609.242 130.908 m\n620.972 130.908 l\n620.972 131.824 l\n609.242 131.824 l\n609.242 130.908 l\n} bind def\n/p1_6f {\nnewpath\ntranslate\n609.242 131.824 m\n620.972 131.824 l\n620.972 132.741 l\n609.242 132.741 l\n609.242 131.824 l\n} bind def\n/p1_70 {\nnewpath\ntranslate\n609.242 132.741 m\n620.972 132.741 l\n620.972 133.657 l\n609.242 133.657 l\n609.242 132.741 l\n} bind def\n/p1_71 {\nnewpath\ntranslate\n609.242 133.657 m\n620.972 133.657 l\n620.972 134.574 l\n609.242 134.574 l\n609.242 133.657 l\n} bind def\n/p1_72 {\nnewpath\ntranslate\n609.242 134.574 m\n620.972 134.574 l\n620.972 135.49 l\n609.242 135.49 l\n609.242 134.574 l\n} bind def\n/p1_73 {\nnewpath\ntranslate\n609.242 135.49 m\n620.972 135.49 l\n620.972 136.406 l\n609.242 136.406 l\n609.242 135.49 l\n} bind def\n/p1_74 {\nnewpath\ntranslate\n609.242 136.406 m\n620.972 136.406 l\n620.972 137.323 l\n609.242 137.323 l\n609.242 136.406 l\n} bind def\n/p1_75 {\nnewpath\ntranslate\n609.242 137.323 m\n620.972 137.323 l\n620.972 138.239 l\n609.242 138.239 l\n609.242 137.323 l\n} bind def\n/p1_76 {\nnewpath\ntranslate\n609.242 138.239 m\n620.972 138.239 l\n620.972 139.156 l\n609.242 139.156 l\n609.242 138.239 l\n} bind def\n/p1_77 {\nnewpath\ntranslate\n609.242 139.156 m\n620.972 139.156 l\n620.972 140.072 l\n609.242 140.072 l\n609.242 139.156 l\n} bind def\n/p1_78 {\nnewpath\ntranslate\n609.242 140.072 m\n620.972 140.072 l\n620.972 140.989 l\n609.242 140.989 l\n609.242 140.072 l\n} bind def\n/p1_79 {\nnewpath\ntranslate\n609.242 140.989 m\n620.972 140.989 l\n620.972 141.905 l\n609.242 141.905 l\n609.242 140.989 l\n} bind def\n/p1_7a {\nnewpath\ntranslate\n609.242 141.905 m\n620.972 141.905 l\n620.972 142.821 l\n609.242 142.821 l\n609.242 141.905 l\n} bind def\n/p1_7b {\nnewpath\ntranslate\n609.242 142.821 m\n620.972 142.821 l\n620.972 143.738 l\n609.242 143.738 l\n609.242 142.821 l\n} bind def\n/p1_7c {\nnewpath\ntranslate\n609.242 143.738 m\n620.972 143.738 l\n620.972 144.654 l\n609.242 144.654 l\n609.242 143.738 l\n} bind def\n/p1_7d {\nnewpath\ntranslate\n609.242 144.654 m\n620.972 144.654 l\n620.972 145.571 l\n609.242 145.571 l\n609.242 144.654 l\n} bind def\n/p1_7e {\nnewpath\ntranslate\n609.242 145.571 m\n620.972 145.571 l\n620.972 146.487 l\n609.242 146.487 l\n609.242 145.571 l\n} bind def\n/p1_7f {\nnewpath\ntranslate\n609.242 146.487 m\n620.972 146.487 l\n620.972 147.404 l\n609.242 147.404 l\n609.242 146.487 l\n} bind def\n/p1_80 {\nnewpath\ntranslate\n609.242 147.404 m\n620.972 147.404 l\n620.972 148.32 l\n609.242 148.32 l\n609.242 147.404 l\n} bind def\n/p1_81 {\nnewpath\ntranslate\n609.242 148.32 m\n620.972 148.32 l\n620.972 149.236 l\n609.242 149.236 l\n609.242 148.32 l\n} bind def\n/p1_82 {\nnewpath\ntranslate\n609.242 149.236 m\n620.972 149.236 l\n620.972 150.153 l\n609.242 150.153 l\n609.242 149.236 l\n} bind def\n/p1_83 {\nnewpath\ntranslate\n609.242 150.153 m\n620.972 150.153 l\n620.972 151.069 l\n609.242 151.069 l\n609.242 150.153 l\n} bind def\n/p1_84 {\nnewpath\ntranslate\n609.242 151.069 m\n620.972 151.069 l\n620.972 151.986 l\n609.242 151.986 l\n609.242 151.069 l\n} bind def\n/p1_85 {\nnewpath\ntranslate\n609.242 151.986 m\n620.972 151.986 l\n620.972 152.902 l\n609.242 152.902 l\n609.242 151.986 l\n} bind def\n/p1_86 {\nnewpath\ntranslate\n609.242 152.902 m\n620.972 152.902 l\n620.972 153.819 l\n609.242 153.819 l\n609.242 152.902 l\n} bind def\n/p1_87 {\nnewpath\ntranslate\n609.242 153.819 m\n620.972 153.819 l\n620.972 154.735 l\n609.242 154.735 l\n609.242 153.819 l\n} bind def\n/p1_88 {\nnewpath\ntranslate\n609.242 154.735 m\n620.972 154.735 l\n620.972 155.651 l\n609.242 155.651 l\n609.242 154.735 l\n} bind def\n/p1_89 {\nnewpath\ntranslate\n609.242 155.651 m\n620.972 155.651 l\n620.972 156.568 l\n609.242 156.568 l\n609.242 155.651 l\n} bind def\n/p1_8a {\nnewpath\ntranslate\n609.242 156.568 m\n620.972 156.568 l\n620.972 157.484 l\n609.242 157.484 l\n609.242 156.568 l\n} bind def\n/p1_8b {\nnewpath\ntranslate\n609.242 157.484 m\n620.972 157.484 l\n620.972 158.401 l\n609.242 158.401 l\n609.242 157.484 l\n} bind def\n/p1_8c {\nnewpath\ntranslate\n609.242 158.401 m\n620.972 158.401 l\n620.972 159.317 l\n609.242 159.317 l\n609.242 158.401 l\n} bind def\n/p1_8d {\nnewpath\ntranslate\n609.242 159.317 m\n620.972 159.317 l\n620.972 160.233 l\n609.242 160.233 l\n609.242 159.317 l\n} bind def\n/p1_8e {\nnewpath\ntranslate\n609.242 160.233 m\n620.972 160.233 l\n620.972 161.15 l\n609.242 161.15 l\n609.242 160.233 l\n} bind def\n/p1_8f {\nnewpath\ntranslate\n609.242 161.15 m\n620.972 161.15 l\n620.972 162.066 l\n609.242 162.066 l\n609.242 161.15 l\n} bind def\n/p1_90 {\nnewpath\ntranslate\n609.242 162.066 m\n620.972 162.066 l\n620.972 162.983 l\n609.242 162.983 l\n609.242 162.066 l\n} bind def\n/p1_91 {\nnewpath\ntranslate\n609.242 162.983 m\n620.972 162.983 l\n620.972 163.899 l\n609.242 163.899 l\n609.242 162.983 l\n} bind def\n/p1_92 {\nnewpath\ntranslate\n609.242 163.899 m\n620.972 163.899 l\n620.972 164.816 l\n609.242 164.816 l\n609.242 163.899 l\n} bind def\n/p1_93 {\nnewpath\ntranslate\n609.242 164.816 m\n620.972 164.816 l\n620.972 165.732 l\n609.242 165.732 l\n609.242 164.816 l\n} bind def\n/p1_94 {\nnewpath\ntranslate\n609.242 165.732 m\n620.972 165.732 l\n620.972 166.648 l\n609.242 166.648 l\n609.242 165.732 l\n} bind def\n/p1_95 {\nnewpath\ntranslate\n609.242 166.648 m\n620.972 166.648 l\n620.972 167.565 l\n609.242 167.565 l\n609.242 166.648 l\n} bind def\n/p1_96 {\nnewpath\ntranslate\n609.242 167.565 m\n620.972 167.565 l\n620.972 168.481 l\n609.242 168.481 l\n609.242 167.565 l\n} bind def\n/p1_97 {\nnewpath\ntranslate\n609.242 168.481 m\n620.972 168.481 l\n620.972 169.398 l\n609.242 169.398 l\n609.242 168.481 l\n} bind def\n/p1_98 {\nnewpath\ntranslate\n609.242 169.398 m\n620.972 169.398 l\n620.972 170.314 l\n609.242 170.314 l\n609.242 169.398 l\n} bind def\n/p1_99 {\nnewpath\ntranslate\n609.242 170.314 m\n620.972 170.314 l\n620.972 171.231 l\n609.242 171.231 l\n609.242 170.314 l\n} bind def\n/p1_9a {\nnewpath\ntranslate\n609.242 171.231 m\n620.972 171.231 l\n620.972 172.147 l\n609.242 172.147 l\n609.242 171.231 l\n} bind def\n/p1_9b {\nnewpath\ntranslate\n609.242 172.147 m\n620.972 172.147 l\n620.972 173.063 l\n609.242 173.063 l\n609.242 172.147 l\n} bind def\n/p1_9c {\nnewpath\ntranslate\n609.242 173.063 m\n620.972 173.063 l\n620.972 173.98 l\n609.242 173.98 l\n609.242 173.063 l\n} bind def\n/p1_9d {\nnewpath\ntranslate\n609.242 173.98 m\n620.972 173.98 l\n620.972 174.896 l\n609.242 174.896 l\n609.242 173.98 l\n} bind def\n/p1_9e {\nnewpath\ntranslate\n609.242 174.896 m\n620.972 174.896 l\n620.972 175.813 l\n609.242 175.813 l\n609.242 174.896 l\n} bind def\n/p1_9f {\nnewpath\ntranslate\n609.242 175.813 m\n620.972 175.813 l\n620.972 176.729 l\n609.242 176.729 l\n609.242 175.813 l\n} bind def\n/p1_a0 {\nnewpath\ntranslate\n609.242 176.729 m\n620.972 176.729 l\n620.972 177.646 l\n609.242 177.646 l\n609.242 176.729 l\n} bind def\n/p1_a1 {\nnewpath\ntranslate\n609.242 177.646 m\n620.972 177.646 l\n620.972 178.562 l\n609.242 178.562 l\n609.242 177.646 l\n} bind def\n/p1_a2 {\nnewpath\ntranslate\n609.242 178.562 m\n620.972 178.562 l\n620.972 179.478 l\n609.242 179.478 l\n609.242 178.562 l\n} bind def\n/p1_a3 {\nnewpath\ntranslate\n609.242 179.478 m\n620.972 179.478 l\n620.972 180.395 l\n609.242 180.395 l\n609.242 179.478 l\n} bind def\n/p1_a4 {\nnewpath\ntranslate\n609.242 180.395 m\n620.972 180.395 l\n620.972 181.311 l\n609.242 181.311 l\n609.242 180.395 l\n} bind def\n/p1_a5 {\nnewpath\ntranslate\n609.242 181.311 m\n620.972 181.311 l\n620.972 182.228 l\n609.242 182.228 l\n609.242 181.311 l\n} bind def\n/p1_a6 {\nnewpath\ntranslate\n609.242 182.228 m\n620.972 182.228 l\n620.972 183.144 l\n609.242 183.144 l\n609.242 182.228 l\n} bind def\n/p1_a7 {\nnewpath\ntranslate\n609.242 183.144 m\n620.972 183.144 l\n620.972 184.06 l\n609.242 184.06 l\n609.242 183.144 l\n} bind def\n/p1_a8 {\nnewpath\ntranslate\n609.242 184.06 m\n620.972 184.06 l\n620.972 184.977 l\n609.242 184.977 l\n609.242 184.06 l\n} bind def\n/p1_a9 {\nnewpath\ntranslate\n609.242 184.977 m\n620.972 184.977 l\n620.972 185.893 l\n609.242 185.893 l\n609.242 184.977 l\n} bind def\n/p1_aa {\nnewpath\ntranslate\n609.242 185.893 m\n620.972 185.893 l\n620.972 186.81 l\n609.242 186.81 l\n609.242 185.893 l\n} bind def\n/p1_ab {\nnewpath\ntranslate\n609.242 186.81 m\n620.972 186.81 l\n620.972 187.726 l\n609.242 187.726 l\n609.242 186.81 l\n} bind def\n/p1_ac {\nnewpath\ntranslate\n609.242 187.726 m\n620.972 187.726 l\n620.972 188.643 l\n609.242 188.643 l\n609.242 187.726 l\n} bind def\n/p1_ad {\nnewpath\ntranslate\n609.242 188.643 m\n620.972 188.643 l\n620.972 189.559 l\n609.242 189.559 l\n609.242 188.643 l\n} bind def\n/p1_ae {\nnewpath\ntranslate\n609.242 189.559 m\n620.972 189.559 l\n620.972 190.475 l\n609.242 190.475 l\n609.242 189.559 l\n} bind def\n/p1_af {\nnewpath\ntranslate\n609.242 190.475 m\n620.972 190.475 l\n620.972 191.392 l\n609.242 191.392 l\n609.242 190.475 l\n} bind def\n/p1_b0 {\nnewpath\ntranslate\n609.242 191.392 m\n620.972 191.392 l\n620.972 192.308 l\n609.242 192.308 l\n609.242 191.392 l\n} bind def\n/p1_b1 {\nnewpath\ntranslate\n609.242 192.308 m\n620.972 192.308 l\n620.972 193.225 l\n609.242 193.225 l\n609.242 192.308 l\n} bind def\n/p1_b2 {\nnewpath\ntranslate\n609.242 193.225 m\n620.972 193.225 l\n620.972 194.141 l\n609.242 194.141 l\n609.242 193.225 l\n} bind def\n/p1_b3 {\nnewpath\ntranslate\n609.242 194.141 m\n620.972 194.141 l\n620.972 195.058 l\n609.242 195.058 l\n609.242 194.141 l\n} bind def\n/p1_b4 {\nnewpath\ntranslate\n609.242 195.058 m\n620.972 195.058 l\n620.972 195.974 l\n609.242 195.974 l\n609.242 195.058 l\n} bind def\n/p1_b5 {\nnewpath\ntranslate\n609.242 195.974 m\n620.972 195.974 l\n620.972 196.89 l\n609.242 196.89 l\n609.242 195.974 l\n} bind def\n/p1_b6 {\nnewpath\ntranslate\n609.242 196.89 m\n620.972 196.89 l\n620.972 197.807 l\n609.242 197.807 l\n609.242 196.89 l\n} bind def\n/p1_b7 {\nnewpath\ntranslate\n609.242 197.807 m\n620.972 197.807 l\n620.972 198.723 l\n609.242 198.723 l\n609.242 197.807 l\n} bind def\n/p1_b8 {\nnewpath\ntranslate\n609.242 198.723 m\n620.972 198.723 l\n620.972 199.64 l\n609.242 199.64 l\n609.242 198.723 l\n} bind def\n/p1_b9 {\nnewpath\ntranslate\n609.242 199.64 m\n620.972 199.64 l\n620.972 200.556 l\n609.242 200.556 l\n609.242 199.64 l\n} bind def\n/p1_ba {\nnewpath\ntranslate\n609.242 200.556 m\n620.972 200.556 l\n620.972 201.473 l\n609.242 201.473 l\n609.242 200.556 l\n} bind def\n/p1_bb {\nnewpath\ntranslate\n609.242 201.473 m\n620.972 201.473 l\n620.972 202.389 l\n609.242 202.389 l\n609.242 201.473 l\n} bind def\n/p1_bc {\nnewpath\ntranslate\n609.242 202.389 m\n620.972 202.389 l\n620.972 203.305 l\n609.242 203.305 l\n609.242 202.389 l\n} bind def\n/p1_bd {\nnewpath\ntranslate\n609.242 203.305 m\n620.972 203.305 l\n620.972 204.222 l\n609.242 204.222 l\n609.242 203.305 l\n} bind def\n/p1_be {\nnewpath\ntranslate\n609.242 204.222 m\n620.972 204.222 l\n620.972 205.138 l\n609.242 205.138 l\n609.242 204.222 l\n} bind def\n/p1_bf {\nnewpath\ntranslate\n609.242 205.138 m\n620.972 205.138 l\n620.972 206.055 l\n609.242 206.055 l\n609.242 205.138 l\n} bind def\n/p1_c0 {\nnewpath\ntranslate\n609.242 206.055 m\n620.972 206.055 l\n620.972 206.971 l\n609.242 206.971 l\n609.242 206.055 l\n} bind def\n/p1_c1 {\nnewpath\ntranslate\n609.242 206.971 m\n620.972 206.971 l\n620.972 207.888 l\n609.242 207.888 l\n609.242 206.971 l\n} bind def\n/p1_c2 {\nnewpath\ntranslate\n609.242 207.888 m\n620.972 207.888 l\n620.972 208.804 l\n609.242 208.804 l\n609.242 207.888 l\n} bind def\n/p1_c3 {\nnewpath\ntranslate\n609.242 208.804 m\n620.972 208.804 l\n620.972 209.72 l\n609.242 209.72 l\n609.242 208.804 l\n} bind def\n/p1_c4 {\nnewpath\ntranslate\n609.242 209.72 m\n620.972 209.72 l\n620.972 210.637 l\n609.242 210.637 l\n609.242 209.72 l\n} bind def\n/p1_c5 {\nnewpath\ntranslate\n609.242 210.637 m\n620.972 210.637 l\n620.972 211.553 l\n609.242 211.553 l\n609.242 210.637 l\n} bind def\n/p1_c6 {\nnewpath\ntranslate\n609.242 211.553 m\n620.972 211.553 l\n620.972 212.47 l\n609.242 212.47 l\n609.242 211.553 l\n} bind def\n/p1_c7 {\nnewpath\ntranslate\n609.242 212.47 m\n620.972 212.47 l\n620.972 213.386 l\n609.242 213.386 l\n609.242 212.47 l\n} bind def\n/p1_c8 {\nnewpath\ntranslate\n609.242 213.386 m\n620.972 213.386 l\n620.972 214.302 l\n609.242 214.302 l\n609.242 213.386 l\n} bind def\n/p1_c9 {\nnewpath\ntranslate\n609.242 214.302 m\n620.972 214.302 l\n620.972 215.219 l\n609.242 215.219 l\n609.242 214.302 l\n} bind def\n/p1_ca {\nnewpath\ntranslate\n609.242 215.219 m\n620.972 215.219 l\n620.972 216.135 l\n609.242 216.135 l\n609.242 215.219 l\n} bind def\n/p1_cb {\nnewpath\ntranslate\n609.242 216.135 m\n620.972 216.135 l\n620.972 217.052 l\n609.242 217.052 l\n609.242 216.135 l\n} bind def\n/p1_cc {\nnewpath\ntranslate\n609.242 217.052 m\n620.972 217.052 l\n620.972 217.968 l\n609.242 217.968 l\n609.242 217.052 l\n} bind def\n/p1_cd {\nnewpath\ntranslate\n609.242 217.968 m\n620.972 217.968 l\n620.972 218.885 l\n609.242 218.885 l\n609.242 217.968 l\n} bind def\n/p1_ce {\nnewpath\ntranslate\n609.242 218.885 m\n620.972 218.885 l\n620.972 219.801 l\n609.242 219.801 l\n609.242 218.885 l\n} bind def\n/p1_cf {\nnewpath\ntranslate\n609.242 219.801 m\n620.972 219.801 l\n620.972 220.717 l\n609.242 220.717 l\n609.242 219.801 l\n} bind def\n/p1_d0 {\nnewpath\ntranslate\n609.242 220.717 m\n620.972 220.717 l\n620.972 221.634 l\n609.242 221.634 l\n609.242 220.717 l\n} bind def\n/p1_d1 {\nnewpath\ntranslate\n609.242 221.634 m\n620.972 221.634 l\n620.972 222.55 l\n609.242 222.55 l\n609.242 221.634 l\n} bind def\n/p1_d2 {\nnewpath\ntranslate\n609.242 222.55 m\n620.972 222.55 l\n620.972 223.467 l\n609.242 223.467 l\n609.242 222.55 l\n} bind def\n/p1_d3 {\nnewpath\ntranslate\n609.242 223.467 m\n620.972 223.467 l\n620.972 224.383 l\n609.242 224.383 l\n609.242 223.467 l\n} bind def\n/p1_d4 {\nnewpath\ntranslate\n609.242 224.383 m\n620.972 224.383 l\n620.972 225.3 l\n609.242 225.3 l\n609.242 224.383 l\n} bind def\n/p1_d5 {\nnewpath\ntranslate\n609.242 225.3 m\n620.972 225.3 l\n620.972 226.216 l\n609.242 226.216 l\n609.242 225.3 l\n} bind def\n/p1_d6 {\nnewpath\ntranslate\n609.242 226.216 m\n620.972 226.216 l\n620.972 227.132 l\n609.242 227.132 l\n609.242 226.216 l\n} bind def\n/p1_d7 {\nnewpath\ntranslate\n609.242 227.132 m\n620.972 227.132 l\n620.972 228.049 l\n609.242 228.049 l\n609.242 227.132 l\n} bind def\n/p1_d8 {\nnewpath\ntranslate\n609.242 228.049 m\n620.972 228.049 l\n620.972 228.965 l\n609.242 228.965 l\n609.242 228.049 l\n} bind def\n/p1_d9 {\nnewpath\ntranslate\n609.242 228.965 m\n620.972 228.965 l\n620.972 229.882 l\n609.242 229.882 l\n609.242 228.965 l\n} bind def\n/p1_da {\nnewpath\ntranslate\n609.242 229.882 m\n620.972 229.882 l\n620.972 230.798 l\n609.242 230.798 l\n609.242 229.882 l\n} bind def\n/p1_db {\nnewpath\ntranslate\n609.242 230.798 m\n620.972 230.798 l\n620.972 231.715 l\n609.242 231.715 l\n609.242 230.798 l\n} bind def\n/p1_dc {\nnewpath\ntranslate\n609.242 231.715 m\n620.972 231.715 l\n620.972 232.631 l\n609.242 232.631 l\n609.242 231.715 l\n} bind def\n/p1_dd {\nnewpath\ntranslate\n609.242 232.631 m\n620.972 232.631 l\n620.972 233.547 l\n609.242 233.547 l\n609.242 232.631 l\n} bind def\n/p1_de {\nnewpath\ntranslate\n609.242 233.547 m\n620.972 233.547 l\n620.972 234.464 l\n609.242 234.464 l\n609.242 233.547 l\n} bind def\n/p1_df {\nnewpath\ntranslate\n609.242 234.464 m\n620.972 234.464 l\n620.972 235.38 l\n609.242 235.38 l\n609.242 234.464 l\n} bind def\n/p1_e0 {\nnewpath\ntranslate\n609.242 235.38 m\n620.972 235.38 l\n620.972 236.297 l\n609.242 236.297 l\n609.242 235.38 l\n} bind def\n/p1_e1 {\nnewpath\ntranslate\n609.242 236.297 m\n620.972 236.297 l\n620.972 237.213 l\n609.242 237.213 l\n609.242 236.297 l\n} bind def\n/p1_e2 {\nnewpath\ntranslate\n609.242 237.213 m\n620.972 237.213 l\n620.972 238.13 l\n609.242 238.13 l\n609.242 237.213 l\n} bind def\n/p1_e3 {\nnewpath\ntranslate\n609.242 238.13 m\n620.972 238.13 l\n620.972 239.046 l\n609.242 239.046 l\n609.242 238.13 l\n} bind def\n/p1_e4 {\nnewpath\ntranslate\n609.242 239.046 m\n620.972 239.046 l\n620.972 239.962 l\n609.242 239.962 l\n609.242 239.046 l\n} bind def\n/p1_e5 {\nnewpath\ntranslate\n609.242 239.962 m\n620.972 239.962 l\n620.972 240.879 l\n609.242 240.879 l\n609.242 239.962 l\n} bind def\n/p1_e6 {\nnewpath\ntranslate\n609.242 240.879 m\n620.972 240.879 l\n620.972 241.795 l\n609.242 241.795 l\n609.242 240.879 l\n} bind def\n/p1_e7 {\nnewpath\ntranslate\n609.242 241.795 m\n620.972 241.795 l\n620.972 242.712 l\n609.242 242.712 l\n609.242 241.795 l\n} bind def\n/p1_e8 {\nnewpath\ntranslate\n609.242 242.712 m\n620.972 242.712 l\n620.972 243.628 l\n609.242 243.628 l\n609.242 242.712 l\n} bind def\n/p1_e9 {\nnewpath\ntranslate\n609.242 243.628 m\n620.972 243.628 l\n620.972 244.544 l\n609.242 244.544 l\n609.242 243.628 l\n} bind def\n/p1_ea {\nnewpath\ntranslate\n609.242 244.544 m\n620.972 244.544 l\n620.972 245.461 l\n609.242 245.461 l\n609.242 244.544 l\n} bind def\n/p1_eb {\nnewpath\ntranslate\n609.242 245.461 m\n620.972 245.461 l\n620.972 246.377 l\n609.242 246.377 l\n609.242 245.461 l\n} bind def\n/p1_ec {\nnewpath\ntranslate\n609.242 246.377 m\n620.972 246.377 l\n620.972 247.294 l\n609.242 247.294 l\n609.242 246.377 l\n} bind def\n/p1_ed {\nnewpath\ntranslate\n609.242 247.294 m\n620.972 247.294 l\n620.972 248.21 l\n609.242 248.21 l\n609.242 247.294 l\n} bind def\n/p1_ee {\nnewpath\ntranslate\n609.242 248.21 m\n620.972 248.21 l\n620.972 249.127 l\n609.242 249.127 l\n609.242 248.21 l\n} bind def\n/p1_ef {\nnewpath\ntranslate\n609.242 249.127 m\n620.972 249.127 l\n620.972 250.043 l\n609.242 250.043 l\n609.242 249.127 l\n} bind def\n/p1_f0 {\nnewpath\ntranslate\n609.242 250.043 m\n620.972 250.043 l\n620.972 250.959 l\n609.242 250.959 l\n609.242 250.043 l\n} bind def\n/p1_f1 {\nnewpath\ntranslate\n609.242 250.959 m\n620.972 250.959 l\n620.972 251.876 l\n609.242 251.876 l\n609.242 250.959 l\n} bind def\n/p1_f2 {\nnewpath\ntranslate\n609.242 251.876 m\n620.972 251.876 l\n620.972 252.792 l\n609.242 252.792 l\n609.242 251.876 l\n} bind def\n/p1_f3 {\nnewpath\ntranslate\n609.242 252.792 m\n620.972 252.792 l\n620.972 253.709 l\n609.242 253.709 l\n609.242 252.792 l\n} bind def\n/p1_f4 {\nnewpath\ntranslate\n609.242 253.709 m\n620.972 253.709 l\n620.972 254.625 l\n609.242 254.625 l\n609.242 253.709 l\n} bind def\n/p1_f5 {\nnewpath\ntranslate\n609.242 254.625 m\n620.972 254.625 l\n620.972 255.542 l\n609.242 255.542 l\n609.242 254.625 l\n} bind def\n/p1_f6 {\nnewpath\ntranslate\n609.242 255.542 m\n620.972 255.542 l\n620.972 256.458 l\n609.242 256.458 l\n609.242 255.542 l\n} bind def\n/p1_f7 {\nnewpath\ntranslate\n609.242 256.458 m\n620.972 256.458 l\n620.972 257.374 l\n609.242 257.374 l\n609.242 256.458 l\n} bind def\n/p1_f8 {\nnewpath\ntranslate\n609.242 257.374 m\n620.972 257.374 l\n620.972 258.291 l\n609.242 258.291 l\n609.242 257.374 l\n} bind def\n/p1_f9 {\nnewpath\ntranslate\n609.242 258.291 m\n620.972 258.291 l\n620.972 259.207 l\n609.242 259.207 l\n609.242 258.291 l\n} bind def\n/p1_fa {\nnewpath\ntranslate\n609.242 259.207 m\n620.972 259.207 l\n620.972 260.124 l\n609.242 260.124 l\n609.242 259.207 l\n} bind def\n/p1_fb {\nnewpath\ntranslate\n609.242 260.124 m\n620.972 260.124 l\n620.972 261.04 l\n609.242 261.04 l\n609.242 260.124 l\n} bind def\n/p1_fc {\nnewpath\ntranslate\n609.242 261.04 m\n620.972 261.04 l\n620.972 261.957 l\n609.242 261.957 l\n609.242 261.04 l\n} bind def\n/p1_fd {\nnewpath\ntranslate\n609.242 261.957 m\n620.972 261.957 l\n620.972 262.873 l\n609.242 262.873 l\n609.242 261.957 l\n} bind def\n/p1_fe {\nnewpath\ntranslate\n609.242 262.873 m\n620.972 262.873 l\n620.972 263.789 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    "path": "multigrid/2dgrid-prolong.eps",
    "content": "%!PS-Adobe-3.0 EPSF-3.0\n%%Title: 2dgrid-prolong.eps\n%%Creator: matplotlib version 1.3.1, http://matplotlib.org/\n%%CreationDate: Sat Mar 28 11:03:48 2015\n%%Orientation: portrait\n%%BoundingBox: 18 108 594 684\n%%EndComments\n%%BeginProlog\n/mpldict 10 dict def\nmpldict begin\n/m { moveto } bind def\n/l { lineto } bind def\n/r { rlineto } bind def\n/c { curveto } bind def\n/cl { closepath } bind def\n/box {\nm\n1 index 0 r\n0 exch r\nneg 0 r\ncl\n} bind def\n/clipbox {\nbox\nclip\nnewpath\n} bind def\n%!PS-Adobe-3.0 Resource-Font\n%%Title: cmr10\n%%Copyright: Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B.\n%%Creator: Converted from TrueType to type 3 by PPR\n25 dict begin\n/_d{bind def}bind def\n/_m{moveto}_d\n/_l{lineto}_d\n/_cl{closepath eofill}_d\n/_c{curveto}_d\n/_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d\n/_e{exec}_d\n/FontName /Cmr10 def\n/PaintType 0 def\n/FontMatrix[.001 0 0 .001 0 0]def\n/FontBBox[-44 -250 1009 750]def\n/FontType 3 def\n/Encoding [ /plus /one ] def\n/FontInfo 10 dict dup begin\n/FamilyName (cmr10) def\n/FullName (cmr10) def\n/Notice (Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. ) def\n/Weight (Regular) def\n/Version (1.1/12-Nov-94) def\n/ItalicAngle 0.0 def\n/isFixedPitch false def\n/UnderlinePosition -133 def\n/UnderlineThickness 20 def\nend readonly def\n/CharStrings 2 dict dup begin\n/plus{{777 0 56 -82 721 583 _sc\n75 230 _m\n69 230 65 232 61 236 _c\n57 240 56 245 56 250 _c\n56 254 57 259 61 263 _c\n65 267 69 270 75 270 _c\n369 270 _l\n369 565 _l\n369 570 371 574 375 578 _c\n379 581 383 583 389 583 _c\n393 583 398 581 402 578 _c\n406 574 409 570 409 565 _c\n409 270 _l\n703 270 _l\n707 270 711 267 715 263 _c\n719 259 721 254 721 250 _c\n721 245 719 240 715 236 _c\n}_e{711 232 707 230 703 230 _c\n409 230 _l\n409 -64 _l\n409 -69 406 -73 402 -77 _c\n398 -80 393 -82 389 -82 _c\n383 -82 379 -80 375 -77 _c\n371 -73 369 -69 369 -64 _c\n369 230 _l\n75 230 _l\n_cl}_e}_d\n/one{500 0 87 0 421 666 _sc\n93 0 _m\n93 35 _l\n176 35 218 45 218 67 _c\n218 592 _l\n183 575 139 567 87 567 _c\n87 602 _l\n168 602 230 623 272 666 _c\n286 666 _l\n288 666 291 665 293 663 _c\n295 661 296 659 296 657 _c\n296 67 _l\n296 45 337 35 421 35 _c\n421 0 _l\n93 0 _l\n_cl}_d\nend readonly def\n\n/BuildGlyph\n {exch begin\n CharStrings exch\n 2 copy known not{pop /.notdef}if\n true 3 1 roll get exec\n end}_d\n\n/BuildChar {\n 1 index /Encoding get exch get\n 1 index /BuildGlyph get exec\n}_d\n\nFontName currentdict end definefont pop\n%!PS-Adobe-3.0 Resource-Font\n%%Title: cmsy10\n%%Copyright: Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B.\n%%Creator: Converted from TrueType to type 3 by PPR\n25 dict begin\n/_d{bind def}bind def\n/_m{moveto}_d\n/_l{lineto}_d\n/_cl{closepath eofill}_d\n/_c{curveto}_d\n/_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d\n/_e{exec}_d\n/FontName /Cmsy10 def\n/PaintType 0 def\n/FontMatrix[.001 0 0 .001 0 0]def\n/FontBBox[-29 -960 1123 779]def\n/FontType 3 def\n/Encoding [ /minus ] def\n/FontInfo 10 dict dup begin\n/FamilyName (cmsy10) def\n/FullName (cmsy10) def\n/Notice (Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. ) def\n/Weight (Regular) def\n/Version (1.1/12-Nov-94) def\n/ItalicAngle 0.0 def\n/isFixedPitch false def\n/UnderlinePosition -133 def\n/UnderlineThickness 20 def\nend readonly def\n/CharStrings 1 dict dup begin\n/minus{777 0 83 230 694 270 _sc\n102 230 _m\n96 230 92 232 88 236 _c\n84 240 83 245 83 250 _c\n83 254 84 259 88 263 _c\n92 267 96 270 102 270 _c\n676 270 _l\n681 270 685 267 689 263 _c\n692 259 694 254 694 250 _c\n694 245 692 240 689 236 _c\n685 232 681 230 676 230 _c\n102 230 _l\n_cl}_d\nend readonly def\n\n/BuildGlyph\n {exch begin\n CharStrings exch\n 2 copy known not{pop /.notdef}if\n true 3 1 roll get exec\n end}_d\n\n/BuildChar {\n 1 index /Encoding get exch get\n 1 index /BuildGlyph get exec\n}_d\n\nFontName currentdict end definefont pop\n%!PS-Adobe-3.0 Resource-Font\n%%Title: cmmi10\n%%Copyright: Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B.\n%%Creator: Converted from TrueType to type 3 by PPR\n25 dict begin\n/_d{bind def}bind def\n/_m{moveto}_d\n/_l{lineto}_d\n/_cl{closepath eofill}_d\n/_c{curveto}_d\n/_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d\n/_e{exec}_d\n/FontName /Cmmi10 def\n/PaintType 0 def\n/FontMatrix[.001 0 0 .001 0 0]def\n/FontBBox[-34 -250 1048 750]def\n/FontType 3 def\n/Encoding [ /phi /comma /i /f /j /c ] def\n/FontInfo 10 dict dup begin\n/FamilyName (cmmi10) def\n/FullName (cmmi10) def\n/Notice (Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. ) def\n/Weight (Regular) def\n/Version (1.1/12-Nov-94) def\n/ItalicAngle 0.0 def\n/isFixedPitch false def\n/UnderlinePosition -133 def\n/UnderlineThickness 20 def\nend readonly def\n/CharStrings 6 dict dup begin\n/phi{{595 0 47 -204 575 694 _sc\n194 -194 _m\n239 -13 _l\n205 -13 173 -5 144 10 _c\n114 25 91 47 73 75 _c\n55 103 47 134 47 168 _c\n47 204 56 238 74 271 _c\n92 304 116 334 147 361 _c\n178 387 211 408 247 423 _c\n283 437 318 445 354 445 _c\n414 688 _l\n415 692 418 694 422 694 _c\n434 694 _l\n436 694 438 692 440 690 _c\n442 688 443 686 443 684 _c\n383 445 _l\n407 445 431 440 455 431 _c\n}_e{479 421 500 408 518 392 _c\n536 376 550 356 560 334 _c\n570 312 575 288 575 263 _c\n575 227 566 192 548 159 _c\n530 126 506 96 476 70 _c\n446 44 412 23 375 9 _c\n337 -5 302 -13 268 -13 _c\n222 -198 _l\n220 -202 218 -204 214 -204 _c\n202 -204 _l\n199 -204 197 -202 196 -200 _c\n194 -198 194 -196 194 -194 _c\n245 12 _m\n347 419 _l\n330 419 312 415 294 408 _c\n275 401 257 392 241 382 _c\n}_e{225 372 210 360 197 347 _c\n180 331 165 311 153 289 _c\n141 267 131 243 125 219 _c\n119 195 116 171 116 147 _c\n116 121 121 98 132 78 _c\n142 58 157 42 177 30 _c\n197 18 219 12 245 12 _c\n275 12 _m\n300 12 327 19 355 33 _c\n383 47 406 64 425 84 _c\n449 108 469 139 484 175 _c\n498 211 506 247 506 284 _c\n506 308 500 331 490 352 _c\n479 372 464 389 444 401 _c\n}_e{424 413 402 419 377 419 _c\n275 12 _l\n_cl}_e}_d\n/comma{277 0 84 -193 205 110 _sc\n99 -179 _m\n99 -176 100 -173 103 -171 _c\n127 -147 145 -120 159 -90 _c\n172 -60 179 -28 179 4 _c\n179 16 _l\n168 5 155 0 139 0 _c\n123 0 110 5 100 16 _c\n89 26 84 39 84 55 _c\n84 71 89 84 100 94 _c\n110 104 123 110 139 110 _c\n163 110 180 99 190 77 _c\n200 55 205 30 205 4 _c\n205 -32 197 -67 183 -101 _c\n168 -134 147 -164 120 -191 _c\n117 -192 115 -193 114 -193 _c\n110 -193 107 -191 104 -188 _c\n100 -185 99 -182 99 -179 _c\n_cl}_d\n/i{{344 0 27 -10 296 661 _sc\n78 72 _m\n78 84 80 97 86 111 _c\n167 326 _l\n175 350 180 370 180 386 _c\n180 406 172 416 158 416 _c\n132 416 110 402 93 375 _c\n75 347 63 317 55 284 _c\n53 280 51 278 47 278 _c\n35 278 _l\n29 278 27 281 27 287 _c\n27 290 _l\n37 330 54 365 76 396 _c\n98 426 126 442 160 442 _c\n184 442 204 434 220 418 _c\n236 402 245 383 245 359 _c\n}_e{245 346 242 333 237 320 _c\n156 105 _l\n146 83 142 63 142 45 _c\n142 25 149 15 165 15 _c\n191 15 212 29 230 57 _c\n247 85 259 115 267 147 _c\n268 151 271 153 275 153 _c\n287 153 _l\n289 153 291 152 293 150 _c\n295 148 296 146 296 144 _c\n296 143 295 142 295 141 _c\n285 103 269 68 247 37 _c\n224 5 196 -10 163 -10 _c\n139 -10 119 -2 103 13 _c\n86 29 78 48 78 72 _c\n}_e{191 606 _m\n191 620 196 632 208 644 _c\n220 655 233 661 247 661 _c\n258 661 267 657 275 650 _c\n282 643 286 634 286 624 _c\n286 609 280 596 268 585 _c\n256 573 243 568 229 568 _c\n218 568 209 571 202 579 _c\n194 586 191 595 191 606 _c\n_cl}_e}_d\n/f{{489 0 53 -204 552 705 _sc\n97 -164 _m\n109 -173 124 -178 142 -178 _c\n166 -178 185 -151 200 -99 _c\n206 -73 223 9 251 151 _c\n297 396 _l\n211 396 _l\n204 396 201 400 201 409 _c\n203 423 208 431 214 431 _c\n303 431 _l\n315 496 _l\n319 516 322 533 326 548 _c\n329 562 332 576 336 588 _c\n340 600 345 614 353 630 _c\n364 652 379 670 399 684 _c\n418 698 439 705 462 705 _c\n476 705 490 702 504 697 _c\n}_e{518 691 529 683 538 673 _c\n547 662 552 649 552 635 _c\n552 617 546 602 535 590 _c\n523 578 510 572 494 572 _c\n483 572 474 575 466 582 _c\n458 588 455 597 455 608 _c\n455 622 460 635 470 646 _c\n480 657 492 663 507 665 _c\n494 674 479 679 461 679 _c\n451 679 441 674 432 665 _c\n423 655 417 645 415 635 _c\n411 617 401 571 387 497 _c\n375 431 _l\n478 431 _l\n484 431 488 426 488 418 _c\n}_e{487 416 486 413 485 409 _c\n484 405 483 401 481 399 _c\n479 397 477 396 475 396 _c\n368 396 _l\n322 152 _l\n316 116 309 80 301 44 _c\n293 8 283 -29 269 -68 _c\n255 -106 238 -139 216 -165 _c\n194 -191 169 -204 140 -204 _c\n117 -204 97 -197 79 -185 _c\n61 -172 53 -155 53 -134 _c\n53 -116 58 -101 69 -89 _c\n80 -77 94 -71 111 -71 _c\n121 -71 130 -74 138 -81 _c\n146 -87 150 -96 150 -107 _c\n}_e{150 -121 144 -134 134 -146 _c\n123 -158 111 -164 97 -164 _c\n_cl}_e}_d\n/j{{411 0 -12 -204 398 661 _sc\n-12 -141 _m\n-12 -125 -6 -111 4 -99 _c\n15 -87 29 -81 45 -81 _c\n55 -81 64 -84 72 -91 _c\n79 -97 83 -106 83 -117 _c\n83 -129 79 -141 71 -151 _c\n63 -161 53 -168 41 -172 _c\n53 -176 64 -178 76 -178 _c\n104 -178 129 -165 150 -139 _c\n170 -113 185 -83 194 -51 _c\n288 324 _l\n292 343 295 359 295 373 _c\n295 401 285 416 265 416 _c\n235 416 210 402 188 376 _c\n}_e{166 349 148 318 135 284 _c\n133 280 131 278 127 278 _c\n115 278 _l\n109 278 107 281 107 287 _c\n107 290 _l\n121 330 142 365 170 396 _c\n198 426 230 442 267 442 _c\n295 442 317 433 335 416 _c\n353 398 362 376 362 350 _c\n362 340 361 331 359 322 _c\n264 -57 _l\n257 -84 244 -109 225 -131 _c\n205 -153 182 -171 156 -184 _c\n129 -197 102 -204 74 -204 _c\n52 -204 32 -198 14 -188 _c\n}_e{-3 -177 -12 -161 -12 -141 _c\n303 606 _m\n303 620 308 633 320 644 _c\n331 655 344 661 359 661 _c\n370 661 379 657 387 650 _c\n394 643 398 634 398 624 _c\n398 609 392 596 380 585 _c\n368 573 355 568 341 568 _c\n330 568 321 571 314 579 _c\n306 586 303 595 303 606 _c\n_cl}_e}_d\n/c{{432 0 40 -10 430 442 _sc\n117 119 _m\n117 90 124 65 139 45 _c\n154 25 175 15 203 15 _c\n243 15 281 24 318 42 _c\n354 60 384 86 407 118 _c\n409 120 411 121 415 121 _c\n418 121 421 119 425 115 _c\n428 111 430 108 430 105 _c\n430 102 429 100 428 99 _c\n404 65 371 38 330 19 _c\n288 0 245 -10 201 -10 _c\n169 -10 140 -2 116 12 _c\n92 26 73 46 60 72 _c\n}_e{46 97 40 126 40 158 _c\n40 203 52 248 77 292 _c\n102 336 135 372 177 400 _c\n219 428 263 442 309 442 _c\n339 442 366 434 390 420 _c\n414 405 426 384 426 356 _c\n426 338 420 322 410 309 _c\n399 296 385 290 367 290 _c\n356 290 347 293 340 300 _c\n332 306 329 315 329 326 _c\n329 342 334 355 346 366 _c\n357 377 370 383 386 383 _c\n388 383 _l\n380 394 368 402 354 408 _c\n}_e{339 413 324 416 308 416 _c\n269 416 235 399 206 366 _c\n176 332 154 292 139 246 _c\n124 200 117 157 117 119 _c\n_cl}_e}_d\nend readonly def\n\n/BuildGlyph\n {exch begin\n CharStrings exch\n 2 copy known not{pop /.notdef}if\n true 3 1 roll get exec\n end}_d\n\n/BuildChar {\n 1 index /Encoding get exch get\n 1 index /BuildGlyph get exec\n}_d\n\nFontName currentdict end definefont pop\nend\n%%EndProlog\nmpldict begin\n18 108 translate\n576 576 0 0 clipbox\ngsave\n0 0 m\n576 0 l\n576 576 l\n0 576 l\ncl\n1.000 setgray\nfill\ngrestore\n1.000 setlinewidth\n1 setlinejoin\n0 setlinecap\n[] 0 setdash\n0.850 setgray\ngsave\n553 553 11.52 11.52 clipbox\n232.704 232.704 m\n232.704 380.16 l\n380.16 380.16 l\n380.16 232.704 l\ncl\ngsave\nfill\ngrestore\nstroke\ngrestore\ngsave\n553 553 11.52 11.52 clipbox\n380.16 232.704 m\n380.16 380.16 l\n527.616 380.16 l\n527.616 232.704 l\ncl\ngsave\nfill\ngrestore\nstroke\ngrestore\ngsave\n553 553 11.52 11.52 clipbox\n85.248 232.704 m\n85.248 380.16 l\n232.704 380.16 l\n232.704 232.704 l\ncl\ngsave\nfill\ngrestore\nstroke\ngrestore\ngsave\n553 553 11.52 11.52 clipbox\n232.704 380.16 m\n232.704 527.616 l\n380.16 527.616 l\n380.16 380.16 l\ncl\ngsave\nfill\ngrestore\nstroke\ngrestore\ngsave\n553 553 11.52 11.52 clipbox\n232.704 85.248 m\n232.704 232.704 l\n380.16 232.704 l\n380.16 85.248 l\ncl\ngsave\nfill\ngrestore\nstroke\ngrestore\n2.000 setlinewidth\n2 setlinecap\n0.000 setgray\ngsave\n553 553 11.52 11.52 clipbox\n48.384 85.248 m\n564.48 85.248 l\nstroke\ngrestore\ngsave\n553 553 11.52 11.52 clipbox\n48.384 232.704 m\n564.48 232.704 l\nstroke\ngrestore\ngsave\n553 553 11.52 11.52 clipbox\n48.384 380.16 m\n564.48 380.16 l\nstroke\ngrestore\ngsave\n553 553 11.52 11.52 clipbox\n48.384 527.616 m\n564.48 527.616 l\nstroke\ngrestore\ngsave\n553 553 11.52 11.52 clipbox\n85.248 48.384 m\n85.248 564.48 l\nstroke\ngrestore\ngsave\n553 553 11.52 11.52 clipbox\n232.704 48.384 m\n232.704 564.48 l\nstroke\ngrestore\ngsave\n553 553 11.52 11.52 clipbox\n380.16 48.384 m\n380.16 564.48 l\nstroke\ngrestore\ngsave\n553 553 11.52 11.52 clipbox\n527.616 48.384 m\n527.616 564.48 l\nstroke\ngrestore\n1.000 setlinewidth\n0 setlinecap\n[6 6] 0 setdash\n0.300 setgray\ngsave\n553 553 11.52 11.52 clipbox\n306.432 232.704 m\n306.432 380.16 l\nstroke\ngrestore\ngsave\n553 553 11.52 11.52 clipbox\n232.704 306.432 m\n380.16 306.432 l\nstroke\ngrestore\n0.000 0.000 1.000 setrgbcolor\ngsave\n431.888000 303.432000 translate\n0.000000 rotate\n/Cmmi10 findfont\n18.0 scalefont\nsetfont\n0.000000 0.540125 moveto\n/phi glyphshow\n\n/Cmmi10 findfont\n12.6 scalefont\nsetfont\n12.961055 8.432813 moveto\n/c glyphshow\n\n10.713867 -4.051562 moveto\n/i glyphshow\n\n/Cmr10 findfont\n12.6 scalefont\nsetfont\n17.263652 -4.051562 moveto\n/plus glyphshow\n\n27.854160 -4.051562 moveto\n/one glyphshow\n\n/Cmmi10 findfont\n12.6 scalefont\nsetfont\n34.154160 -4.051562 moveto\n/comma glyphshow\n\n37.886855 -4.051562 moveto\n/j glyphshow\n\n\ngrestore\ngsave\n284.432000 450.888000 translate\n0.000000 rotate\n/Cmmi10 findfont\n18.0 scalefont\nsetfont\n0.000000 0.540125 moveto\n/phi glyphshow\n\n/Cmmi10 findfont\n12.6 scalefont\nsetfont\n12.961055 8.432813 moveto\n/c glyphshow\n\n10.713867 -4.051562 moveto\n/i glyphshow\n\n15.051270 -4.051562 moveto\n/comma glyphshow\n\n18.783965 -4.051562 moveto\n/j glyphshow\n\n/Cmr10 findfont\n12.6 scalefont\nsetfont\n26.176621 -4.051562 moveto\n/plus glyphshow\n\n36.767129 -4.051562 moveto\n/one glyphshow\n\n\ngrestore\ngsave\n138.476000 303.432000 translate\n0.000000 rotate\n/Cmmi10 findfont\n18.0 scalefont\nsetfont\n0.000000 0.540125 moveto\n/phi glyphshow\n\n/Cmmi10 findfont\n12.6 scalefont\nsetfont\n12.961055 8.432813 moveto\n/c glyphshow\n\n10.713867 -4.051562 moveto\n/i glyphshow\n\n/Cmsy10 findfont\n12.6 scalefont\nsetfont\n15.051270 -4.051562 moveto\n/minus glyphshow\n\n/Cmr10 findfont\n12.6 scalefont\nsetfont\n24.839648 -4.051562 moveto\n/one glyphshow\n\n/Cmmi10 findfont\n12.6 scalefont\nsetfont\n31.139648 -4.051562 moveto\n/comma glyphshow\n\n34.872344 -4.051562 moveto\n/j glyphshow\n\n\ngrestore\ngsave\n285.932000 155.976000 translate\n0.000000 rotate\n/Cmmi10 findfont\n18.0 scalefont\nsetfont\n0.000000 0.540125 moveto\n/phi glyphshow\n\n/Cmmi10 findfont\n12.6 scalefont\nsetfont\n12.961055 8.432813 moveto\n/c glyphshow\n\n10.713867 -4.051562 moveto\n/i glyphshow\n\n15.051270 -4.051562 moveto\n/comma glyphshow\n\n18.783965 -4.051562 moveto\n/j glyphshow\n\n/Cmsy10 findfont\n12.6 scalefont\nsetfont\n23.964238 -4.051562 moveto\n/minus glyphshow\n\n/Cmr10 findfont\n12.6 scalefont\nsetfont\n33.752617 -4.051562 moveto\n/one glyphshow\n\n\ngrestore\n0.000 setgray\ngsave\n145.976000 33.638400 translate\n0.000000 rotate\n/Cmmi10 findfont\n16.0 scalefont\nsetfont\n0.000000 0.343750 moveto\n/i glyphshow\n\n/Cmsy10 findfont\n16.0 scalefont\nsetfont\n5.507812 0.343750 moveto\n/minus glyphshow\n\n/Cmr10 findfont\n16.0 scalefont\nsetfont\n17.937500 0.343750 moveto\n/one glyphshow\n\n\ngrestore\ngsave\n303.432000 33.638400 translate\n0.000000 rotate\n/Cmmi10 findfont\n16.0 scalefont\nsetfont\n0.000000 0.421875 moveto\n/i glyphshow\n\n\ngrestore\ngsave\n438.888000 33.638400 translate\n0.000000 rotate\n/Cmmi10 findfont\n16.0 scalefont\nsetfont\n0.000000 0.343750 moveto\n/i glyphshow\n\n/Cmr10 findfont\n16.0 scalefont\nsetfont\n8.317187 0.343750 moveto\n/plus glyphshow\n\n21.767187 0.343750 moveto\n/one glyphshow\n\n\ngrestore\ngsave\n33.638400 155.233813 translate\n0.000000 rotate\n/Cmmi10 findfont\n16.0 scalefont\nsetfont\n0.000000 0.343750 moveto\n/j glyphshow\n\n/Cmsy10 findfont\n16.0 scalefont\nsetfont\n6.578125 0.343750 moveto\n/minus glyphshow\n\n/Cmr10 findfont\n16.0 scalefont\nsetfont\n19.007812 0.343750 moveto\n/one glyphshow\n\n\ngrestore\ngsave\n33.638400 302.689813 translate\n0.000000 rotate\n/Cmmi10 findfont\n16.0 scalefont\nsetfont\n0.000000 0.421875 moveto\n/j glyphshow\n\n\ngrestore\ngsave\n33.638400 450.145813 translate\n0.000000 rotate\n/Cmmi10 findfont\n16.0 scalefont\nsetfont\n0.000000 0.343750 moveto\n/j glyphshow\n\n/Cmr10 findfont\n16.0 scalefont\nsetfont\n9.387500 0.343750 moveto\n/plus glyphshow\n\n22.837500 0.343750 moveto\n/one glyphshow\n\n\ngrestore\n0.000 0.000 1.000 setrgbcolor\ngsave\n294.432000 303.432000 translate\n0.000000 rotate\n/Cmmi10 findfont\n18.0 scalefont\nsetfont\n0.000000 0.540125 moveto\n/phi glyphshow\n\n/Cmmi10 findfont\n12.6 scalefont\nsetfont\n12.961055 8.432813 moveto\n/c glyphshow\n\n10.713867 -4.051562 moveto\n/i glyphshow\n\n15.051270 -4.051562 moveto\n/comma glyphshow\n\n18.783965 -4.051562 moveto\n/j glyphshow\n\n\ngrestore\n1.000 0.000 0.000 setrgbcolor\ngsave\n254.068000 263.568000 translate\n0.000000 rotate\n/Cmmi10 findfont\n18.0 scalefont\nsetfont\n0.000000 0.226063 moveto\n/phi glyphshow\n\n/Cmmi10 findfont\n12.6 scalefont\nsetfont\n12.961055 8.118750 moveto\n/f glyphshow\n\n/Cmsy10 findfont\n12.6 scalefont\nsetfont\n10.713867 -4.365625 moveto\n/minus glyphshow\n\n20.502246 -4.365625 moveto\n/minus glyphshow\n\n\ngrestore\ngsave\n252.568000 337.796000 translate\n0.000000 rotate\n/Cmmi10 findfont\n18.0 scalefont\nsetfont\n0.000000 0.226062 moveto\n/phi glyphshow\n\n/Cmmi10 findfont\n12.6 scalefont\nsetfont\n12.961055 8.118750 moveto\n/f glyphshow\n\n/Cmsy10 findfont\n12.6 scalefont\nsetfont\n10.713867 -4.365625 moveto\n/minus glyphshow\n\n/Cmr10 findfont\n12.6 scalefont\nsetfont\n22.714629 -4.365625 moveto\n/plus glyphshow\n\n\ngrestore\ngsave\n326.296000 264.068000 translate\n0.000000 rotate\n/Cmmi10 findfont\n18.0 scalefont\nsetfont\n0.000000 0.226062 moveto\n/phi glyphshow\n\n/Cmmi10 findfont\n12.6 scalefont\nsetfont\n12.961055 8.118750 moveto\n/f glyphshow\n\n/Cmr10 findfont\n12.6 scalefont\nsetfont\n12.926250 -4.365625 moveto\n/plus glyphshow\n\n/Cmsy10 findfont\n12.6 scalefont\nsetfont\n23.516758 -4.365625 moveto\n/minus glyphshow\n\n\ngrestore\ngsave\n324.796000 337.796000 translate\n0.000000 rotate\n/Cmmi10 findfont\n18.0 scalefont\nsetfont\n0.000000 0.226062 moveto\n/phi glyphshow\n\n/Cmmi10 findfont\n12.6 scalefont\nsetfont\n12.961055 8.118750 moveto\n/f glyphshow\n\n/Cmr10 findfont\n12.6 scalefont\nsetfont\n12.926250 -4.365625 moveto\n/plus glyphshow\n\n25.729141 -4.365625 moveto\n/plus glyphshow\n\n\ngrestore\n\nend\nshowpage\n"
  },
  {
    "path": "multigrid/GNUmakefile",
    "content": "EPStoPDF = epstopdf\n\nALL: multigrid.pdf\n\n\nmultigrid.pdf: main.tex multigrid.tex\n\tpdflatex -jobname=multigrid main.tex < /dev/null\n\tbibtex multigrid.aux\n\tpdflatex -jobname=multigrid main.tex < /dev/null\n\tpdflatex -jobname=multigrid main.tex < /dev/null\n\n\nclean:\n\t$(RM) *.aux *.log *.dvi *.bbl *.blg \n\t$(RM) *~\n\n.PHONY: clean\n"
  },
  {
    "path": "multigrid/main.tex",
    "content": "\\documentclass[11pt]{article}\n\n% margins\n\\usepackage[margin=0.75in]{geometry}\n\n% figures\n\\usepackage{graphicx}\n\n% font\n\\usepackage{mathpazo}\n\n\\usepackage{helvet}\n\n% coloring\n\\usepackage{color}\n\\definecolor{mygray}{gray}{0.5}\n\n% footer\n\\usepackage{fancyhdr}\n\\pagestyle{fancy}\n\\fancyfoot[LO,LE]{\\footnotesize \\sffamily \\color{mygray} M.\\ Zingale---Notes on multigrid}\n\\fancyfoot[RO,RE]{\\footnotesize \\sffamily \\color{mygray}  (\\today)}\n\\fancyfoot[CO,CE]{\\thepage}\n\\fancyhead{}\n\\renewcommand{\\headrulewidth}{0.0pt}\n\\renewcommand{\\footrulewidth}{0.0pt}\n\n% captions\n\\usepackage{caption}\n\\renewcommand{\\captionfont}{\\footnotesize}\n\\renewcommand{\\captionlabelfont}{\\footnotesize}\n\\setlength{\\captionmargin}{5em}\n\n\n\\usepackage{sectsty}\n\\allsectionsfont{\\sffamily}\n\n\n\\begin{document}\n\n\\begin{center}\n{\\LARGE \\textsf{\\textbf{\nNotes on multigrid}}\n}\n\\end{center}\n\n\\input multigrid\n\n\\bibliographystyle{plain}\n\\bibliography{../refs}\n\n\n\\end{document}\n"
  },
  {
    "path": "multigrid/mg-converge.eps",
    "content": "%!PS-Adobe-3.0 EPSF-3.0\n%%Title: mg-converge.eps\n%%Creator: matplotlib version 1.3.1, http://matplotlib.org/\n%%CreationDate: Mon Jun 30 21:07:18 2014\n%%Orientation: portrait\n%%BoundingBox: 18 180 594 612\n%%EndComments\n%%BeginProlog\n/mpldict 11 dict def\nmpldict begin\n/m { moveto } bind def\n/l { lineto } bind def\n/r { rlineto } bind def\n/c { curveto } bind def\n/cl { closepath } bind def\n/box {\nm\n1 index 0 r\n0 exch r\nneg 0 r\ncl\n} bind def\n/clipbox {\nbox\nclip\nnewpath\n} bind def\n%!PS-Adobe-3.0 Resource-Font\n%%Title: cmmi10\n%%Copyright: Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B.\n%%Creator: Converted from TrueType to type 3 by PPR\n25 dict begin\n/_d{bind def}bind def\n/_m{moveto}_d\n/_l{lineto}_d\n/_cl{closepath eofill}_d\n/_c{curveto}_d\n/_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d\n/_e{exec}_d\n/FontName /Cmmi10 def\n/PaintType 0 def\n/FontMatrix[.001 0 0 .001 0 0]def\n/FontBBox[-34 -250 1048 750]def\n/FontType 3 def\n/Encoding [ /x ] def\n/FontInfo 10 dict dup begin\n/FamilyName (cmmi10) def\n/FullName (cmmi10) def\n/Notice (Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. ) def\n/Weight (Regular) def\n/Version (1.1/12-Nov-94) def\n/ItalicAngle 0.0 def\n/isFixedPitch false def\n/UnderlinePosition -133 def\n/UnderlineThickness 20 def\nend readonly def\n/CharStrings 1 dict dup begin\n/x{{571 0 31 -10 526 442 _sc\n78 29 _m\n90 19 106 15 128 15 _c\n148 15 167 25 183 45 _c\n199 65 210 87 216 111 _c\n261 288 _l\n268 320 272 342 272 354 _c\n272 370 267 385 258 397 _c\n248 409 235 416 219 416 _c\n198 416 178 409 160 396 _c\n141 382 125 365 113 345 _c\n100 325 91 304 86 284 _c\n84 280 82 278 78 278 _c\n66 278 _l\n60 278 58 281 58 287 _c\n58 290 _l\n}_e{64 314 75 338 91 362 _c\n107 386 126 405 149 420 _c\n171 434 195 442 221 442 _c\n245 442 267 435 287 422 _c\n307 409 321 391 329 369 _c\n340 389 354 407 372 421 _c\n390 435 409 442 431 442 _c\n445 442 460 439 475 434 _c\n490 429 502 421 512 411 _c\n521 401 526 388 526 372 _c\n526 355 520 340 509 328 _c\n498 316 484 310 468 310 _c\n457 310 448 313 441 320 _c\n}_e{433 327 430 336 430 346 _c\n430 360 434 372 444 383 _c\n454 393 465 400 479 402 _c\n467 411 450 416 429 416 _c\n407 416 389 406 373 386 _c\n357 366 346 344 340 320 _c\n296 143 _l\n288 116 285 94 285 77 _c\n285 60 289 45 299 33 _c\n309 21 322 15 338 15 _c\n369 15 397 28 421 56 _c\n445 84 462 114 470 147 _c\n471 151 474 153 478 153 _c\n490 153 _l\n492 153 494 152 496 150 _c\n}_e{498 148 499 146 499 144 _c\n499 143 498 142 498 141 _c\n488 101 468 66 438 36 _c\n408 5 374 -10 336 -10 _c\n311 -10 289 -3 269 9 _c\n249 21 236 39 228 62 _c\n217 42 202 25 184 11 _c\n166 -3 146 -10 126 -10 _c\n111 -10 96 -7 81 -3 _c\n66 1 54 9 45 19 _c\n35 29 31 43 31 59 _c\n31 75 36 89 47 102 _c\n57 114 71 121 88 121 _c\n}_e{98 121 107 117 115 111 _c\n123 104 127 95 127 85 _c\n127 71 122 58 113 48 _c\n103 38 92 31 78 29 _c\n_cl}_e}_d\nend readonly def\n\n/BuildGlyph\n {exch begin\n CharStrings exch\n 2 copy known not{pop /.notdef}if\n true 3 1 roll get exec\n end}_d\n\n/BuildChar {\n 1 index /Encoding get exch get\n 1 index /BuildGlyph get exec\n}_d\n\nFontName currentdict end definefont pop\n%!PS-Adobe-3.0 Resource-Font\n%%Title: cmsy10\n%%Copyright: Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B.\n%%Creator: Converted from TrueType to type 3 by PPR\n25 dict begin\n/_d{bind def}bind def\n/_m{moveto}_d\n/_l{lineto}_d\n/_cl{closepath eofill}_d\n/_c{curveto}_d\n/_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d\n/_e{exec}_d\n/FontName /Cmsy10 def\n/PaintType 0 def\n/FontMatrix[.001 0 0 .001 0 0]def\n/FontBBox[-29 -960 1123 779]def\n/FontType 3 def\n/Encoding [ /O ] def\n/FontInfo 10 dict dup begin\n/FamilyName (cmsy10) def\n/FullName (cmsy10) def\n/Notice (Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. ) def\n/Weight (Regular) def\n/Version (1.1/12-Nov-94) def\n/ItalicAngle 0.0 def\n/isFixedPitch false def\n/UnderlinePosition -133 def\n/UnderlineThickness 20 def\nend readonly def\n/CharStrings 1 dict dup begin\n/O{{796 0 57 -21 777 705 _sc\n291 -21 _m\n217 -21 160 4 119 56 _c\n77 108 57 171 57 247 _c\n57 345 89 438 155 526 _c\n170 546 192 570 222 598 _c\n251 626 281 650 311 671 _c\n341 691 366 703 388 705 _c\n393 705 _l\n397 703 400 700 400 697 _c\n400 694 398 691 396 687 _c\n392 682 375 669 346 648 _c\n316 627 300 615 297 613 _c\n266 586 240 556 218 524 _c\n194 489 176 452 164 412 _c\n}_e{152 372 146 331 146 289 _c\n146 244 154 202 170 164 _c\n186 125 211 94 244 71 _c\n277 48 317 37 363 37 _c\n411 37 455 49 496 73 _c\n536 97 571 129 599 168 _c\n627 207 648 251 664 299 _c\n680 347 688 393 688 439 _c\n688 475 682 509 671 540 _c\n660 571 643 596 620 616 _c\n596 636 567 646 531 646 _c\n516 646 502 643 489 637 _c\n467 625 446 604 426 574 _c\n}_e{406 543 390 512 377 480 _c\n370 468 359 457 344 449 _c\n329 440 315 435 301 434 _c\n296 434 _l\n294 434 293 435 291 437 _c\n289 438 289 440 289 442 _c\n289 443 289 445 290 447 _c\n320 522 361 582 413 626 _c\n441 650 470 669 501 683 _c\n532 697 564 705 597 705 _c\n603 705 _l\n643 705 676 694 702 674 _c\n728 653 747 626 759 592 _c\n771 558 777 521 777 482 _c\n777 438 771 396 759 354 _c\n}_e{747 312 731 273 710 237 _c\n688 201 662 167 632 136 _c\n600 105 566 78 531 55 _c\n495 31 457 13 417 0 _c\n376 -14 336 -21 296 -21 _c\n291 -21 _l\n_cl}_e}_d\nend readonly def\n\n/BuildGlyph\n {exch begin\n CharStrings exch\n 2 copy known not{pop /.notdef}if\n true 3 1 roll get exec\n end}_d\n\n/BuildChar {\n 1 index /Encoding get exch get\n 1 index /BuildGlyph get exec\n}_d\n\nFontName currentdict end definefont pop\n%!PS-Adobe-3.0 Resource-Font\n%%Title: Bitstream Vera Sans\n%%Copyright: Copyright (c) 2003 by Bitstream, Inc. All Rights Reserved.\n%%Creator: Converted from TrueType to type 3 by PPR\n25 dict begin\n/_d{bind def}bind def\n/_m{moveto}_d\n/_l{lineto}_d\n/_cl{closepath eofill}_d\n/_c{curveto}_d\n/_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d\n/_e{exec}_d\n/FontName /BitstreamVeraSans-Roman def\n/PaintType 0 def\n/FontMatrix[.001 0 0 .001 0 0]def\n/FontBBox[-183 -236 1287 928]def\n/FontType 3 def\n/Encoding [ /space /hyphen /zero /one /two /three /four /five /six /seven /eight /L /M /N /a /b /c /d /e /f /g /i /l /m /n /o /r /s /t /u /v ] def\n/FontInfo 10 dict dup begin\n/FamilyName (Bitstream Vera Sans) def\n/FullName (Bitstream Vera Sans) def\n/Notice (Copyright (c) 2003 by Bitstream, Inc. All Rights Reserved. Bitstream Vera is a trademark of Bitstream, Inc.) def\n/Weight (Roman) def\n/Version (Release 1.10) def\n/ItalicAngle 0.0 def\n/isFixedPitch false def\n/UnderlinePosition -213 def\n/UnderlineThickness 143 def\nend readonly def\n/CharStrings 31 dict dup begin\n/space{318 0 0 0 0 0 _sc\n}_d\n/hyphen{361 0 49 234 312 314 _sc\n49 314 _m\n312 314 _l\n312 234 _l\n49 234 _l\n49 314 _l\n_cl}_d\n/zero{636 0 66 -13 570 742 _sc\n318 664 _m\n267 664 229 639 203 589 _c\n177 539 165 464 165 364 _c\n165 264 177 189 203 139 _c\n229 89 267 64 318 64 _c\n369 64 407 89 433 139 _c\n458 189 471 264 471 364 _c\n471 464 458 539 433 589 _c\n407 639 369 664 318 664 _c\n318 742 _m\n399 742 461 709 505 645 _c\n548 580 570 486 570 364 _c\n570 241 548 147 505 83 _c\n461 19 399 -13 318 -13 _c\n236 -13 173 19 130 83 _c\n87 147 66 241 66 364 _c\n66 486 87 580 130 645 _c\n173 709 236 742 318 742 _c\n_cl}_d\n/one{636 0 110 0 544 729 _sc\n124 83 _m\n285 83 _l\n285 639 _l\n110 604 _l\n110 694 _l\n284 729 _l\n383 729 _l\n383 83 _l\n544 83 _l\n544 0 _l\n124 0 _l\n124 83 _l\n_cl}_d\n/two{{636 0 73 0 536 742 _sc\n192 83 _m\n536 83 _l\n536 0 _l\n73 0 _l\n73 83 _l\n110 121 161 173 226 239 _c\n290 304 331 346 348 365 _c\n380 400 402 430 414 455 _c\n426 479 433 504 433 528 _c\n433 566 419 598 392 622 _c\n365 646 330 659 286 659 _c\n255 659 222 653 188 643 _c\n154 632 117 616 78 594 _c\n78 694 _l\n118 710 155 722 189 730 _c\n223 738 255 742 284 742 _c\n}_e{359 742 419 723 464 685 _c\n509 647 532 597 532 534 _c\n532 504 526 475 515 449 _c\n504 422 484 390 454 354 _c\n446 344 420 317 376 272 _c\n332 227 271 164 192 83 _c\n_cl}_e}_d\n/three{{636 0 76 -13 556 742 _sc\n406 393 _m\n453 383 490 362 516 330 _c\n542 298 556 258 556 212 _c\n556 140 531 84 482 45 _c\n432 6 362 -13 271 -13 _c\n240 -13 208 -10 176 -4 _c\n144 1 110 10 76 22 _c\n76 117 _l\n103 101 133 89 166 81 _c\n198 73 232 69 268 69 _c\n330 69 377 81 409 105 _c\n441 129 458 165 458 212 _c\n458 254 443 288 413 312 _c\n383 336 341 349 287 349 _c\n}_e{202 349 _l\n202 430 _l\n291 430 _l\n339 430 376 439 402 459 _c\n428 478 441 506 441 543 _c\n441 580 427 609 401 629 _c\n374 649 336 659 287 659 _c\n260 659 231 656 200 650 _c\n169 644 135 635 98 623 _c\n98 711 _l\n135 721 170 729 203 734 _c\n235 739 266 742 296 742 _c\n370 742 429 725 473 691 _c\n517 657 539 611 539 553 _c\n539 513 527 479 504 451 _c\n481 423 448 403 406 393 _c\n_cl}_e}_d\n/four{636 0 49 0 580 729 _sc\n378 643 _m\n129 254 _l\n378 254 _l\n378 643 _l\n352 729 _m\n476 729 _l\n476 254 _l\n580 254 _l\n580 172 _l\n476 172 _l\n476 0 _l\n378 0 _l\n378 172 _l\n49 172 _l\n49 267 _l\n352 729 _l\n_cl}_d\n/five{{636 0 77 -13 549 729 _sc\n108 729 _m\n495 729 _l\n495 646 _l\n198 646 _l\n198 467 _l\n212 472 227 476 241 478 _c\n255 480 270 482 284 482 _c\n365 482 429 459 477 415 _c\n525 370 549 310 549 234 _c\n549 155 524 94 475 51 _c\n426 8 357 -13 269 -13 _c\n238 -13 207 -10 175 -6 _c\n143 -1 111 6 77 17 _c\n77 116 _l\n106 100 136 88 168 80 _c\n199 72 232 69 267 69 _c\n}_e{323 69 368 83 401 113 _c\n433 143 450 183 450 234 _c\n450 284 433 324 401 354 _c\n368 384 323 399 267 399 _c\n241 399 214 396 188 390 _c\n162 384 135 375 108 363 _c\n108 729 _l\n_cl}_e}_d\n/six{{636 0 70 -13 573 742 _sc\n330 404 _m\n286 404 251 388 225 358 _c\n199 328 186 286 186 234 _c\n186 181 199 139 225 109 _c\n251 79 286 64 330 64 _c\n374 64 409 79 435 109 _c\n461 139 474 181 474 234 _c\n474 286 461 328 435 358 _c\n409 388 374 404 330 404 _c\n526 713 _m\n526 623 _l\n501 635 476 644 451 650 _c\n425 656 400 659 376 659 _c\n310 659 260 637 226 593 _c\n}_e{192 549 172 482 168 394 _c\n187 422 211 444 240 459 _c\n269 474 301 482 336 482 _c\n409 482 467 459 509 415 _c\n551 371 573 310 573 234 _c\n573 159 550 99 506 54 _c\n462 9 403 -13 330 -13 _c\n246 -13 181 19 137 83 _c\n92 147 70 241 70 364 _c\n70 479 97 571 152 639 _c\n206 707 280 742 372 742 _c\n396 742 421 739 447 735 _c\n472 730 498 723 526 713 _c\n_cl}_e}_d\n/seven{636 0 82 0 551 729 _sc\n82 729 _m\n551 729 _l\n551 687 _l\n286 0 _l\n183 0 _l\n432 646 _l\n82 646 _l\n82 729 _l\n_cl}_d\n/eight{{636 0 68 -13 568 742 _sc\n318 346 _m\n271 346 234 333 207 308 _c\n180 283 167 249 167 205 _c\n167 161 180 126 207 101 _c\n234 76 271 64 318 64 _c\n364 64 401 76 428 102 _c\n455 127 469 161 469 205 _c\n469 249 455 283 429 308 _c\n402 333 365 346 318 346 _c\n219 388 _m\n177 398 144 418 120 447 _c\n96 476 85 511 85 553 _c\n85 611 105 657 147 691 _c\n188 725 245 742 318 742 _c\n}_e{390 742 447 725 489 691 _c\n530 657 551 611 551 553 _c\n551 511 539 476 515 447 _c\n491 418 459 398 417 388 _c\n464 377 501 355 528 323 _c\n554 291 568 251 568 205 _c\n568 134 546 80 503 43 _c\n459 5 398 -13 318 -13 _c\n237 -13 175 5 132 43 _c\n89 80 68 134 68 205 _c\n68 251 81 291 108 323 _c\n134 355 171 377 219 388 _c\n183 544 _m\n183 506 194 476 218 455 _c\n}_e{242 434 275 424 318 424 _c\n360 424 393 434 417 455 _c\n441 476 453 506 453 544 _c\n453 582 441 611 417 632 _c\n393 653 360 664 318 664 _c\n275 664 242 653 218 632 _c\n194 611 183 582 183 544 _c\n_cl}_e}_d\n/L{557 0 98 0 552 729 _sc\n98 729 _m\n197 729 _l\n197 83 _l\n552 83 _l\n552 0 _l\n98 0 _l\n98 729 _l\n_cl}_d\n/M{863 0 98 0 765 729 _sc\n98 729 _m\n245 729 _l\n431 233 _l\n618 729 _l\n765 729 _l\n765 0 _l\n669 0 _l\n669 640 _l\n481 140 _l\n382 140 _l\n194 640 _l\n194 0 _l\n98 0 _l\n98 729 _l\n_cl}_d\n/N{748 0 98 0 650 729 _sc\n98 729 _m\n231 729 _l\n554 119 _l\n554 729 _l\n650 729 _l\n650 0 _l\n517 0 _l\n194 610 _l\n194 0 _l\n98 0 _l\n98 729 _l\n_cl}_d\n/a{{613 0 60 -13 522 560 _sc\n343 275 _m\n270 275 220 266 192 250 _c\n164 233 150 205 150 165 _c\n150 133 160 107 181 89 _c\n202 70 231 61 267 61 _c\n317 61 357 78 387 114 _c\n417 149 432 196 432 255 _c\n432 275 _l\n343 275 _l\n522 312 _m\n522 0 _l\n432 0 _l\n432 83 _l\n411 49 385 25 355 10 _c\n325 -5 287 -13 243 -13 _c\n187 -13 142 2 109 33 _c\n76 64 60 106 60 159 _c\n}_e{60 220 80 266 122 298 _c\n163 329 224 345 306 345 _c\n432 345 _l\n432 354 _l\n432 395 418 427 391 450 _c\n364 472 326 484 277 484 _c\n245 484 215 480 185 472 _c\n155 464 127 453 100 439 _c\n100 522 _l\n132 534 164 544 195 550 _c\n226 556 256 560 286 560 _c\n365 560 424 539 463 498 _c\n502 457 522 395 522 312 _c\n_cl}_e}_d\n/b{{635 0 91 -13 580 760 _sc\n487 273 _m\n487 339 473 390 446 428 _c\n418 466 381 485 334 485 _c\n286 485 249 466 222 428 _c\n194 390 181 339 181 273 _c\n181 207 194 155 222 117 _c\n249 79 286 61 334 61 _c\n381 61 418 79 446 117 _c\n473 155 487 207 487 273 _c\n181 464 _m\n199 496 223 520 252 536 _c\n281 552 316 560 356 560 _c\n422 560 476 533 518 481 _c\n559 428 580 359 580 273 _c\n}_e{580 187 559 117 518 65 _c\n476 13 422 -13 356 -13 _c\n316 -13 281 -5 252 10 _c\n223 25 199 49 181 82 _c\n181 0 _l\n91 0 _l\n91 760 _l\n181 760 _l\n181 464 _l\n_cl}_e}_d\n/c{{550 0 55 -13 488 560 _sc\n488 526 _m\n488 442 _l\n462 456 437 466 411 473 _c\n385 480 360 484 334 484 _c\n276 484 230 465 198 428 _c\n166 391 150 339 150 273 _c\n150 206 166 154 198 117 _c\n230 80 276 62 334 62 _c\n360 62 385 65 411 72 _c\n437 79 462 90 488 104 _c\n488 21 _l\n462 9 436 0 410 -5 _c\n383 -10 354 -13 324 -13 _c\n242 -13 176 12 128 64 _c\n}_e{79 115 55 185 55 273 _c\n55 362 79 432 128 483 _c\n177 534 244 560 330 560 _c\n358 560 385 557 411 551 _c\n437 545 463 537 488 526 _c\n_cl}_e}_d\n/d{{635 0 55 -13 544 760 _sc\n454 464 _m\n454 760 _l\n544 760 _l\n544 0 _l\n454 0 _l\n454 82 _l\n435 49 411 25 382 10 _c\n353 -5 319 -13 279 -13 _c\n213 -13 159 13 117 65 _c\n75 117 55 187 55 273 _c\n55 359 75 428 117 481 _c\n159 533 213 560 279 560 _c\n319 560 353 552 382 536 _c\n411 520 435 496 454 464 _c\n148 273 _m\n148 207 161 155 188 117 _c\n215 79 253 61 301 61 _c\n}_e{348 61 385 79 413 117 _c\n440 155 454 207 454 273 _c\n454 339 440 390 413 428 _c\n385 466 348 485 301 485 _c\n253 485 215 466 188 428 _c\n161 390 148 339 148 273 _c\n_cl}_e}_d\n/e{{615 0 55 -13 562 560 _sc\n562 296 _m\n562 252 _l\n149 252 _l\n153 190 171 142 205 110 _c\n238 78 284 62 344 62 _c\n378 62 412 66 444 74 _c\n476 82 509 95 541 113 _c\n541 28 _l\n509 14 476 3 442 -3 _c\n408 -9 373 -13 339 -13 _c\n251 -13 182 12 131 62 _c\n80 112 55 181 55 268 _c\n55 357 79 428 127 481 _c\n175 533 241 560 323 560 _c\n397 560 455 536 498 489 _c\n}_e{540 441 562 377 562 296 _c\n472 322 _m\n471 371 457 410 431 440 _c\n404 469 368 484 324 484 _c\n274 484 234 469 204 441 _c\n174 413 156 373 152 322 _c\n472 322 _l\n_cl}_e}_d\n/f{352 0 23 0 371 760 _sc\n371 760 _m\n371 685 _l\n285 685 _l\n253 685 230 678 218 665 _c\n205 652 199 629 199 595 _c\n199 547 _l\n347 547 _l\n347 477 _l\n199 477 _l\n199 0 _l\n109 0 _l\n109 477 _l\n23 477 _l\n23 547 _l\n109 547 _l\n109 585 _l\n109 645 123 690 151 718 _c\n179 746 224 760 286 760 _c\n371 760 _l\n_cl}_d\n/g{{635 0 55 -207 544 560 _sc\n454 280 _m\n454 344 440 395 414 431 _c\n387 467 349 485 301 485 _c\n253 485 215 467 188 431 _c\n161 395 148 344 148 280 _c\n148 215 161 165 188 129 _c\n215 93 253 75 301 75 _c\n349 75 387 93 414 129 _c\n440 165 454 215 454 280 _c\n544 68 _m\n544 -24 523 -93 482 -139 _c\n440 -184 377 -207 292 -207 _c\n260 -207 231 -204 203 -200 _c\n175 -195 147 -188 121 -178 _c\n}_e{121 -91 _l\n147 -105 173 -115 199 -122 _c\n225 -129 251 -133 278 -133 _c\n336 -133 380 -117 410 -87 _c\n439 -56 454 -10 454 52 _c\n454 96 _l\n435 64 411 40 382 24 _c\n353 8 319 0 279 0 _c\n211 0 157 25 116 76 _c\n75 127 55 195 55 280 _c\n55 364 75 432 116 483 _c\n157 534 211 560 279 560 _c\n319 560 353 552 382 536 _c\n411 520 435 496 454 464 _c\n454 547 _l\n544 547 _l\n}_e{544 68 _l\n_cl}_e}_d\n/i{278 0 94 0 184 760 _sc\n94 547 _m\n184 547 _l\n184 0 _l\n94 0 _l\n94 547 _l\n94 760 _m\n184 760 _l\n184 646 _l\n94 646 _l\n94 760 _l\n_cl}_d\n/l{278 0 94 0 184 760 _sc\n94 760 _m\n184 760 _l\n184 0 _l\n94 0 _l\n94 760 _l\n_cl}_d\n/m{{974 0 91 0 889 560 _sc\n520 442 _m\n542 482 569 511 600 531 _c\n631 550 668 560 711 560 _c\n767 560 811 540 842 500 _c\n873 460 889 403 889 330 _c\n889 0 _l\n799 0 _l\n799 327 _l\n799 379 789 418 771 444 _c\n752 469 724 482 686 482 _c\n639 482 602 466 575 435 _c\n548 404 535 362 535 309 _c\n535 0 _l\n445 0 _l\n445 327 _l\n445 379 435 418 417 444 _c\n398 469 369 482 331 482 _c\n}_e{285 482 248 466 221 435 _c\n194 404 181 362 181 309 _c\n181 0 _l\n91 0 _l\n91 547 _l\n181 547 _l\n181 462 _l\n201 495 226 520 255 536 _c\n283 552 317 560 357 560 _c\n397 560 430 550 458 530 _c\n486 510 506 480 520 442 _c\n_cl}_e}_d\n/n{634 0 91 0 549 560 _sc\n549 330 _m\n549 0 _l\n459 0 _l\n459 327 _l\n459 379 448 417 428 443 _c\n408 469 378 482 338 482 _c\n289 482 251 466 223 435 _c\n195 404 181 362 181 309 _c\n181 0 _l\n91 0 _l\n91 547 _l\n181 547 _l\n181 462 _l\n202 494 227 519 257 535 _c\n286 551 320 560 358 560 _c\n420 560 468 540 500 501 _c\n532 462 549 405 549 330 _c\n_cl}_d\n/o{612 0 55 -13 557 560 _sc\n306 484 _m\n258 484 220 465 192 427 _c\n164 389 150 338 150 273 _c\n150 207 163 156 191 118 _c\n219 80 257 62 306 62 _c\n354 62 392 80 420 118 _c\n448 156 462 207 462 273 _c\n462 337 448 389 420 427 _c\n392 465 354 484 306 484 _c\n306 560 _m\n384 560 445 534 490 484 _c\n534 433 557 363 557 273 _c\n557 183 534 113 490 63 _c\n445 12 384 -13 306 -13 _c\n227 -13 165 12 121 63 _c\n77 113 55 183 55 273 _c\n55 363 77 433 121 484 _c\n165 534 227 560 306 560 _c\n_cl}_d\n/r{411 0 91 0 411 560 _sc\n411 463 _m\n401 469 390 473 378 476 _c\n366 478 353 480 339 480 _c\n288 480 249 463 222 430 _c\n194 397 181 350 181 288 _c\n181 0 _l\n91 0 _l\n91 547 _l\n181 547 _l\n181 462 _l\n199 495 224 520 254 536 _c\n284 552 321 560 365 560 _c\n371 560 378 559 386 559 _c\n393 558 401 557 411 555 _c\n411 463 _l\n_cl}_d\n/s{{521 0 54 -13 472 560 _sc\n443 531 _m\n443 446 _l\n417 458 391 468 364 475 _c\n336 481 308 485 279 485 _c\n234 485 200 478 178 464 _c\n156 450 145 430 145 403 _c\n145 382 153 366 169 354 _c\n185 342 217 330 265 320 _c\n296 313 _l\n360 299 405 279 432 255 _c\n458 230 472 195 472 151 _c\n472 100 452 60 412 31 _c\n372 1 316 -13 246 -13 _c\n216 -13 186 -10 154 -5 _c\n}_e{122 0 89 8 54 20 _c\n54 113 _l\n87 95 120 82 152 74 _c\n184 65 216 61 248 61 _c\n290 61 323 68 346 82 _c\n368 96 380 117 380 144 _c\n380 168 371 187 355 200 _c\n339 213 303 226 247 238 _c\n216 245 _l\n160 257 119 275 95 299 _c\n70 323 58 356 58 399 _c\n58 450 76 490 112 518 _c\n148 546 200 560 268 560 _c\n301 560 332 557 362 552 _c\n391 547 418 540 443 531 _c\n}_e{_cl}_e}_d\n/t{392 0 27 0 368 702 _sc\n183 702 _m\n183 547 _l\n368 547 _l\n368 477 _l\n183 477 _l\n183 180 _l\n183 135 189 106 201 94 _c\n213 81 238 75 276 75 _c\n368 75 _l\n368 0 _l\n276 0 _l\n206 0 158 13 132 39 _c\n106 65 93 112 93 180 _c\n93 477 _l\n27 477 _l\n27 547 _l\n93 547 _l\n93 702 _l\n183 702 _l\n_cl}_d\n/u{634 0 85 -13 543 547 _sc\n85 216 _m\n85 547 _l\n175 547 _l\n175 219 _l\n175 167 185 129 205 103 _c\n225 77 255 64 296 64 _c\n344 64 383 79 411 110 _c\n439 141 453 183 453 237 _c\n453 547 _l\n543 547 _l\n543 0 _l\n453 0 _l\n453 84 _l\n431 50 405 26 377 10 _c\n348 -5 315 -13 277 -13 _c\n214 -13 166 6 134 45 _c\n101 83 85 140 85 216 _c\n_cl}_d\n/v{592 0 30 0 562 547 _sc\n30 547 _m\n125 547 _l\n296 88 _l\n467 547 _l\n562 547 _l\n357 0 _l\n235 0 _l\n30 547 _l\n_cl}_d\nend readonly def\n\n/BuildGlyph\n {exch begin\n CharStrings exch\n 2 copy known not{pop /.notdef}if\n true 3 1 roll get exec\n end}_d\n\n/BuildChar {\n 1 index /Encoding get exch get\n 1 index /BuildGlyph get exec\n}_d\n\nFontName currentdict end definefont pop\n%!PS-Adobe-3.0 Resource-Font\n%%Title: cmr10\n%%Copyright: Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B.\n%%Creator: Converted from TrueType to type 3 by PPR\n25 dict begin\n/_d{bind def}bind def\n/_m{moveto}_d\n/_l{lineto}_d\n/_cl{closepath eofill}_d\n/_c{curveto}_d\n/_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d\n/_e{exec}_d\n/FontName /Cmr10 def\n/PaintType 0 def\n/FontMatrix[.001 0 0 .001 0 0]def\n/FontBBox[-44 -250 1009 750]def\n/FontType 3 def\n/Encoding [ /Delta /two /parenright /parenleft ] def\n/FontInfo 10 dict dup begin\n/FamilyName (cmr10) def\n/FullName (cmr10) def\n/Notice (Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. ) def\n/Weight (Regular) def\n/Version (1.1/12-Nov-94) def\n/ItalicAngle 0.0 def\n/isFixedPitch false def\n/UnderlinePosition -133 def\n/UnderlineThickness 20 def\nend readonly def\n/CharStrings 4 dict dup begin\n/Delta{833 0 46 0 786 716 _sc\n54 0 _m\n48 0 46 2 46 8 _c\n46 8 46 9 46 9 _c\n46 9 47 10 47 11 _c\n391 705 _l\n394 712 400 716 409 716 _c\n423 716 _l\n432 716 438 712 441 705 _c\n785 11 _l\n785 10 786 9 786 9 _c\n786 9 786 8 786 8 _c\n786 2 783 0 778 0 _c\n54 0 _l\n120 80 _m\n647 80 _l\n384 611 _l\n120 80 _l\n_cl}_d\n/two{{500 0 50 0 449 666 _sc\n50 0 _m\n50 27 _l\n50 28 50 30 52 32 _c\n207 204 _l\n230 229 249 250 264 268 _c\n278 285 293 305 307 327 _c\n321 349 333 372 341 396 _c\n349 419 354 444 354 470 _c\n354 497 349 523 339 548 _c\n329 573 314 593 294 608 _c\n274 623 249 631 221 631 _c\n192 631 166 622 143 605 _c\n119 587 103 565 94 537 _c\n96 537 100 538 105 538 _c\n}_e{119 538 132 533 143 523 _c\n153 513 159 500 159 484 _c\n159 468 153 455 143 445 _c\n132 434 119 429 105 429 _c\n89 429 76 434 66 445 _c\n55 456 50 469 50 484 _c\n50 508 54 532 64 554 _c\n73 576 86 595 104 613 _c\n122 630 142 643 164 652 _c\n186 661 210 666 236 666 _c\n274 666 309 658 342 642 _c\n375 626 401 603 420 573 _c\n439 543 449 509 449 470 _c\n}_e{449 441 442 414 430 388 _c\n417 362 401 338 381 317 _c\n361 295 336 271 305 244 _c\n274 217 254 199 244 190 _c\n131 81 _l\n227 81 _l\n274 81 313 81 345 82 _c\n377 82 394 84 396 86 _c\n404 94 412 125 420 178 _c\n449 178 _l\n421 0 _l\n50 0 _l\n_cl}_e}_d\n/parenright{{388 0 56 -249 291 750 _sc\n65 -249 _m\n59 -249 56 -246 56 -240 _c\n56 -237 56 -235 58 -234 _c\n172 -122 229 38 229 250 _c\n229 462 172 623 60 733 _c\n57 734 56 737 56 741 _c\n56 743 57 745 59 747 _c\n61 749 63 750 65 750 _c\n74 750 _l\n76 750 77 749 79 748 _c\n127 710 167 664 199 610 _c\n231 556 254 498 269 437 _c\n283 376 291 314 291 250 _c\n291 202 287 156 279 110 _c\n}_e{271 64 258 20 241 -24 _c\n223 -68 201 -108 175 -146 _c\n148 -184 116 -217 79 -247 _c\n77 -248 76 -249 74 -249 _c\n65 -249 _l\n_cl}_e}_d\n/parenleft{{388 0 97 -249 332 750 _sc\n310 -247 _m\n272 -217 240 -184 214 -146 _c\n187 -108 165 -67 148 -25 _c\n131 17 118 62 110 109 _c\n101 155 97 202 97 250 _c\n97 298 101 345 110 392 _c\n118 438 131 483 149 527 _c\n166 570 188 611 215 649 _c\n242 687 274 720 310 748 _c\n310 749 311 750 315 750 _c\n324 750 _l\n326 750 327 749 329 747 _c\n331 745 332 743 332 741 _c\n332 738 331 736 330 735 _c\n}_e{297 703 270 668 248 632 _c\n226 596 209 557 196 516 _c\n183 474 174 432 168 388 _c\n162 344 159 298 159 250 _c\n159 38 215 -122 329 -232 _c\n331 -234 332 -236 332 -240 _c\n332 -242 331 -244 329 -246 _c\n327 -248 326 -249 324 -249 _c\n315 -249 _l\n311 -249 310 -248 310 -247 _c\n_cl}_e}_d\nend readonly def\n\n/BuildGlyph\n {exch begin\n CharStrings exch\n 2 copy known not{pop /.notdef}if\n true 3 1 roll get exec\n end}_d\n\n/BuildChar {\n 1 index /Encoding get exch get\n 1 index /BuildGlyph get exec\n}_d\n\nFontName currentdict end definefont pop\nend\n%%EndProlog\nmpldict begin\n18 180 translate\n576 432 0 0 clipbox\ngsave\n0 0 m\n576 0 l\n576 432 l\n0 432 l\ncl\n1.000 setgray\nfill\ngrestore\ngsave\n58.6553 50.49 m\n552.168 50.49 l\n552.168 403.24 l\n58.6553 403.24 l\ncl\n1.000 setgray\nfill\ngrestore\n/p0_0 {\nnewpath\ntranslate\n0 -2.23607 m\n0.593012 -2.23607 1.16182 -2.00046 1.58114 -1.58114 c\n2.00046 -1.16182 2.23607 -0.593012 2.23607 0 c\n2.23607 0.593012 2.00046 1.16182 1.58114 1.58114 c\n1.16182 2.00046 0.593012 2.23607 0 2.23607 c\n-0.593012 2.23607 -1.16182 2.00046 -1.58114 1.58114 c\n-2.00046 1.16182 -2.23607 0.593012 -2.23607 0 c\n-2.23607 -0.593012 -2.00046 -1.16182 -1.58114 -1.58114 c\n-1.16182 -2.00046 -0.593012 -2.23607 0 -2.23607 c\ncl\n} bind def\n1.000 setlinewidth\n1 setlinejoin\n0 setlinecap\n[] 0 setdash\n1.000 0.000 0.000 setrgbcolor\ngsave\n493.5 352.8 58.66 50.49 clipbox\n109.023 360.639 p0_0\ngsave\nfill\ngrestore\nstroke\ngrestore\ngsave\n493.5 352.8 58.66 50.49 clipbox\n183.304 318.166 p0_0\ngsave\nfill\ngrestore\nstroke\ngrestore\ngsave\n493.5 352.8 58.66 50.49 clipbox\n257.585 275.691 p0_0\ngsave\nfill\ngrestore\nstroke\ngrestore\ngsave\n493.5 352.8 58.66 50.49 clipbox\n331.866 233.216 p0_0\ngsave\nfill\ngrestore\nstroke\ngrestore\ngsave\n493.5 352.8 58.66 50.49 clipbox\n406.148 190.74 p0_0\ngsave\nfill\ngrestore\nstroke\ngrestore\ngsave\n493.5 352.8 58.66 50.49 clipbox\n480.429 148.265 p0_0\ngsave\nfill\ngrestore\nstroke\ngrestore\n2 setlinecap\n0.000 setgray\ngsave\n493.5 352.8 58.66 50.49 clipbox\n109.023 360.642 m\n183.304 318.166 l\n257.585 275.691 l\n331.866 233.216 l\n406.148 190.74 l\n480.429 148.265 l\nstroke\ngrestore\n0.500 setlinewidth\n0 setlinecap\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 4 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n58.6553 50.49 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -4 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n58.6553 403.24 o\ngrestore\ngsave\n46.655275 34.990000 translate\n0.000000 rotate\n/BitstreamVeraSans-Roman findfont\n12.0 scalefont\nsetfont\n0.000000 0.575000 moveto\n/one glyphshow\n\n7.634766 0.575000 moveto\n/zero glyphshow\n\n/BitstreamVeraSans-Roman findfont\n8.4 scalefont\nsetfont\n15.269531 6.875000 moveto\n/one glyphshow\n\n\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 4 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n305.412 50.49 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -4 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n305.412 403.24 o\ngrestore\ngsave\n293.411685 34.990000 translate\n0.000000 rotate\n/BitstreamVeraSans-Roman findfont\n12.0 scalefont\nsetfont\n0.000000 0.465625 moveto\n/one glyphshow\n\n7.634766 0.465625 moveto\n/zero glyphshow\n\n/BitstreamVeraSans-Roman findfont\n8.4 scalefont\nsetfont\n15.269531 6.765625 moveto\n/two glyphshow\n\n\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 4 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n552.168 50.49 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -4 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n552.168 403.24 o\ngrestore\ngsave\n540.168094 34.990000 translate\n0.000000 rotate\n/BitstreamVeraSans-Roman findfont\n12.0 scalefont\nsetfont\n0.000000 0.465625 moveto\n/one glyphshow\n\n7.634766 0.465625 moveto\n/zero glyphshow\n\n/BitstreamVeraSans-Roman findfont\n8.4 scalefont\nsetfont\n15.269531 6.765625 moveto\n/three glyphshow\n\n\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n132.936 50.49 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n132.936 403.24 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n176.388 50.49 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n176.388 403.24 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n207.217 50.49 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n207.217 403.24 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n231.131 50.49 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n231.131 403.24 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n250.669 50.49 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n250.669 403.24 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n267.189 50.49 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n267.189 403.24 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n281.499 50.49 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n281.499 403.24 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n294.121 50.49 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n294.121 403.24 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n379.693 50.49 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n379.693 403.24 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n423.144 50.49 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n423.144 403.24 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n453.974 50.49 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n453.974 403.24 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n477.887 50.49 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n477.887 403.24 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n497.425 50.49 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n497.425 403.24 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n513.945 50.49 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n513.945 403.24 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n528.255 50.49 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -2 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n528.255 403.24 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 2 l\ngsave\n0.000 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setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n58.6553 392.312 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n552.168 392.312 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n58.6553 396.403 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n552.168 396.403 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n58.6553 400.012 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-2 0 l\ngsave\n0.000 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    "content": "%!PS-Adobe-3.0 EPSF-3.0\n%%Title: mg_error_vs_cycle.eps\n%%Creator: matplotlib version 1.2.0, http://matplotlib.org/\n%%CreationDate: Mon Apr  8 13:48:56 2013\n%%Orientation: portrait\n%%BoundingBox: 18 180 594 612\n%%EndComments\n%%BeginProlog\n/mpldict 10 dict def\nmpldict begin\n/m { moveto } bind def\n/l { lineto } bind def\n/r { rlineto } bind def\n/c { curveto } bind def\n/cl { closepath } bind def\n/box {\nm\n1 index 0 r\n0 exch r\nneg 0 r\ncl\n} bind def\n/clipbox {\nbox\nclip\nnewpath\n} bind def\n%!PS-Adobe-3.0 Resource-Font\n%%Title: cmsy10\n%%Copyright: Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B.\n%%Creator: Converted from TrueType to type 3 by PPR\n25 dict begin\n/_d{bind def}bind def\n/_m{moveto}_d\n/_l{lineto}_d\n/_cl{closepath eofill}_d\n/_c{curveto}_d\n/_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d\n/_e{exec}_d\n/FontName /Cmsy10 def\n/PaintType 0 def\n/FontMatrix[.001 0 0 .001 0 0]def\n/FontBBox[-29 -960 1123 779]def\n/FontType 3 def\n/Encoding [ /bar ] def\n/FontInfo 10 dict dup begin\n/FamilyName (cmsy10) def\n/FullName (cmsy10) def\n/Notice (Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. ) def\n/Weight (Regular) def\n/Version (1.1/12-Nov-94) def\n/ItalicAngle 0.0 def\n/isFixedPitch false def\n/UnderlinePosition -133 def\n/UnderlineThickness 20 def\nend readonly def\n/CharStrings 1 dict dup begin\n/bar{277 0 119 -249 159 750 _sc\n119 -231 _m\n119 732 _l\n119 737 121 741 125 745 _c\n129 748 133 750 139 750 _c\n143 750 148 748 152 745 _c\n156 741 159 737 159 732 _c\n159 -231 _l\n159 -236 156 -240 152 -244 _c\n148 -247 143 -249 139 -249 _c\n133 -249 129 -247 125 -244 _c\n121 -240 119 -236 119 -231 _c\n_cl}_d\nend readonly def\n\n/BuildGlyph\n {exch begin\n CharStrings exch\n 2 copy known not{pop /.notdef}if\n true 3 1 roll get exec\n end}_d\n\n/BuildChar {\n 1 index /Encoding get exch get\n 1 index /BuildGlyph get exec\n}_d\n\nFontName currentdict end definefont pop\n%!PS-Adobe-3.0 Resource-Font\n%%Title: DejaVu Sans\n%%Copyright: Copyright (c) 2003 by Bitstream, Inc. All Rights Reserved. Copyright (c) 2006 by Tavmjong Bah. All Rights Reserved. DejaVu changes are in public domain \n%%Creator: Converted from TrueType to type 3 by PPR\n25 dict begin\n/_d{bind def}bind def\n/_m{moveto}_d\n/_l{lineto}_d\n/_cl{closepath eofill}_d\n/_c{curveto}_d\n/_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d\n/_e{exec}_d\n/FontName /DejaVuSans def\n/PaintType 0 def\n/FontMatrix[.001 0 0 .001 0 0]def\n/FontBBox[-1021 -415 1681 1167]def\n/FontType 3 def\n/Encoding [ /space /numbersign /hyphen /zero /one /two /three /four /five /six /seven /eight /nine /L /V /c /e /f /l /m /n /o /r /s /y ] def\n/FontInfo 10 dict dup begin\n/FamilyName (DejaVu Sans) def\n/FullName (DejaVu Sans) def\n/Notice (Copyright (c) 2003 by Bitstream, Inc. All Rights Reserved. Copyright (c) 2006 by Tavmjong Bah. All Rights Reserved. DejaVu changes are in public domain ) def\n/Weight (Book) def\n/Version (Version 2.33) def\n/ItalicAngle 0.0 def\n/isFixedPitch false def\n/UnderlinePosition -130 def\n/UnderlineThickness 90 def\nend readonly def\n/CharStrings 25 dict dup begin\n/space{318 0 0 0 0 0 _sc\n}_d\n/numbersign{{838 0 77 0 761 718 _sc\n511 440 _m\n369 440 _l\n328 277 _l\n471 277 _l\n511 440 _l\n438 718 _m\n387 515 _l\n530 515 _l\n581 718 _l\n659 718 _l\n609 515 _l\n761 515 _l\n761 440 _l\n590 440 _l\n550 277 _l\n705 277 _l\n705 202 _l\n531 202 _l\n480 0 _l\n402 0 _l\n452 202 _l\n309 202 _l\n259 0 _l\n180 0 _l\n231 202 _l\n77 202 _l\n77 277 _l\n249 277 _l\n290 440 _l\n133 440 _l\n}_e{133 515 _l\n309 515 _l\n359 718 _l\n438 718 _l\n_cl}_e}_d\n/hyphen{361 0 49 234 312 314 _sc\n49 314 _m\n312 314 _l\n312 234 _l\n49 234 _l\n49 314 _l\n_cl}_d\n/zero{636 0 66 -13 570 742 _sc\n318 664 _m\n267 664 229 639 203 589 _c\n177 539 165 464 165 364 _c\n165 264 177 189 203 139 _c\n229 89 267 64 318 64 _c\n369 64 407 89 433 139 _c\n458 189 471 264 471 364 _c\n471 464 458 539 433 589 _c\n407 639 369 664 318 664 _c\n318 742 _m\n399 742 461 709 505 645 _c\n548 580 570 486 570 364 _c\n570 241 548 147 505 83 _c\n461 19 399 -13 318 -13 _c\n236 -13 173 19 130 83 _c\n87 147 66 241 66 364 _c\n66 486 87 580 130 645 _c\n173 709 236 742 318 742 _c\n_cl}_d\n/one{636 0 110 0 544 729 _sc\n124 83 _m\n285 83 _l\n285 639 _l\n110 604 _l\n110 694 _l\n284 729 _l\n383 729 _l\n383 83 _l\n544 83 _l\n544 0 _l\n124 0 _l\n124 83 _l\n_cl}_d\n/two{{636 0 73 0 536 742 _sc\n192 83 _m\n536 83 _l\n536 0 _l\n73 0 _l\n73 83 _l\n110 121 161 173 226 239 _c\n290 304 331 346 348 365 _c\n380 400 402 430 414 455 _c\n426 479 433 504 433 528 _c\n433 566 419 598 392 622 _c\n365 646 330 659 286 659 _c\n255 659 222 653 188 643 _c\n154 632 117 616 78 594 _c\n78 694 _l\n118 710 155 722 189 730 _c\n223 738 255 742 284 742 _c\n}_e{359 742 419 723 464 685 _c\n509 647 532 597 532 534 _c\n532 504 526 475 515 449 _c\n504 422 484 390 454 354 _c\n446 344 420 317 376 272 _c\n332 227 271 164 192 83 _c\n_cl}_e}_d\n/three{{636 0 76 -13 556 742 _sc\n406 393 _m\n453 383 490 362 516 330 _c\n542 298 556 258 556 212 _c\n556 140 531 84 482 45 _c\n432 6 362 -13 271 -13 _c\n240 -13 208 -10 176 -4 _c\n144 1 110 10 76 22 _c\n76 117 _l\n103 101 133 89 166 81 _c\n198 73 232 69 268 69 _c\n330 69 377 81 409 105 _c\n441 129 458 165 458 212 _c\n458 254 443 288 413 312 _c\n383 336 341 349 287 349 _c\n}_e{202 349 _l\n202 430 _l\n291 430 _l\n339 430 376 439 402 459 _c\n428 478 441 506 441 543 _c\n441 580 427 609 401 629 _c\n374 649 336 659 287 659 _c\n260 659 231 656 200 650 _c\n169 644 135 635 98 623 _c\n98 711 _l\n135 721 170 729 203 734 _c\n235 739 266 742 296 742 _c\n370 742 429 725 473 691 _c\n517 657 539 611 539 553 _c\n539 513 527 479 504 451 _c\n481 423 448 403 406 393 _c\n_cl}_e}_d\n/four{636 0 49 0 580 729 _sc\n378 643 _m\n129 254 _l\n378 254 _l\n378 643 _l\n352 729 _m\n476 729 _l\n476 254 _l\n580 254 _l\n580 172 _l\n476 172 _l\n476 0 _l\n378 0 _l\n378 172 _l\n49 172 _l\n49 267 _l\n352 729 _l\n_cl}_d\n/five{{636 0 77 -13 549 729 _sc\n108 729 _m\n495 729 _l\n495 646 _l\n198 646 _l\n198 467 _l\n212 472 227 476 241 478 _c\n255 480 270 482 284 482 _c\n365 482 429 459 477 415 _c\n525 370 549 310 549 234 _c\n549 155 524 94 475 51 _c\n426 8 357 -13 269 -13 _c\n238 -13 207 -10 175 -6 _c\n143 -1 111 6 77 17 _c\n77 116 _l\n106 100 136 88 168 80 _c\n199 72 232 69 267 69 _c\n}_e{323 69 368 83 401 113 _c\n433 143 450 183 450 234 _c\n450 284 433 324 401 354 _c\n368 384 323 399 267 399 _c\n241 399 214 396 188 390 _c\n162 384 135 375 108 363 _c\n108 729 _l\n_cl}_e}_d\n/six{{636 0 70 -13 573 742 _sc\n330 404 _m\n286 404 251 388 225 358 _c\n199 328 186 286 186 234 _c\n186 181 199 139 225 109 _c\n251 79 286 64 330 64 _c\n374 64 409 79 435 109 _c\n461 139 474 181 474 234 _c\n474 286 461 328 435 358 _c\n409 388 374 404 330 404 _c\n526 713 _m\n526 623 _l\n501 635 476 644 451 650 _c\n425 656 400 659 376 659 _c\n310 659 260 637 226 593 _c\n}_e{192 549 172 482 168 394 _c\n187 422 211 444 240 459 _c\n269 474 301 482 336 482 _c\n409 482 467 459 509 415 _c\n551 371 573 310 573 234 _c\n573 159 550 99 506 54 _c\n462 9 403 -13 330 -13 _c\n246 -13 181 19 137 83 _c\n92 147 70 241 70 364 _c\n70 479 97 571 152 639 _c\n206 707 280 742 372 742 _c\n396 742 421 739 447 735 _c\n472 730 498 723 526 713 _c\n_cl}_e}_d\n/seven{636 0 82 0 551 729 _sc\n82 729 _m\n551 729 _l\n551 687 _l\n286 0 _l\n183 0 _l\n432 646 _l\n82 646 _l\n82 729 _l\n_cl}_d\n/eight{{636 0 68 -13 568 742 _sc\n318 346 _m\n271 346 234 333 207 308 _c\n180 283 167 249 167 205 _c\n167 161 180 126 207 101 _c\n234 76 271 64 318 64 _c\n364 64 401 76 428 102 _c\n455 127 469 161 469 205 _c\n469 249 455 283 429 308 _c\n402 333 365 346 318 346 _c\n219 388 _m\n177 398 144 418 120 447 _c\n96 476 85 511 85 553 _c\n85 611 105 657 147 691 _c\n188 725 245 742 318 742 _c\n}_e{390 742 447 725 489 691 _c\n530 657 551 611 551 553 _c\n551 511 539 476 515 447 _c\n491 418 459 398 417 388 _c\n464 377 501 355 528 323 _c\n554 291 568 251 568 205 _c\n568 134 546 80 503 43 _c\n459 5 398 -13 318 -13 _c\n237 -13 175 5 132 43 _c\n89 80 68 134 68 205 _c\n68 251 81 291 108 323 _c\n134 355 171 377 219 388 _c\n183 544 _m\n183 506 194 476 218 455 _c\n}_e{242 434 275 424 318 424 _c\n360 424 393 434 417 455 _c\n441 476 453 506 453 544 _c\n453 582 441 611 417 632 _c\n393 653 360 664 318 664 _c\n275 664 242 653 218 632 _c\n194 611 183 582 183 544 _c\n_cl}_e}_d\n/nine{{636 0 63 -13 566 742 _sc\n110 15 _m\n110 105 _l\n134 93 159 84 185 78 _c\n210 72 235 69 260 69 _c\n324 69 374 90 408 134 _c\n442 178 462 244 468 334 _c\n448 306 424 284 396 269 _c\n367 254 335 247 300 247 _c\n226 247 168 269 126 313 _c\n84 357 63 417 63 494 _c\n63 568 85 628 129 674 _c\n173 719 232 742 306 742 _c\n390 742 455 709 499 645 _c\n543 580 566 486 566 364 _c\n}_e{566 248 538 157 484 89 _c\n429 21 356 -13 264 -13 _c\n239 -13 214 -10 189 -6 _c\n163 -2 137 5 110 15 _c\n306 324 _m\n350 324 385 339 411 369 _c\n437 399 450 441 450 494 _c\n450 546 437 588 411 618 _c\n385 648 350 664 306 664 _c\n262 664 227 648 201 618 _c\n175 588 162 546 162 494 _c\n162 441 175 399 201 369 _c\n227 339 262 324 306 324 _c\n_cl}_e}_d\n/L{557 0 98 0 552 729 _sc\n98 729 _m\n197 729 _l\n197 83 _l\n552 83 _l\n552 0 _l\n98 0 _l\n98 729 _l\n_cl}_d\n/V{684 0 8 0 676 729 _sc\n286 0 _m\n8 729 _l\n111 729 _l\n342 115 _l\n573 729 _l\n676 729 _l\n398 0 _l\n286 0 _l\n_cl}_d\n/c{{550 0 55 -13 488 560 _sc\n488 526 _m\n488 442 _l\n462 456 437 466 411 473 _c\n385 480 360 484 334 484 _c\n276 484 230 465 198 428 _c\n166 391 150 339 150 273 _c\n150 206 166 154 198 117 _c\n230 80 276 62 334 62 _c\n360 62 385 65 411 72 _c\n437 79 462 90 488 104 _c\n488 21 _l\n462 9 436 0 410 -5 _c\n383 -10 354 -13 324 -13 _c\n242 -13 176 12 128 64 _c\n}_e{79 115 55 185 55 273 _c\n55 362 79 432 128 483 _c\n177 534 244 560 330 560 _c\n358 560 385 557 411 551 _c\n437 545 463 537 488 526 _c\n_cl}_e}_d\n/e{{615 0 55 -13 562 560 _sc\n562 296 _m\n562 252 _l\n149 252 _l\n153 190 171 142 205 110 _c\n238 78 284 62 344 62 _c\n378 62 412 66 444 74 _c\n476 82 509 95 541 113 _c\n541 28 _l\n509 14 476 3 442 -3 _c\n408 -9 373 -13 339 -13 _c\n251 -13 182 12 131 62 _c\n80 112 55 181 55 268 _c\n55 357 79 428 127 481 _c\n175 533 241 560 323 560 _c\n397 560 455 536 498 489 _c\n}_e{540 441 562 377 562 296 _c\n472 322 _m\n471 371 457 410 431 440 _c\n404 469 368 484 324 484 _c\n274 484 234 469 204 441 _c\n174 413 156 373 152 322 _c\n472 322 _l\n_cl}_e}_d\n/f{352 0 23 0 371 760 _sc\n371 760 _m\n371 685 _l\n285 685 _l\n253 685 230 678 218 665 _c\n205 652 199 629 199 595 _c\n199 547 _l\n347 547 _l\n347 477 _l\n199 477 _l\n199 0 _l\n109 0 _l\n109 477 _l\n23 477 _l\n23 547 _l\n109 547 _l\n109 585 _l\n109 645 123 690 151 718 _c\n179 746 224 760 286 760 _c\n371 760 _l\n_cl}_d\n/l{278 0 94 0 184 760 _sc\n94 760 _m\n184 760 _l\n184 0 _l\n94 0 _l\n94 760 _l\n_cl}_d\n/m{{974 0 91 0 889 560 _sc\n520 442 _m\n542 482 569 511 600 531 _c\n631 550 668 560 711 560 _c\n767 560 811 540 842 500 _c\n873 460 889 403 889 330 _c\n889 0 _l\n799 0 _l\n799 327 _l\n799 379 789 418 771 444 _c\n752 469 724 482 686 482 _c\n639 482 602 466 575 435 _c\n548 404 535 362 535 309 _c\n535 0 _l\n445 0 _l\n445 327 _l\n445 379 435 418 417 444 _c\n398 469 369 482 331 482 _c\n}_e{285 482 248 466 221 435 _c\n194 404 181 362 181 309 _c\n181 0 _l\n91 0 _l\n91 547 _l\n181 547 _l\n181 462 _l\n201 495 226 520 255 536 _c\n283 552 317 560 357 560 _c\n397 560 430 550 458 530 _c\n486 510 506 480 520 442 _c\n_cl}_e}_d\n/n{634 0 91 0 549 560 _sc\n549 330 _m\n549 0 _l\n459 0 _l\n459 327 _l\n459 379 448 417 428 443 _c\n408 469 378 482 338 482 _c\n289 482 251 466 223 435 _c\n195 404 181 362 181 309 _c\n181 0 _l\n91 0 _l\n91 547 _l\n181 547 _l\n181 462 _l\n202 494 227 519 257 535 _c\n286 551 320 560 358 560 _c\n420 560 468 540 500 501 _c\n532 462 549 405 549 330 _c\n_cl}_d\n/o{612 0 55 -13 557 560 _sc\n306 484 _m\n258 484 220 465 192 427 _c\n164 389 150 338 150 273 _c\n150 207 163 156 191 118 _c\n219 80 257 62 306 62 _c\n354 62 392 80 420 118 _c\n448 156 462 207 462 273 _c\n462 337 448 389 420 427 _c\n392 465 354 484 306 484 _c\n306 560 _m\n384 560 445 534 490 484 _c\n534 433 557 363 557 273 _c\n557 183 534 113 490 63 _c\n445 12 384 -13 306 -13 _c\n227 -13 165 12 121 63 _c\n77 113 55 183 55 273 _c\n55 363 77 433 121 484 _c\n165 534 227 560 306 560 _c\n_cl}_d\n/r{411 0 91 0 411 560 _sc\n411 463 _m\n401 469 390 473 378 476 _c\n366 478 353 480 339 480 _c\n288 480 249 463 222 430 _c\n194 397 181 350 181 288 _c\n181 0 _l\n91 0 _l\n91 547 _l\n181 547 _l\n181 462 _l\n199 495 224 520 254 536 _c\n284 552 321 560 365 560 _c\n371 560 378 559 386 559 _c\n393 558 401 557 411 555 _c\n411 463 _l\n_cl}_d\n/s{{521 0 54 -13 472 560 _sc\n443 531 _m\n443 446 _l\n417 458 391 468 364 475 _c\n336 481 308 485 279 485 _c\n234 485 200 478 178 464 _c\n156 450 145 430 145 403 _c\n145 382 153 366 169 354 _c\n185 342 217 330 265 320 _c\n296 313 _l\n360 299 405 279 432 255 _c\n458 230 472 195 472 151 _c\n472 100 452 60 412 31 _c\n372 1 316 -13 246 -13 _c\n216 -13 186 -10 154 -5 _c\n}_e{122 0 89 8 54 20 _c\n54 113 _l\n87 95 120 82 152 74 _c\n184 65 216 61 248 61 _c\n290 61 323 68 346 82 _c\n368 96 380 117 380 144 _c\n380 168 371 187 355 200 _c\n339 213 303 226 247 238 _c\n216 245 _l\n160 257 119 275 95 299 _c\n70 323 58 356 58 399 _c\n58 450 76 490 112 518 _c\n148 546 200 560 268 560 _c\n301 560 332 557 362 552 _c\n391 547 418 540 443 531 _c\n}_e{_cl}_e}_d\n/y{592 0 30 -207 562 547 _sc\n322 -50 _m\n296 -114 271 -157 247 -177 _c\n223 -197 191 -207 151 -207 _c\n79 -207 _l\n79 -132 _l\n132 -132 _l\n156 -132 175 -126 189 -114 _c\n203 -102 218 -75 235 -31 _c\n251 9 _l\n30 547 _l\n125 547 _l\n296 119 _l\n467 547 _l\n562 547 _l\n322 -50 _l\n_cl}_d\nend readonly def\n\n/BuildGlyph\n {exch begin\n CharStrings exch\n 2 copy known not{pop /.notdef}if\n true 3 1 roll get exec\n end}_d\n\n/BuildChar {\n 1 index /Encoding get exch get\n 1 index /BuildGlyph get exec\n}_d\n\nFontName currentdict end definefont pop\n%!PS-Adobe-3.0 Resource-Font\n%%Title: cmmi10\n%%Copyright: Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B.\n%%Creator: Converted from TrueType to type 3 by PPR\n25 dict begin\n/_d{bind def}bind def\n/_m{moveto}_d\n/_l{lineto}_d\n/_cl{closepath eofill}_d\n/_c{curveto}_d\n/_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d\n/_e{exec}_d\n/FontName /Cmmi10 def\n/PaintType 0 def\n/FontMatrix[.001 0 0 .001 0 0]def\n/FontBBox[-34 -250 1048 750]def\n/FontType 3 def\n/Encoding [ /e /r ] def\n/FontInfo 10 dict dup begin\n/FamilyName (cmmi10) def\n/FullName (cmmi10) def\n/Notice (Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. ) def\n/Weight (Regular) def\n/Version (1.1/12-Nov-94) def\n/ItalicAngle 0.0 def\n/isFixedPitch false def\n/UnderlinePosition -133 def\n/UnderlineThickness 20 def\nend readonly def\n/CharStrings 2 dict dup begin\n/e{{465 0 44 -10 430 442 _sc\n201 -10 _m\n169 -10 141 -2 117 14 _c\n93 30 75 53 63 81 _c\n50 109 44 138 44 170 _c\n44 217 56 262 80 304 _c\n104 346 136 379 177 404 _c\n218 429 262 442 309 442 _c\n337 442 362 434 384 419 _c\n405 404 416 383 416 356 _c\n416 317 400 288 370 270 _c\n339 251 304 239 266 235 _c\n228 231 185 229 139 229 _c\n136 229 _l\n125 189 120 155 120 127 _c\n}_e{120 97 126 71 140 49 _c\n154 26 175 15 203 15 _c\n243 15 281 24 318 42 _c\n354 60 384 86 407 118 _c\n409 120 411 121 415 121 _c\n418 121 421 119 425 115 _c\n428 111 430 108 430 105 _c\n430 102 429 100 428 99 _c\n404 65 371 38 330 19 _c\n288 0 245 -10 201 -10 _c\n142 255 _m\n180 255 216 256 249 260 _c\n282 264 312 273 339 287 _c\n365 301 379 324 379 355 _c\n}_e{379 367 375 378 368 387 _c\n361 396 352 403 342 408 _c\n331 413 320 416 308 416 _c\n280 416 254 408 232 393 _c\n209 377 190 357 176 333 _c\n161 308 150 282 142 255 _c\n_cl}_e}_d\n/r{{451 0 27 -10 434 442 _sc\n77 17 _m\n77 21 77 24 78 26 _c\n153 324 _l\n157 342 160 359 160 373 _c\n160 401 150 416 131 416 _c\n110 416 94 403 84 378 _c\n74 353 64 322 55 284 _c\n55 282 54 280 52 279 _c\n50 278 48 278 47 278 _c\n35 278 _l\n33 278 31 279 29 281 _c\n27 283 27 286 27 288 _c\n34 317 41 342 48 362 _c\n54 382 65 400 79 417 _c\n}_e{93 433 110 442 132 442 _c\n155 442 175 435 193 422 _c\n210 408 221 391 225 369 _c\n241 391 260 408 282 422 _c\n304 435 328 442 354 442 _c\n375 442 394 435 410 423 _c\n426 411 434 394 434 373 _c\n434 356 428 341 418 329 _c\n407 316 393 310 376 310 _c\n365 310 356 313 349 320 _c\n341 326 338 335 338 346 _c\n338 360 343 373 354 385 _c\n364 397 377 403 391 403 _c\n}_e{380 411 367 416 352 416 _c\n323 416 298 405 277 385 _c\n256 365 237 339 220 308 _c\n149 22 _l\n147 12 142 5 134 -1 _c\n126 -7 117 -10 107 -10 _c\n99 -10 92 -7 86 -3 _c\n80 1 77 8 77 17 _c\n_cl}_e}_d\nend readonly def\n\n/BuildGlyph\n {exch begin\n CharStrings exch\n 2 copy known not{pop /.notdef}if\n true 3 1 roll get exec\n end}_d\n\n/BuildChar {\n 1 index /Encoding get exch get\n 1 index /BuildGlyph get exec\n}_d\n\nFontName currentdict end definefont pop\nend\n%%EndProlog\nmpldict begin\n18 180 translate\n576 432 0 0 clipbox\ngsave\n0 0 m\n576 0 l\n576 432 l\n0 432 l\ncl\n1.000 setgray\nfill\ngrestore\ngsave\n71.7907 49.86 m\n555.053 49.86 l\n555.053 408.645 l\n71.7907 408.645 l\ncl\n1.000 setgray\nfill\ngrestore\n1.000 setlinewidth\n1 setlinejoin\n2 setlinecap\n[] 0 setdash\n0.000 setgray\ngsave\n483.3 358.8 71.79 49.86 clipbox\n112.063 344.172 m\n152.334 315.976 l\n192.606 272.567 l\n232.878 238.85 l\n273.15 245.051 l\n313.422 245.276 l\n353.693 245.168 l\n393.965 245.167 l\n434.237 245.168 l\n474.509 245.168 l\n514.781 245.168 l\nstroke\ngrestore\n0 setlinecap\n[6 6] 0 setdash\ngsave\n483.3 358.8 71.79 49.86 clipbox\n112.063 403.564 m\n152.334 367.185 l\n192.606 333.024 l\n232.878 299.795 l\n273.15 267.108 l\n313.422 234.738 l\n353.693 202.589 l\n393.965 170.586 l\n434.237 138.691 l\n474.509 106.838 l\n514.781 75.2145 l\nstroke\ngrestore\n0.500 setlinewidth\n[] 0 setdash\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 4 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 49.86 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -4 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 408.645 o\ngrestore\n/DejaVuSans findfont\n12.000 scalefont\nsetfont\ngsave\n68.767262 36.781875 translate\n0.000000 rotate\n0.000000 0.171875 m /zero glyphshow\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 4 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n152.334 49.86 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -4 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n152.334 408.645 o\ngrestore\ngsave\n149.553083 36.953750 translate\n0.000000 rotate\n0.000000 0.000000 m /two glyphshow\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 4 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n232.878 49.86 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -4 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n232.878 408.645 o\ngrestore\ngsave\n229.690466 37.110000 translate\n0.000000 rotate\n0.000000 0.000000 m /four glyphshow\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 4 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n313.422 49.86 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -4 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n313.422 408.645 o\ngrestore\ngsave\n310.405975 36.781875 translate\n0.000000 rotate\n0.000000 0.171875 m /six glyphshow\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 4 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n393.965 49.86 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -4 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n393.965 408.645 o\ngrestore\ngsave\n390.965233 36.781875 translate\n0.000000 rotate\n0.000000 0.171875 m /eight glyphshow\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 4 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n474.509 49.86 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -4 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n474.509 408.645 o\ngrestore\ngsave\n467.922929 36.781875 translate\n0.000000 rotate\n0.000000 0.171875 m /one glyphshow\n7.634766 0.171875 m /zero glyphshow\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 4 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 49.86 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n0 -4 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 408.645 o\ngrestore\ngsave\n548.669688 36.953750 translate\n0.000000 rotate\n0.000000 0.000000 m /one glyphshow\n7.634766 0.000000 m /two glyphshow\ngrestore\ngsave\n274.562225 20.156875 translate\n0.000000 rotate\n0.000000 2.500000 m /numbersign glyphshow\n10.054688 2.500000 m /space glyphshow\n13.869141 2.500000 m /o glyphshow\n21.210938 2.500000 m /f glyphshow\n25.435547 2.500000 m /space glyphshow\n29.250000 2.500000 m /V glyphshow\n36.708984 2.500000 m /hyphen glyphshow\n41.039062 2.500000 m /c glyphshow\n47.636719 2.500000 m /y glyphshow\n54.738281 2.500000 m /c glyphshow\n61.335938 2.500000 m /l glyphshow\n64.669922 2.500000 m /e glyphshow\n72.052734 2.500000 m /s glyphshow\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n4 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 49.86 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-4 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 49.86 o\ngrestore\ngsave\n36.790699 42.360000 translate\n0.000000 rotate\n/DejaVuSans findfont\n12.0 scalefont\nsetfont\n0.000000 1.828125 moveto\n/one glyphshow\n\n7.634766 1.828125 moveto\n/zero glyphshow\n\n/DejaVuSans findfont\n8.4 scalefont\nsetfont\n15.269531 8.765625 moveto\n/hyphen glyphshow\n\n18.300586 8.765625 moveto\n/one glyphshow\n\n23.644922 8.765625 moveto\n/two glyphshow\n\n\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n4 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 82.4768 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-4 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 82.4768 o\ngrestore\ngsave\n36.790699 74.976818 translate\n0.000000 rotate\n/DejaVuSans findfont\n12.0 scalefont\nsetfont\n0.000000 1.937500 moveto\n/one glyphshow\n\n7.634766 1.937500 moveto\n/zero glyphshow\n\n/DejaVuSans findfont\n8.4 scalefont\nsetfont\n15.269531 8.875000 moveto\n/hyphen glyphshow\n\n18.300586 8.875000 moveto\n/one glyphshow\n\n23.644922 8.875000 moveto\n/one glyphshow\n\n\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n4 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 115.094 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-4 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 115.094 o\ngrestore\ngsave\n36.790699 107.593636 translate\n0.000000 rotate\n/DejaVuSans findfont\n12.0 scalefont\nsetfont\n0.000000 1.828125 moveto\n/one glyphshow\n\n7.634766 1.828125 moveto\n/zero glyphshow\n\n/DejaVuSans findfont\n8.4 scalefont\nsetfont\n15.269531 8.765625 moveto\n/hyphen glyphshow\n\n18.300586 8.765625 moveto\n/one glyphshow\n\n23.644922 8.765625 moveto\n/zero glyphshow\n\n\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n4 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 147.71 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-4 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 147.71 o\ngrestore\ngsave\n42.790699 140.210455 translate\n0.000000 rotate\n/DejaVuSans findfont\n12.0 scalefont\nsetfont\n0.000000 1.828125 moveto\n/one glyphshow\n\n7.634766 1.828125 moveto\n/zero glyphshow\n\n/DejaVuSans findfont\n8.4 scalefont\nsetfont\n15.269531 8.765625 moveto\n/hyphen glyphshow\n\n18.300586 8.765625 moveto\n/nine glyphshow\n\n\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n4 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 180.327 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-4 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 180.327 o\ngrestore\ngsave\n42.790699 172.827273 translate\n0.000000 rotate\n/DejaVuSans findfont\n12.0 scalefont\nsetfont\n0.000000 1.828125 moveto\n/one glyphshow\n\n7.634766 1.828125 moveto\n/zero glyphshow\n\n/DejaVuSans findfont\n8.4 scalefont\nsetfont\n15.269531 8.765625 moveto\n/hyphen glyphshow\n\n18.300586 8.765625 moveto\n/eight glyphshow\n\n\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n4 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 212.944 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-4 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 212.944 o\ngrestore\ngsave\n42.790699 205.444091 translate\n0.000000 rotate\n/DejaVuSans findfont\n12.0 scalefont\nsetfont\n0.000000 1.937500 moveto\n/one glyphshow\n\n7.634766 1.937500 moveto\n/zero glyphshow\n\n/DejaVuSans findfont\n8.4 scalefont\nsetfont\n15.269531 8.875000 moveto\n/hyphen glyphshow\n\n18.300586 8.875000 moveto\n/seven glyphshow\n\n\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n4 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 245.561 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-4 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 245.561 o\ngrestore\ngsave\n42.790699 238.060909 translate\n0.000000 rotate\n/DejaVuSans findfont\n12.0 scalefont\nsetfont\n0.000000 1.828125 moveto\n/one glyphshow\n\n7.634766 1.828125 moveto\n/zero glyphshow\n\n/DejaVuSans findfont\n8.4 scalefont\nsetfont\n15.269531 8.765625 moveto\n/hyphen glyphshow\n\n18.300586 8.765625 moveto\n/six glyphshow\n\n\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n4 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 278.178 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-4 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 278.178 o\ngrestore\ngsave\n42.790699 270.677727 translate\n0.000000 rotate\n/DejaVuSans findfont\n12.0 scalefont\nsetfont\n0.000000 1.937500 moveto\n/one glyphshow\n\n7.634766 1.937500 moveto\n/zero glyphshow\n\n/DejaVuSans findfont\n8.4 scalefont\nsetfont\n15.269531 8.875000 moveto\n/hyphen glyphshow\n\n18.300586 8.875000 moveto\n/five glyphshow\n\n\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n4 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 310.795 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-4 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 310.795 o\ngrestore\ngsave\n42.790699 303.294545 translate\n0.000000 rotate\n/DejaVuSans findfont\n12.0 scalefont\nsetfont\n0.000000 1.937500 moveto\n/one glyphshow\n\n7.634766 1.937500 moveto\n/zero glyphshow\n\n/DejaVuSans findfont\n8.4 scalefont\nsetfont\n15.269531 8.875000 moveto\n/hyphen glyphshow\n\n18.300586 8.875000 moveto\n/four glyphshow\n\n\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n4 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 343.411 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-4 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 343.411 o\ngrestore\ngsave\n42.790699 335.911364 translate\n0.000000 rotate\n/DejaVuSans findfont\n12.0 scalefont\nsetfont\n0.000000 1.828125 moveto\n/one glyphshow\n\n7.634766 1.828125 moveto\n/zero glyphshow\n\n/DejaVuSans findfont\n8.4 scalefont\nsetfont\n15.269531 8.765625 moveto\n/hyphen glyphshow\n\n18.300586 8.765625 moveto\n/three glyphshow\n\n\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n4 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 376.028 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-4 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 376.028 o\ngrestore\ngsave\n42.790699 368.528182 translate\n0.000000 rotate\n/DejaVuSans findfont\n12.0 scalefont\nsetfont\n0.000000 1.828125 moveto\n/one glyphshow\n\n7.634766 1.828125 moveto\n/zero glyphshow\n\n/DejaVuSans findfont\n8.4 scalefont\nsetfont\n15.269531 8.765625 moveto\n/hyphen glyphshow\n\n18.300586 8.765625 moveto\n/two glyphshow\n\n\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n4 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 408.645 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-4 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 408.645 o\ngrestore\ngsave\n42.790699 401.145000 translate\n0.000000 rotate\n/DejaVuSans findfont\n12.0 scalefont\nsetfont\n0.000000 1.937500 moveto\n/one glyphshow\n\n7.634766 1.937500 moveto\n/zero glyphshow\n\n/DejaVuSans findfont\n8.4 scalefont\nsetfont\n15.269531 8.875000 moveto\n/hyphen glyphshow\n\n18.300586 8.875000 moveto\n/one glyphshow\n\n\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 59.6786 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 59.6786 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 69.4973 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 69.4973 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 75.2408 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 75.2408 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 79.3159 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 79.3159 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 92.2955 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 92.2955 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 102.114 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 102.114 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 107.858 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 107.858 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 111.933 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 111.933 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 124.912 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 124.912 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 134.731 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 134.731 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 140.474 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 140.474 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 144.55 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 144.55 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 157.529 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 157.529 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 167.348 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 167.348 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 173.091 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 173.091 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 177.166 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 177.166 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 190.146 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 190.146 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 199.965 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 199.965 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 205.708 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 205.708 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 209.783 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n555.053 209.783 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind def\n71.7907 222.763 o\ngrestore\ngsave\n/o {\ngsave\nnewpath\ntranslate\n0.5 setlinewidth\n1 setlinejoin\n0 setlinecap\n0 0 m\n-2 0 l\ngsave\n0.000 setgray\nfill\ngrestore\nstroke\ngrestore\n} bind 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    "content": "\\label{ch:multigrid}\n\n\\section{Elliptic equations}\n\nThe simplest elliptic PDE is {\\em Laplace's equation}:\n\\begin{equation}\n\\nabla^2 \\phi = 0\n\\end{equation}\nOnly slightly more complex is {\\em Poisson's equation} (Laplace + a source term):\n\\begin{equation}\n\\nabla^2 \\phi = f\n\\end{equation}\nThese equations can arise in electrostatics (for the electric potential),\nsolving for the gravitational potential from a mass distribution, or\nenforcing a divergence constraint on a vector field (we'll see this\nwhen we consider incompressible flow).\n\nAnother common elliptic equation is the {\\em Helmholtz equation}:\n\\begin{equation}\n(\\alpha - \\nabla \\cdot \\beta \\nabla) \\phi = f\n\\end{equation}\nA Helmholtz equation can arise, for example, from a time-dependent\nequation (like diffusion) by discretizing in time.\n\nNotice that there is no time-dependence in any of these equations.\nThe quantity $\\phi$ is specified instantaneously in the domain subject to\nboundary conditions.  This makes the solution methods very different then\nwhat we saw for hyperbolic problems.\n\n\n\\section{\\label{elliptic:sec:fft} Fourier Method}\n\nA direct way of solving a constant-coefficient elliptic equation is\nusing Fourier transforms.  Using a general Fourier transform (which we\nconsider here) works only for periodic boundary conditions, but other\nbasis functions (e.g., all sines or all cosines) can be used for other\nboundary conditions. \\MarginPar{ref?}\n\nConsider the Poisson equation:\n\\begin{equation}\n\\nabla^2 \\phi = f\n\\end{equation}\nWe will difference this in a second-order accurate fashion---see\nFigure~\\ref{fig:mg:laplacian}.  \n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{laplacian}\n\\caption[Data centerings for the discrete Laplacian]{\\label{fig:mg:laplacian} The centerings of the first and second\nderivatives for a standard Laplacian discretization.  Our data, $\\phi$,\nis cell-centered.  The first-derivatives, $d\\phi/dx$, are edge-centered, and the\nsecond-derivative, $d^2\\phi/dx^2$, is cell-centered.}\n\\end{figure}\n%\nIn 1-d, the Laplacian is just the second-derivative.  If our solution\nis defined at cell-centers, then we first compute the first-derivative\non cell edges:\n\\begin{align}\n\\left . \\frac{d\\phi}{dx} \\right |_{i-\\myhalf} &= \\frac{\\phi_i - \\phi_{i-1}}{\\dx}\\\\\n\\left . \\frac{d\\phi}{dx} \\right |_{i+\\myhalf} &= \\frac{\\phi_{i+1} - \\phi_i}{\\dx}\n\\end{align}\nThese are second-order accurate on the interface.  We can then compute\nthe second-derivative at the cell-center by differencing these edge values:\n\\begin{equation}\n\\left . \\frac{d^2\\phi}{dx^2} \\right |_i = \\frac{ d\\phi/dx |_{i+\\myhalf} - d\\phi/dx |_{i-\\myhalf}}{\\dx}\n\\end{equation}\n\nThe extension to 2-d is straightforward.  Thinking of the Laplacian as $\\nabla^2\n\\phi$ = $\\nabla \\cdot \\nabla \\phi$, we first compute the gradient of\n$\\phi$ on edges:\n\\begin{align}\n[\\nabla \\phi \\cdot \\hat{x}]_{i+\\myhalf,j} &= \\frac{\\phi_{i+1,j} - \\phi_{i,j}}{\\dx} \\\\\n[\\nabla \\phi \\cdot \\hat{y}]_{i,j+\\myhalf} &= \\frac{\\phi_{i,j+1} - \\phi_{i,j}}{\\dy} \n\\end{align}\nAgain, since this is defined on edges, this represents a centered\ndifference, and is therefore second-order accurate.  We then\ndifference the edge-centered gradients to the center to get the\nLaplacian at cell-centers:\n\\begin{align}\n[\\nabla^2 \\phi]_{i,j} &=\n   \\frac{[\\nabla \\phi \\cdot \\hat{x}]_{i+\\myhalf,j} -\n         [\\nabla \\phi \\cdot \\hat{x}]_{i-\\myhalf,j}}{\\dx} +\n   \\frac{[\\nabla \\phi \\cdot \\hat{y}]_{i,j+\\myhalf} -\n         [\\nabla \\phi \\cdot \\hat{y}]_{i,j-\\myhalf}}{\\dy} \\nonumber\\\\\n%\n  &= \\frac{\\phi_{i+1,j} - 2\\phi_{i,j} + \\phi_{i-1,j}}{\\dx^2} +\n     \\frac{\\phi_{i,j+1} - 2\\phi_{i,j} + \\phi_{i,j-1}}{\\dy^2} = f_{i,j}\n\\end{align}        \nAgain, since we used a centered-difference of the edge values, this\nexpression is second-order accurate.  This is the standard {\\em\n  5-point stencil} for the 2-d Laplacian\\footnote{There are other\n  possible second-order accurate stencils, including a 9-point stencil\n  in 2-d, that are less commonly used.}.\n\nWe now assume that we have an FFT subroutine (see\n\\S~\\ref{sec:intro:ffts}) that can take our discrete real-space data,\n$\\phi_{i,j}$ and return the discrete Fourier coefficients,\n$\\Phi_{k_x,k_y}$, and likewise for the source term:\n\\begin{equation}\n\\Phi_{k_x,k_y} = \\mathcal{F}(\\phi_{i,j}) \\quad\nF_{k_x,k_y} = \\mathcal{F}(f_{i,j})\n\\end{equation}\nThe power of the Fourier method is that derivatives in real space are\nmultiplications in Fourier space, which makes the solution process in\nFourier space straightforward.\n\nWe now express $\\phi_{i,j}$ and $f_{i,j}$ as sums over their Fourier\ncomponents.  Here we define $M$ as the number of grid points in the\n$x$-direction and $N$ as the number of grid points in the\n$y$-direction.  As before, we are using $i$ as the grid index, we\nwill use $\\imag$ as the imaginary unit:\n\\begin{align}\n\\phi_{i,j} &= \\frac{1}{MN} \\sum_{k_x = 0}^{M-1} \\sum_{k_y = 0}^{N-1}\n  \\Phi_{k_x,k_y} e^{2\\pi\\imag i k_x/M} e^{2\\pi\\imag j k_y/N} \\\\\nf_{i,j} &= \\frac{1}{MN} \\sum_{k_x = 0}^{M-1} \\sum_{k_y = 0}^{N-1}\n  F_{k_x,k_y} e^{2\\pi\\imag i k_x/M} e^{2\\pi\\imag j k_y/N} \n\\end{align}\n\nInserting these into the differenced equation, we have:\n\\begin{align}\n\\frac{1}{MN}\\sum_{k_x=0}^{M-1}\\sum_{k_y=0}^{N-1}\n\\biggl \\{\n&\\frac{\\Phi_{k_x,k_y}}{\\dx^2} e^{2\\pi\\imag j k_y/N}\n  \\left [ e^{2\\pi\\imag (i+1) k_x/M} -2 e^{2\\pi\\imag i k_x/M} +\n         e^{2\\pi\\imag (i-1) k_x/M} \\right ] + \\nonumber \\\\\n&\\frac{\\Phi_{k_x,k_y}}{\\dy^2} e^{2\\pi\\imag i k_x/M}\n  \\left [ e^{2\\pi\\imag (j+1) k_y/N} -2 e^{2\\pi\\imag j k_y/N} +\n         e^{2\\pi\\imag (j-1) k_y/N} \\right ] \\biggr \\} = \\nonumber \\\\\n\\frac{1}{MN}\\sum_{k_x=0}^{M-1}\\sum_{k_y=0}^{N-1}\n & F_{k_x,k_y} e^{2\\pi\\imag i k_x/M} e^{2\\pi\\imag j k_y/N}\n\\end{align}\n\nWe can bring the righthand side into the sums on the left, and we can\nthen look at just a single $(k_x,k_y)$ term in the series:\n\\begin{align}\n  e^{2\\pi\\imag i k_x/M} e^{2\\pi\\imag j k_y/N}\n  \\biggl \\{\n  &\\frac{\\Phi_{k_x,k_x}}{\\dx^2}\n  \\left [e^{2\\pi\\imag k_x/M} + e^{-2\\pi\\imag k_x/M} - 2 \\right ] + \\nonumber \\\\\n  &\\frac{\\Phi_{k_x,k_x}}{\\dy^2}\n  \\left [e^{2\\pi\\imag k_y/N} + e^{-2\\pi\\imag k_y/N} - 2 \\right ] \n  - F_{k_x,k_y}\n  \\biggr \\} = 0\n\\end{align}\nSimplifying, we have:\n\\begin{equation}\n  \\Phi_{k_x,k_y} = \\frac{1}{2}\\frac{F_{k_x,k_y}}\n      {\\left [\\cos(2\\pi k_x/M) - 1 \\right ] \\Delta x^{-2} +\n        \\left [\\cos(2\\pi k_y/N) - 1 \\right ] \\Delta y^{-2}}\n      \\label{eq:FFTsol}\n\\end{equation}\nThis is the algebraic solution to the Poisson equation in Fourier (frequency)\nspace.  Once we evaluate this, we can get the real-space solution\nby doing the inverse transform:\n\\begin{equation}\n\\phi_{i,j} = \\mathcal{F}^{-1}(\\Phi_{k_x,k_y})\n\\end{equation}\n\n\nWe can test this technique with the source term:\n\\begin{align}\nf =\\, & 8\\pi^2\\cos(4\\pi y) \\left [\\cos(4\\pi x) - \\sin(4\\pi x) \\right ] - \\nonumber \\\\\n    & 16\\pi^2 \\left [ \\sin(4\\pi x)\\cos(2\\pi y)^2 + \\sin(2\\pi x)^2 \\cos(4\\pi y) \\right ]\n    \\label{eq:mg:fftsource}\n\\end{align}\nwhich has the analytic solution\\footnote{Note: throughout this\nchapter, we devise test problems by picking a function\nthat meets the desired boundary conditions and then inserting\nit into the analytic equation we are solving to find the righthand\nside}:\n\\begin{equation}\n\\phi = \\sin(2\\pi x)^2 \\cos(4\\pi y) + \\sin(4\\pi x)\\cos(2\\pi y)^2\n\\end{equation}\nNote that this solution has the required periodic behavior.\nFigure~\\ref{fig:mg:fftpoisson} shows the solution.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.53\\linewidth]{poisson_fft}\n\\includegraphics[width=0.44\\linewidth]{fft-poisson-converge}\n\\caption[FFT solution to the Poisson\n  equation]{\\label{fig:mg:fftpoisson} (left) Solution to the Poisson\n  equation on a $64^2$ grid with source from\n  Eq.~\\ref{eq:mg:fftsource}. (right) Error vs.\\ the true solution as a\n  function of resolution for the Fourier method, showing second-order\n  convergence.\n  \\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/elliptic/poisson_fft.py}{poisson\\_fft.py}}}\n\\end{figure}\n\n\nThe main downside of this approach is that, because we solve for a\nsingle component independently (Eq.~\\ref{eq:FFTsol}), this only works\nfor linear problems with constant coefficients.  This makes it an\nexcellent choice for cosmological problems solving the gravitational\nPoisson equation with periodic boundaries on all sides of the domain\n(see, e.g., \\cite{heitmann:2008}).  However, for a problem like:\n\\begin{equation}\n  \\nabla \\cdot (\\beta \\nabla \\phi) = f\n\\end{equation}\nthere would be ``cross-talk'' between the Fourier modes of $\\beta$ and\n$\\phi$, and we would not be able to solve for a single mode of\n$\\Phi_{k_x,k_y}$ independently.  We discuss more general methods that\nwork for these forms next.\n\n\n\\section{Relaxation}\n\nRelaxation is an iterative technique, and as we will see shortly, it\nprovides the basis for the multigrid technique.  \n\nConsider Poisson's equation, again differenced as:\n\\begin{equation}\n\\frac{\\phi_{i+1,j} - 2 \\phi_{i,j} + \\phi_{i-1,j}}{\\dx^2} +\n\\frac{\\phi_{i,j+1} - 2 \\phi_{i,j} + \\phi_{i,j-1}}{\\dy^2} = f_{i,j}\n\\end{equation}\nFor each zone $(i,j)$, we couple in the zones $\\pm 1$ in $x$ and $\\pm\n1$ in $y$.  For the moment, consider the case where $\\dx = \\dy$.  If\nwe solve this discretized equation for $\\phi_{i,j}$, then we have:\n\\begin{equation}\n\\phi_{i,j} = \\frac{1}{4} (\\phi_{i+1,j} + \\phi_{i-1,j} + \n                          \\phi_{i,j+1} + \\phi_{i,j-1} - \\dx^2 f_{i,j} )\n\\end{equation}\nA similar expression exists for every zone in our domain, coupling all\nthe zones together.  We can't separate the solution of $\\phi_{i,j}$\nfor the neighboring zones, but instead can apply an iterative\ntechnique called {\\em relaxation} (also sometimes called {\\em\n  smoothing} because generally speaking the solution to elliptic\nequations is a smooth function) to find the solution for $\\phi$\neverywhere.  \n\nImagine an initial guess to $\\phi$: $\\phi_{i,j}^{(0)}$.\nWe can improve that guess by using our difference equation to define a\nnew value of $\\phi$, $\\phi_{i,j}^{(1)}$:\n\\begin{equation}\n\\phi_{i,j}^{(1)} = \\frac{1}{4} (\\phi_{i+1,j}^{(0)} + \\phi_{i-1,j}^{(0)} + \n                                \\phi_{i,j+1}^{(0)} + \\phi_{i,j-1}^{(0)} - \n                                 \\dx^2 f_{i,j} )\n\\end{equation}\nor generally, the $k+1$ iteration will see:\n\\begin{equation}\n\\phi_{i,j}^{(k+1)} = \\frac{1}{4} (\\phi_{i+1,j}^{(k)} + \\phi_{i-1,j}^{(k)} + \n                                  \\phi_{i,j+1}^{(k)} + \\phi_{i,j-1}^{(k)} - \n                                   \\dx^2 f_{i,j} )\n\\end{equation}\nThis will (slowly) converge to the true solution\\footnote{Formally,\n  convergence is only guaranteed if the matrix in our linear system is\n  {\\em diagonally dominant}.  The Laplacian used here is not quite\n  diagonally domainant, but these methods still converge.}, since each\nzone is coupled to each other zone (and to the boundary values that we\nneed to specify---more on that in a moment).  This form of relaxation\nis called {\\em Jacobi iteration}.  To implement this, you need two\ncopies of $\\phi$---the old iteration value and the new iteration\nvalue.\n\nAn alternate way to do the relaxation is to update $\\phi_{i,j}$ in\nplace, as soon as the new value is known.  Thus the neighboring cells\nwill see a mix of the old and new solutions.  We can express this in-place\nupdating as:\n\\begin{equation}\n\\phi_{i,j} \\leftarrow \\frac{1}{4} (\\phi_{i+1,j} + \\phi_{i-1,j} + \n                                   \\phi_{i,j+1} + \\phi_{i,j-1} - \n                                   \\dx^2 f_{i,j} )\n\\end{equation}\nThis only requires a single copy of $\\phi$ to be stored.  This\ntechnique is called {\\em Gauss-Seidel iteration}.  A host of other\nrelaxation methods exist, including linear combinations of these two.\nAn excellent discussion of these approaches, and their\nstrengths and weaknesses is given in \\cite{multigridtutorial}.\n\nNext consider the Helmholz equation with constant coefficients:\n\\begin{equation}\n(\\alpha - \\beta \\nabla^2) \\phi = f\n\\end{equation}\nWe can discretize this as:\n\\begin{equation}\n\\alpha \\phi_{i,j} - \\beta \\left ( \n    \\frac{\\phi_{i+1,j} - 2 \\phi_{i,j} + \\phi_{i-1,j}}{\\dx^2} +\n    \\frac{\\phi_{i,j+1} - 2 \\phi_{i,j} + \\phi_{i,j-1}}{\\dy^2} \\right )\n= f_{i,j}\n\\end{equation}\nand the update of $\\phi_{i,j}$ through relaxation is:\n\\begin{equation}\n\\phi_{i,j} \\leftarrow\n     \\left . \\left ( f_{i,j} + \\frac{\\beta}{\\dx^2} \\phi_{i+1,j}\n                             + \\frac{\\beta}{\\dx^2} \\phi_{i-1,j}\n                             + \\frac{\\beta}{\\dy^2} \\phi_{i,j+1}\n                             + \\frac{\\beta}{\\dy^2} \\phi_{i,j-1} \\right ) \n\\middle / \\left ( \\alpha + \\frac{2 \\beta}{\\dx^2} + \\frac{2 \\beta}{\\dy^2} \\right ) \\right .\n\\end{equation}\nNotice that if $\\alpha = 0$, $\\beta = -1$, and $\\dx = \\dy$, we \nrecover the relaxation expression for Poisson's equation from above.\n\n\n\\subsection{Boundary conditions}\n\nWhen using a cell-centered grid, no points fall exactly on the\nboundary, so we need to use ghost cells to specify boundary\nconditions.  A single ghost cell is sufficient for the 5-point stencil\nused here.  The common types of boundary conditions are {\\em\n  Dirichlet} (specified value on the boundary), {\\em Neumann}\n(specified first derivative on the boundary), and periodic.  Some\nrestrictions apply (see discuss this later, in\n\\S~\\ref{sec:multigrid:solvability}).\n\n% created by figures/multigrid/fv-fd_bnd.py\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=\\linewidth]{fv-fd_grid_bc}\n\\caption[Node-centered vs.\\ cell-centered data at boundaries]{\\label{mg:fig:bcs} \n  A node-centered (top) and cell-centered (bottom) finite difference\n  grid showing the data and domain boundaries.  Notice that for the \n  cell-centered grid, there is no data point precisely at the boundary.}\n\\end{figure}\n\nConsider Dirichlet boundary conditions, specifying values $\\phi_l$ on\nthe left and $\\phi_r$ on the right boundaries.\\footnote{If the value,\n  $\\phi_l$ or $\\phi_r$ is zero, we call this a {\\em homogeneous\n    boundary condition}.  Otherwise we call it an {\\em inhomogeneous\n    boundary condition}} We'll label the first zone inside the domain,\nat the left boundary, $\\mathrm{lo}$, and the last zone inside the\ndomain, at the right boundary, $\\mathrm{hi}$---see\nFigure~\\ref{mg:fig:bcs}.  To second order, we can average the zone values on \neither side of the interface to get the boundary condition:\n\\begin{eqnarray}\n\\phi_l &=& \\frac{1}{2} ( \\phi_\\mathrm{lo} + \\phi_\\mathrm{lo-1} ) \\\\\n\\phi_r &=& \\frac{1}{2} ( \\phi_\\mathrm{hi} + \\phi_\\mathrm{hi+1} )\n\\end{eqnarray}\nThis then tells us that the values we need to assign to the ghost cells are:\n\\begin{eqnarray}\n\\label{eq:bc_inhomo_dir}\n\\phi_\\mathrm{lo-1} &=& 2 \\phi_l - \\phi_\\mathrm{lo} \\\\\n\\phi_\\mathrm{hi+1} &=& 2 \\phi_r - \\phi_\\mathrm{hi}\n\\end{eqnarray}\n\nIf we instead consider Neumann boundary conditions, we specify values\nof the derivative on the boundaries: $\\phi_x |_l$ on the left and\n$\\phi_x |_r$ on the right.  We note that a single difference across\nthe boundary is second-order accurate on the boundary (it is a\ncentered-difference there), so to second-order:\n\\begin{eqnarray}\n\\phi_x |_l &=& \\frac{\\phi_\\mathrm{lo} - \\phi_\\mathrm{lo-1}}{\\dx} \\\\\n\\phi_x |_r &=& \\frac{\\phi_\\mathrm{hi+1} - \\phi_\\mathrm{hi}}{\\dx}\n\\end{eqnarray}\nThis then tells us that the ghost cells are filled as:\n\\begin{eqnarray}\n\\label{eq:bc_inhomo_neum}\n\\phi_\\mathrm{lo-1} &=& \\phi_\\mathrm{lo} - \\dx \\, \\phi_x |_l \\\\\n\\phi_\\mathrm{hi+1} &=& \\phi_\\mathrm{hi} + \\dx \\, \\phi_x |_r\n\\end{eqnarray}\n\n\n\\subsection{Residual and true error}\n\nThe {\\em residual error} is a measure of how well our discrete solution\nsatisfies the discretized equation.  For the Poisson equation, we\ncan the residual as:\n\\begin{equation}\nr_{i,j} = f_{i,j} - (L \\phi)_{i,j} \n\\end{equation}\nand the residual error as:\n\\begin{equation}\n\\epsilon^{(r)} = \\| r \\|\n\\end{equation}\nwhere $L$ represents our discretized Laplacian.  Note that $r$ is the\nerror with respect to the discrete form of the equation.  The true\nerror is the measure of how well our discrete solution approximates\nthe true solution.  If $\\phi^\\mathrm{true}$ satisfies $\\nabla^2\n\\phi^\\mathrm{true} = f$, then the true error in each zone is\n\\begin{equation}\ne_{i,j} = \\phi^\\mathrm{true}(x_i,y_j) - \\phi_{i,j} \n\\end{equation}\nand\n\\begin{equation}\n\\epsilon^\\mathrm{true} = \\| e_{i,j} \\|\n\\end{equation}\n\nWe can make $\\epsilon^{(r)}$ approach machine precision by performing\nmore and more relaxation iterations, but after some point, this will\nno longer improve $\\epsilon^\\mathrm{true}$.  The only way to improve\n$\\epsilon^\\mathrm{true}$ is to make $\\dx$ and $\\dy$ smaller.\nIn practice we do not know the true solution so we cannot compute\n$\\epsilon^\\mathrm{true}$ and will instead have to rely on\n$\\epsilon^{(r)}$ to monitor our error.\n\nNote that since our operator is linear,\n\\begin{equation}\nL e = L\\phi^\\mathrm{true} - L\\phi = f - L\\phi = r\n\\end{equation}\nso the error in our solution obeys a Poisson equation with the residual\nas the source---we'll see this in the next section.\n\n\\subsection{Norms}\n\nThere are several different norms that are typically used in defining\nerrors on the grid.  The $L_\\infty$ norm (or `inf'-norm) is just the\nmaximum error on the grid:\n\\begin{equation}\n\\|e\\|_\\infty = \\max_{i,j} \\{ |e_{i,j}| \\}\n\\end{equation}\nThis will pick up on local errors.  \n\nThe $L_1$ norm and $L_2$ norms are more global. \n\\begin{eqnarray}\n\\|e\\|_1 &=& \\frac{1}{N} \\sum_{i,j} |e_{i,j} | \\\\\n\\|e\\|_2 &=& \\left ( \\frac{1}{N} \\sum_{i,j} |e_{i,j} |^2 \\right )^{1/2}\n\\end{eqnarray}\nGenerally, the measure in $L_2$ falls between $L_\\infty$ and $L_1$.\nRegardless of the norm used, if the problem converges, it should\nconverge in all norms.  For reference, the AMReX\nlibrary\\footnote{\\url{https://github.com/amrex-codes}} uses $L_\\infty$\nin its multigrid solvers.\n\n\n\n\\subsection{Performance}\n\n\n% this figure can be created with figures/multigrid/smooth-separate.py\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{smooth-error}\n\\caption[Convergence as a function of number of iterations using Gauss-Seidel relaxation]{\\label{fig:smootherror} Gauss-Seidel relaxation applied to\n  $\\phi_{xx} = \\sin(x)$ with $\\phi(0) = \\phi(1) = 0$.  Shown are the\n  L2 norm of the error compared with the true solution (solid lines)\n  and the L2 norm of the residual (dotted lines) for 3 different\n  resolutions (16, 32, and 64 zones). \\\\\n  \\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/multigrid/smooth.py}{smooth.py}}}\n\\end{figure}\n\nConsider the simple Poisson problem\\footnote{This is the test problem\n  used throughout {\\em A Multigrid Tutorial} \\cite{multigridtutorial}.\n  We use the same problem here to allow for easy comparison to the\n  discussions in that text.} on $x \\in [0,1]$:\n\\begin{equation}\n\\phi_{xx} = \\sin(x), \\qquad \\phi(0) = \\phi(1) = 0\n\\end{equation}\nThe analytic solution to this is simply \n\\begin{equation}\n\\phi^a(x) = -\\sin(x) + x \\sin(1)\n\\end{equation}\nWe can perform smoothing and compute both the error against the\nanalytic solution (the `true' error), $e \\equiv \\| \\phi^a(x_i) - \\phi_i \\|_2$ and the\nresidual error, $\\| r_i \\|_2$.  Figure~\\ref{fig:smootherror} shows these\nerrors as a function of the number of smoothing iterations for 3\ndifferent resolutions.\n\nNotice that the true error stalls at a relatively high value---this is\nthe truncation error of the method.  From one resolution to the next,\nthe true error changes as $\\dx^2$, indicating that we are\nconverging as our method should.  No additional amount of smoothing\nwill change this---we are getting the best answer to the problem we\ncan with our choice of discretization.\nIn contrast, the residual error decreases to machine precision\nlevels---this is indicating that our solution is an exact solution to\nthe discrete equation (to roundoff-error).\nIn practice, we can only monitor the residual error, not the true\nerror, and we hope that small residual error implies a small true\nerror.\n\nFigure~\\ref{fig:smoothnorms} shows the error with respect to the true\nsolution and of the residual for pure smoothing in three different\nnorms.  The overall behavior is qualitative similar regardless of the\nchoice of norm.\n\n% this figure can be created with figures/multigrid/smooth-separate.py\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{smooth-error-norms}\n\\caption[Convergence of smoothing in different norms]{\\label{fig:smoothnorms} \n  Gauss-Seidel relaxation applied to $\\phi_{xx} = \\sin(x)$ with\n  $\\phi(0) = \\phi(1) = 0$.  This is like figure \\ref{fig:smootherror}, but\n  now we show the error in 3 different norms: $L_1$, $L_2$, and $L_\\infty$.\n  \\\\ \\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/multigrid/smooth-norms.py}{smooth-norms.py}}}\n\\end{figure}\n\nTo demonstrate the influence of the boundary conditions,\nFigure~\\ref{fig:smooth-badbcs} shows the norm of the true error for\nthe same problem, but this time with a naive implementation of the\nboundary conditions---simply initializing the ghost cell to the\nboundary value, instead of averaging to the interface:\n\\begin{align}\n\\phi_\\mathrm{lo-1} &= \\phi_\\mathrm{lo} \\\\\n\\phi_\\mathrm{hi+1} &= \\phi_\\mathrm{hi} \n\\end{align}\nWe see that with this mistake at the boundaries, the error of the\nentire solution is affected, and we get only first-order convergence\nwith resolution (this can be seen by looking at the spacing of the\ncurves).  The previous solution, with the correct BCs is shown for\nreference, and shows second-order convergence.  It is important to\nnote that because every zone is linked to every other zone (and the\nboundary) in elliptic problems, an error at the boundary can pollute the\nglobal solution.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{smooth-badBCs}\n\\caption[Convergence of smoothing in first-order BCs]{\\label{fig:smooth-badbcs} \n  The same problem as in figure \\ref{fig:smootherror}, but\n  now we done with a naive first-order boundary conditions---just initializing\n  the ghost cell to the boundary value (solid lines).  We see that this \n  achieves only first-order convergence in the true error.  The correct\n  second-order implmentation is shown as the dotted lines.\n  \\\\ \\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/multigrid/smooth-badbcs.py}{smooth-badbcs.py}}}\n\\end{figure}\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{smooth-multimode}\n\\caption[Smoothing of different wavenumbers]{\\label{fig:mg:smooth}\n  Error in the solution to $\\phi'' = 0$ given an initial guess with 3\n  different wavenumbers of noise.  A 128 zone grid was used.  The\n  different curves are different numbers of smoothing\n  iterations. \\\\ \\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/multigrid/smooth-modes.py}{smooth-modes.py}}}\n\\end{figure}\n\n\\subsection{Frequency/wavelength-dependent error}\n\nWe can think of the error in the solution as a superposition of high\n(short) and low (long) frequency (wavelength) modes.  Smoothing works\nreally well to eliminate the short wavelength noise quickly, but many\niterations are needed to remove the long wavelength noise (see\nFigure~\\ref{fig:smooth}).  A very important concept to understand here\nis that when we talk about long wavelength error, we are expressing it\nin terms of the number of zones across the feature, not as physical\nlength.  We can get an intuitive feel for this behavior by thinking\nabout how smoothing works.  Every zone is coupled to every other zone,\nand we can think about each zone seeing one more zone away for each\niteration.  When the error is short wavelength, that means that there\nare only a few zones across it, and after a few iterations, all of the\nzones have seen the short wavelength error, and can eliminate it.  For\na very long wavelength error, many iterations will be needed until the\nsmoothing couples one zone to another that is a wavelength away.\n\nThis behavior suggests that if we could represent our problem on a\ncoarser grid, the error will now be of shorter wavelength, and\nsmoothing will be more efficient.  This is the core idea behind\nmultigrid, which we see next.\n\n\\begin{exercise}[Smoothing the 1-d Laplace equation]\n{Implement 1-d smoothing for the Laplace equation on\n  cc-grid.  Use an initial guess for the solution:\n  \\begin{equation}\n    \\label{eq:mg:phimodes}\n  \\phi_0(x) = \\frac{1}{3} ( \\sin(2\\pi x) + \\sin(2\\pi \\, 8 x) + \\sin(2\\pi \\, 16 x) )\n  \\end{equation}\n  on a 128 zone grid with Dirichlet boundary conditions.  This initial\n  guess has both high-frequency and low-frequency noise.  Observe that\n  the high-frequency stuff goes after only a few smoothing iterations,\n  but many iterations are needed to remove the low-frequency noise.\n  You should see something like Figure~\\ref{fig:smooth}.\n}\n\\end{exercise}\n\nFigure~\\ref{fig:mg:smooth} illustrates this behavior.  We are solving\nthe Laplace equation, $\\phi^{\\prime\\prime} = 0$ in 1-d on $[0,1]$ with\nhomogeneous Dirichlet boundary conditions.  The solution is simply\n$\\phi = 0$.  We use Eq.~\\ref{eq:mg:phimodes} as an initial guess---this is a superposition\nof 3 different modes.  We see that the after just a few iterations, the \nshortest wavelength mode, $\\sin(2\\pi 16x)$ is no longer apparent in the error,\nand just the two longer wavelength modes dominate.  By 100 iterations,\nthe error appears to be only due to the longest wavelength mode, $\\sin(2\\pi x)$.\nEven after 1000 iterations, we still see this mode in the error.  This \ndemonstrates that the longest wavelength modes in the error take the\nlongest to smooth away.\n\n\\section{Multigrid}\n\nThe text {\\em A Multigrid Tutorial}~\\cite{multigridtutorial} provides\nan excellent introduction to the mechanics of multigrid.  The\ndiscussion here differs mainly in that we are dealing with\ncell-centered/finite-volume data.  We already saw that the treatment\nof boundary conditions is more complicated because we do not have a\npoint directly on the boundary.  The other complication comes in\ntransferring the data from a fine grid to a coarser grid, and back.\nBefore we discuss the multigrid technique, we'll understand how\nthe operations between grids work.\n\n\n\\subsection{Prolongation and restriction on cell-centered grids}\n\nMultigrid relies on transferring the problem up and down a hierarchy of\ngrids. \nThe process of moving the data from the fine grid to the coarse grid\nis called {\\em restriction}.  The reverse process, moving data from\nthe coarse grid to the fine grid is called {\\em prolongation}.  If the\ndata on our grid is a conserved quantity, we want restriction and prolongation to\nconserve the amount of stuff when transitioning data between the fine and coarse grids.\n\n\n\\subsubsection{1-d}\n\nConsider a 1-d finite-volume/cell-centered finite-difference grid\nshown in Figure~\\ref{fig:mg:1dgrid-prolong}---we see a fine grid and the\ncorresponding coarser grid.\n%\n% this figure can be created by figures/multigrid/fvrestrict.py\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{fvrestrict}\n\\caption[The geometry for 1-d\n  prolongation and restriction]{\\label{fig:mg:1dgrid-prolong} A fine grid and\n  corresponding underlying coarse grid.}\n\\end{figure}\n%\nIf $\\phi$ represents a density, then\nconservation requires:\n\\begin{equation}\n\\phi^c_j = \\frac{1}{\\dx^c} (\\dx^f \\phi^f_i + \\dx^f \\phi^f_{i+1} )\n\\end{equation}\nor, for a jump in 2 in resolution ($\\dx^c = 2 \\dx^f$), \n\\begin{equation}\n\\phi^c_j = \\frac{1}{2} (\\phi^f_i + \\phi^f_{i+1} )\n\\end{equation}\nThis latter form appears as a simple average to the interface of the\ntwo fine cells / center of the corresponding coarse cell.\n\nThe simplest type of \nprolongation is simply {\\em direct injection}:\n\\begin{align}\n\\phi^f_i &= \\phi^c_j \\\\\n\\phi^f_{i+1} &= \\phi^c_j \n\\end{align}\nA higher-order method is to do linear reconstruction of the coarse\ndata and then average under the profile, e.g., \n\\begin{equation}\n\\phi(x) = \\frac{\\phi^c_{j+1} - \\phi^c_{j-1}}{2\\dx^c} (x - x_j^c) + \\phi_j^c\n\\end{equation}\n\nTo second-order, we can find the values of $\\phi^f_i$ and\n$\\phi^f_{i+1}$  by evaluating $\\phi(x)$ at the $x$-coordinate corresponding to their\ncell-centers,\n\\begin{align}\nx^f_i &= x^c_j - \\frac{\\dx^c}{4} \\\\\nx^f_{i+1} &= x^c_j + \\frac{\\dx^c}{4}\n\\end{align}\ngiving\n\\begin{align}\n\\phi^f_i     &= \\phi^c_j - \\frac{1}{8} (\\phi^c_{j+1} - \\phi^c_{j-1}) \\\\\n\\phi^f_{i+1} &= \\phi^c_j + \\frac{1}{8} (\\phi^c_{j+1} - \\phi^c_{j-1}) \n\\end{align}\nNotice that this is conservative, since $\\dx^f (\\phi^f_i + \\phi^f_{i+1}) = \\dx^c \\phi^c_j$.\n\n\\subsubsection{2-d}\n\n% this figure can be created by figures/multigrid/2dgrid-mg.py\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{2dgrid-prolong}\n\\caption[The geometry for 2-d\n  prolongation and restriction]{\\label{fig:2dgrid-prolong} Four fine cells and the\n  underlying coarse grid.  For prolongation, the fine cells in red are\n  initialized from a coarse parent.  The gray coarse cells are used in\n  the reconstruction of the coarse data.  For restriction, the fine\n  cells are averaged to the underlying coarse cell.}\n\\end{figure}\n\n\nRestriction from the fine grid to the coarse grid is straightforward.\nSince the fine cells are perfectly enclosed by a single coarse cell,\nwe simply average:\n\\begin{equation}\n\\phi_{i,j}^c = \\frac{1}{4} ( \\phi_{--}^f + \\phi_{+-}^f +\n                             \\phi_{-+}^f + \\phi_{++}^f )\n\\end{equation}\n\nProlongation requires us to reconstruct the coarse data and use\nthis reconstruction to determine what the fine cell values are.  For\ninstance, a linear reconstruction of the coarse data in $x$ and $y$ is:\n\\begin{equation}\n\\phi(x,y) = \\frac{m_x}{\\dx^c} (x - x_i^c) + \n            \\frac{m_y}{\\dy^c} (y - y_j^c) + \\phi_{i,j}^c\n\\end{equation}\nwith slopes:\n\\begin{eqnarray}\nm_x &=& \\frac{1}{2}({\\phi_{i+1,j}^c - \\phi_{i-1,j}^c}) \\\\\nm_y &=& \\frac{1}{2}({\\phi_{i,j+1}^c - \\phi_{i,j-1}^c})\n\\end{eqnarray}\n%\nWhen averaged over the coarse cell, $\\phi(x,y)$ recovers the average,\n$\\phi_{i,j}^c$ in that cell (this means that our interpolant is\nconservative).  We can evaluate the value in the fine cells by\nevaluating $\\phi(x,y)$ at the center of the fine cells,\n\\begin{eqnarray}\nx_\\pm^f &=& x_i^c \\pm \\frac{\\dx^c}{4} \\\\\ny_\\pm^f &=& y_j^c \\pm \\frac{\\dy^c}{4} \\\\\n\\end{eqnarray}\nThis gives\n\\begin{equation}\n\\phi_{\\pm\\pm}^f = \\phi_{i,j}^c \\pm \\frac{1}{4}m_x \\pm \\frac{1}{4}m_y\n\\end{equation}\n(Note: you would get the same expression if you averaged $\\phi(x,y)$ over\nthe fine cell.)\n\nThere are other options for prolongation and restriction, both of\nhigher and lower order accuracy.  However, the methods above seem to\nwork well.\n\n\n\\subsection{Multigrid cycles}\n\nThe basic idea of\nmultigrid\\footnote{ In these discussions, we use {\\em multigrid} to\n  mean {\\em geometric multigrid}, where the coarsening is done to the\n  grid geometry directly.  The alternative is {\\em algebraic\n    multigrid}, where it is the structure of the matrix in the linear\n  system itself that is coarsened.}  is to smooth a little on the\ncurrent grid solving $L\\phi = f$, compute the residual, $r$, then {\\em\n  restrict} $r$ to a coarser grid and smooth on that grid solving $Le\n= r$, restrict again, $\\ldots$.  Once you reach a sufficiently coarse\ngrid, the problem solved exactly.  Then the data is moved up to the\nfiner grids, a process called {\\em prolongation}.  The error on the\ncoarse grid, $e$, is prolonged to the finer grid.  This error is then\nused to correct the solution on the finer grid, some smoothing is\ndone, and then the data is prolonged up again.\n\nNote: on the coarse grids, you are not solving the original system,\nbut rather an error equation.  If the boundary conditions in the\noriginal system are inhomogeneous, the boundary conditions for the\nerror equations are now homogeneous.  This must be understood by\nany ghost cell filling routines.\n\nThere are many different forms of the multigrid process.  The simplest \nis called the {\\em V-cycle}.  Here you start of the fine grid, restrict\ndown to the coarsest, solve, and then prolong back up to the finest. \nThe flow looks like a `V'.  You continue with additional V-cycles\nuntil the residual error is smaller than your tolerance.\n\n\n\n\\subsection{Bottom solver}\n\n% this figure can be created with figures/multigrid/mgtower.py\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{mgtower}\n\\caption[A multigrid hierarchy]{\\label{fig:mgtower} Illustration of\n  the hierarchy of grids leading to the coarsest 2-zone grid (in\n  one-dimension).  Each grid has a single ghost cell to accommodate\n  boundary conditions.}\n\\end{figure}\n\nOnce the grid is sufficiently coarse, the linear system is small\nenough to be solved directly.  This is the bottom solver operation.\nIn the most ideal case, where the finest grid is some power of 2, $N_x\n= N_y = 2^n$, then the multigrid procedure can continue down until a\n$2\\times 2$ grid is created (Figure~\\ref{fig:mgtower} illustrates this idea\nfor a one-dimensional grid).  This is the coarsest grid upon which one\ncan still impose boundary conditions.  With this small grid, just\ndoing additional smoothing is sufficient enough to `solve' the\nproblem.  No fancy bottom solver is needed.\n\nFor a general rectangular grid or one that is not a power of 2, the\ncoarsest grid will likely be larger.  For the general case, a linear\nsystem solver like conjugate gradient (or a variant) is used on the\ncoarsest grid.\n\n\\subsection{Boundary conditions throughout the hierarchy}\n\nThe general inhomogeneous boundary conditions from\nEqs.~\\ref{eq:bc_inhomo_dir} and \\ref{eq:bc_inhomo_neum} apply to the\nfinest level.  But because we are solving the residual equation of the\ncoarsest levels in the multigrid hierarchy, the boundary conditions on\n$Le = r$ are all homogeneous (but of the same type, Dirichlet,\nNeumann, or periodic, as the fine level).\n\nImplementing these boundary conditions in your multigrid solver means\nthat you will have separate actions for the fine level (where\ninhomogeneous boundaries may apply) and the coarser levels (where you\nwill always be homogeneous).\n\nAn alternate way to enforce boundary conditions is via {\\em boundary\n  charges}.  For inhomogeneous boundary conditions, boundary charges\ncan be used to convert the BCs to homogeneous BCs.  This has the\nadvantage of allowing the ghost cell filling routines only deal with\nthe homogeneous case.\n\nConsider the one-dimensional Poisson equation, near the left boundary\nour discretized equation appears as:\n\\begin{equation}\n\\frac{\\phi_\\mathrm{lo-1} - 2\\phi_\\mathrm{lo} + \\phi_\\mathrm{lo+1}}{\\dx^2}\n = f_\\mathrm{lo}\n\\end{equation}\nInhomogeneous BCs at the left boundary would give the condition:\n\\begin{equation}\n\\phi_\\mathrm{lo-1} = 2 \\phi_l - \\phi_\\mathrm{lo}\n\\end{equation}\nSubstituting this into the discrete equation, we have:\n\\begin{equation}\n\\frac{2 \\phi_l - \\phi_\\mathrm{lo} - 2\\phi_\\mathrm{lo} + \\phi_\\mathrm{lo+1}}{\\dx^2}\n = f_\\mathrm{lo}\n\\end{equation}\nBringing the boundary condition value over to the RHS, we see\n\\begin{equation}\n\\frac{- 3\\phi_\\mathrm{lo} + \\phi_\\mathrm{lo+1}}{\\dx^2}\n = f_\\mathrm{lo} - \\frac{2\\phi_l}{\\dx^2}\n\\end{equation}\nNow the left side looks precisely like the differenced Poisson equation\nwith homogeneous Dirichlet BCs.  The RHS has an additional `charge' that\ncaptures the boundary value.  By modifying the source term, $f$, in the\nmultigrid solver to include this charge, we can use the homogeneous \nghost cell filling routines throughout the multigrid algorithm.\nThis technique is discussed a bit in~\\cite{colellanotes}.\n\nNote that the form of the boundary charge will depend on the form of the\nelliptic equation---the expressions derived above apply only for\n$\\nabla^2 \\phi = f$.\n\n\n\n\n\\subsection{Stopping criteria}\n\nRepeated V-cycles are done until:\n\\begin{equation}\n\\| r \\| < \\epsilon \\|f\\|\n\\end{equation}\non the finest grid, for some user-input tolerance, $\\epsilon$.  Here,\n$\\|f\\|$ is called the {\\em source norm}.  If $\\|f\\| = 0$, then we stop\nwhen\n\\begin{equation}\n\\| r \\| < \\epsilon \n\\end{equation}\nPicking the tolerance $\\epsilon$ is sometimes problem-dependent, and\ngenerally speaking, a problem with a large number of zones will require\na looser tolerance.\n\nThe general rule-of-thumb is that each V-cycle should reduce your\nresidual by about 1 order of magnitude.  It is important that your\nbottom solver solves the coarse problem to a tolerance of $10^{-3}$ or\n$10^{-4}$ in order for the solver to converge.  Figure~\\ref{fig:mgerror}\nshows the true and residual errors for $\\phi_{xx} = \\sin(x)$ as a function\nof V-cycle number, illustrating the expected performance.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.85\\linewidth]{mg_error_vs_cycle}\n\\caption[Error in solution as a function of multigrid V-cycle\n  number]{\\label{fig:mgerror} Error in the multigrid solution to our\n  model problem ($\\phi_{xx} = \\sin(x)$) as a function of V-cycle.  We\n  see that the true error, $\\|e\\|$ stalls at truncation error while\n  the residual error, $\\|r\\|$ reaches roundoff error, the same\n  behavior as seen with smoothing alone (as expected). \\\\\n  \\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/multigrid/mg_test.py}{mg\\_test.py}}}\n\\end{figure}\n\nThe overall convergence of the multigrid algorithm is limited by the\ndiscretization of the Laplacian operator used and the implementation\nof the boundary conditions.  Figure~\\ref{fig:mg:convergence} shows\nthe error in the solution as the number of zones is increased---demonstrating\nsecond-order convergence for our implementation.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.85\\linewidth]{mg-converge}\n\\caption[Convergence of the multigrid\n  algorithm]{\\label{fig:mg:convergence} Convergence of the multigrid\n  algorithm. \\\\ \n  \\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/multigrid/mg_converge.py}{mg\\_converge.py}}}\n\\end{figure}\n\n\n\\section{Solvability}\n\\label{sec:multigrid:solvability}\n\nFor $\\nabla^2 \\phi = f$ with periodic or\nNeumann boundaries all around, the sum of $f$ must equal $0$\notherwise the solution will not converge.  Instead, we will simply\nfind the solution increase each V-cycle.  This is seen as follows:\n\\begin{equation}\n  \\int_\\Omega f d\\Omega = \\int_\\Omega \\nabla^2 \\phi d\\Omega =\n  \\int_{\\partial \\Omega} \\nabla \\phi \\cdot n dS = 0\n\\end{equation}\n\nFor all homogeneous Neumann boundaries, we have $\\nabla \\phi \\cdot dS\n= 0$ by construction, so that integral is zero, requiring that the\nsource integrate to zero.  If the Neumann boundaries are inhomogeneous,\nthere is still a solvability condition on $f$ based on the sum on\nthe boundary values.\n\nA simple example of solvability is:\n\\begin{equation}\n\\phi^{\\prime\\prime} = 0\n\\end{equation}\non $[a, b]$ with \n\\begin{align}\n\\phi^\\prime(a) &= A \\\\\n\\phi^\\prime(b) &= B\n\\end{align}\nWe can integrate $\\phi^{\\prime\\prime} = 0$ to get $\\phi(x) = \\alpha x\n+ \\beta$ where $\\alpha$ and $\\beta$ are integration constants.  But\nnote that this is just a straight line, with a single slope, $\\alpha$,\nso it is not possible to specify two unique slopes, $A$ and $B$ at the\nboundary unless there is a non-zero source term.\n\nFor all periodic boundaries, we have $\\nabla \\phi\n|_\\mathrm{left} = -\\nabla \\phi |_\\mathrm{right}$ on the left and right\nboundaries by definition of the periodicity (and similarly for the top\nand bottom).  Again this implies that $f$ must integrate to zero.\n\nSometimes, with periodic boundary conditions all around, you need to\nenforce that $f$ integrate to zero numerically to test convergence.\nThis is discussed in \\S~\\ref{sec:lm:periodicbcs}.\n\n\n\n\\section{Going Further}\n\n\\label{sec:multigrid:other}\n\n\\subsection{Red-black Ordering}\n\nWhen using domain decomposition to spread the problem across parallel\nprocessors, the smoothing is often done as {\\em red-black\n  Gauss-Seidel}.  In this ordering, you imagine the grid to be a\ncheckerboard (see Figure~\\ref{fig:rb}).  In the first Gauss-Seidel\npass you update the red squares and in the second, the black squares.\nThe advantage is that when updating the red, you can be sure that none\nof the zones you depend on (the neighboring black zones) will change.\nThis makes the decomposition parallel.  Note: this works for the\nstandard 5-point Laplacian.  If you are doing some other operator with\na different stencil, then this decomposition may no longer hold.\n\n% this figure can be created by figures/multigrid/red_black.py\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.6\\linewidth]{rb}\n  \\caption[Red-black ordering of zones]\n{\\label{fig:rb} The red-black ordering of zones.}\n\\end{figure}\n\n\n\n\n\n\\subsection{More General Elliptic Equations}\n\nThe most general {\\em second-order} elliptic equation takes the form:\n\\begin{equation}\n  \\alpha \\phi + \\nabla \\cdot (\\beta \\nabla \\phi) +\n  \\gamma \\cdot \\nabla \\phi + \\nabla \\cdot (\\zeta \\phi) = f\n\\end{equation}\nHere, $\\gamma$ and $\\zeta$ are vectors.  Solving a general elliptic\nequation of this form can be accomplished with multigrid using the\nsame basic ideas here.  The main change is that the smoothing\nalgorithm and the construction of the residual will need to discretize\nthe more general operator, and these coefficients will need to\nbe restricted to the coarser grids (some on edges).  This is explored in\n\\S~\\ref{sec:lm:vcelliptic} for a variable-coefficient Poisson equation:\n \\begin{equation}\n \\nabla \\cdot (\\beta \\nabla \\phi) = f\n \\end{equation}\n\\ifdefined\\debugmode\nand in \\S~\\ref{sec:rad:generalelliptic} for an equation with $\\zeta = 0$.\n\\fi\n"
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    "content": "\n\\section{Integrating Multiphysics}\n\nConsider a system whose evolution depends on several different\nphysical processes, represented by the operators $A$, $D$,\n$R$ (advection, diffusion, and reactions, respectively).\n\\begin{equation}\n\\phi_t = -A(\\phi) + D(\\phi) + R(\\phi)\n\\end{equation}\nOne way to solve this system is to discretize each of the operators in\nspace.  For instance the discrete advection operator, $[A(\\phi)]_i$\nmight use the ideas on piecewise linear reconstruction techniques\ndiscussed in chapter~\\ref{ch:advection}, $[D(\\phi)]_i$ can use the\ndiscrete form of the Laplacian from chapter~\\ref{ch:diffusion}, and\n$[R(\\phi)]_i$ may be an algebraic relation.  This leaves us with an ordinary differential equation for the time-evolution of $\\phi$,\n\\begin{equation}\n\\frac{d\\phi_i}{dt} = -[A(\\phi)]_i + [D(\\phi)]_i + [R(\\phi)]_i\n\\end{equation}\nwhich can be solve using standard ODE techniques.  This is the {\\em\n  method of lines} technique we saw with advection\n(\\S~\\ref{adv:sec:mol_2d}) and compressible hydrodynamics\n(\\S~\\ref{sec:comp:mol}), and can be a powerful technique to solve PDEs\nor systems of PDEs with multiple physics operators\n\nA difficulty arises if these processes each have different timescales\nassociated with them.  For instance, reactions may be vigorous and\nrequire a small timestep to accurately capture the changes, but the\nadvection is slow.  Or, recall that the timestep limiter for explicit\ndiffusion scales as $\\Delta x^2$ while explicit advection scales as\n$\\Delta x$, so these processes could demand very different timescale\nfor evolution.  Therefore, we don't want to use the same timestep for\nall the processes, since that will needlessly make things\ncomputationally expensive.  {\\em Operator splitting} solves for the\neffects of each operator separately, using whichever timestep (and\ntime-discretization, e.g., explicit or implicit) is most suited to the\noperation.  The result of one operation is used as the input to the\nnext\\footnote{This directly parallels the dimensional splitting\n  approach we saw for advection in \\S~\\ref{adv:sec:dimensionalsplit}}.\nThe downside of this approach is that the operations may not be well\ncoupled.\n\n\n\\section{Ex: diffusion-reaction}\n\\label{sec:multiphys:diffreact}\n\nConsider a diffusion-reaction equation:\n\\begin{equation}\n\\phi_t = \\kappa \\phi_{xx} + \\frac{1}{\\tau} R(\\phi)\n\\end{equation}\nThis can be thought of as a simple model for a combustion flame, and\ncan propagate a front.  It is often the case that the reactions are\nstiff, and require a smaller timestep then the diffusion part.  In\nfact, we may want to use an implicit integration method designed for\nstiff ODEs for the reaction part, but use a standard explicit method\nfor the diffusion part.  This requires operator splitting.\n\nWe can use Strang splitting \\cite{strang} to make the integration\nsecond-order accurate overall:\n\\begin{equation}\n\\phi^{n+1} = R_{\\Delta t/2} D_{\\Delta t} R_{\\Delta t/2} \\phi^n\n\\end{equation}\nwhere $R_{\\Delta t/2}$ represents reacting for a step of $\\Delta t/2$\nand $D_{\\Delta t}$ represents diffusing for a step of $\\Delta t$.  In\neach case, these operators act as if the other were not present, but\nthey see the effect of the previous operation on the input $\\phi$.  \nNote that no explicit source terms describing one process appear in the other\nprocess's update.  The procedure for updating appears as:\n\\begin{enumerate}\n\\item {\\em Evolve reaction ODE system for $\\Delta t/2$}\n\n  Define $\\phi^\\star$ as the the solution to the ODE:\n   \\begin{equation}\n     \\frac{d\\phi}{dt} = \\frac{1}{\\tau} R(\\phi), \\quad \n   \\phi(t^n) = \\phi^n, \\quad t \\in [t^n, t^{n+\\myhalf}]\n   \\end{equation}\n\n\\item {\\em Solve the diffusion equation for $\\Delta t$ with an\n           implicit Crank-Nicolson discretization}\n   %\n   \\begin{equation}\n     \\frac{\\phi^{\\star\\star} - \\phi^\\star}{\\Delta t} =\n      \\frac{1}{2} (D(\\phi^\\star) + D(\\phi^{\\star\\star}))\n   \\end{equation}\n\n   Note that the starting point is $\\phi^\\star$.\n\n\\item {\\em Evolve reaction ODE system for $\\Delta t/2$}\n\n  Define $\\phi^{n+1}$ as the the solution to the ODE:\n   \\begin{equation}\n     \\text{define } \\phi^{n+1}: \\, \\frac{d\\phi}{dt} = \\frac{1}{\\tau} R(\\phi), \\quad \n   \\phi(t^{n+\\myhalf}) = \\phi^{\\star\\star}, \\quad t \\in [t^{n+\\myhalf}, t^{n+1}]\n   \\end{equation}\n\n\\end{enumerate}\n\nConsider a simple reaction source\n\\begin{equation}\n  \\label{eq:multiphysics:react}\n  R(\\phi) = \\frac{1}{4} \\phi (1 - \\phi)\n\\end{equation}\nThis is called a KPP reaction source.  Here $\\phi$ can be thought of\nas a progress variable that varies between pure ash ($\\phi = 0$) and\npure fuel ($\\phi = 1$).  Figure~\\ref{fig:diffreact} shows the solution\nto our diffusion-reaction equation with 256 zones, $\\kappa = 0.1$,\n$\\tau = 1.0$ at several times.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=5.0in]{flame_seq}\n\\caption[Solution to the diffusion-reaction equation]\n  {\\label{fig:diffreact} Solution to the diffusion-reaction equation\n   with 256 zones, and $\\kappa = 0.1$, $\\tau = 1.0$.  The lines shown\n   are spaced 8.0 time-units apart.  We see the initial smoothed tophat\n   profile giving rise to a traveling front. \\\\\n   \\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/multiphysics/diffusion-reaction.py}{diffusion-reaction.py}}\n   }\n\\end{figure}\n\n The solution in this case is a wave with speed $S = \\sqrt{\\kappa/\\tau}$\n and thickness $\\delta = \\sqrt{\\kappa \\tau}$ (see \\cite{vladimirova2006} for\n some details of this system).\n\n\\begin{exercise}[Diffusion-reaction system]\n Solve the the difusion reaction equation with the source given in \n Eq.~\\ref{eq:multiphyiscs:react}.  Note that you should begin with\n some smooth initial conditions---if they are too sharp than the \n C-N discretization will cause jagged features to appear.  \n Compare your results to Figure~\\ref{fig:diffreact}.\n\\end{exercise}\n\n\n\\section{Ex: advection-diffusion}\n\n\\label{ch:multiphysics:sec:adburgers}\n\nThe viscous Burgers' equation appears as:\n\\begin{equation}\nu_t + u u_x = \\epsilon u_{xx}\n\\end{equation}\nThis admits shocks and rarefactions just like the inviscid form we\nstudied in Chapter~\\ref{ch:burgers}, but now the viscosity can act to\nsmooth out the shock---instead of being infinitely thin, it will have\na physical width.\n\nAs we saw earlier, there are efficient, accurate methods for handling\nthe advective parts explicitly, but for diffusion, we often want to \nsolve it implicitly.  We can split the solution up, but couple the \ntwo processes together to make a method that is overall second-order\naccurate in time.  We write our equation as:\n\\begin{equation}\nu_t + A(u) = D(u)\n\\end{equation}\nwith $A(u) = [\\frac{1}{2} u^2]_x$ and $D(u) = \\epsilon u_{xx}$.  Then our update \nappears in two steps.\n\\begin{enumerate}\n\\item {\\em Find the advective update over the timestep}:\n   We use an approximation of the diffusion term at time-level $n$, $D(u^n)$\n   as a source in the construction of the interface states for the \n   advective part.  Once the interface states, $u_{i+\\myhalf}^{n+\\myhalf}$ are\n   known, we construct the advective update term as:\n   \\begin{equation}\n   A_i^{n+\\myhalf} = \n     \\frac{\\left [ \\frac{1}{2} \\left (u_{i+\\myhalf}^{n+\\myhalf}\\right)^2\\right ] -\n           \\left [ \\frac{1}{2} \\left (u_{i-\\myhalf}^{n+\\myhalf}\\right)^2\\right ]}\n          {\\Delta x}\n    \\end{equation}\n\n\\item {\\em Solve the diffusion equation with the advective source}:\n    We use a Crank-Nicolson discretization of the diffusion part of \n    our equation, with the advective update term appearing as a source.\n    \\begin{equation}\n    \\frac{u^{n+1} - u^n}{\\Delta t} = \n        \\frac{1}{2}D(u^n) + \\frac{1}{2}D(u^{n+1}) - A^{n+\\myhalf}\n    \\end{equation}\n    \n    This is a linear system that can be solved as a tridiagonal matrix\n    or with multigrid.  The result of this step is that $u^{n+1}$ is\n    updated with both the advection and diffusion terms.\n\n\\end{enumerate}\n\nBecause the diffusion is done implicitly, the timestep constraint (for\nstability) for this solve is due to the advective portion only.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=5in]{burgersvisc}\n\\caption[Viscous Burgers' equation solution]\n  {\\label{fig:viscburger} Solution to the viscous Burgers' equation\n  with a variety of different viscosities.  The initial conditions was\n  a single wavelength of a sine wave for $x \\in [1/3,2/3]$, and $u = 1$\n  otherwise. \\\\\n  \\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/multiphysics/burgersvisc.py}{burgersvisc.py}}}\n\\end{figure}\n\nFor step 1, the addition of the explicit diffusion source requires\na small change to the method we used to predict the interface states.\n\n\\begin{eqnarray}\nu^{n+1}_{i+\\myhalf,L} &=& u^n_i + \\frac{\\Delta x}{2} \\frac{\\partial u}{\\partial x}\n                        + \\frac{\\Delta t}{2} \\frac{\\partial u}{\\partial t} + \\ldots \\\\\n                &=& u^n_i + \\frac{\\Delta x}{2} \\frac{\\partial u}{\\partial x}\n                        + \\frac{\\Delta t}{2} \\left (-u_i \\frac{\\partial u}{\\partial x} + D(u^n_i) \\right ) + \\ldots \\\\\n                &=& u^n_i + \\frac{\\Delta x}{2} \\left ( 1 - \\frac{\\Delta t}{\\Delta x}u_i \\right ) \\frac{\\partial u}{\\partial x} + {\\color{red} \\frac{\\Delta t}{2} D(u^n) } + \\ldots\n\\end{eqnarray}\nhere the source term (shown in red) incorporates the effects of the\ndiffusion on the prediction of the states for advection.  This entered\ninto our states when we replaced $\\partial u/\\partial t$ with our PDE\nequation.  The spatial derivative, $\\partial u/\\partial x$ is replaced\nby a monotonized difference, and the method then proceeds as with the\nregular Burgers' equation.  The Riemann problem is unchanged from the\ninviscid case.\n\nFigure~\\ref{fig:viscburger} shows the solution of the viscous Burgers'\nequation for shock initial conditions with different amounts of\nviscosity.  Notice that the effect of the viscosity is to smooth the\nshock profile, but the shock position itself agrees between the cases.\n\n\n\\subsection{Convergence without an analytic solution}\n\nAssessing the convergence of this is more difficult than the tests we\nlooked at before, since there is no analytic solution to compare to.\nIn this case, we can compare a very high-resolution solution and take\nthis to be the {\\em right} answer / benchmark solution, and then\ncompare coarser simulations to a coarsened version of the benchmark solution.\nThis is done in Figure~\\ref{fig:multiphysics:burgersconverge}.  The\nfine zones of the benchmark are averaged into a corresponding coarse grid zone\nto produce a coarsened representation of the benchmark, and the norm \nof the error with the coarse simulation is computed.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=5.0in]{burgersvisc_converge}\n\\caption[Convergence of the viscous Burgers' equation]\n  {\\label{fig:multiphysics:burgersconverge} Convergence of the Burgers'\n    solution with $\\epsilon = 0.005$.  To test convergence a run with 4096\n    zones was done as a benchmark solution.  We see second-order\n    convergence for the higher-resolution runs in this sample.\\\\\n   \\hydroexdoit{\\href{https://github.com/zingale/hydro_examples/blob/master/multiphysics/burgersvisc_converge.py}{burgersvisc\\_converge.py}}\n   }\n\\end{figure}\n"
  },
  {
    "path": "mysymbols.tex",
    "content": "\\newcommand{\\cfl}{{\\mathcal{C}}}\n\n% velocity\n\\newcommand{\\Ub}{{\\bf U}}\n\\newcommand{\\gb}{{\\bf g}}\n\\newcommand{\\Gb}{{\\bf G}}\n\\newcommand{\\Hb}{{\\bf H}}\n\n% systems\n\\newcommand{\\Ab}{{\\bf A}}\n\\newcommand{\\Uc}{{\\,\\boldsymbol{\\mathcal{U}}}}\n\\newcommand{\\Rc}{{\\,\\boldsymbol{\\mathcal{R}}}}\n\\newcommand{\\Fb}{{\\bf F}}\n\\newcommand{\\qb}{{\\bf q}}\n\n% reactions\n\\newcommand{\\omegadot}{{\\dot{\\omega}}}\n\\newcommand{\\Hnuc}{{H_\\mathrm{nuc}}}\n\n% diffusion\n\\newcommand{\\kth}{{k_\\mathrm{th}}}\n\n% eigenvectors\n\\newcommand{\\evm}{{(-)}}\n\\newcommand{\\evz}{{(\\circ)}}\n\\newcommand{\\evzs}[1]{{(\\circ,#1)}}\n\\newcommand{\\evp}{{(+)}}\n\\newcommand{\\enu}{{(\\nu)}}\n\n\\newcommand{\\lb}{{\\bf l}}\n\\newcommand{\\rb}{{\\bf r}}\n\\newcommand{\\Lb}{{\\bf L}}\n\\newcommand{\\Rb}{{\\bf R}}\n\\newcommand{\\Lambdab}{{\\boldsymbol{\\Lambda}}}\n\n\\newcommand{\\wb}{{\\bf w}}\n\n% directional exponents\n\\newcommand{\\exx}{{(x)}}\n\\newcommand{\\eyy}{{(y)}}\n\n% Riemann and compressible\n\\newcommand{\\riemann}{{\\mathcal{R}}}\n\\newcommand{\\Ic}{{\\mathcal{I}}}\n\n% derivative shortcuts\n\\newcommand{\\ddx}[1]{{\\frac{{\\partial#1}}{\\partial x}}}\n\\newcommand{\\ddy}[1]{{\\frac{{\\partial#1}}{\\partial y}}}\n\\newcommand{\\ddz}[1]{{\\frac{{\\partial#1}}{\\partial z}}}\n\\newcommand{\\ddr}[1]{{\\frac{{\\partial#1}}{\\partial r}}}\n\\newcommand{\\ddt}[1]{{\\frac{{\\partial#1}}{\\partial t}}}\n\\newcommand{\\DDt}[1]{{\\frac{{D#1}}{D t}}}\n\n% F-V shortcuts\n\\newcommand{\\favg}[1]{{\\langle #1\\rangle}}\n\n\\newcommand{\\dx}{\\Delta x}\n\\newcommand{\\dy}{\\Delta y}\n\\newcommand{\\dt}{\\Delta t}\n\n% limited slopes\n\\newcommand{\\Dux}{\\overline{\\Delta u}^{(x)}}\n\\newcommand{\\Duy}{\\overline{\\Delta u}^{(y)}}\n\n\\newcommand{\\Dvx}{\\overline{\\Delta v}^{(x)}}\n\\newcommand{\\Dvy}{\\overline{\\Delta v}^{(y)}}\n\n\n\\newcommand{\\imag}{I}\n"
  },
  {
    "path": "pde-classes/pde-classes.tex",
    "content": "\\section{Introduction}\n\nPartial differential equations (PDEs) are usually grouped into one of\nthree different classes: {\\em hyperbolic}, {\\em parabolic}, or {\\em\n  elliptic}.  You can find the precise mathematical definition of\nthese classifications in most books on PDEs, but this formal\ndefinition is not very intuitive or useful.  Instead, it is helpful to\nlook at some prototypical examples of each type of PDE.\n\nWhen we are solving multiphysics problems, we will see that our \nsystem of PDEs spans these different types.  Nevertheless, we will\nlook at solutions methods for each type separately first, and then\nuse what we learn to solve more complex systems of equations.\n\n\\section{Hyperbolic PDEs}\n\nThe canonical hyperbolic PDE is the wave equation:\n\\begin{equation}\n  \\frac{\\partial^2 \\phi}{\\partial t^2} = c^2 \\frac{\\partial^2 \\phi}{\\partial x^2}\n\\end{equation}   \nThe general solution to this is traveling waves in either direction:\n\\begin{equation}\n\\label{eq:pde:wavesol}\n  \\phi(x,t) = \\alpha f_0(x - ct) + \\beta g_0(x + ct)\n\\end{equation}\nHere $f_0$ and $g_0$ are set by the initial\nconditions, and the solution propagates $f_0$ to the right and $g_0$ to\nthe left at a speed $c$.\n\n\\begin{exercise}[Wave equation]\n{Show by substitution that Eq.~\\ref{eq:pde:wavesol} is a solution\nto the wave equation}\n\\end{exercise}\n\nA simple first-order hyperbolic PDE is the linear advection equation:\n\\begin{equation}\na_t + u a_x = 0\n\\end{equation}\nThis simply propagates any initial profile to the right at the speed\n$u$.  We will use linear advection as our model equation for numerical\nmethods for hyperbolic PDEs.\n\nA system of first-order hyperbolic PDEs takes the form:\n\\begin{equation}\n{\\bf a}_t + \\Ab {\\bf a}_x = 0\n\\end{equation}\nwhere ${\\bf a} = (a_0, a_1, \\ldots a_{N-1})^\\intercal$ and $\\Ab$ is a matrix.\nThis system is hyperbolic if the eigenvalues of $A$ are real (see\n\\cite{leveque:2002} for an excellent introduction).\n\nAn important concept for hyperbolic PDEs are {\\em\n  characteristics}---these are curves in a space-time diagram along\nwhich the solution is constant.  Associated with these curves is a\nspeed---this is the wave speed at which information on how the\nsolution changes is communicated.  For a linear PDE (or system of\nPDEs), these will tell you everything you need to know about the\nsolution.\n\n\n\\section{Elliptic PDEs}\n\nThe canonical elliptic PDE is the Poisson equation:\n\\begin{equation}\n  \\nabla^2 \\phi = f\n\\end{equation}\nNote that there is no time-variable here.  This is a pure boundary\nvalue problem.  The solution, $\\phi$ is determined completely by the\nsource, $f$, and the boundary conditions.\n\nIn contrast to the hyperbolic case, there is no propagation of\ninformation here.  The potential, $\\phi$, is known instantaneously\neverywhere in the domain.  For astrophysical flows, this commonly\narises as the Poisson equation describing the gravitational potential.\n\n\n\n\\section{Parabolic PDEs}\n\nThe canonical parabolic PDE is the heat equation:\n\\begin{equation}\n\\label{eq:pde:heat}\n  \\frac{\\partial \\phi}{\\partial t} = k \\frac{\\partial^2 f}{\\partial x^2}\n\\end{equation}\nThis has aspects of both hyperbolic and elliptic PDEs.\n\nThe heat equation represents diffusion---an initially sharp feature\nwill spread out into a smoother profile on a timescale that depends on\nthe coefficient $k$.  We'll encounter parabolic equations for thermal\ndiffusion and other types of diffusion (like species, mass), and with\nviscosity.\n\n\n\\begin{exercise}[Diffusion timescale]\n{Using dimensional analysis, estimate the characteristic timescale for\n  diffusion from Eq.~\\ref{eq:pde:heat}.}\n\\end{exercise}\n"
  },
  {
    "path": "preface/preface.tex",
    "content": "\nThis text started as a set of notes to help new students at Stony\nBrook University working on projects in computational astrophysics.\nThey focus on the common methods used in computational hydrodynamics\nfor astrophysical flows and are written at a level appropriate for\nupper-level undergraduates.  Problems integrated in the text help\ndemonstrate the core ideas.  An underlying principle is that source\ncode is provided for all the methods described here (including all the\nfigures).  This allows the reader to explore the routines themselves.\n\nThese notes are very much a work in progress, and new chapters will be\nadded with time.  The page size is formatted for easy reading\non a tablet or for 2-up printing in a landscape orientation on\nletter-sized paper.\n\nThis text is part of the Open Astrophysics Bookshelf.\nContributions to these notes are welcomed.  The \\LaTeX\\ source\nfor these notes is available online on github at: \\\\[0.25em]\n%\n\\url{https://github.com/Open-Astrophysics-Bookshelf/numerical_exercises} \\\\[0.25em]\n%\nSimply fork the notes, hack away, and submit a pull-request to add\nyour contributions.  All contributions will be acknowledged in the text.\n\n\nA PDF version of the notes is always available\nat: \\\\[0.25em]\n%\n\\url{https://open-astrophysics-bookshelf.github.io/numerical_exercises/CompHydroTutorial.pdf} \\\\[0.25em]\n%\nThese notes are updated at irregular intervals, usually when I have a\nnew student working with me, or if I am using them for a course.\n\nThe source (usually python) for all the figures is also contained in\nthe main git repo.  The line drawings of the grids are done using the\nclasses in\n\\href{https://github.com/Open-Astrophysics-Bookshelf/numerical_exercises/blob/master/grid_plot.py}{{\\tt\n    grid\\_plot.py}}.  This needs to be in your {\\tt PYTHONPATH} if you\nwish to run the scripts.\n\nThe best way to understand the methods described here is to run\nthem for yourself.  There are several sets of example codes that\ngo along with these notes:\n\n\\begin{enumerate}\n\\item \\hydroex\\ is a set of simple 1-d, standalone python scripts\n  that illustrate some of the basic solvers.  Many of the figures\n  in these notes were created using these codes---the relevant\n  script will be noted in the figure caption.\n\n  You can get this set of scripts from github at:\\\\\n  \\url{https://github.com/zingale/hydro_examples/}\n\n  References to the scripts in \\hydroex\\ are shown throughout\n  the text as: \\\\[0.5em]\n  \\hydroexdoit{scriptname} \\\\[0.5em]\n  Clicking on the name of the script will bring up the source code\n  to the script (on github) in your web browser.\n\n  More details on the codes available in \\hydroex\\ are described\n  in Appendix~\\ref{app:hydroex}.\n\n\\item\n  The \\pyro\\ code~\\cite{pyro} is a 2-d simulation code with\n  solvers for advection, diffusion, compressible and incompressible\n  hydrodynamics, as well as multigrid.  A gray flux-limited diffusion\n  radiation hydrodynamics solver is\n  in development.  \\pyro\\ is designed with clarity in mind and to make\n  experimentation easy.\n\n  You can download \\pyro\\ at: \\\\\n  \\url{https://github.com/python-hydro/pyro2/}\n\n  A brief overview of \\pyro\\ is given in Appendix~\\ref{app:pyro},\n  and more information can be found at: \\\\\n\\url{http://python-hydro.github.io/pyro2/}\n\n\\item \\hydrooned\\ is a simple one-dimensional compressible\n  hydrodynamics code that implements the piecewise parabolic method\n  from Chapter~\\ref{ch:compressible}.  It can be obtained from\n  github at:\\\\\n  \\url{https://github.com/zingale/hydro1d/}\n\n  Details on it are given in\n  Appendix~\\ref{app:hydro1d}.\n\\end{enumerate}\n\nWherever possible we try to use standardized notation for physical\nquantities, as listed in Table~\\ref{table:sym}.\n\nThese notes benefited {\\em immensely} from numerous conversations and\nan ongoing collaboration with Ann Almgren, John Bell, Andy Nonaka, \\&\nWeiqun Zhang---pretty much everything I know about projection methods\ncomes from working with them.  Discussions with Alan Calder, Sean\nCouch, Max Katz, Chris Malone, and Doug Swesty have also been\ninfluential in the presentation of these notes.\n\nIf you find errors, please e-mail me at michael.zingale@stonybrook.edu,\nor issue a pull request to the git repo noted above.\n\n\n\n\\begin{flushright}\nMichael Zingale \\\\\nStony Brook University\n\\end{flushright}\n\n\\clearpage\n\n\\input symbols/symbols\n\n\\clearpage\n\n\\section*{Authorship}\n\n\\subsection*{Primary Author}\n\nMichael Zingale (Stony Brook)\n\n\n\\subsection*{Contributions}\n\nThank you to the\nfollowing people for pointing out typos or confusing remarks in the text:\n\\begin{itemize}\n\\item Chen-Hung\n\\item Rixin Li (Arizona)\n\\item Zhi Li (Shanghai Astronomical Observatory)\n\\item Chris Malone\n\\item Sai Praneeth (Waterloo)\n\\item Donald Willcox (Stony Brook)\n\\end{itemize}\n\nMaterial on WENO schemes was contributed by Ian Hawke (Southampton).\n\nSee the git log for full details on contributions.  All contributions\nvia pull-requests will be acknowledged here.\n"
  },
  {
    "path": "public-codes/NOTES",
    "content": "Describe the features of public codes:\n\n\nAthena\n\nCastro\n\nEnzo\n\nFlash\n\nMaestro\n\npyro\n\npluto\n\n\n\n\ngive details on:\n\n -- solver\n -- griding (including subcycling?)\n -- language\n -- EOS\n -- other physics\n -- species\n -- particles\n -- visualization tools\n -- gravity\n -- other details\n -- diffusion\n -- radiation\n\n\n\n"
  },
  {
    "path": "pyro/pyro.tex",
    "content": "\\label{app:pyro}\n\n%% \\begin{center}\n%% \\includegraphics[width=0.125\\linewidth]{pyro-sm}\n%% \\end{center}\n\n\n\\section{Introduction}\n\n\\pyro\\ is a simple two-dimensional code that implements the core\nsolvers described in these notes.  It is written primarily in python,\nwith some kernels in Fortran.  Clarity of the methods is emphasized\nover performance.\n\n\n\\section{Getting \\pyro}\n\n\\pyro\\ can be downloaded from its github repository, \\url{https://github.com/python-hydro/pyro2} as:\n\\begin{verbatim}\ngit clone https://github.com/python-hydro/pyro2\n\\end{verbatim}\n\n\\pyro\\ uses the {\\tt h5py}, {\\sf matplotlib}, and {\\sf numpy} libraries.\nSeveral routines are written in Fortran and need to be compiled.  The\nscript {\\tt mk.sh} will build the packages.  This requires that {\\tt\nf2py} is installed.  By default, \\pyro\\ will use python 3.  \nThe following sequence will setup \\pyro:\n\\begin{itemize}\n\\item Set the {\\tt PYRO\\_HOME} environment variable to point to the\n  main {\\tt pyro2/} directory, e.g.,\n  \\begin{verbatim}\n  export PYRO_HOME=/path/to/pyro2\n  \\end{verbatim}\n\n\\item Set the {\\tt PYTHONPATH} environment variable to point to the\n  main {\\tt pyro2/} directory, e.g.,\n  \\begin{verbatim}\n  export PYTHONPATH=\"$PYTHONPATH:$PYRO_HOME\"\n  \\end{verbatim}\n\n\\item Build the Fortran source using the {\\tt mk.sh} script.  In the\n  {\\tt pyro2/} directory, simply do\n  \\begin{verbatim}\n  ./mk.sh\n  \\end{verbatim}\n\\end{itemize}\n\n\n\n\\section{\\pyro\\ solvers}\n\n\\pyro\\ offers the following 2-d solvers:\n\\begin{itemize}\n\\item {\\em advection}: an unsplit, second-order method for linear advection,\n  following the ideas from Chapter~\\ref{ch:advection}.\n\n\\item {\\em advection\\_rk}: an alternate advection algorithm using\n  method-of-lines time integration, again following the ideas\n  from Chapter~\\ref{ch:advection}.\n\n\\item {\\em compressible}: an unsplit, second-order compressible\n  hydrodynamics solver using the piecewise linear reconstruction\n  discussed in Chapter~\\ref{ch:compressible}.\n\n\\item {\\em compressible\\_rk}: an alternate compressible algorithm\n  using method-of-lines time integration, using the piecewise\n  linear reconstruction discussed in Chapter~\\ref{ch:compressible}.\n\n\\item {\\em diffusion}: a second-order implicit diffusion solver, based\n  on the ideas from Chapter~\\ref{ch:diffusion}.\n\n\\item {\\em incompressible}: a second-order incompressible hydrodynamics\n  solver using a cell-centered approximate projection, as discussed\n  in Chapter~\\ref{ch:incompressible}.\n\n\\item {\\em lm\\_atm}: a low-Mach number hydrodynamics solver for\n  atmospheric flows, as discussed in S~\\ref{sec:lm:atm}\n  \n\\item {\\em multigrid}: a multigrid solver for constant-coefficient Helmholtz\n  elliptic equations.  This follows the ideas from Chapter~\\ref{ch:multigrid},\n  and is used by the diffusion and incompressible solvers.\n\n\\end{itemize}\n\n\\section{\\pyro 's structure}\n\nThe structure of the code and descriptions of the various runtime\nparameters is found on the pyro webpage,\n\\url{http://python-hydro.github.io/pyro2/}, and described\nin \\cite{pyro}.  Here we provide a summary.\n\nThe grid structure is managed by the {\\tt patch.Grid2d} class.  Data\nis that lives on the grid is contained in a {\\tt\n  patch.CellCenterData2d} object.  Methods are available to provide\naccess to the data and fill the ghost cells.  When accessing the data\non the grid, a {\\tt array\\_indexer.ArrayIndexer} object is returned.\nThis is a subclass of the normal NumPy {\\tt ndarray} that provides\nadditional methods that are useful for interacting with finite-volume\ndata.  In particular, for an object {\\tt a}, you can do {\\tt a.ip(1)}\nto mimic $a_{i+1,j}$.  The Jupyter notebook\n\\href{https://github.com/python-hydro/pyro2/blob/master/mesh/mesh-examples.ipynb}{mesh-examples.ipynb}\nwalks through these classes to demonstrate how they work.\n\nEach \\pyro\\ solver is its own python module.  All but the multigrid\nsolver represent time-dependent problems.  Each of these provide a\n{\\tt Simulation} class that provides the routines necessary to\ninitialize the solver, determine the timestep, and evolve the solution\nin time.  Each solver has one or more sets of initial conditions\ndefined in the solver's {\\tt problems/} directory.\n\nAll time-dependent problems are run through the {\\tt pyro.py} script.\nThe general form is:\n\\begin{verbatim}\n./pyro.py solver problem inputs\n\\end{verbatim}\n where {\\em solver} is one of \\{{\\tt advection}, {\\tt advection\\_rk},\n {\\tt compressible}, {\\tt compressible\\_rk}, {\\tt diffusion}, {\\tt\n   incompressible}, {\\tt lm\\_atm}\\}, {\\em problem} is one of the\n problems defined in the solver's {\\tt problems/} directory, and {\\em\n   inputs} is the name of an input file that defines the values of\n runtime parameter.\n\nThe possible runtime parameters and their defaults are defined in the\n{\\tt \\_defaults} files in the main directory and each solver and\nproblem directory.  Note that the inputs file need not be in the \n{\\tt pyro2/} directory.  The solver's {\\tt problems/} directory\nwill also be checked.\n\n\\section{Running \\pyro}\n\nA simple Gaussian advection simulation is provided by the advection\n{\\em smooth} problem.  This is run as:\n\\begin{verbatim}\n./pyro.py advection smooth inputs.smooth\n\\end{verbatim}\nAs this is run, the solution will be visualized at each step,\nshowing the progression of the simulation.  \n\n\nA list of the problems available for each solver is given in\nTable~\\ref{tab:problems}.  For the multigrid solver, there are scripts\navailable in the {\\tt multigrid/} directory that illustrate its use.\n\n\n\\section{Output and visualization}\n\n\\pyro\\ uses HDF5 for I/O\\footnote{Previously a simple python {\\tt\n    pickle} was done, but this was not very portable.}  The simple\nscript {\\tt plot.py} in the {\\tt pyro/} directory can read in the \ndata and plot the output.  It will determine which solver was used\nfor the run and do the visualization using that solver's methods.\n\n\n\\section{Testing}\n\nThe script {\\tt test.py} provides an interface to \\pyro's unit and\nregression tests.  Simply running it was {\\tt ./test.py} will\nexecutate all of the tests.  The unit tests are enabled through the\n{\\sf pytest} framework.\n\n\n\\begin{sidewaystable*}\n\\centering\n\\renewcommand{\\arraystretch}{1.2}\n\\begin{tabular}{lp{1in}lp{4.0in}}\n\\hline\n{\\bf solver} & {\\bf section} & {\\bf problem} \n   & {\\bf problem description} \\\\\n\\hline\n%\n\\multirow{2}{*}{\\tt advection}\n   & \\multirow{2}{*}{\\S~\\ref{adv:sec:unsplit_2d}}\n   & {\\tt smooth} & advect a smooth Gaussian profile \\\\\n   & & {\\tt tophat} & advect a discontinuous tophat profile \\\\\n\\hline\n%\n\\multirow{1}{*}{\\tt advection\\_rk} \n   & \\multirow{1}{*}{\\S~\\ref{adv:sec:mol_2d}}\n   & \\multicolumn{2}{l}{same as {\\tt advection}} \\\\\n\\hline\n%\n\\multirow{7}{*}{\\tt compressible} \n   & \\multirow{7}{*}{Ch.~\\ref{ch:compressible}}\n   & {\\tt advect} & a Gaussian profile in the density field advected diagonally (in constant pressure background) \\\\\n   & & {\\tt bubble} & a buoyant bubble in a stratified atmosphere \\\\\n   & & {\\tt kh} & setup a shear layer to drive Kelvin-Helmholtz instabilities \\\\\n   & & {\\tt quad} & 2-d Riemann problem based on \\cite{schulz-rinne:1993}  \\\\\n   & & {\\tt rt}  & a simple Rayleigh-Taylor instability\\\\\n   & & {\\tt sedov} & the classic Sedov-Taylor blast wave \\\\\n   & & {\\tt sod} & the Sod shock tube problem  \\\\\n\\hline\n%\n\\multirow{1}{*}{\\tt compressible\\_rk} \n   & \\multirow{1}{*}{Ch.~\\ref{ch:compressible}}\n   & \\multicolumn{2}{l}{same as {\\tt compressible}} \\\\\n\\hline\n%\n{\\tt diffusion} & \\S~\\ref{diff:sec:implicit_mg} & {\\tt gaussian} \n   & diffuse an initial Gaussian profile \\\\\n\\hline\n%\n\\multirow{2}{*}{\\tt incompressible} \n   & \\multirow{2}{*}{Ch.~\\ref{ch:incompressible}}\n   & {\\tt converge} & A simple incompressible problem with known\n                      analytic solution.\\\\\n   & & {\\tt shear} & a doubly-periodic shear layer \\\\\n\\hline\n%\n\\multirow{1}{*}{\\tt lm\\_atm} \n   & \\multirow{1}{*}{Ch.~\\ref{sec:lm:atm}}\n   & {\\tt bubble} & a buoyant bubble in a stratified atmosphere \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{tab:problems} \\pyro\\ solvers and their distributed problems.  The relevant section in these notes that describes the implementation is also given.}\n\\renewcommand{\\arraystretch}{1.0}\n\\end{sidewaystable*}\n\n\n\n\n"
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o\ngrestore\ngsave\n11.270000 213.009687 translate\n90.000000 rotate\n0.000000 0.000000 m /e glyphshow\n7.382812 0.000000 m /r glyphshow\n12.066406 0.000000 m /r glyphshow\n16.750000 0.000000 m /o glyphshow\n24.091797 0.000000 m /r glyphshow\ngrestore\n1.000 setlinewidth\ngsave\n49.77 414 m\n482.4 414 l\nstroke\ngrestore\ngsave\n482.4 40.41 m\n482.4 414 l\nstroke\ngrestore\ngsave\n49.77 40.41 m\n482.4 40.41 l\nstroke\ngrestore\ngsave\n49.77 40.41 m\n49.77 414 l\nstroke\ngrestore\n\nend\nshowpage\n"
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    "content": "%!PS-Adobe-3.0 EPSF-3.0\n%%Title: mg_general_inhomogeneous_test.eps\n%%Creator: matplotlib version 1.3.1, http://matplotlib.org/\n%%CreationDate: Wed Mar 25 08:57:33 2015\n%%Orientation: portrait\n%%BoundingBox: -54 252 666 540\n%%EndComments\n%%BeginProlog\n/mpldict 8 dict def\nmpldict begin\n/m { moveto } bind def\n/l { lineto } bind def\n/r { rlineto } bind def\n/c { curveto } bind def\n/cl { closepath } bind def\n/box {\nm\n1 index 0 r\n0 exch r\nneg 0 r\ncl\n} bind def\n/clipbox {\nbox\nclip\nnewpath\n} bind def\n%!PS-Adobe-3.0 Resource-Font\n%%Title: Bitstream Vera Sans\n%%Copyright: Copyright (c) 2003 by Bitstream, Inc. All Rights Reserved.\n%%Creator: Converted from TrueType to type 3 by PPR\n25 dict begin\n/_d{bind def}bind def\n/_m{moveto}_d\n/_l{lineto}_d\n/_cl{closepath eofill}_d\n/_c{curveto}_d\n/_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d\n/_e{exec}_d\n/FontName /BitstreamVeraSans-Roman def\n/PaintType 0 def\n/FontMatrix[.001 0 0 .001 0 0]def\n/FontBBox[-183 -236 1287 928]def\n/FontType 3 def\n/Encoding [ /space /period /zero /one /two /three /four /five /six /seven /eight /nine /equal /e /n /o /r /x /y ] def\n/FontInfo 10 dict dup begin\n/FamilyName (Bitstream Vera Sans) def\n/FullName (Bitstream Vera Sans) def\n/Notice (Copyright (c) 2003 by Bitstream, Inc. All Rights Reserved. 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_l\n383 83 _l\n544 83 _l\n544 0 _l\n124 0 _l\n124 83 _l\n_cl}_d\n/two{{636 0 73 0 536 742 _sc\n192 83 _m\n536 83 _l\n536 0 _l\n73 0 _l\n73 83 _l\n110 121 161 173 226 239 _c\n290 304 331 346 348 365 _c\n380 400 402 430 414 455 _c\n426 479 433 504 433 528 _c\n433 566 419 598 392 622 _c\n365 646 330 659 286 659 _c\n255 659 222 653 188 643 _c\n154 632 117 616 78 594 _c\n78 694 _l\n118 710 155 722 189 730 _c\n223 738 255 742 284 742 _c\n}_e{359 742 419 723 464 685 _c\n509 647 532 597 532 534 _c\n532 504 526 475 515 449 _c\n504 422 484 390 454 354 _c\n446 344 420 317 376 272 _c\n332 227 271 164 192 83 _c\n_cl}_e}_d\n/three{{636 0 76 -13 556 742 _sc\n406 393 _m\n453 383 490 362 516 330 _c\n542 298 556 258 556 212 _c\n556 140 531 84 482 45 _c\n432 6 362 -13 271 -13 _c\n240 -13 208 -10 176 -4 _c\n144 1 110 10 76 22 _c\n76 117 _l\n103 101 133 89 166 81 _c\n198 73 232 69 268 69 _c\n330 69 377 81 409 105 _c\n441 129 458 165 458 212 _c\n458 254 443 288 413 312 _c\n383 336 341 349 287 349 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_c\n}_e{323 69 368 83 401 113 _c\n433 143 450 183 450 234 _c\n450 284 433 324 401 354 _c\n368 384 323 399 267 399 _c\n241 399 214 396 188 390 _c\n162 384 135 375 108 363 _c\n108 729 _l\n_cl}_e}_d\n/six{{636 0 70 -13 573 742 _sc\n330 404 _m\n286 404 251 388 225 358 _c\n199 328 186 286 186 234 _c\n186 181 199 139 225 109 _c\n251 79 286 64 330 64 _c\n374 64 409 79 435 109 _c\n461 139 474 181 474 234 _c\n474 286 461 328 435 358 _c\n409 388 374 404 330 404 _c\n526 713 _m\n526 623 _l\n501 635 476 644 451 650 _c\n425 656 400 659 376 659 _c\n310 659 260 637 226 593 _c\n}_e{192 549 172 482 168 394 _c\n187 422 211 444 240 459 _c\n269 474 301 482 336 482 _c\n409 482 467 459 509 415 _c\n551 371 573 310 573 234 _c\n573 159 550 99 506 54 _c\n462 9 403 -13 330 -13 _c\n246 -13 181 19 137 83 _c\n92 147 70 241 70 364 _c\n70 479 97 571 152 639 _c\n206 707 280 742 372 742 _c\n396 742 421 739 447 735 _c\n472 730 498 723 526 713 _c\n_cl}_e}_d\n/seven{636 0 82 0 551 729 _sc\n82 729 _m\n551 729 _l\n551 687 _l\n286 0 _l\n183 0 _l\n432 646 _l\n82 646 _l\n82 729 _l\n_cl}_d\n/eight{{636 0 68 -13 568 742 _sc\n318 346 _m\n271 346 234 333 207 308 _c\n180 283 167 249 167 205 _c\n167 161 180 126 207 101 _c\n234 76 271 64 318 64 _c\n364 64 401 76 428 102 _c\n455 127 469 161 469 205 _c\n469 249 455 283 429 308 _c\n402 333 365 346 318 346 _c\n219 388 _m\n177 398 144 418 120 447 _c\n96 476 85 511 85 553 _c\n85 611 105 657 147 691 _c\n188 725 245 742 318 742 _c\n}_e{390 742 447 725 489 691 _c\n530 657 551 611 551 553 _c\n551 511 539 476 515 447 _c\n491 418 459 398 417 388 _c\n464 377 501 355 528 323 _c\n554 291 568 251 568 205 _c\n568 134 546 80 503 43 _c\n459 5 398 -13 318 -13 _c\n237 -13 175 5 132 43 _c\n89 80 68 134 68 205 _c\n68 251 81 291 108 323 _c\n134 355 171 377 219 388 _c\n183 544 _m\n183 506 194 476 218 455 _c\n}_e{242 434 275 424 318 424 _c\n360 424 393 434 417 455 _c\n441 476 453 506 453 544 _c\n453 582 441 611 417 632 _c\n393 653 360 664 318 664 _c\n275 664 242 653 218 632 _c\n194 611 183 582 183 544 _c\n_cl}_e}_d\n/nine{{636 0 63 -13 566 742 _sc\n110 15 _m\n110 105 _l\n134 93 159 84 185 78 _c\n210 72 235 69 260 69 _c\n324 69 374 90 408 134 _c\n442 178 462 244 468 334 _c\n448 306 424 284 396 269 _c\n367 254 335 247 300 247 _c\n226 247 168 269 126 313 _c\n84 357 63 417 63 494 _c\n63 568 85 628 129 674 _c\n173 719 232 742 306 742 _c\n390 742 455 709 499 645 _c\n543 580 566 486 566 364 _c\n}_e{566 248 538 157 484 89 _c\n429 21 356 -13 264 -13 _c\n239 -13 214 -10 189 -6 _c\n163 -2 137 5 110 15 _c\n306 324 _m\n350 324 385 339 411 369 _c\n437 399 450 441 450 494 _c\n450 546 437 588 411 618 _c\n385 648 350 664 306 664 _c\n262 664 227 648 201 618 _c\n175 588 162 546 162 494 _c\n162 441 175 399 201 369 _c\n227 339 262 324 306 324 _c\n_cl}_e}_d\n/equal{838 0 106 172 732 454 _sc\n106 454 _m\n732 454 _l\n732 372 _l\n106 372 _l\n106 454 _l\n106 255 _m\n732 255 _l\n732 172 _l\n106 172 _l\n106 255 _l\n_cl}_d\n/e{{615 0 55 -13 562 560 _sc\n562 296 _m\n562 252 _l\n149 252 _l\n153 190 171 142 205 110 _c\n238 78 284 62 344 62 _c\n378 62 412 66 444 74 _c\n476 82 509 95 541 113 _c\n541 28 _l\n509 14 476 3 442 -3 _c\n408 -9 373 -13 339 -13 _c\n251 -13 182 12 131 62 _c\n80 112 55 181 55 268 _c\n55 357 79 428 127 481 _c\n175 533 241 560 323 560 _c\n397 560 455 536 498 489 _c\n}_e{540 441 562 377 562 296 _c\n472 322 _m\n471 371 457 410 431 440 _c\n404 469 368 484 324 484 _c\n274 484 234 469 204 441 _c\n174 413 156 373 152 322 _c\n472 322 _l\n_cl}_e}_d\n/n{634 0 91 0 549 560 _sc\n549 330 _m\n549 0 _l\n459 0 _l\n459 327 _l\n459 379 448 417 428 443 _c\n408 469 378 482 338 482 _c\n289 482 251 466 223 435 _c\n195 404 181 362 181 309 _c\n181 0 _l\n91 0 _l\n91 547 _l\n181 547 _l\n181 462 _l\n202 494 227 519 257 535 _c\n286 551 320 560 358 560 _c\n420 560 468 540 500 501 _c\n532 462 549 405 549 330 _c\n_cl}_d\n/o{612 0 55 -13 557 560 _sc\n306 484 _m\n258 484 220 465 192 427 _c\n164 389 150 338 150 273 _c\n150 207 163 156 191 118 _c\n219 80 257 62 306 62 _c\n354 62 392 80 420 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bind def\n/p0_47 {\nnewpath\ntranslate\n293.116 99.1513 m\n304.837 99.1513 l\n304.837 100.067 l\n293.116 100.067 l\n293.116 99.1513 l\n} bind def\n/p0_48 {\nnewpath\ntranslate\n293.116 100.067 m\n304.837 100.067 l\n304.837 100.983 l\n293.116 100.983 l\n293.116 100.067 l\n} bind def\n/p0_49 {\nnewpath\ntranslate\n293.116 100.983 m\n304.837 100.983 l\n304.837 101.898 l\n293.116 101.898 l\n293.116 100.983 l\n} bind def\n/p0_4a {\nnewpath\ntranslate\n293.116 101.898 m\n304.837 101.898 l\n304.837 102.814 l\n293.116 102.814 l\n293.116 101.898 l\n} bind def\n/p0_4b {\nnewpath\ntranslate\n293.116 102.814 m\n304.837 102.814 l\n304.837 103.73 l\n293.116 103.73 l\n293.116 102.814 l\n} bind def\n/p0_4c {\nnewpath\ntranslate\n293.116 103.73 m\n304.837 103.73 l\n304.837 104.646 l\n293.116 104.646 l\n293.116 103.73 l\n} bind def\n/p0_4d {\nnewpath\ntranslate\n293.116 104.646 m\n304.837 104.646 l\n304.837 105.561 l\n293.116 105.561 l\n293.116 104.646 l\n} bind def\n/p0_4e {\nnewpath\ntranslate\n293.116 105.561 m\n304.837 105.561 l\n304.837 106.477 l\n293.116 106.477 l\n293.116 105.561 l\n} bind def\n/p0_4f {\nnewpath\ntranslate\n293.116 106.477 m\n304.837 106.477 l\n304.837 107.393 l\n293.116 107.393 l\n293.116 106.477 l\n} bind def\n/p0_50 {\nnewpath\ntranslate\n293.116 107.393 m\n304.837 107.393 l\n304.837 108.309 l\n293.116 108.309 l\n293.116 107.393 l\n} bind def\n/p0_51 {\nnewpath\ntranslate\n293.116 108.309 m\n304.837 108.309 l\n304.837 109.224 l\n293.116 109.224 l\n293.116 108.309 l\n} bind def\n/p0_52 {\nnewpath\ntranslate\n293.116 109.224 m\n304.837 109.224 l\n304.837 110.14 l\n293.116 110.14 l\n293.116 109.224 l\n} bind def\n/p0_53 {\nnewpath\ntranslate\n293.116 110.14 m\n304.837 110.14 l\n304.837 111.056 l\n293.116 111.056 l\n293.116 110.14 l\n} bind def\n/p0_54 {\nnewpath\ntranslate\n293.116 111.056 m\n304.837 111.056 l\n304.837 111.971 l\n293.116 111.971 l\n293.116 111.056 l\n} bind def\n/p0_55 {\nnewpath\ntranslate\n293.116 111.971 m\n304.837 111.971 l\n304.837 112.887 l\n293.116 112.887 l\n293.116 111.971 l\n} bind def\n/p0_56 {\nnewpath\ntranslate\n293.116 112.887 m\n304.837 112.887 l\n304.837 113.803 l\n293.116 113.803 l\n293.116 112.887 l\n} bind def\n/p0_57 {\nnewpath\ntranslate\n293.116 113.803 m\n304.837 113.803 l\n304.837 114.719 l\n293.116 114.719 l\n293.116 113.803 l\n} bind def\n/p0_58 {\nnewpath\ntranslate\n293.116 114.719 m\n304.837 114.719 l\n304.837 115.634 l\n293.116 115.634 l\n293.116 114.719 l\n} bind def\n/p0_59 {\nnewpath\ntranslate\n293.116 115.634 m\n304.837 115.634 l\n304.837 116.55 l\n293.116 116.55 l\n293.116 115.634 l\n} bind def\n/p0_5a {\nnewpath\ntranslate\n293.116 116.55 m\n304.837 116.55 l\n304.837 117.466 l\n293.116 117.466 l\n293.116 116.55 l\n} bind def\n/p0_5b {\nnewpath\ntranslate\n293.116 117.466 m\n304.837 117.466 l\n304.837 118.381 l\n293.116 118.381 l\n293.116 117.466 l\n} bind def\n/p0_5c {\nnewpath\ntranslate\n293.116 118.381 m\n304.837 118.381 l\n304.837 119.297 l\n293.116 119.297 l\n293.116 118.381 l\n} bind def\n/p0_5d {\nnewpath\ntranslate\n293.116 119.297 m\n304.837 119.297 l\n304.837 120.213 l\n293.116 120.213 l\n293.116 119.297 l\n} bind def\n/p0_5e {\nnewpath\ntranslate\n293.116 120.213 m\n304.837 120.213 l\n304.837 121.129 l\n293.116 121.129 l\n293.116 120.213 l\n} bind def\n/p0_5f {\nnewpath\ntranslate\n293.116 121.129 m\n304.837 121.129 l\n304.837 122.044 l\n293.116 122.044 l\n293.116 121.129 l\n} bind def\n/p0_60 {\nnewpath\ntranslate\n293.116 122.044 m\n304.837 122.044 l\n304.837 122.96 l\n293.116 122.96 l\n293.116 122.044 l\n} bind def\n/p0_61 {\nnewpath\ntranslate\n293.116 122.96 m\n304.837 122.96 l\n304.837 123.876 l\n293.116 123.876 l\n293.116 122.96 l\n} bind def\n/p0_62 {\nnewpath\ntranslate\n293.116 123.876 m\n304.837 123.876 l\n304.837 124.791 l\n293.116 124.791 l\n293.116 123.876 l\n} bind def\n/p0_63 {\nnewpath\ntranslate\n293.116 124.791 m\n304.837 124.791 l\n304.837 125.707 l\n293.116 125.707 l\n293.116 124.791 l\n} bind def\n/p0_64 {\nnewpath\ntranslate\n293.116 125.707 m\n304.837 125.707 l\n304.837 126.623 l\n293.116 126.623 l\n293.116 125.707 l\n} bind def\n/p0_65 {\nnewpath\ntranslate\n293.116 126.623 m\n304.837 126.623 l\n304.837 127.539 l\n293.116 127.539 l\n293.116 126.623 l\n} bind def\n/p0_66 {\nnewpath\ntranslate\n293.116 127.539 m\n304.837 127.539 l\n304.837 128.454 l\n293.116 128.454 l\n293.116 127.539 l\n} bind def\n/p0_67 {\nnewpath\ntranslate\n293.116 128.454 m\n304.837 128.454 l\n304.837 129.37 l\n293.116 129.37 l\n293.116 128.454 l\n} bind def\n/p0_68 {\nnewpath\ntranslate\n293.116 129.37 m\n304.837 129.37 l\n304.837 130.286 l\n293.116 130.286 l\n293.116 129.37 l\n} bind def\n/p0_69 {\nnewpath\ntranslate\n293.116 130.286 m\n304.837 130.286 l\n304.837 131.202 l\n293.116 131.202 l\n293.116 130.286 l\n} bind def\n/p0_6a {\nnewpath\ntranslate\n293.116 131.202 m\n304.837 131.202 l\n304.837 132.117 l\n293.116 132.117 l\n293.116 131.202 l\n} bind def\n/p0_6b {\nnewpath\ntranslate\n293.116 132.117 m\n304.837 132.117 l\n304.837 133.033 l\n293.116 133.033 l\n293.116 132.117 l\n} bind def\n/p0_6c {\nnewpath\ntranslate\n293.116 133.033 m\n304.837 133.033 l\n304.837 133.949 l\n293.116 133.949 l\n293.116 133.033 l\n} bind def\n/p0_6d {\nnewpath\ntranslate\n293.116 133.949 m\n304.837 133.949 l\n304.837 134.864 l\n293.116 134.864 l\n293.116 133.949 l\n} bind def\n/p0_6e {\nnewpath\ntranslate\n293.116 134.864 m\n304.837 134.864 l\n304.837 135.78 l\n293.116 135.78 l\n293.116 134.864 l\n} bind def\n/p0_6f {\nnewpath\ntranslate\n293.116 135.78 m\n304.837 135.78 l\n304.837 136.696 l\n293.116 136.696 l\n293.116 135.78 l\n} bind def\n/p0_70 {\nnewpath\ntranslate\n293.116 136.696 m\n304.837 136.696 l\n304.837 137.612 l\n293.116 137.612 l\n293.116 136.696 l\n} bind def\n/p0_71 {\nnewpath\ntranslate\n293.116 137.612 m\n304.837 137.612 l\n304.837 138.527 l\n293.116 138.527 l\n293.116 137.612 l\n} bind def\n/p0_72 {\nnewpath\ntranslate\n293.116 138.527 m\n304.837 138.527 l\n304.837 139.443 l\n293.116 139.443 l\n293.116 138.527 l\n} bind def\n/p0_73 {\nnewpath\ntranslate\n293.116 139.443 m\n304.837 139.443 l\n304.837 140.359 l\n293.116 140.359 l\n293.116 139.443 l\n} bind def\n/p0_74 {\nnewpath\ntranslate\n293.116 140.359 m\n304.837 140.359 l\n304.837 141.274 l\n293.116 141.274 l\n293.116 140.359 l\n} bind def\n/p0_75 {\nnewpath\ntranslate\n293.116 141.274 m\n304.837 141.274 l\n304.837 142.19 l\n293.116 142.19 l\n293.116 141.274 l\n} bind def\n/p0_76 {\nnewpath\ntranslate\n293.116 142.19 m\n304.837 142.19 l\n304.837 143.106 l\n293.116 143.106 l\n293.116 142.19 l\n} bind def\n/p0_77 {\nnewpath\ntranslate\n293.116 143.106 m\n304.837 143.106 l\n304.837 144.022 l\n293.116 144.022 l\n293.116 143.106 l\n} bind def\n/p0_78 {\nnewpath\ntranslate\n293.116 144.022 m\n304.837 144.022 l\n304.837 144.937 l\n293.116 144.937 l\n293.116 144.022 l\n} bind def\n/p0_79 {\nnewpath\ntranslate\n293.116 144.937 m\n304.837 144.937 l\n304.837 145.853 l\n293.116 145.853 l\n293.116 144.937 l\n} bind def\n/p0_7a {\nnewpath\ntranslate\n293.116 145.853 m\n304.837 145.853 l\n304.837 146.769 l\n293.116 146.769 l\n293.116 145.853 l\n} bind def\n/p0_7b {\nnewpath\ntranslate\n293.116 146.769 m\n304.837 146.769 l\n304.837 147.685 l\n293.116 147.685 l\n293.116 146.769 l\n} bind def\n/p0_7c {\nnewpath\ntranslate\n293.116 147.685 m\n304.837 147.685 l\n304.837 148.6 l\n293.116 148.6 l\n293.116 147.685 l\n} bind def\n/p0_7d {\nnewpath\ntranslate\n293.116 148.6 m\n304.837 148.6 l\n304.837 149.516 l\n293.116 149.516 l\n293.116 148.6 l\n} bind def\n/p0_7e {\nnewpath\ntranslate\n293.116 149.516 m\n304.837 149.516 l\n304.837 150.432 l\n293.116 150.432 l\n293.116 149.516 l\n} bind def\n/p0_7f {\nnewpath\ntranslate\n293.116 150.432 m\n304.837 150.432 l\n304.837 151.347 l\n293.116 151.347 l\n293.116 150.432 l\n} bind def\n/p0_80 {\nnewpath\ntranslate\n293.116 151.347 m\n304.837 151.347 l\n304.837 152.263 l\n293.116 152.263 l\n293.116 151.347 l\n} bind def\n/p0_81 {\nnewpath\ntranslate\n293.116 152.263 m\n304.837 152.263 l\n304.837 153.179 l\n293.116 153.179 l\n293.116 152.263 l\n} bind def\n/p0_82 {\nnewpath\ntranslate\n293.116 153.179 m\n304.837 153.179 l\n304.837 154.095 l\n293.116 154.095 l\n293.116 153.179 l\n} bind def\n/p0_83 {\nnewpath\ntranslate\n293.116 154.095 m\n304.837 154.095 l\n304.837 155.01 l\n293.116 155.01 l\n293.116 154.095 l\n} bind def\n/p0_84 {\nnewpath\ntranslate\n293.116 155.01 m\n304.837 155.01 l\n304.837 155.926 l\n293.116 155.926 l\n293.116 155.01 l\n} bind def\n/p0_85 {\nnewpath\ntranslate\n293.116 155.926 m\n304.837 155.926 l\n304.837 156.842 l\n293.116 156.842 l\n293.116 155.926 l\n} bind def\n/p0_86 {\nnewpath\ntranslate\n293.116 156.842 m\n304.837 156.842 l\n304.837 157.757 l\n293.116 157.757 l\n293.116 156.842 l\n} bind def\n/p0_87 {\nnewpath\ntranslate\n293.116 157.757 m\n304.837 157.757 l\n304.837 158.673 l\n293.116 158.673 l\n293.116 157.757 l\n} bind def\n/p0_88 {\nnewpath\ntranslate\n293.116 158.673 m\n304.837 158.673 l\n304.837 159.589 l\n293.116 159.589 l\n293.116 158.673 l\n} bind def\n/p0_89 {\nnewpath\ntranslate\n293.116 159.589 m\n304.837 159.589 l\n304.837 160.505 l\n293.116 160.505 l\n293.116 159.589 l\n} bind def\n/p0_8a {\nnewpath\ntranslate\n293.116 160.505 m\n304.837 160.505 l\n304.837 161.42 l\n293.116 161.42 l\n293.116 160.505 l\n} bind def\n/p0_8b {\nnewpath\ntranslate\n293.116 161.42 m\n304.837 161.42 l\n304.837 162.336 l\n293.116 162.336 l\n293.116 161.42 l\n} bind def\n/p0_8c {\nnewpath\ntranslate\n293.116 162.336 m\n304.837 162.336 l\n304.837 163.252 l\n293.116 163.252 l\n293.116 162.336 l\n} bind def\n/p0_8d {\nnewpath\ntranslate\n293.116 163.252 m\n304.837 163.252 l\n304.837 164.168 l\n293.116 164.168 l\n293.116 163.252 l\n} bind def\n/p0_8e {\nnewpath\ntranslate\n293.116 164.168 m\n304.837 164.168 l\n304.837 165.083 l\n293.116 165.083 l\n293.116 164.168 l\n} bind def\n/p0_8f {\nnewpath\ntranslate\n293.116 165.083 m\n304.837 165.083 l\n304.837 165.999 l\n293.116 165.999 l\n293.116 165.083 l\n} bind def\n/p0_90 {\nnewpath\ntranslate\n293.116 165.999 m\n304.837 165.999 l\n304.837 166.915 l\n293.116 166.915 l\n293.116 165.999 l\n} bind def\n/p0_91 {\nnewpath\ntranslate\n293.116 166.915 m\n304.837 166.915 l\n304.837 167.83 l\n293.116 167.83 l\n293.116 166.915 l\n} bind def\n/p0_92 {\nnewpath\ntranslate\n293.116 167.83 m\n304.837 167.83 l\n304.837 168.746 l\n293.116 168.746 l\n293.116 167.83 l\n} bind def\n/p0_93 {\nnewpath\ntranslate\n293.116 168.746 m\n304.837 168.746 l\n304.837 169.662 l\n293.116 169.662 l\n293.116 168.746 l\n} bind def\n/p0_94 {\nnewpath\ntranslate\n293.116 169.662 m\n304.837 169.662 l\n304.837 170.578 l\n293.116 170.578 l\n293.116 169.662 l\n} bind def\n/p0_95 {\nnewpath\ntranslate\n293.116 170.578 m\n304.837 170.578 l\n304.837 171.493 l\n293.116 171.493 l\n293.116 170.578 l\n} bind def\n/p0_96 {\nnewpath\ntranslate\n293.116 171.493 m\n304.837 171.493 l\n304.837 172.409 l\n293.116 172.409 l\n293.116 171.493 l\n} bind def\n/p0_97 {\nnewpath\ntranslate\n293.116 172.409 m\n304.837 172.409 l\n304.837 173.325 l\n293.116 173.325 l\n293.116 172.409 l\n} bind def\n/p0_98 {\nnewpath\ntranslate\n293.116 173.325 m\n304.837 173.325 l\n304.837 174.24 l\n293.116 174.24 l\n293.116 173.325 l\n} bind def\n/p0_99 {\nnewpath\ntranslate\n293.116 174.24 m\n304.837 174.24 l\n304.837 175.156 l\n293.116 175.156 l\n293.116 174.24 l\n} bind def\n/p0_9a {\nnewpath\ntranslate\n293.116 175.156 m\n304.837 175.156 l\n304.837 176.072 l\n293.116 176.072 l\n293.116 175.156 l\n} bind def\n/p0_9b {\nnewpath\ntranslate\n293.116 176.072 m\n304.837 176.072 l\n304.837 176.988 l\n293.116 176.988 l\n293.116 176.072 l\n} bind def\n/p0_9c {\nnewpath\ntranslate\n293.116 176.988 m\n304.837 176.988 l\n304.837 177.903 l\n293.116 177.903 l\n293.116 176.988 l\n} bind def\n/p0_9d {\nnewpath\ntranslate\n293.116 177.903 m\n304.837 177.903 l\n304.837 178.819 l\n293.116 178.819 l\n293.116 177.903 l\n} bind def\n/p0_9e {\nnewpath\ntranslate\n293.116 178.819 m\n304.837 178.819 l\n304.837 179.735 l\n293.116 179.735 l\n293.116 178.819 l\n} bind def\n/p0_9f {\nnewpath\ntranslate\n293.116 179.735 m\n304.837 179.735 l\n304.837 180.65 l\n293.116 180.65 l\n293.116 179.735 l\n} bind def\n/p0_a0 {\nnewpath\ntranslate\n293.116 180.65 m\n304.837 180.65 l\n304.837 181.566 l\n293.116 181.566 l\n293.116 180.65 l\n} bind def\n/p0_a1 {\nnewpath\ntranslate\n293.116 181.566 m\n304.837 181.566 l\n304.837 182.482 l\n293.116 182.482 l\n293.116 181.566 l\n} bind def\n/p0_a2 {\nnewpath\ntranslate\n293.116 182.482 m\n304.837 182.482 l\n304.837 183.398 l\n293.116 183.398 l\n293.116 182.482 l\n} bind def\n/p0_a3 {\nnewpath\ntranslate\n293.116 183.398 m\n304.837 183.398 l\n304.837 184.313 l\n293.116 184.313 l\n293.116 183.398 l\n} bind def\n/p0_a4 {\nnewpath\ntranslate\n293.116 184.313 m\n304.837 184.313 l\n304.837 185.229 l\n293.116 185.229 l\n293.116 184.313 l\n} bind def\n/p0_a5 {\nnewpath\ntranslate\n293.116 185.229 m\n304.837 185.229 l\n304.837 186.145 l\n293.116 186.145 l\n293.116 185.229 l\n} bind def\n/p0_a6 {\nnewpath\ntranslate\n293.116 186.145 m\n304.837 186.145 l\n304.837 187.061 l\n293.116 187.061 l\n293.116 186.145 l\n} bind def\n/p0_a7 {\nnewpath\ntranslate\n293.116 187.061 m\n304.837 187.061 l\n304.837 187.976 l\n293.116 187.976 l\n293.116 187.061 l\n} bind def\n/p0_a8 {\nnewpath\ntranslate\n293.116 187.976 m\n304.837 187.976 l\n304.837 188.892 l\n293.116 188.892 l\n293.116 187.976 l\n} bind def\n/p0_a9 {\nnewpath\ntranslate\n293.116 188.892 m\n304.837 188.892 l\n304.837 189.808 l\n293.116 189.808 l\n293.116 188.892 l\n} bind def\n/p0_aa {\nnewpath\ntranslate\n293.116 189.808 m\n304.837 189.808 l\n304.837 190.723 l\n293.116 190.723 l\n293.116 189.808 l\n} bind def\n/p0_ab {\nnewpath\ntranslate\n293.116 190.723 m\n304.837 190.723 l\n304.837 191.639 l\n293.116 191.639 l\n293.116 190.723 l\n} bind def\n/p0_ac {\nnewpath\ntranslate\n293.116 191.639 m\n304.837 191.639 l\n304.837 192.555 l\n293.116 192.555 l\n293.116 191.639 l\n} bind def\n/p0_ad {\nnewpath\ntranslate\n293.116 192.555 m\n304.837 192.555 l\n304.837 193.471 l\n293.116 193.471 l\n293.116 192.555 l\n} bind def\n/p0_ae {\nnewpath\ntranslate\n293.116 193.471 m\n304.837 193.471 l\n304.837 194.386 l\n293.116 194.386 l\n293.116 193.471 l\n} bind def\n/p0_af {\nnewpath\ntranslate\n293.116 194.386 m\n304.837 194.386 l\n304.837 195.302 l\n293.116 195.302 l\n293.116 194.386 l\n} bind def\n/p0_b0 {\nnewpath\ntranslate\n293.116 195.302 m\n304.837 195.302 l\n304.837 196.218 l\n293.116 196.218 l\n293.116 195.302 l\n} bind def\n/p0_b1 {\nnewpath\ntranslate\n293.116 196.218 m\n304.837 196.218 l\n304.837 197.133 l\n293.116 197.133 l\n293.116 196.218 l\n} bind def\n/p0_b2 {\nnewpath\ntranslate\n293.116 197.133 m\n304.837 197.133 l\n304.837 198.049 l\n293.116 198.049 l\n293.116 197.133 l\n} bind def\n/p0_b3 {\nnewpath\ntranslate\n293.116 198.049 m\n304.837 198.049 l\n304.837 198.965 l\n293.116 198.965 l\n293.116 198.049 l\n} bind def\n/p0_b4 {\nnewpath\ntranslate\n293.116 198.965 m\n304.837 198.965 l\n304.837 199.881 l\n293.116 199.881 l\n293.116 198.965 l\n} bind def\n/p0_b5 {\nnewpath\ntranslate\n293.116 199.881 m\n304.837 199.881 l\n304.837 200.796 l\n293.116 200.796 l\n293.116 199.881 l\n} bind def\n/p0_b6 {\nnewpath\ntranslate\n293.116 200.796 m\n304.837 200.796 l\n304.837 201.712 l\n293.116 201.712 l\n293.116 200.796 l\n} bind def\n/p0_b7 {\nnewpath\ntranslate\n293.116 201.712 m\n304.837 201.712 l\n304.837 202.628 l\n293.116 202.628 l\n293.116 201.712 l\n} bind def\n/p0_b8 {\nnewpath\ntranslate\n293.116 202.628 m\n304.837 202.628 l\n304.837 203.544 l\n293.116 203.544 l\n293.116 202.628 l\n} bind def\n/p0_b9 {\nnewpath\ntranslate\n293.116 203.544 m\n304.837 203.544 l\n304.837 204.459 l\n293.116 204.459 l\n293.116 203.544 l\n} bind def\n/p0_ba {\nnewpath\ntranslate\n293.116 204.459 m\n304.837 204.459 l\n304.837 205.375 l\n293.116 205.375 l\n293.116 204.459 l\n} bind def\n/p0_bb {\nnewpath\ntranslate\n293.116 205.375 m\n304.837 205.375 l\n304.837 206.291 l\n293.116 206.291 l\n293.116 205.375 l\n} bind def\n/p0_bc {\nnewpath\ntranslate\n293.116 206.291 m\n304.837 206.291 l\n304.837 207.206 l\n293.116 207.206 l\n293.116 206.291 l\n} bind def\n/p0_bd {\nnewpath\ntranslate\n293.116 207.206 m\n304.837 207.206 l\n304.837 208.122 l\n293.116 208.122 l\n293.116 207.206 l\n} bind def\n/p0_be {\nnewpath\ntranslate\n293.116 208.122 m\n304.837 208.122 l\n304.837 209.038 l\n293.116 209.038 l\n293.116 208.122 l\n} bind def\n/p0_bf {\nnewpath\ntranslate\n293.116 209.038 m\n304.837 209.038 l\n304.837 209.954 l\n293.116 209.954 l\n293.116 209.038 l\n} bind def\n/p0_c0 {\nnewpath\ntranslate\n293.116 209.954 m\n304.837 209.954 l\n304.837 210.869 l\n293.116 210.869 l\n293.116 209.954 l\n} bind def\n/p0_c1 {\nnewpath\ntranslate\n293.116 210.869 m\n304.837 210.869 l\n304.837 211.785 l\n293.116 211.785 l\n293.116 210.869 l\n} bind def\n/p0_c2 {\nnewpath\ntranslate\n293.116 211.785 m\n304.837 211.785 l\n304.837 212.701 l\n293.116 212.701 l\n293.116 211.785 l\n} bind def\n/p0_c3 {\nnewpath\ntranslate\n293.116 212.701 m\n304.837 212.701 l\n304.837 213.616 l\n293.116 213.616 l\n293.116 212.701 l\n} bind def\n/p0_c4 {\nnewpath\ntranslate\n293.116 213.616 m\n304.837 213.616 l\n304.837 214.532 l\n293.116 214.532 l\n293.116 213.616 l\n} bind def\n/p0_c5 {\nnewpath\ntranslate\n293.116 214.532 m\n304.837 214.532 l\n304.837 215.448 l\n293.116 215.448 l\n293.116 214.532 l\n} bind def\n/p0_c6 {\nnewpath\ntranslate\n293.116 215.448 m\n304.837 215.448 l\n304.837 216.364 l\n293.116 216.364 l\n293.116 215.448 l\n} bind def\n/p0_c7 {\nnewpath\ntranslate\n293.116 216.364 m\n304.837 216.364 l\n304.837 217.279 l\n293.116 217.279 l\n293.116 216.364 l\n} bind def\n/p0_c8 {\nnewpath\ntranslate\n293.116 217.279 m\n304.837 217.279 l\n304.837 218.195 l\n293.116 218.195 l\n293.116 217.279 l\n} bind def\n/p0_c9 {\nnewpath\ntranslate\n293.116 218.195 m\n304.837 218.195 l\n304.837 219.111 l\n293.116 219.111 l\n293.116 218.195 l\n} bind def\n/p0_ca {\nnewpath\ntranslate\n293.116 219.111 m\n304.837 219.111 l\n304.837 220.027 l\n293.116 220.027 l\n293.116 219.111 l\n} bind def\n/p0_cb {\nnewpath\ntranslate\n293.116 220.027 m\n304.837 220.027 l\n304.837 220.942 l\n293.116 220.942 l\n293.116 220.027 l\n} bind def\n/p0_cc {\nnewpath\ntranslate\n293.116 220.942 m\n304.837 220.942 l\n304.837 221.858 l\n293.116 221.858 l\n293.116 220.942 l\n} bind def\n/p0_cd {\nnewpath\ntranslate\n293.116 221.858 m\n304.837 221.858 l\n304.837 222.774 l\n293.116 222.774 l\n293.116 221.858 l\n} bind def\n/p0_ce {\nnewpath\ntranslate\n293.116 222.774 m\n304.837 222.774 l\n304.837 223.689 l\n293.116 223.689 l\n293.116 222.774 l\n} bind def\n/p0_cf {\nnewpath\ntranslate\n293.116 223.689 m\n304.837 223.689 l\n304.837 224.605 l\n293.116 224.605 l\n293.116 223.689 l\n} bind def\n/p0_d0 {\nnewpath\ntranslate\n293.116 224.605 m\n304.837 224.605 l\n304.837 225.521 l\n293.116 225.521 l\n293.116 224.605 l\n} bind def\n/p0_d1 {\nnewpath\ntranslate\n293.116 225.521 m\n304.837 225.521 l\n304.837 226.437 l\n293.116 226.437 l\n293.116 225.521 l\n} bind def\n/p0_d2 {\nnewpath\ntranslate\n293.116 226.437 m\n304.837 226.437 l\n304.837 227.352 l\n293.116 227.352 l\n293.116 226.437 l\n} bind def\n/p0_d3 {\nnewpath\ntranslate\n293.116 227.352 m\n304.837 227.352 l\n304.837 228.268 l\n293.116 228.268 l\n293.116 227.352 l\n} bind def\n/p0_d4 {\nnewpath\ntranslate\n293.116 228.268 m\n304.837 228.268 l\n304.837 229.184 l\n293.116 229.184 l\n293.116 228.268 l\n} bind def\n/p0_d5 {\nnewpath\ntranslate\n293.116 229.184 m\n304.837 229.184 l\n304.837 230.099 l\n293.116 230.099 l\n293.116 229.184 l\n} bind def\n/p0_d6 {\nnewpath\ntranslate\n293.116 230.099 m\n304.837 230.099 l\n304.837 231.015 l\n293.116 231.015 l\n293.116 230.099 l\n} bind def\n/p0_d7 {\nnewpath\ntranslate\n293.116 231.015 m\n304.837 231.015 l\n304.837 231.931 l\n293.116 231.931 l\n293.116 231.015 l\n} bind def\n/p0_d8 {\nnewpath\ntranslate\n293.116 231.931 m\n304.837 231.931 l\n304.837 232.847 l\n293.116 232.847 l\n293.116 231.931 l\n} bind def\n/p0_d9 {\nnewpath\ntranslate\n293.116 232.847 m\n304.837 232.847 l\n304.837 233.762 l\n293.116 233.762 l\n293.116 232.847 l\n} bind def\n/p0_da {\nnewpath\ntranslate\n293.116 233.762 m\n304.837 233.762 l\n304.837 234.678 l\n293.116 234.678 l\n293.116 233.762 l\n} bind def\n/p0_db {\nnewpath\ntranslate\n293.116 234.678 m\n304.837 234.678 l\n304.837 235.594 l\n293.116 235.594 l\n293.116 234.678 l\n} bind def\n/p0_dc {\nnewpath\ntranslate\n293.116 235.594 m\n304.837 235.594 l\n304.837 236.509 l\n293.116 236.509 l\n293.116 235.594 l\n} bind def\n/p0_dd {\nnewpath\ntranslate\n293.116 236.509 m\n304.837 236.509 l\n304.837 237.425 l\n293.116 237.425 l\n293.116 236.509 l\n} bind def\n/p0_de {\nnewpath\ntranslate\n293.116 237.425 m\n304.837 237.425 l\n304.837 238.341 l\n293.116 238.341 l\n293.116 237.425 l\n} bind def\n/p0_df {\nnewpath\ntranslate\n293.116 238.341 m\n304.837 238.341 l\n304.837 239.257 l\n293.116 239.257 l\n293.116 238.341 l\n} bind def\n/p0_e0 {\nnewpath\ntranslate\n293.116 239.257 m\n304.837 239.257 l\n304.837 240.172 l\n293.116 240.172 l\n293.116 239.257 l\n} bind def\n/p0_e1 {\nnewpath\ntranslate\n293.116 240.172 m\n304.837 240.172 l\n304.837 241.088 l\n293.116 241.088 l\n293.116 240.172 l\n} bind def\n/p0_e2 {\nnewpath\ntranslate\n293.116 241.088 m\n304.837 241.088 l\n304.837 242.004 l\n293.116 242.004 l\n293.116 241.088 l\n} bind def\n/p0_e3 {\nnewpath\ntranslate\n293.116 242.004 m\n304.837 242.004 l\n304.837 242.92 l\n293.116 242.92 l\n293.116 242.004 l\n} bind def\n/p0_e4 {\nnewpath\ntranslate\n293.116 242.92 m\n304.837 242.92 l\n304.837 243.835 l\n293.116 243.835 l\n293.116 242.92 l\n} bind def\n/p0_e5 {\nnewpath\ntranslate\n293.116 243.835 m\n304.837 243.835 l\n304.837 244.751 l\n293.116 244.751 l\n293.116 243.835 l\n} bind def\n/p0_e6 {\nnewpath\ntranslate\n293.116 244.751 m\n304.837 244.751 l\n304.837 245.667 l\n293.116 245.667 l\n293.116 244.751 l\n} bind def\n/p0_e7 {\nnewpath\ntranslate\n293.116 245.667 m\n304.837 245.667 l\n304.837 246.582 l\n293.116 246.582 l\n293.116 245.667 l\n} bind def\n/p0_e8 {\nnewpath\ntranslate\n293.116 246.582 m\n304.837 246.582 l\n304.837 247.498 l\n293.116 247.498 l\n293.116 246.582 l\n} bind def\n/p0_e9 {\nnewpath\ntranslate\n293.116 247.498 m\n304.837 247.498 l\n304.837 248.414 l\n293.116 248.414 l\n293.116 247.498 l\n} bind def\n/p0_ea {\nnewpath\ntranslate\n293.116 248.414 m\n304.837 248.414 l\n304.837 249.33 l\n293.116 249.33 l\n293.116 248.414 l\n} bind def\n/p0_eb {\nnewpath\ntranslate\n293.116 249.33 m\n304.837 249.33 l\n304.837 250.245 l\n293.116 250.245 l\n293.116 249.33 l\n} bind def\n/p0_ec {\nnewpath\ntranslate\n293.116 250.245 m\n304.837 250.245 l\n304.837 251.161 l\n293.116 251.161 l\n293.116 250.245 l\n} bind def\n/p0_ed {\nnewpath\ntranslate\n293.116 251.161 m\n304.837 251.161 l\n304.837 252.077 l\n293.116 252.077 l\n293.116 251.161 l\n} bind def\n/p0_ee {\nnewpath\ntranslate\n293.116 252.077 m\n304.837 252.077 l\n304.837 252.992 l\n293.116 252.992 l\n293.116 252.077 l\n} bind def\n/p0_ef {\nnewpath\ntranslate\n293.116 252.992 m\n304.837 252.992 l\n304.837 253.908 l\n293.116 253.908 l\n293.116 252.992 l\n} bind def\n/p0_f0 {\nnewpath\ntranslate\n293.116 253.908 m\n304.837 253.908 l\n304.837 254.824 l\n293.116 254.824 l\n293.116 253.908 l\n} bind def\n/p0_f1 {\nnewpath\ntranslate\n293.116 254.824 m\n304.837 254.824 l\n304.837 255.74 l\n293.116 255.74 l\n293.116 254.824 l\n} bind def\n/p0_f2 {\nnewpath\ntranslate\n293.116 255.74 m\n304.837 255.74 l\n304.837 256.655 l\n293.116 256.655 l\n293.116 255.74 l\n} bind def\n/p0_f3 {\nnewpath\ntranslate\n293.116 256.655 m\n304.837 256.655 l\n304.837 257.571 l\n293.116 257.571 l\n293.116 256.655 l\n} bind def\n/p0_f4 {\nnewpath\ntranslate\n293.116 257.571 m\n304.837 257.571 l\n304.837 258.487 l\n293.116 258.487 l\n293.116 257.571 l\n} bind def\n/p0_f5 {\nnewpath\ntranslate\n293.116 258.487 m\n304.837 258.487 l\n304.837 259.403 l\n293.116 259.403 l\n293.116 258.487 l\n} bind def\n/p0_f6 {\nnewpath\ntranslate\n293.116 259.403 m\n304.837 259.403 l\n304.837 260.318 l\n293.116 260.318 l\n293.116 259.403 l\n} bind def\n/p0_f7 {\nnewpath\ntranslate\n293.116 260.318 m\n304.837 260.318 l\n304.837 261.234 l\n293.116 261.234 l\n293.116 260.318 l\n} bind def\n/p0_f8 {\nnewpath\ntranslate\n293.116 261.234 m\n304.837 261.234 l\n304.837 262.15 l\n293.116 262.15 l\n293.116 261.234 l\n} bind def\n/p0_f9 {\nnewpath\ntranslate\n293.116 262.15 m\n304.837 262.15 l\n304.837 263.065 l\n293.116 263.065 l\n293.116 262.15 l\n} bind def\n/p0_fa {\nnewpath\ntranslate\n293.116 263.065 m\n304.837 263.065 l\n304.837 263.981 l\n293.116 263.981 l\n293.116 263.065 l\n} bind def\n/p0_fb {\nnewpath\ntranslate\n293.116 263.981 m\n304.837 263.981 l\n304.837 264.897 l\n293.116 264.897 l\n293.116 263.981 l\n} bind def\n/p0_fc {\nnewpath\ntranslate\n293.116 264.897 m\n304.837 264.897 l\n304.837 265.813 l\n293.116 265.813 l\n293.116 264.897 l\n} bind def\n/p0_fd {\nnewpath\ntranslate\n293.116 265.813 m\n304.837 265.813 l\n304.837 266.728 l\n293.116 266.728 l\n293.116 265.813 l\n} bind def\n/p0_fe {\nnewpath\ntranslate\n293.116 266.728 m\n304.837 266.728 l\n304.837 267.644 l\n293.116 267.644 l\n293.116 266.728 l\n} bind def\n/p0_ff {\nnewpath\ntranslate\n293.116 267.644 m\n304.837 267.644 l\n304.837 268.56 l\n293.116 268.56 l\n293.116 267.644 l\n} bind def\ngsave\n11.72 234.4 293.1 34.13 clipbox\n0 0 p0_0\n0.000 0.000 0.500 setrgbcolor\nfill\ngrestore\ngsave\n11.72 234.4 293.1 34.13 clipbox\n0 0 p0_1\n0.000 0.000 0.518 setrgbcolor\nfill\ngrestore\ngsave\n11.72 234.4 293.1 34.13 clipbox\n0 0 p0_2\n0.000 0.000 0.536 setrgbcolor\nfill\ngrestore\ngsave\n11.72 234.4 293.1 34.13 clipbox\n0 0 p0_3\n0.000 0.000 0.553 setrgbcolor\nfill\ngrestore\ngsave\n11.72 234.4 293.1 34.13 clipbox\n0 0 p0_4\n0.000 0.000 0.571 setrgbcolor\nfill\ngrestore\ngsave\n11.72 234.4 293.1 34.13 clipbox\n0 0 p0_5\n0.000 0.000 0.589 setrgbcolor\nfill\ngrestore\ngsave\n11.72 234.4 293.1 34.13 clipbox\n0 0 p0_6\n0.000 0.000 0.607 setrgbcolor\nfill\ngrestore\ngsave\n11.72 234.4 293.1 34.13 clipbox\n0 0 p0_7\n0.000 0.000 0.625 setrgbcolor\nfill\ngrestore\ngsave\n11.72 234.4 293.1 34.13 clipbox\n0 0 p0_8\n0.000 0.000 0.643 setrgbcolor\nfill\ngrestore\ngsave\n11.72 234.4 293.1 34.13 clipbox\n0 0 p0_9\n0.000 0.000 0.660 setrgbcolor\nfill\ngrestore\ngsave\n11.72 234.4 293.1 34.13 clipbox\n0 0 p0_a\n0.000 0.000 0.678 setrgbcolor\nfill\ngrestore\ngsave\n11.72 234.4 293.1 34.13 clipbox\n0 0 p0_b\n0.000 0.000 0.696 setrgbcolor\nfill\ngrestore\ngsave\n11.72 234.4 293.1 34.13 clipbox\n0 0 p0_c\n0.000 0.000 0.714 setrgbcolor\nfill\ngrestore\ngsave\n11.72 234.4 293.1 34.13 clipbox\n0 0 p0_d\n0.000 0.000 0.732 setrgbcolor\nfill\ngrestore\ngsave\n11.72 234.4 293.1 34.13 clipbox\n0 0 p0_e\n0.000 0.000 0.750 setrgbcolor\nfill\ngrestore\ngsave\n11.72 234.4 293.1 34.13 clipbox\n0 0 p0_f\n0.000 0.000 0.767 setrgbcolor\nfill\ngrestore\ngsave\n11.72 234.4 293.1 34.13 clipbox\n0 0 p0_10\n0.000 0.000 0.785 setrgbcolor\nfill\ngrestore\ngsave\n11.72 234.4 293.1 34.13 clipbox\n0 0 p0_11\n0.000 0.000 0.803 setrgbcolor\nfill\ngrestore\ngsave\n11.72 234.4 293.1 34.13 clipbox\n0 0 p0_12\n0.000 0.000 0.821 setrgbcolor\nfill\ngrestore\ngsave\n11.72 234.4 293.1 34.13 clipbox\n0 0 p0_13\n0.000 0.000 0.839 setrgbcolor\nfill\ngrestore\ngsave\n11.72 234.4 293.1 34.13 clipbox\n0 0 p0_14\n0.000 0.000 0.857 setrgbcolor\nfill\ngrestore\ngsave\n11.72 234.4 293.1 34.13 clipbox\n0 0 p0_15\n0.000 0.000 0.874 setrgbcolor\nfill\ngrestore\ngsave\n11.72 234.4 293.1 34.13 clipbox\n0 0 p0_16\n0.000 0.000 0.892 setrgbcolor\nfill\ngrestore\ngsave\n11.72 234.4 293.1 34.13 clipbox\n0 0 p0_17\n0.000 0.000 0.910 setrgbcolor\nfill\ngrestore\ngsave\n11.72 234.4 293.1 34.13 clipbox\n0 0 p0_18\n0.000 0.000 0.928 setrgbcolor\nfill\ngrestore\ngsave\n11.72 234.4 293.1 34.13 clipbox\n0 0 p0_19\n0.000 0.000 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m\n635.407 93.6566 l\n635.407 94.5724 l\n623.686 94.5724 l\n623.686 93.6566 l\n} bind def\n/p1_42 {\nnewpath\ntranslate\n623.686 94.5724 m\n635.407 94.5724 l\n635.407 95.4881 l\n623.686 95.4881 l\n623.686 94.5724 l\n} bind def\n/p1_43 {\nnewpath\ntranslate\n623.686 95.4881 m\n635.407 95.4881 l\n635.407 96.4038 l\n623.686 96.4038 l\n623.686 95.4881 l\n} bind def\n/p1_44 {\nnewpath\ntranslate\n623.686 96.4038 m\n635.407 96.4038 l\n635.407 97.3195 l\n623.686 97.3195 l\n623.686 96.4038 l\n} bind def\n/p1_45 {\nnewpath\ntranslate\n623.686 97.3195 m\n635.407 97.3195 l\n635.407 98.2353 l\n623.686 98.2353 l\n623.686 97.3195 l\n} bind def\n/p1_46 {\nnewpath\ntranslate\n623.686 98.2353 m\n635.407 98.2353 l\n635.407 99.151 l\n623.686 99.151 l\n623.686 98.2353 l\n} bind def\n/p1_47 {\nnewpath\ntranslate\n623.686 99.151 m\n635.407 99.151 l\n635.407 100.067 l\n623.686 100.067 l\n623.686 99.151 l\n} bind def\n/p1_48 {\nnewpath\ntranslate\n623.686 100.067 m\n635.407 100.067 l\n635.407 100.982 l\n623.686 100.982 l\n623.686 100.067 l\n} bind def\n/p1_49 {\nnewpath\ntranslate\n623.686 100.982 m\n635.407 100.982 l\n635.407 101.898 l\n623.686 101.898 l\n623.686 100.982 l\n} bind def\n/p1_4a {\nnewpath\ntranslate\n623.686 101.898 m\n635.407 101.898 l\n635.407 102.814 l\n623.686 102.814 l\n623.686 101.898 l\n} bind def\n/p1_4b {\nnewpath\ntranslate\n623.686 102.814 m\n635.407 102.814 l\n635.407 103.73 l\n623.686 103.73 l\n623.686 102.814 l\n} bind def\n/p1_4c {\nnewpath\ntranslate\n623.686 103.73 m\n635.407 103.73 l\n635.407 104.645 l\n623.686 104.645 l\n623.686 103.73 l\n} bind def\n/p1_4d {\nnewpath\ntranslate\n623.686 104.645 m\n635.407 104.645 l\n635.407 105.561 l\n623.686 105.561 l\n623.686 104.645 l\n} bind def\n/p1_4e {\nnewpath\ntranslate\n623.686 105.561 m\n635.407 105.561 l\n635.407 106.477 l\n623.686 106.477 l\n623.686 105.561 l\n} bind def\n/p1_4f {\nnewpath\ntranslate\n623.686 106.477 m\n635.407 106.477 l\n635.407 107.393 l\n623.686 107.393 l\n623.686 106.477 l\n} bind def\n/p1_50 {\nnewpath\ntranslate\n623.686 107.393 m\n635.407 107.393 l\n635.407 108.308 l\n623.686 108.308 l\n623.686 107.393 l\n} bind def\n/p1_51 {\nnewpath\ntranslate\n623.686 108.308 m\n635.407 108.308 l\n635.407 109.224 l\n623.686 109.224 l\n623.686 108.308 l\n} bind def\n/p1_52 {\nnewpath\ntranslate\n623.686 109.224 m\n635.407 109.224 l\n635.407 110.14 l\n623.686 110.14 l\n623.686 109.224 l\n} bind def\n/p1_53 {\nnewpath\ntranslate\n623.686 110.14 m\n635.407 110.14 l\n635.407 111.055 l\n623.686 111.055 l\n623.686 110.14 l\n} bind def\n/p1_54 {\nnewpath\ntranslate\n623.686 111.055 m\n635.407 111.055 l\n635.407 111.971 l\n623.686 111.971 l\n623.686 111.055 l\n} bind def\n/p1_55 {\nnewpath\ntranslate\n623.686 111.971 m\n635.407 111.971 l\n635.407 112.887 l\n623.686 112.887 l\n623.686 111.971 l\n} bind def\n/p1_56 {\nnewpath\ntranslate\n623.686 112.887 m\n635.407 112.887 l\n635.407 113.803 l\n623.686 113.803 l\n623.686 112.887 l\n} bind def\n/p1_57 {\nnewpath\ntranslate\n623.686 113.803 m\n635.407 113.803 l\n635.407 114.718 l\n623.686 114.718 l\n623.686 113.803 l\n} bind def\n/p1_58 {\nnewpath\ntranslate\n623.686 114.718 m\n635.407 114.718 l\n635.407 115.634 l\n623.686 115.634 l\n623.686 114.718 l\n} bind def\n/p1_59 {\nnewpath\ntranslate\n623.686 115.634 m\n635.407 115.634 l\n635.407 116.55 l\n623.686 116.55 l\n623.686 115.634 l\n} bind def\n/p1_5a {\nnewpath\ntranslate\n623.686 116.55 m\n635.407 116.55 l\n635.407 117.465 l\n623.686 117.465 l\n623.686 116.55 l\n} bind def\n/p1_5b {\nnewpath\ntranslate\n623.686 117.465 m\n635.407 117.465 l\n635.407 118.381 l\n623.686 118.381 l\n623.686 117.465 l\n} bind def\n/p1_5c {\nnewpath\ntranslate\n623.686 118.381 m\n635.407 118.381 l\n635.407 119.297 l\n623.686 119.297 l\n623.686 118.381 l\n} bind def\n/p1_5d {\nnewpath\ntranslate\n623.686 119.297 m\n635.407 119.297 l\n635.407 120.213 l\n623.686 120.213 l\n623.686 119.297 l\n} bind def\n/p1_5e {\nnewpath\ntranslate\n623.686 120.213 m\n635.407 120.213 l\n635.407 121.128 l\n623.686 121.128 l\n623.686 120.213 l\n} bind def\n/p1_5f {\nnewpath\ntranslate\n623.686 121.128 m\n635.407 121.128 l\n635.407 122.044 l\n623.686 122.044 l\n623.686 121.128 l\n} bind def\n/p1_60 {\nnewpath\ntranslate\n623.686 122.044 m\n635.407 122.044 l\n635.407 122.96 l\n623.686 122.96 l\n623.686 122.044 l\n} bind def\n/p1_61 {\nnewpath\ntranslate\n623.686 122.96 m\n635.407 122.96 l\n635.407 123.876 l\n623.686 123.876 l\n623.686 122.96 l\n} bind def\n/p1_62 {\nnewpath\ntranslate\n623.686 123.876 m\n635.407 123.876 l\n635.407 124.791 l\n623.686 124.791 l\n623.686 123.876 l\n} bind def\n/p1_63 {\nnewpath\ntranslate\n623.686 124.791 m\n635.407 124.791 l\n635.407 125.707 l\n623.686 125.707 l\n623.686 124.791 l\n} bind def\n/p1_64 {\nnewpath\ntranslate\n623.686 125.707 m\n635.407 125.707 l\n635.407 126.623 l\n623.686 126.623 l\n623.686 125.707 l\n} bind def\n/p1_65 {\nnewpath\ntranslate\n623.686 126.623 m\n635.407 126.623 l\n635.407 127.538 l\n623.686 127.538 l\n623.686 126.623 l\n} bind def\n/p1_66 {\nnewpath\ntranslate\n623.686 127.538 m\n635.407 127.538 l\n635.407 128.454 l\n623.686 128.454 l\n623.686 127.538 l\n} bind def\n/p1_67 {\nnewpath\ntranslate\n623.686 128.454 m\n635.407 128.454 l\n635.407 129.37 l\n623.686 129.37 l\n623.686 128.454 l\n} bind def\n/p1_68 {\nnewpath\ntranslate\n623.686 129.37 m\n635.407 129.37 l\n635.407 130.286 l\n623.686 130.286 l\n623.686 129.37 l\n} bind def\n/p1_69 {\nnewpath\ntranslate\n623.686 130.286 m\n635.407 130.286 l\n635.407 131.201 l\n623.686 131.201 l\n623.686 130.286 l\n} bind def\n/p1_6a {\nnewpath\ntranslate\n623.686 131.201 m\n635.407 131.201 l\n635.407 132.117 l\n623.686 132.117 l\n623.686 131.201 l\n} bind def\n/p1_6b {\nnewpath\ntranslate\n623.686 132.117 m\n635.407 132.117 l\n635.407 133.033 l\n623.686 133.033 l\n623.686 132.117 l\n} bind def\n/p1_6c {\nnewpath\ntranslate\n623.686 133.033 m\n635.407 133.033 l\n635.407 133.949 l\n623.686 133.949 l\n623.686 133.033 l\n} bind def\n/p1_6d {\nnewpath\ntranslate\n623.686 133.949 m\n635.407 133.949 l\n635.407 134.864 l\n623.686 134.864 l\n623.686 133.949 l\n} bind def\n/p1_6e {\nnewpath\ntranslate\n623.686 134.864 m\n635.407 134.864 l\n635.407 135.78 l\n623.686 135.78 l\n623.686 134.864 l\n} bind def\n/p1_6f {\nnewpath\ntranslate\n623.686 135.78 m\n635.407 135.78 l\n635.407 136.696 l\n623.686 136.696 l\n623.686 135.78 l\n} bind def\n/p1_70 {\nnewpath\ntranslate\n623.686 136.696 m\n635.407 136.696 l\n635.407 137.611 l\n623.686 137.611 l\n623.686 136.696 l\n} bind def\n/p1_71 {\nnewpath\ntranslate\n623.686 137.611 m\n635.407 137.611 l\n635.407 138.527 l\n623.686 138.527 l\n623.686 137.611 l\n} bind def\n/p1_72 {\nnewpath\ntranslate\n623.686 138.527 m\n635.407 138.527 l\n635.407 139.443 l\n623.686 139.443 l\n623.686 138.527 l\n} bind def\n/p1_73 {\nnewpath\ntranslate\n623.686 139.443 m\n635.407 139.443 l\n635.407 140.359 l\n623.686 140.359 l\n623.686 139.443 l\n} bind def\n/p1_74 {\nnewpath\ntranslate\n623.686 140.359 m\n635.407 140.359 l\n635.407 141.274 l\n623.686 141.274 l\n623.686 140.359 l\n} bind def\n/p1_75 {\nnewpath\ntranslate\n623.686 141.274 m\n635.407 141.274 l\n635.407 142.19 l\n623.686 142.19 l\n623.686 141.274 l\n} bind def\n/p1_76 {\nnewpath\ntranslate\n623.686 142.19 m\n635.407 142.19 l\n635.407 143.106 l\n623.686 143.106 l\n623.686 142.19 l\n} bind def\n/p1_77 {\nnewpath\ntranslate\n623.686 143.106 m\n635.407 143.106 l\n635.407 144.021 l\n623.686 144.021 l\n623.686 143.106 l\n} bind def\n/p1_78 {\nnewpath\ntranslate\n623.686 144.021 m\n635.407 144.021 l\n635.407 144.937 l\n623.686 144.937 l\n623.686 144.021 l\n} bind def\n/p1_79 {\nnewpath\ntranslate\n623.686 144.937 m\n635.407 144.937 l\n635.407 145.853 l\n623.686 145.853 l\n623.686 144.937 l\n} bind def\n/p1_7a {\nnewpath\ntranslate\n623.686 145.853 m\n635.407 145.853 l\n635.407 146.769 l\n623.686 146.769 l\n623.686 145.853 l\n} bind def\n/p1_7b {\nnewpath\ntranslate\n623.686 146.769 m\n635.407 146.769 l\n635.407 147.684 l\n623.686 147.684 l\n623.686 146.769 l\n} bind def\n/p1_7c {\nnewpath\ntranslate\n623.686 147.684 m\n635.407 147.684 l\n635.407 148.6 l\n623.686 148.6 l\n623.686 147.684 l\n} bind def\n/p1_7d {\nnewpath\ntranslate\n623.686 148.6 m\n635.407 148.6 l\n635.407 149.516 l\n623.686 149.516 l\n623.686 148.6 l\n} bind def\n/p1_7e {\nnewpath\ntranslate\n623.686 149.516 m\n635.407 149.516 l\n635.407 150.432 l\n623.686 150.432 l\n623.686 149.516 l\n} bind def\n/p1_7f {\nnewpath\ntranslate\n623.686 150.432 m\n635.407 150.432 l\n635.407 151.347 l\n623.686 151.347 l\n623.686 150.432 l\n} bind def\n/p1_80 {\nnewpath\ntranslate\n623.686 151.347 m\n635.407 151.347 l\n635.407 152.263 l\n623.686 152.263 l\n623.686 151.347 l\n} bind def\n/p1_81 {\nnewpath\ntranslate\n623.686 152.263 m\n635.407 152.263 l\n635.407 153.179 l\n623.686 153.179 l\n623.686 152.263 l\n} bind def\n/p1_82 {\nnewpath\ntranslate\n623.686 153.179 m\n635.407 153.179 l\n635.407 154.094 l\n623.686 154.094 l\n623.686 153.179 l\n} bind def\n/p1_83 {\nnewpath\ntranslate\n623.686 154.094 m\n635.407 154.094 l\n635.407 155.01 l\n623.686 155.01 l\n623.686 154.094 l\n} bind def\n/p1_84 {\nnewpath\ntranslate\n623.686 155.01 m\n635.407 155.01 l\n635.407 155.926 l\n623.686 155.926 l\n623.686 155.01 l\n} bind def\n/p1_85 {\nnewpath\ntranslate\n623.686 155.926 m\n635.407 155.926 l\n635.407 156.842 l\n623.686 156.842 l\n623.686 155.926 l\n} bind def\n/p1_86 {\nnewpath\ntranslate\n623.686 156.842 m\n635.407 156.842 l\n635.407 157.757 l\n623.686 157.757 l\n623.686 156.842 l\n} bind def\n/p1_87 {\nnewpath\ntranslate\n623.686 157.757 m\n635.407 157.757 l\n635.407 158.673 l\n623.686 158.673 l\n623.686 157.757 l\n} bind def\n/p1_88 {\nnewpath\ntranslate\n623.686 158.673 m\n635.407 158.673 l\n635.407 159.589 l\n623.686 159.589 l\n623.686 158.673 l\n} bind def\n/p1_89 {\nnewpath\ntranslate\n623.686 159.589 m\n635.407 159.589 l\n635.407 160.505 l\n623.686 160.505 l\n623.686 159.589 l\n} bind def\n/p1_8a {\nnewpath\ntranslate\n623.686 160.505 m\n635.407 160.505 l\n635.407 161.42 l\n623.686 161.42 l\n623.686 160.505 l\n} bind def\n/p1_8b {\nnewpath\ntranslate\n623.686 161.42 m\n635.407 161.42 l\n635.407 162.336 l\n623.686 162.336 l\n623.686 161.42 l\n} bind def\n/p1_8c {\nnewpath\ntranslate\n623.686 162.336 m\n635.407 162.336 l\n635.407 163.252 l\n623.686 163.252 l\n623.686 162.336 l\n} bind def\n/p1_8d {\nnewpath\ntranslate\n623.686 163.252 m\n635.407 163.252 l\n635.407 164.167 l\n623.686 164.167 l\n623.686 163.252 l\n} bind def\n/p1_8e {\nnewpath\ntranslate\n623.686 164.167 m\n635.407 164.167 l\n635.407 165.083 l\n623.686 165.083 l\n623.686 164.167 l\n} bind def\n/p1_8f {\nnewpath\ntranslate\n623.686 165.083 m\n635.407 165.083 l\n635.407 165.999 l\n623.686 165.999 l\n623.686 165.083 l\n} bind def\n/p1_90 {\nnewpath\ntranslate\n623.686 165.999 m\n635.407 165.999 l\n635.407 166.915 l\n623.686 166.915 l\n623.686 165.999 l\n} bind def\n/p1_91 {\nnewpath\ntranslate\n623.686 166.915 m\n635.407 166.915 l\n635.407 167.83 l\n623.686 167.83 l\n623.686 166.915 l\n} bind def\n/p1_92 {\nnewpath\ntranslate\n623.686 167.83 m\n635.407 167.83 l\n635.407 168.746 l\n623.686 168.746 l\n623.686 167.83 l\n} bind def\n/p1_93 {\nnewpath\ntranslate\n623.686 168.746 m\n635.407 168.746 l\n635.407 169.662 l\n623.686 169.662 l\n623.686 168.746 l\n} bind def\n/p1_94 {\nnewpath\ntranslate\n623.686 169.662 m\n635.407 169.662 l\n635.407 170.577 l\n623.686 170.577 l\n623.686 169.662 l\n} bind def\n/p1_95 {\nnewpath\ntranslate\n623.686 170.577 m\n635.407 170.577 l\n635.407 171.493 l\n623.686 171.493 l\n623.686 170.577 l\n} bind def\n/p1_96 {\nnewpath\ntranslate\n623.686 171.493 m\n635.407 171.493 l\n635.407 172.409 l\n623.686 172.409 l\n623.686 171.493 l\n} bind def\n/p1_97 {\nnewpath\ntranslate\n623.686 172.409 m\n635.407 172.409 l\n635.407 173.325 l\n623.686 173.325 l\n623.686 172.409 l\n} bind def\n/p1_98 {\nnewpath\ntranslate\n623.686 173.325 m\n635.407 173.325 l\n635.407 174.24 l\n623.686 174.24 l\n623.686 173.325 l\n} bind def\n/p1_99 {\nnewpath\ntranslate\n623.686 174.24 m\n635.407 174.24 l\n635.407 175.156 l\n623.686 175.156 l\n623.686 174.24 l\n} bind def\n/p1_9a {\nnewpath\ntranslate\n623.686 175.156 m\n635.407 175.156 l\n635.407 176.072 l\n623.686 176.072 l\n623.686 175.156 l\n} bind def\n/p1_9b {\nnewpath\ntranslate\n623.686 176.072 m\n635.407 176.072 l\n635.407 176.988 l\n623.686 176.988 l\n623.686 176.072 l\n} bind def\n/p1_9c {\nnewpath\ntranslate\n623.686 176.988 m\n635.407 176.988 l\n635.407 177.903 l\n623.686 177.903 l\n623.686 176.988 l\n} bind def\n/p1_9d {\nnewpath\ntranslate\n623.686 177.903 m\n635.407 177.903 l\n635.407 178.819 l\n623.686 178.819 l\n623.686 177.903 l\n} bind def\n/p1_9e {\nnewpath\ntranslate\n623.686 178.819 m\n635.407 178.819 l\n635.407 179.735 l\n623.686 179.735 l\n623.686 178.819 l\n} bind def\n/p1_9f {\nnewpath\ntranslate\n623.686 179.735 m\n635.407 179.735 l\n635.407 180.65 l\n623.686 180.65 l\n623.686 179.735 l\n} bind def\n/p1_a0 {\nnewpath\ntranslate\n623.686 180.65 m\n635.407 180.65 l\n635.407 181.566 l\n623.686 181.566 l\n623.686 180.65 l\n} bind def\n/p1_a1 {\nnewpath\ntranslate\n623.686 181.566 m\n635.407 181.566 l\n635.407 182.482 l\n623.686 182.482 l\n623.686 181.566 l\n} bind def\n/p1_a2 {\nnewpath\ntranslate\n623.686 182.482 m\n635.407 182.482 l\n635.407 183.398 l\n623.686 183.398 l\n623.686 182.482 l\n} bind def\n/p1_a3 {\nnewpath\ntranslate\n623.686 183.398 m\n635.407 183.398 l\n635.407 184.313 l\n623.686 184.313 l\n623.686 183.398 l\n} bind def\n/p1_a4 {\nnewpath\ntranslate\n623.686 184.313 m\n635.407 184.313 l\n635.407 185.229 l\n623.686 185.229 l\n623.686 184.313 l\n} bind def\n/p1_a5 {\nnewpath\ntranslate\n623.686 185.229 m\n635.407 185.229 l\n635.407 186.145 l\n623.686 186.145 l\n623.686 185.229 l\n} bind def\n/p1_a6 {\nnewpath\ntranslate\n623.686 186.145 m\n635.407 186.145 l\n635.407 187.061 l\n623.686 187.061 l\n623.686 186.145 l\n} bind def\n/p1_a7 {\nnewpath\ntranslate\n623.686 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{\nnewpath\ntranslate\n623.686 207.206 m\n635.407 207.206 l\n635.407 208.122 l\n623.686 208.122 l\n623.686 207.206 l\n} bind def\n/p1_be {\nnewpath\ntranslate\n623.686 208.122 m\n635.407 208.122 l\n635.407 209.038 l\n623.686 209.038 l\n623.686 208.122 l\n} bind def\n/p1_bf {\nnewpath\ntranslate\n623.686 209.038 m\n635.407 209.038 l\n635.407 209.954 l\n623.686 209.954 l\n623.686 209.038 l\n} bind def\n/p1_c0 {\nnewpath\ntranslate\n623.686 209.954 m\n635.407 209.954 l\n635.407 210.869 l\n623.686 210.869 l\n623.686 209.954 l\n} bind def\n/p1_c1 {\nnewpath\ntranslate\n623.686 210.869 m\n635.407 210.869 l\n635.407 211.785 l\n623.686 211.785 l\n623.686 210.869 l\n} bind def\n/p1_c2 {\nnewpath\ntranslate\n623.686 211.785 m\n635.407 211.785 l\n635.407 212.701 l\n623.686 212.701 l\n623.686 211.785 l\n} bind def\n/p1_c3 {\nnewpath\ntranslate\n623.686 212.701 m\n635.407 212.701 l\n635.407 213.617 l\n623.686 213.617 l\n623.686 212.701 l\n} bind def\n/p1_c4 {\nnewpath\ntranslate\n623.686 213.617 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bind def\n/p1_d3 {\nnewpath\ntranslate\n623.686 227.352 m\n635.407 227.352 l\n635.407 228.268 l\n623.686 228.268 l\n623.686 227.352 l\n} bind def\n/p1_d4 {\nnewpath\ntranslate\n623.686 228.268 m\n635.407 228.268 l\n635.407 229.184 l\n623.686 229.184 l\n623.686 228.268 l\n} bind def\n/p1_d5 {\nnewpath\ntranslate\n623.686 229.184 m\n635.407 229.184 l\n635.407 230.1 l\n623.686 230.1 l\n623.686 229.184 l\n} bind def\n/p1_d6 {\nnewpath\ntranslate\n623.686 230.1 m\n635.407 230.1 l\n635.407 231.015 l\n623.686 231.015 l\n623.686 230.1 l\n} bind def\n/p1_d7 {\nnewpath\ntranslate\n623.686 231.015 m\n635.407 231.015 l\n635.407 231.931 l\n623.686 231.931 l\n623.686 231.015 l\n} bind def\n/p1_d8 {\nnewpath\ntranslate\n623.686 231.931 m\n635.407 231.931 l\n635.407 232.847 l\n623.686 232.847 l\n623.686 231.931 l\n} bind def\n/p1_d9 {\nnewpath\ntranslate\n623.686 232.847 m\n635.407 232.847 l\n635.407 233.762 l\n623.686 233.762 l\n623.686 232.847 l\n} bind def\n/p1_da {\nnewpath\ntranslate\n623.686 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l\n623.686 241.088 l\n623.686 240.173 l\n} bind def\n/p1_e2 {\nnewpath\ntranslate\n623.686 241.088 m\n635.407 241.088 l\n635.407 242.004 l\n623.686 242.004 l\n623.686 241.088 l\n} bind def\n/p1_e3 {\nnewpath\ntranslate\n623.686 242.004 m\n635.407 242.004 l\n635.407 242.92 l\n623.686 242.92 l\n623.686 242.004 l\n} bind def\n/p1_e4 {\nnewpath\ntranslate\n623.686 242.92 m\n635.407 242.92 l\n635.407 243.835 l\n623.686 243.835 l\n623.686 242.92 l\n} bind def\n/p1_e5 {\nnewpath\ntranslate\n623.686 243.835 m\n635.407 243.835 l\n635.407 244.751 l\n623.686 244.751 l\n623.686 243.835 l\n} bind def\n/p1_e6 {\nnewpath\ntranslate\n623.686 244.751 m\n635.407 244.751 l\n635.407 245.667 l\n623.686 245.667 l\n623.686 244.751 l\n} bind def\n/p1_e7 {\nnewpath\ntranslate\n623.686 245.667 m\n635.407 245.667 l\n635.407 246.583 l\n623.686 246.583 l\n623.686 245.667 l\n} bind def\n/p1_e8 {\nnewpath\ntranslate\n623.686 246.583 m\n635.407 246.583 l\n635.407 247.498 l\n623.686 247.498 l\n623.686 246.583 l\n} bind def\n/p1_e9 {\nnewpath\ntranslate\n623.686 247.498 m\n635.407 247.498 l\n635.407 248.414 l\n623.686 248.414 l\n623.686 247.498 l\n} bind def\n/p1_ea {\nnewpath\ntranslate\n623.686 248.414 m\n635.407 248.414 l\n635.407 249.33 l\n623.686 249.33 l\n623.686 248.414 l\n} bind def\n/p1_eb {\nnewpath\ntranslate\n623.686 249.33 m\n635.407 249.33 l\n635.407 250.246 l\n623.686 250.246 l\n623.686 249.33 l\n} bind def\n/p1_ec {\nnewpath\ntranslate\n623.686 250.246 m\n635.407 250.246 l\n635.407 251.161 l\n623.686 251.161 l\n623.686 250.246 l\n} bind def\n/p1_ed {\nnewpath\ntranslate\n623.686 251.161 m\n635.407 251.161 l\n635.407 252.077 l\n623.686 252.077 l\n623.686 251.161 l\n} bind def\n/p1_ee {\nnewpath\ntranslate\n623.686 252.077 m\n635.407 252.077 l\n635.407 252.993 l\n623.686 252.993 l\n623.686 252.077 l\n} bind def\n/p1_ef {\nnewpath\ntranslate\n623.686 252.993 m\n635.407 252.993 l\n635.407 253.908 l\n623.686 253.908 l\n623.686 252.993 l\n} bind def\n/p1_f0 {\nnewpath\ntranslate\n623.686 253.908 m\n635.407 253.908 l\n635.407 254.824 l\n623.686 254.824 l\n623.686 253.908 l\n} bind def\n/p1_f1 {\nnewpath\ntranslate\n623.686 254.824 m\n635.407 254.824 l\n635.407 255.74 l\n623.686 255.74 l\n623.686 254.824 l\n} bind def\n/p1_f2 {\nnewpath\ntranslate\n623.686 255.74 m\n635.407 255.74 l\n635.407 256.656 l\n623.686 256.656 l\n623.686 255.74 l\n} bind def\n/p1_f3 {\nnewpath\ntranslate\n623.686 256.656 m\n635.407 256.656 l\n635.407 257.571 l\n623.686 257.571 l\n623.686 256.656 l\n} bind def\n/p1_f4 {\nnewpath\ntranslate\n623.686 257.571 m\n635.407 257.571 l\n635.407 258.487 l\n623.686 258.487 l\n623.686 257.571 l\n} bind def\n/p1_f5 {\nnewpath\ntranslate\n623.686 258.487 m\n635.407 258.487 l\n635.407 259.403 l\n623.686 259.403 l\n623.686 258.487 l\n} bind def\n/p1_f6 {\nnewpath\ntranslate\n623.686 259.403 m\n635.407 259.403 l\n635.407 260.318 l\n623.686 260.318 l\n623.686 259.403 l\n} bind def\n/p1_f7 {\nnewpath\ntranslate\n623.686 260.318 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  {
    "path": "radiation/radiation.tex",
    "content": "\\label{ch:radiation}\n\n\\section{Equations of Radiation Hydrodynamics}\n\n\\subsection{Reference Frames}\n\n\\subsection{Angular Approximations / Moments}\n\n\\subsection{Closures}\n\n\n\n\\section{Hyperbolic System}\n\n\n\n\\section{Parabolic System}\n\n\\subsection{General Elliptic Solver}\n\n\\label{sec:rad:generalelliptic}\n\nFor the gray radiation hydrodynamics system, we need to solve a\nlinear system that takes the form:\n\\begin{equation}\n  \\alpha \\phi + \\nabla \\cdot \\beta \\nabla \\phi + \\gamma \\cdot \\nabla \\phi = f\n\\end{equation}\n(see Zhang et al., Eq. 45). \nWe can discretize this for a cell-centered $\\phi$ to second-order as:\n\\begin{align}\n  \\alpha_{i,j} \\phi_{i,j} &+\n  \\frac{(\\beta \\nabla \\phi)_{i+1/2,j} -\n        (\\beta \\nabla \\phi)_{i-1/2,j}}{\\Delta x} +\n  \\frac{(\\beta \\nabla \\phi)_{i,j+1/2} -\n        (\\beta \\nabla \\phi)_{i,j-1/2}}{\\Delta y} \\nonumber \\\\\n  &+\n  \\gamma^{(x)}_{i,j} \\frac{\\phi_{i+1,j} - \\phi_{i-1,j}}{2\\Delta x} +\n  \\gamma^{(y)}_{i,j} \\frac{\\phi_{i,j+1} - \\phi_{i,j-1}}{2\\Delta y} = f_{i,j}\n\\end{align}\nwhere we decompose the vector, $\\gamma$ as $\\gamma = \\gamma^{(x)}\\hat{x} + \\gamma^{(y)}\\hat{y}$.  Expanding the gradients:\n\\begin{align}\n\\label{eq:general_L}\n\\alpha_{i,j} \\phi_{i,j} &+\n  \\frac{\\beta_{i+1/2,j} (\\phi_{i+1,j} - \\phi_{i,j}) -\n        \\beta_{i-1/2,j} (\\phi_{i,j} - \\phi_{i-1,j})}{\\Delta x^2} \\nonumber \\\\\n &+\n  \\frac{\\beta_{i,j+1/2} (\\phi_{i,j+1} - \\phi_{i,j}) -\n        \\beta_{i,j-1/2} (\\phi_{i,j} - \\phi_{i,j-1})}{\\Delta y^2} \\nonumber \\\\\n  &+\n  \\gamma^{(x)}_{i,j} \\frac{\\phi_{i+1,j} - \\phi_{i-1,j}}{2\\Delta x} +\n  \\gamma^{(y)}_{i,j} \\frac{\\phi_{i,j+1} - \\phi_{i,j-1}}{2\\Delta y} = f_{i,j}\n\\end{align}\n\nThere are several different linear system algorithms people use to solve\nthese types of systems.  Since we have already developed a multigrid\nsolver, we will add equations of this type to the multigrid framework.\n\nDefining $\\tilde{\\beta}_{i\\pm1/2,j} \\equiv \\beta_{i\\pm1/2,j}/\\Delta x^2$,\n$\\tilde{\\beta}_{i,j\\pm1/2} \\equiv \\beta_{i,j\\pm1/2}/\\Delta y^2$,\n$\\tilde{\\gamma}^{(x)}_{i,j} = \\gamma^{(x)}_{i,j}/(2\\Delta x)$,\nand $\\tilde{\\gamma}^{(y)}_{i,j} = \\gamma^{(y)}_{i,j}/(2\\Delta y)$, we have\nas an update to $\\phi_{i,j}$,\n\\begin{align}\n\\label{eq:general_mg_smooth}\n\\phi_{i,j} = \\frac{1}{D_{i,j}} \\bigg [ f_{i,j}\n  &-(\\tilde{\\beta}_{i+1/2,j} + \\tilde{\\gamma}^{(x)}_{i,j}) \\phi_{i+1,j}\n   -(\\tilde{\\beta}_{i-1/2,j} - \\tilde{\\gamma}^{(x)}_{i,j}) \\phi_{i-1,j}\n  \\nonumber \\\\\n  &-(\\tilde{\\beta}_{i,j+1/2} + \\tilde{\\gamma}^{(y)}_{i,j}) \\phi_{i,j+1}\n   -(\\tilde{\\beta}_{i,j-1/2} - \\tilde{\\gamma}^{(y)}_{i,j}) \\phi_{i,j-1}\n   \\bigg ]\n\\end{align}\nwith\n\\begin{equation}\nD_{i,j} = \\alpha_{i,j} - \\tilde{\\beta}_{i+1/2,j} - \\tilde{\\beta}_{i-1/2,j}\n                       - \\tilde{\\beta}_{i,j+1/2} - \\tilde{\\beta}_{i,j-1/2}\n\\end{equation}            \n\nThe remaining details of the multigrid solver are unchanged.  The boundary\nconditions are implemented in the same way as done for Poisson's equation.\nFor radiation, we will need to implement inhomogeneous boundary conditions,\nwhich we can do via the ghost cell filling using Eqs.~\\ref{eq:bc_inhomo_dir}\nand \\ref{eq:bc_inhomo_neum}.  Note that because of this general form\nof the equation, it is no longer simple to do the boundary charge method\ndescribed in \\S~\\ref{sec:multigrid:other}, so we must modify the ghost\ncell fill routines explicitly.\n\nThe only changes to the core solver are in the smoothing function, which\nimplements Eq.~\\ref{eq:general_mg_smooth} and in the residual, which\nconstructs our operator as discretized in Eq.~\\ref{eq:general_L}.\n\nTo test this solver out, we can define a test problem by picking functional\nforms of $\\alpha$, $\\beta$, and $\\gamma$, and a solution, $\\phi$, with\ndesired boundary conditions and find the resulting $f$ matching:\n\\begin{equation}\n  \\label{eq:general_elliptic}\n\\alpha \\phi + \\nabla \\cdot (\\beta \\nabla \\phi) + \\gamma \\cdot \\nabla \\phi = f\n\\end{equation}\nLet's find a set of coefficients and righthand side for which\n\\begin{equation}\n\\phi = \\cos(\\pi x/2) \\cos(\\pi y/2)\n\\end{equation}\nis a solution on $[0,1]\\times[0,1]$.  This would satisfy the Dirichlet\nboundary conditions:\n\\begin{align}\n\\phi |_{x=0} &= \\cos(\\pi y/2) \\\\\n\\phi |_{x=1} &= 0 \\\\\n\\phi |_{y=0} &= \\cos(\\pi x/2) \\\\\n\\phi |_{y=1} &= 0\n\\end{align}\nFor the coefficients, we choose:\non  with homogeneous Dirichlet boundary conditions.  We\nuse\n\\begin{align}\n\\alpha &= 10 \\\\\n\\beta  &= xy+1 \\\\\n\\gamma &= \\hat{x} + \\hat{y}\n\\end{align}\nThis gives\n\\begin{align}\n  \\label{eq:general_elliptic_rhs}\nf = &- \\frac{\\pi}{2} (x + 1)\n   \\sin{\\left (\\frac{\\pi y}{2} \\right )} \\cos{\\left (\\frac{\\pi x}{2} \\right )}\n - \\frac{\\pi}{2} (y + 1)\n   \\sin{\\left (\\frac{\\pi x}{2} \\right )} \\cos{\\left (\\frac{\\pi y}{2} \\right )}\n    \\nonumber \\\\\n   &+ \\left(10- \\frac{x y+1}{2} \\pi^{2}\\right)\n    \\cos{\\left (\\frac{\\pi x}{2} \\right )} \\cos{\\left (\\frac{\\pi y}{2} \\right )}\n\\end{align}\n\nFigure~\\ref{fig:general_mg_converge} shows the convergence of multigrid\nfor this problem, and figure~\\ref{fig:general_mg_solution} shows the solution.\nWe see second order convergence, as expected with this discretization.\n\n\\begin{figure}\n  \\centering\n\\includegraphics[width=0.6\\linewidth]{mg_general_inhomogeneous_converge}\n\\caption[Convergence of the general elliptic multigrid\n  solver]{\\label{fig:general_mg_converge} Convergence of the multigrid\n  solver on our test problem $\\alpha \\phi + \\nabla \\cdot (\\beta \\nabla \\phi) +\n  \\gamma \\cdot \\nabla \\phi = f$ with inhomogeneous Dirichlet boundary conditions.}\n\\end{figure}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\linewidth]{mg_general_inhomogeneous_test}\n  \\caption[Solution of a general elliptic equation]\n          {\\label{fig:general_mg_solution} The solution to our general\n            elliptic text problem, Eqs.~\\ref{eq:general_elliptic} to\n            \\ref{eq:general_elliptic_rhs} with inhomogeneous boundary\n            conditions on a $512^2$ grid.  This test can be run in \\pyro\\\n            {\\tt multigrid/test\\_mg\\_general\\_inhomogeneous.py}.}\n\\end{figure}\n"
  },
  {
    "path": "reactive_flow/notes.txt",
    "content": "-- mass fractions, partial density\n\n-- ODEs for reaction network\n\n-- number density\n\n-- coupling (Strang splitting vs. SDC)\n\n-- flames\n\n-- detonations\n\n\n"
  },
  {
    "path": "reactive_flow/reactive_flow.tex",
    "content": "\n\\section{Introduction}\n\nMany astrophysical problems involve modeling nuclear (or chemical)\nreactions coupled with hydrodynamics.  Astrophysical reacting flows\nshare a lot of similarities with terrestrial combustion, including\nboth subsonic (deflagrations) and supersonic (detonations) burning\nfronts.  Nice discussions of the physics of reacting flows can be\nfound in, e.g., \\cite{oranboris, peters:2000}\n\n\nFor multifluid flows, the Euler equations are\naugmented with continuity equations for each of the nuclear (or chemical) species.  If we denote the density of species $k$ as\n$\\rho_k$, then conservation of mass for each species individually\nimplies:\n\\begin{equation}\n\\label{react:eq:partialdensitycont}\n\\ddt{\\rho_k} + \\nabla \\cdot (\\rho_k \\Ub) = 0\n\\end{equation}\nSumming Eq.~\\ref{react:eq:partialdensitycont} over $k$ gives us back\nthe mass contunity equation, since $\\sum_k \\rho_k = \\rho$.  The\n$\\rho_k$ are sometimes called {\\em partial densities}.  Note that\nusing both Eq.~\\ref{react:eq:partialdensitycont} \nand the normal mass continuity equation for $\\rho$ overspecifies \nthe system.\n\nIt is often\neasier to define a mass fraction of species $k$, $X_k$, as\n\\begin{equation}\nX_k = \\frac{\\rho_k}{\\rho}\n\\end{equation}\nand it is easy to see that $\\sum_k X_k = 1$, and \n\\begin{equation}\n\\label{react:eq:partialrho}\n\\frac{\\partial (\\rho X_k)}{\\partial t} + \\nabla \\cdot  (\\rho X_k \\Ub) = 0\n\\end{equation}\n\n\\begin{exercise}[Species advection]\nUsing the continuity equation, show that we can write\nEq.~\\ref{react:eq:partialrho} as an advection equation:\n\\begin{equation}\n\\frac{\\partial X_k}{\\partial t} + \\Ub \\cdot \\nabla X_k = 0\n\\end{equation}\nor \n\\begin{equation}\n\\DDt{X_k} = 0\n\\end{equation}\n\\end{exercise}\n\nThis latter form says that as a fluid element advects with the flow,\nits composition does not change.  However, reactions can turn one\nspecies into another, so for reacting flow, $DX_k/Dt \\ne 0$.  If we\ndefine the creation rate for species $k$ as $\\omegadot_k$, then we\nhave:\n\\begin{equation}\n\\DDt{X_k} = \\omegadot_k\n\\end{equation}\nSince $\\sum_k X_k = 1$, it must be that $\\sum_k \\omegadot_k = 0$.\n\nReactions will also release (or consume) energy, due to the change in\nnuclear or chemical binding energy.  This energy will be a source to\nthe internal energy.  In particular, our first law of thermodynamics\nnow has a source:\\MarginPar{this assumes chemical\n  equilibrium}\n\\begin{equation}\nT ds = de + p d \\left( \\frac{1}{\\rho}\\right ) = q_\\mathrm{react}\n\\end{equation}\nwhere $q_\\mathrm{react}$ is the specific energy release due to\nreactions (energy / mass).  Following a fluid element, this takes the\nform:\n\\begin{equation}\nT\\DDt{s} = \\DDt{e} + p \\DDt{(1/\\rho)} = \\Hnuc\n\\end{equation}\nWhere now $\\Hnuc$ is the time-rate of energy release per unit mass\n($dq_\\mathrm{react}/dt$).\n\nAn additional energy source that we often consider along with reactions\nis thermal diffusion.  Fick's law says that the heat flux is proportional\nto $\\nabla T$, so we write:\n\\begin{equation}\nF_\\mathrm{cond} = -\\kth \\nabla T\n\\end{equation}\nas the conductive heat flux.  Here the `$-$' indicates that heat flows\nfrom hot to cold and $\\kth$ is the thermal conductivity.  See\n\\S~\\ref{sec:diffusion} for more discussion.\n\nThe conservative Euler equations with reactive and diffusive source terms\nappear as:\n\\begin{align}\n\\ddt{\\rho} + \\nabla \\cdot (\\rho \\Ub) &= 0 \\\\\n\\ddt{\\rho X_k} + \\nabla \\cdot (\\rho \\Ub X_k) &= \\rho \\omegadot_k \\\\\n\\ddt{(\\rho \\Ub)} + \\nabla \\cdot (\\rho \\Ub \\Ub) + \\nabla p &= 0 \\\\\n\\ddt{(\\rho E)} + \\nabla \\cdot (\\rho E \\Ub + p \\Ub ) &= \\nabla \\cdot (\\kth \\nabla T) + \\rho \\Hnuc\n\\end{align}\nWe'll write the heat sources simply as $H$, \n\\begin{equation}\nH = \\Hnuc + \\frac{1}{\\rho} \\nabla \\cdot \\kth \\nabla T\n\\end{equation}\n\nIn primitive form, the derivation of the pressure equation is a\ncomplex.  Previously, we found the pressure evolution in the absence\nof sources by differentiating pressure along streamlines\n(Eq.~\\ref{eq:euler:presnosource}):\n\\begin{equation}\n\\frac{Dp}{Dt} = \\left . \\frac{\\partial p}{\\partial \\rho} \\right |_s\n     \\DDt{\\rho} + \n     \\left . \\frac{\\partial p}{\\partial s} \\right |_\\rho\n     \\DDt{s}\n\\end{equation}\nwhere, as before, we recognize that $\\Gamma_1 \\equiv \\partial \\log p/\\partial \\log \\rho |_s$.\nNow, however, we have an entropy source, so $Ds/Dt \\ne 0$.  \nThe Maxwell relations\\footnote{see, for instance, \\cite{shu}, for some discussion\non these relations} tell us that\n\\begin{equation}\n\\partial p/\\partial s |_\\rho = \\rho^2 \\partial T/\\partial \\rho |_s\n\\end{equation}\nThis gives us the form:\n\\begin{equation}\n\\frac{Dp}{Dt} = \\frac{p}{\\rho} \\Gamma_1 \\frac{D\\rho}{Dt} +\n     \\left . \\rho^2 \\frac{\\partial T}{\\partial \\rho} \\right |_s\n     \\frac{H}{T}\n\\end{equation}\n\nWe need an expression for $\\partial T/\\partial \\rho |_s$ in terms of\nderivatives of $\\rho$ and $T$ (since that is what our equations of\nstate typically provide).  Consider $\\Gamma_1$, with $p = p(\\rho, T)$:\n\\begin{equation}\n  \\Gamma_1 = \\frac{\\rho}{p} \\left . \\frac{dp}{d\\rho} \\right |_s\n  = \\frac{\\rho}{p} \\left [ p_\\rho + p_T \\left . \\frac{dT}{d\\rho} \\right |_s \\right ]\n\\end{equation}\nwhere we use the shorthand $p_\\rho \\equiv \\partial p/\\partial \\rho |_T$ and\n$p_T \\equiv \\partial p/\\partial T |_\\rho$.  This tells us that\n\\begin{equation}\n  \\left . \\frac{\\partial T}{\\partial \\rho} \\right |_s = \\frac{p}{\\rho p_T} (\\Gamma_1 - \\chi_\\rho)\n\\end{equation}\nwhere $\\chi_\\rho \\equiv \\partial \\log p / \\partial \\log \\rho |_T$.  To further simplify this,\nwe use some standard relations for general equations of state\n\\begin{equation}\n\\frac{\\Gamma_1}{\\chi_\\rho} = \\frac{c_p}{c_v}\n\\end{equation}\nand\n\\begin{equation}\n  c_p - c_v = \\frac{p}{\\rho T} \\frac{\\chi_T^2}{\\chi_\\rho}\n\\end{equation}\nwith $\\chi_T \\equiv \\partial \\log p / \\partial \\log T |_\\rho$\n(see, e.g., \\cite{HKT} for both of these relations).  These allow us to write\n\\begin{equation}\n  \\left . \\frac{\\partial T}{\\partial \\rho} \\right |_s = \\frac{p\\chi_T}{\\rho^2 c_v}\n\\end{equation}\n\nPutting this all together, we have:\n\\begin{equation}\n\\frac{Dp}{Dt} = \\frac{p}{\\rho}\\Gamma_1  \\frac{D\\rho}{Dt}\n   + \\frac{p \\chi_T}{c_v T} H\n\\end{equation}\nand using the continuity equation, $D\\rho/Dt = -\\rho \\nabla \\cdot \\Ub$, \nwe finally have:\n\\begin{equation}\n\\label{eq:reacting:p}\n\\ddt{p} + \\Ub \\cdot \\nabla p + \\Gamma_1 p \\nabla \\cdot \\Ub = \\Gamma_1 p \\sigma H\n\\end{equation}\nwhere we defined\n\\begin{equation}\n\\sigma = \\frac{\\partial p/\\partial T |_\\rho}{\\rho c_p \\partial p/\\partial \\rho |_T}\n\\end{equation}\nThe $\\sigma$ notation follows from \\cite{ABRZ:II}.\n\nThe primitive variable is then\n\\begin{align}\n\\ddt{\\rho} + \\Ub \\cdot \\nabla \\rho + \\rho \\nabla \\cdot \\Ub &= 0\\\\\n\\ddt{\\Ub} + \\Ub \\cdot\\nabla \\Ub + \\frac{1}{\\rho} \\nabla p &= 0\\\\\n\\ddt{p} + \\Ub \\cdot \\nabla p + \\Gamma_1 p \\nabla \\cdot \\Ub &= \\Gamma_1 p \\sigma H \\\\\n\\ddt{X_k} + \\Ub \\cdot \\nabla {X_k} &= \\omegadot_k \n\\end{align}\n\nFinally, we can derive a temperature evolution equation following the \nmethod from \\S~\\ref{sec:euler:otherthermo}.  We find\n\\begin{equation}\nc_v \\DDt{T} = \\left ( \\left . \\frac{\\partial e}{\\partial \\rho} \\right |_T - \\frac{p}{\\rho} \\right ) \\nabla \\cdot \\Ub\n  + H\n\\end{equation}\nor\n\\begin{equation}\nc_p \\DDt{T} = \\left ( \\frac{1}{\\rho} - \\left . \\frac{\\partial h}{\\partial p} \\right |_T \\right ) \\DDt{p}\n + H\n\\end{equation}\ndepending on whether we start with $e$ or $h$.  \n\n\n\\section{Operator splitting approach}\n\nUsing operator splitting, we separate the equations into hydrodynamics\nand reactive parts and we do these operations in turn.  Following the\nideas from \\S~\\ref{sec:multiphys:diffreact}, we perform the update\nin stages.  If we denote the reaction update operator as $R_{\\Delta t}$\nand the hydrodynamics operator as $A_{\\Delta t}$, then our update \nappears as:\n\\begin{equation}\n  \\Uc^{n+1} = R_{\\Delta t/2} A_{\\Delta t} R_{\\Delta t/2} \\Uc^n\n\\end{equation}\n\nTo make it more explicit, we write out system as:\n\\begin{equation}\n\\ddt{\\Uc} + \\nabla \\cdot \\Fb(\\Uc) = \\Rc(\\Uc)\n\\end{equation}\nwhere $\\Rc = (0, \\omegadot_k, 0, \\rho \\Hnuc)^\\intercal$ is the vector\nof reactive sources.  The advance through $\\Delta t$ in time is:\n\\begin{enumerate}\n\\item {\\em Evolve reaction part for $\\Delta t/2$}.\n\n  We define $\\Uc_i^\\star$ as the result of integrating\n  \\begin{equation}\n    \\frac{d\\Uc_i}{dt} = \\Rc(\\Uc_i)\n  \\end{equation}\n  over $t \\in [t^n, t^{n+\\myhalf}]$ with initial conditions, \n  $\\Uc_i(t^n) = \\Uc_i^n$.\n\n\\item {\\em Solve the hydrodynamics portion for $\\Delta t$, beginning\n    with the state $\\Uc^\\star$}\n   %\n   \\begin{equation}\n     \\frac{\\Uc^{\\star\\star}_i - \\Uc^\\star_i}{\\Delta t} =\n      \\frac{\\Fb^\\exx(\\Uc^{\\star,n+\\myhalf}_{i-\\myhalf}) - \\Fb^\\exx(\\Uc^{\\star,n+\\myhalf}_{i+\\myhalf})}{\\Delta x}\n   \\end{equation}\n  where $\\Uc^{\\star,n+\\myhalf}_{i-\\myhalf}$ is the predict interface\n  state (using the ideas from \\S~\\ref{sec:onedrecon}), beginning with\n  the state $\\Uc^\\star$.  Note that $\\Rc$ does not appear here\n  explicitly.\n\n  Alternately, we\n  could use the MOL hydrodynamics update here, but the basic idea will\n  be the same---the hydrodynamics does not explicitly see the reaction\n  terms.\n\n\\item {\\em Evolve reaction part for $\\Delta t/2$}\n\n  We reach the final update, $\\Uc^{n+1}$, by integrating\n  \\begin{equation}\n    \\frac{d\\Uc}{dt} = \\Rc(\\Uc)\n  \\end{equation}\n  over $t \\in [t^{\\myhalf}, t^{n+1}]$ with initial conditions\n    $\\Uc(t^{n+\\myhalf}) = \\Uc^{\\star\\star}$.\n\n\\end{enumerate}\n\n\n\n\\subsection{Adding species to hydrodynamics}\n\nIn the hydrodynamics algorithm, we neglect the diffusion and reaction terms.  Their\neffects are incorporated implicitly by performing the hydrodynamics\nupdate starting with the state that has already experienced burning.\n\\begin{align}\n\\ddt{\\rho} + \\nabla \\cdot (\\rho \\Ub) &= 0 \\\\\n\\ddt{(\\rho X_k)} + \\nabla \\cdot (\\rho \\Ub X_k) &= 0 \\\\\n\\ddt{(\\rho \\Ub)} + \\nabla \\cdot (\\rho \\Ub \\Ub) + \\nabla p &= 0 \\\\\n\\ddt{(\\rho E)} + \\nabla \\cdot (\\rho E \\Ub + p \\Ub ) &= 0\n\\end{align}\n\n\n\nWhen we now consider our primitive variables: $\\qb = (\\rho, u, p, X_k)$,\nwe find\n\\begin{equation}\n\\Ab(\\qb) = \\left ( \\begin{array}{cccc} u  & \\rho     & 0      &  0\\\\\n                                  0  &  u       & 1/\\rho &  0\\\\\n                                  0  & \\Gamma_1 p & u      &  0\\\\\n                                  0  & 0        & 0      & u \\end{array} \\right )\n\\end{equation}\nThere are now 4 eigenvalues, with the new one also being simply $u$.\nThis says that the species simply advect with the flow.  The right\neigenvectors are now:\n\\begin{equation}\n\\rb^\\evm = \\left ( \\begin{array}{c} 1 \\\\ -c/\\rho \\\\ c^2 \\\\ 0\\end{array} \\right )\n%\n\\qquad\n\\rb^\\evz = \\left ( \\begin{array}{c} 1 \\\\ 0 \\\\ 0  \\\\ 0\\end{array} \\right )\n%\n\\qquad\n\\rb^\\evzs{X} = \\left ( \\begin{array}{c} 0 \\\\ 0  \\\\ 0 \\\\ 1 \\end{array} \\right )\n%\n\\qquad\n\\rb^\\evp = \\left ( \\begin{array}{c} 1 \\\\ c/\\rho \\\\ c^2 \\\\ 0 \\end{array} \\right )\n\\end{equation}\ncorresponding to $\\lambda^\\evm = u -c$, $\\lambda^\\evz = u$,\n$\\lambda^\\evzs{X} = u$, and $\\lambda^\\evp = u + c$.  We see that for\nthe species, the only non-zero element is for one of the $u$\neigenvectors ($\\rb^\\evzs{X}$).  This means that $X_k$ only jumps over\nthis middle wave.  In the Riemann solver then, there is no `star'\nstate for the species, it just jumps across the contact wave.\n\nTo add species into the solver, you simply need to reconstruct $X_k$\nas described in Chapter~\\ref{ch:compressible} to find the interface values.  If we \nare doing characteristic tracing, then we use this new $\\Ab(\\qb)$\nand associated eigenvectors.  We then solve the Riemann problem, with $X_k$ on\nthe interface taken as simply the left or right state depending on the\nsign of the contact wave speed, and do the conservative update for\n$\\rho X_k$ using the species flux.\n\n\nOne issue that can arise with species is that even if $\\sum_k X_k = 1$\ninitially, after the update, that may no longer be true.  There are a\nvariety of ways to handle this:\n\\begin{itemize}\n\\item You can update the species, $(\\rho X_k)$ to the new time and then\ndefine the density to be $\\rho = \\sum_k (\\rho X_k)$---this means that\nyou are not relying on the value of the density from the mass continuity\nequation itself.\n\n\\item You can force the interface states of $X_k$ to sum to 1.  Because\nthe limiting is non-linear, this is where problems can arise.  If the\ninterface values of $X_k$ are forced to sum to 1 (by renormalizing), then\nthe updated cell-centered value of $X_k$ will as well.  This is the\napproach discussed in \\cite{plewamuller:1999}.\n\n\\item You can design the limiting procedure to preserve the summation\nproperty.  This approach is sometimes taken in the combustion field.\nFor piecewise linear reconstruction, this can be obtained by computing\nthe limited slopes of all the species, and taking the most restrictive\nslope and applying this same slope to all the species.\n\\end{itemize}\n\n\n\\subsection{Integrating the reaction network}\n\nThe reactive portion of our operator split system is:\n\\begin{align}\n\\frac{dX_k}{dt} &= \\omegadot_k \\\\\n\\frac{d(\\rho e)}{dt} &= \\rho \\Hnuc\n\\end{align}\nAs written, this system is not closed without the equation of state,\nsince $\\omegadot_k = \\omegadot_k(\\rho, T, X_k)$, and likewise for\n$\\Hnuc$.  Our temperature evolution equation reduces to\n\\begin{equation}\nc_x \\frac{dT}{dt} = \\Hnuc\n\\end{equation}\nwhere $c_x$ is $c_v$ or $c_p$, depending on the physical effects we are\ntrying to capture.\n\\if DEBUG\n\\footnote{formally, there should be a composition term here too, of the\nform\n\\[\n\\sum_k \\left . \\frac{\\partial e}{\\partial X_k} \\right |_{\\rho,T} \\frac{dX_k}{dt}  =\n      \\sum_k \\mu_k \\frac{dX_k}{dt} \n\\]\nwhere $\\mu_k$ is the chemical potential.  But this term is zero if we\nare in chemical equilibrium, which is usually assumed (but perhaps\nshould be checked) for astrophysical flows.  }\n\\fi\nThis difference only arises because we are neglecting the hydrodynamic\nportions of the temperature evolution during the reaction step.\n\nMany simulation codes neglect the evolution of temperature during the\nreaction step (see, e.g., \\cite{flash}).\n\nAn excellent introduction to integrating astrophysical nuclear\nreaction networks is found in \\cite{timmes_nets}.  The main issue with\nreaction networks is that they tend to be stiff, which means that\nimplicit integration techniques are needed.  There are a number of\nwell-tested, freely available implicit ODE integrators\n(e.g.~\\cite{vode}).\n\nFor stiff systems, you need to supply both a righthand side routine\nand a Jacobian routine\\footnote{although this could be calculated via\n  finite-differences if needed}.\n\n\n\\subsection{Incorporating explicit diffusion}\n\n\n\n\\section{Burning modes}\n\n\n\\subsection{Convective burning}\n\n\n\\subsection{Deflagrations}\n\nA deflagration or flame is a subsonic burning front that propagates\nvia a balance of diffusion and reactions.  Thermal diffusion raises\nthe temperature of fuel ahead of the flame to ignition and the energy\nrelease heats the ash, keeping the thermal gradient needed for\ndiffusion.  In terrestrial flames, species diffusion can be important\nas well, but this is usually negligible in astrophysical flames.\n\nBecause the burning front is subsonic, sound waves can communicate\nacross the flame and therefore the fuel and ash are at the same\npressure.  The temperature jump across the flame means that the\ndensity will drop behind the flame to preserve the constant pressure.\nOur temperature evolution equation, taking $Dp/Dt = 0$, is\n\\begin{equation}\n\\rho c_p \\DDt{T} = \\nabla \\cdot \\kth \\nabla T + \\rho \\Hnuc\n\\end{equation}\n\nWe can estimate the timescale for diffusion by neglecting the reactions\nand writing\n\\begin{equation}\n\\DDt{T} = \\frac{1}{\\rho c_p} \\nabla \\cdot \\kth \\nabla T \\approx \\mathcal{D} \\nabla^2 T\n\\end{equation}\nwhere $\\mathcal{D} = \\kth/(\\rho c_p)$ is the diffusion coefficient\n(assuming $\\rho c_p$ is spatially constant). Dimensional analysis suggests that the \ndiffusion timescale is then\n\\begin{equation}\nt_\\mathrm{diff} \\approx \\frac{\\delta^2}{\\mathcal{D}}\n\\end{equation}\nwhere $\\delta$ is the width of the region where diffusion takes\nplace---we'll identify this as the flame thickness.\n\nSimilarly, we can estimate the reaction timescale by considering\n\\begin{equation}\n\\rho c_p \\DDt{T} = \\rho \\Hnuc\n\\end{equation}\nand writing\n\\begin{equation}\nt_\\mathrm{burn} \\approx \\frac{c_p T}{\\Hnuc}\n\\end{equation}\nWe can alternately expand $\\Hnuc$ in terms of temperature\nif we wish to capture some of the temperature sensitivity.\n\nEquating these timescales gives\n\\begin{equation}\n\\delta \\sim \\left ( \\frac{\\kth T}{\\rho \\Hnuc} \\right )^{1/2}\n\\end{equation}\nwhich is an estimate of the flame thickness.  We can define the flame\nspeed as the time it takes to burn through this thickness:\n\\begin{equation}\nv_f \\sim \\frac{\\delta}{t_\\mathrm{burn}} \\sim \\left ( \\frac{\\kth \\Hnuc}{\\rho c_p T} \\right )^{1/2}\n\\end{equation}\nWe note the main dependences are:\n\\begin{equation}\nv_f \\sim \\sqrt{\\kth \\Hnuc} \\qquad \\delta \\sim \\sqrt{\\frac{\\kth}{\\Hnuc}}\n\\end{equation}\n\n\\subsection{Detonations}\n\n"
  },
  {
    "path": "refs.bib",
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LeVeque},\n  publisher = {SIAM, Society of Industrial and Applied Mathematics },\n  series = {Classics in Applied Mathematics},\n  title = {Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems},\n  year = 2007\n}\n\n\n@ARTICLE{plewamuller:1999,\n   author = {{Plewa}, T. and {M{\\\"u}ller}, E.},\n    title = \"{The consistent multi-fluid advection method}\",\n  journal = {Astron Astrophys},\n   eprint = {arXiv:astro-ph/9807241},\n keywords = {HYDRODYNAMICS, NUCLEAR REACTIONS, NUCLEOSYNTHESIS, ABUNDANCES, METHODS: NUMERICAL, STARS: SUPERNOVAE: GENERAL},\n     year = 1999,\n    month = feb,\n   volume = 342,\n    pages = {179-191},\n   adsurl = {http://adsabs.harvard.edu/abs/1999A%26A...342..179P},\n  adsnote = {Provided by the SAO/NASA Astrophysics Data System}\n}\n\n@ARTICLE{bryan:1995,\n   author = {{Bryan}, G.~L. and {Norman}, M.~L. and {Stone}, J.~M. and {Cen}, R. and\n{Ostriker}, J.~P.},\n    title = \"{A piecewise parabolic method for cosmological hydrodynamics}\",\n  journal = {Computer Physics Communications},\n     year = 1995,\n    month = aug,\n   volume = 89,\n    pages = {149-168},\n      doi = {10.1016/0010-4655(94)00191-4},\n   adsurl = {http://adsabs.harvard.edu/abs/1995CoPhC..89..149B},\n  adsnote = {Provided by the SAO/NASA Astrophysics Data System}\n}\n\n\n@book{multigridtutorial,\n  author = {Briggs, W.~L. and Henson, V-E. and McCormick, S.~F.},\n  publisher = {SIAM},\n  title = {A Mutigrid Tutorial, 2nd Ed.},\n  year = 2000\n}\n\n\n@book{colellanotes,\n  author = {Colella, P. and Puckett, E.~G.},\n  publisher = {unpublished manuscript},\n  title = {Modern Numerical Methods for Fluid Flow},\n  note = {obtained from http://www.amath.unc.edu/Faculty/minion/class/puckett/}\n}\n\n@ARTICLE{BCG,\n   author = {{Bell}, J.~B. and {Colella}, P. and {Glaz}, H.~M.},\n    title = \"{A Second Order Projection Method for the Incompressible Navier-Stokes Equations}\",\n  journal = {J Comput Phys},\n     year = 1989,\n    month = dec,\n   volume = 85,\n    pages = {257},\n      doi = {10.1016/0021-9991(89)90151-4},\n   adsurl = {http://adsabs.harvard.edu/abs/1989JCoPh..85..257B},\n  adsnote = {Provided by the SAO/NASA Astrophysics Data System}\n}\n\n\n@INPROCEEDINGS{BCH,\n        AUTHOR = {J. B. Bell and P. Colella and L. H. Howell},\n        TITLE = {An efficient second-order projection method\n         for viscous incompressible flow},\n        YEAR = {1991},\n        BOOKTITLE = {Proceedings of the Tenth AIAA Computational\n                     Fluid Dynamics Conference},\n        MONTH = Jun,\n        ORGANIZATION = {AIAA},\n        PAGES = {360--367},\n        note = {see also: https://seesar.lbl.gov/anag/publications/colella/A\\_2\\_10.pdf}\n}\n\n@ARTICLE{ABS,\n        TITLE = {A numerical method for the incompressible {N}avier-{S}tokes\n        equations based on an approximate projection},\n        AUTHOR = {A. S. Almgren and J. B. Bell and W. G. Szymczak},\n        YEAR = {1996},\n        VOLUME = {17},\n        NUMBER = {2},\n        MONTH = Mar,\n        JOURNAL = {SIAM J Sci Comput},\n        PAGES = {358--369}\n}\n\n@ARTICLE{chorin:1968,\n        AUTHOR = {A. J. Chorin},\n        TITLE = {Numerical Solution of the {N}avier-{S}tokes Equations},\n        JOURNAL = {Math. Comp.},\n        VOLUME = {22},\n        PAGES = {745--762},\n        YEAR = {1968}\n}\n\n@ARTICLE{MartinColella,\n   author = {{Martin}, D.~F. and {Colella}, P.},\n    title = \"{A Cell-Centered Adaptive Projection Method for the Incompressible Euler Equations}\",\n  journal = {Journal of Computational Physics},\n     year = 2000,\n    month = sep,\n   volume = 163,\n    pages = {271-312},\n      doi = {10.1006/jcph.2000.6575},\n   adsurl = {http://adsabs.harvard.edu/abs/2000JCoPh.163..271M},\n  adsnote = {Provided by the SAO/NASA Astrophysics Data System}\n}\n\n@TECHREPORT{rider,\n    author = {{Rider}, W.~J. },\n    title = {Approximate Projection Methods for Incompressible Flow: Implementation, Variants and Robustness.},\n    institution = {LANL UNCLASSIFIED REPORT LA-UR-94-2000, LOS ALAMOS NATIONAL LABORATORY},\n    year = {1995}\n}\n\n\n@ARTICLE{ABC,\nauthor =   {A. S. Almgren and J. B. Bell and W. Y. Crutchfield},\ntitle =    {Approximate Projection Methods: Part {I}. {I}nviscid Analysis},\njournal =  {SIAM J Sci Comput},\nvolume={22},\nnumber={4},\npages={1139--59},\nyear =     {2000}\n}\n\n\n@article{BellMarcus,\ntitle = \"A second-order projection method for variable-density flows\",\njournal = JCP,\nvolume = \"101\",\nnumber = \"2\",\npages = \"334 - 348\",\nyear = \"1992\",\nnote = \"\",\nissn = \"0021-9991\",\ndoi = \"10.1016/0021-9991(92)90011-M\",\nurl = \"http://www.sciencedirect.com/science/article/pii/002199919290011M\",\nauthor = \"Bell, J.~B. and Marcus, D.~L.\"\n}\n\n\n@Article{SNpaper,\n  author  =      {J.~B. Bell and M.~S. Day and C.~A. Rendleman and\n                  S.~E. Woosley and M.~A. Zingale},\n  title   =      {Adaptive low {M}ach number simulations of nuclear flame microphysics},\n  journal =      {J Comput Phys},\n  year    =      {2004},\n  volume  =      {195},\n  number  =      {2},\n  pages   =      {677-694}\n}\n\n@Article{pember-flame,\n  author =       {R. B. Pember and L. H. Howell and J. B. Bell and\n                  P. Colella and W. Y. Crutchfield and W. A. Fiveland\n                  and J. P. Jessee},\n  title =        {An Adaptive Projection Method for Unsteady\n                  Low-{M}ach Number Combustion},\n  journal =      {Comb. Sci. Tech.},\n  volume =       {140},\n  pages =        {123--168},\n  year =         {1998},\n}\n\n@ARTICLE{DayBell:2000,\n  author={M. S. Day and J. B. Bell},\n  title={Numerical simulation of laminar reacting flows with complex chemistry},\n  journal={Combust. Theory Modelling},\n  volume={4},\n  number={4},\n  pages={535-556},\n  year={2000}\n }\n\n@ARTICLE{minion:1996,\n   author = {{Minion}, M.~L.},\n    title = \"{A Projection Method for Locally Refined Grids}\",\n  journal = JCP,\n     year = 1996,\n   volume = 127,\n    pages = {158--177}\n}\n\n@ARTICLE{malone:2011,\n   author = {{Malone}, C.~M. and {Nonaka}, A. and {Almgren}, A.~S. and {Bell}, J.~B. and\n{Zingale}, M.},\n    title = \"{Multidimensional Modeling of Type I X-ray Bursts. I. Two-dimensional Convection Prior to the Outburst of a Pure $^{4}$He Accretor}\",\n  journal = APJ,\narchivePrefix = \"arXiv\",\n   eprint = {1012.0609},\n primaryClass = \"astro-ph.HE\",\n keywords = {convection, hydrodynamics, methods: numerical, stars: neutron, X-rays: bursts},\n     year = 2011,\n    month = feb,\n   volume = 728,\n      eid = {118},\n    pages = {118},\n      doi = {10.1088/0004-637X/728/2/118},\n   adsurl = {http://adsabs.harvard.edu/abs/2011ApJ...728..118M},\n  adsnote = {Provided by the SAO/NASA Astrophysics Data System}\n}\n\n@ARTICLE{hse,\n   author = {{Zingale}, M. and {Dursi}, L.~J. and {ZuHone}, J. and {Calder}, A.~C. and\n{Fryxell}, B. and {Plewa}, T. and {Truran}, J.~W. and {Caceres}, A. and\n{Olson}, K. and {Ricker}, P.~M. and {Riley}, K. and {Rosner}, R. and\n{Siegel}, A. and {Timmes}, F.~X. and {Vladimirova}, N.},\n    title = \"{Mapping Initial Hydrostatic Models in Godunov Codes}\",\n  journal = APJ:sup,\n   eprint = {arXiv:astro-ph/0208031},\n keywords = {Hydrodynamics, Methods: Numerical, Stellar Dynamics},\n     year = 2002,\n    month = dec,\n   volume = 143,\n    pages = {539-565},\n      doi = {10.1086/342754},\n   adsurl = {http://adsabs.harvard.edu/abs/2002ApJS..143..539Z},\n  adsnote = {Provided by the SAO/NASA Astrophysics Data System}\n}\n\n\n@ARTICLE{BDS,\n   author = {J. B. Bell and C. N. Dawson and G. R. Shubin},\n    title = {An unsplit, higher order {G}odunov method for scalar conservation l\naws in\n             multiple dimensions},\n  journal = JCP,\n  volume  = {74},\n     year = {1988},\n  pages   = {1-24}\n}\n\n\n@article{quadBDS,\n   author = {S. May and A. J. Nonaka and A. S. Almgren and J. B. Bell},\n   title = {An Unsplit, Higher Order {G}odunov Method Using Quadratic\n           Reconstruction for Advection in Multiple Dimensions},\n     year ={2011},\n  journal =\"{Communications in Applied Mathematics and Computational Science}\",\n  volume = 6,\n  number = 1\n}\n\n@Book{laney,\n  author =       {C. B. 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J. Berger and P. Colella},\n  title =        {Local Adaptive Mesh Refinement for Shock Hydrodynamics},\n  journal =      JCP,\n  volume =       {82},\n  number =       {1},\n  pages =        {64--84},\n  month =        May,\n  year =         {1989},\n}\n\n@article{goldberg:1991,\n    author = {David Goldberg},\n    title = {What Every Computer Scientist Should Know About Floating Point Arithmetic},\n    journal = {ACM Computing Surveys},\n    year = {1991},\n    volume = {23},\n    number = {1},\n    pages = {5--48}\n}\n\n\n@article{strang,\n     jstor_articletype = {research-article},\n     title = {On the Construction and Comparison of Difference Schemes},\n     author = {Strang, G.},\n     journal = {SIAM Journal on Numerical Analysis},\n     jstor_issuetitle = {},\n     volume = {5},\n     number = {3},\n     jstor_formatteddate = {Sep., 1968},\n     pages = {pp. 506-517},\n     url = {http://www.jstor.org/stable/2949700},\n     ISSN = {00361429},\n     abstract = {},\n     language = {English},\n     year = {1968},\n     publisher = {Society for Industrial and Applied Mathematics},\n     copyright = {Copyright © 1968 Society for Industrial and Applied Mathematics},\n}\n\n@ARTICLE{ABRZ:I,\n   author = {A. S. Almgren and J. B. Bell and C. A. Rendleman and M. Zingale},\n    title = \"{Low Mach Number Modeling of Type Ia Supernovae. I. Hydrodynamics}\",\n  journal = APJ,\n     year = 2006,\n   volume = 637,\n    pages = {922-936},\n    month = feb\n}\n\n@ARTICLE{ABRZ:II,\n   author = {A. S. Almgren and J. B. Bell and C. A. Rendleman and M. Zingale},\n    title = \"{Low Mach Number Modeling of Type Ia Supernovae. II. Energy Evolution}\",\n  journal = APJ,\n     year = 2006,\n   volume = 649,\n    pages = {927-938},\n    month = oct\n}\n\n@ARTICLE{ABNZ:III,\n   author = {{Almgren}, A.~S. and {Bell}, J.~B. and {Nonaka}, A. and\n             {Zingale}, M.},\n    title = \"{Low Mach Number Modeling of Type Ia Supernovae. III. Reactions}\",\n  journal = APJ,\n     year = 2008,\n   volume = 684,\n    pages = {449-470},\n      doi = {10.1086/590321},\n   adsurl = {http://adsabs.harvard.edu/abs/2008ApJ...684..449A},\n}\n\n\n@ARTICLE{SPH,\n   author = {{Monaghan}, J.~J.},\n    title = \"{An introduction to SPH}\",\n  journal = {Computer Physics Communications},\n     year = 1988,\n    month = jan,\n   volume = 48,\n    pages = {89-96},\n      doi = {10.1016/0010-4655(88)90026-4},\n   adsurl = {http://adsabs.harvard.edu/abs/1988CoPhC..48...89M},\n  adsnote = {Provided by the SAO/NASA Astrophysics Data System}\n}\n\n@ARTICLE{arepo,\n   author = {{Springel}, V.},\n    title = \"{E pur si muove: Galilean-invariant cosmological hydrodynamical simulations on a moving mesh}\",\n  journal = MNRAS,\narchivePrefix = \"arXiv\",\n   eprint = {0901.4107},\n primaryClass = \"astro-ph.CO\",\n keywords = {methods: numerical, galaxies: interactions, cosmology: dark matter},\n     year = 2010,\n    month = jan,\n   volume = 401,\n    pages = {791-851},\n      doi = {10.1111/j.1365-2966.2009.15715.x},\n   adsurl = {http://adsabs.harvard.edu/abs/2010MNRAS.401..791S},\n  adsnote = {Provided by the SAO/NASA Astrophysics Data System}\n}\n\n\n@ARTICLE{schulz-rinne:1993,\n  author = {{Schulz-Rinne}, C.~W. and {Collins}, J.~P. and {Glaz}, H.~M.},\n  title = \"{Numerical Solution of the Riemann Problem for Two-Dimensional\n            Gas Dynamics}\",\n  journal = {SIAM J Sci Comput},\n  volume = {14},\n  number = {6},\n  year = {1993},\n  pages = {1394-1414}\n}\n\n@ARTICLE{pyro,\n  author = {{Zingale}, M.},\n  title = \"{\\pyro: A teaching code for computational astrophysical hydrodynamics}\",\n  journal = {Astronomy and Computing},\n  year = 2014,\n  note = \"accepted for publication\"\n}\n\n@ARTICLE{byrnehindmarsh,\n  author = {{Byrne}, G.~D. and {Hindmarsh}, A.~C.},\n  title = \"{Stiff ODE Solvers: A Review of Current and Coming Attractions}\",\n   year = 1986,\n  journal = {UCRL-94297 Preprint}\n}\n\n@ARTICLE{almgren:2000,\n        AUTHOR = {A. Almgren},\n        TITLE = {A New Look at the Pseudo-incompressible Solution to\n                 {L}amb's problem of Hydrostatic Adjustment},\n        JOURNAL = JAS,\n        VOLUME = {57},\n        MONTH = Apr,\n        YEAR = {2000},\n        PAGES = {995--998}\n}\n\n\n@ARTICLE{KP:2012,\n   author = {Rupert Klein and Olivier Pauluis},\n    title = {Thermodynamic Consistency of a Pseudoincompressible Approximation for General Equations of State},\n  journal = JAS,\n    month = {March},\n     year = 2012\n}\n\n@ARTICLE{VLBWZ:2013,\n   author = {Geoffrey M. Vasil and Daniel Lecoanet and Benjamin P. Brown and Toby S. Wood and Ellen G. Zweibel},\n    title = {Energy Conservation and Gravity Waves in Sound-proof Treatments of Stellar Interiors. II.\n             Lagrangian Constrained Analysis},\n  journal = APJ,\n  volume = 773,\n  pages = {169-},\n     year = 2013\n}\n\n\n@ARTICLE{multilevel,\n   author = {A. Nonaka and A. S. Almgren and J. B. Bell and M. J. Lijewski and C. M. Malone and M. Zingale},\n    title = {{\\tt MAESTRO}: An Adaptive Low Mach Number Hydrodynamics Algorithm for Stellar Flows},\n  journal = APJ:sup,\n     year = {2010},\n   volume = {188},\n    pages = {358-383},\n}\n\n\n@ARTICLE{mccorquodalecolella,\n   author = {{McCorquodale}, P. and {Colella}, P.},\n  title = \"{A high-order finite-volume method for conservation laws on\n                  locally refined grids}\",\n   journal = {Communication in Applied Mathematics and Computational Science},\n    year = {2011},\n   volume = {6},\n  number = {1},\n  pages = {1--25}\n}\n\n@ARTICLE{timmes_networks,\n   author = {{Timmes}, F.~X.},\n    title = \"{Integration of Nuclear Reaction Networks for Stellar Hydrodynamics}\",\n  journal = APJ:sup,\n keywords = {HYDRODYNAMICS, METHODS: NUMERICAL, NUCLEAR REACTIONS, NUCLEOSYNTHESIS, ABUNDANCES, STARS: INTERIORS, Hydrodynamics, Methods: Numerical, Nuclear Reactions, Nucleosynthesis, Abundances, Stars: Interiors},\n     year = 1999,\n    month = sep,\n   volume = 124,\n    pages = {241-263},\n      doi = {10.1086/313257},\n   adsurl = {http://adsabs.harvard.edu/abs/1999ApJS..124..241T},\n  adsnote = {Provided by the SAO/NASA Astrophysics Data System}\n}\n\n@BOOK{HKT,\n  author = {{Hansen}, C.~J. and {Kawaler}, S.~D. and {Trimble}, V.},\n  title = \"{Stellar interiors : physical principles, structure, and evolution}\",\n  keywords = {STELLAR INTERIORS, STELLAR STRUCTURE, STELLAR EVOLUTION},\n  ooktitle = {Stellar interiors : physical principles, structure, and evolution, 2nd ed., by C.J.~Hansen, S.D.~Kawaler, and V.~Trimble.~New York: Springer-Verlag, 2004.},\n  year = 2004,\n  adsurl = {http://adsabs.harvard.edu/abs/2004sipp.book.....H},\n  adsnote = {Provided by the SAO/NASA Astrophysics Data System}\n }\n\n@Book{pang,\n  author = {{Pang}, T.},\n  title = {An Introduction to Computational Physics, 2nd Edition},\n  publisher = {Cambridge},\n  year = {2006}\n}\n\n@article{vladimirova2006,\nauthor = { Natalia Vladimirova and V.~Gregory Weirs and Lenya Ryzhik},\ntitle = \"{Flame capturing with an advection-reaction-diffusion model}\",\njournal = {Combustion Theory and Modelling},\nvolume = {10},\nnumber = {5},\npages = {727-747},\nyear = {2006},\ndoi = {10.1080/13647830500464146},\nURL = {http://dx.doi.org/10.1080/13647830500464146},\neprint = {http://dx.doi.org/10.1080/13647830500464146}\n}\n\n@Book{cg,\n  author =       {J. P. Cox and R. T. Giuli},\n  title =        {Principles of Stellar Structure},\n  publisher =    {Cambridge University Press},\n  year =         {1968},\n  OPTkey =       {},\n  volume =       {1},\n  OPTnumber =    {},\n  OPTseries =    {},\n  OPTaddress =   {},\n  OPTedition =   {},\n  OPTmonth =     {},\n  OPTnote =      {},\n  OPTannote =    {}\n}\n\n@article{titarevtoro,\ntitle = \"Finite-volume \\{WENO\\} schemes for three-dimensional conservation laws \",\njournal = \"Journal of Computational Physics \",\nvolume = \"201\",\nnumber = \"1\",\npages = \"238 - 260\",\nyear = \"2004\",\nnote = \"\",\nissn = \"0021-9991\",\ndoi = \"http://dx.doi.org/10.1016/j.jcp.2004.05.015\",\nurl = \"http://www.sciencedirect.com/science/article/pii/S0021999104002281\",\nauthor = \"V.A. Titarev and E.F. Toro\",\nkeywords = \"High-order schemes\",\nkeywords = \"Weighted essentially non-oscillatory\",\nkeywords = \"HLLC flux\",\nkeywords = \"FORCE flux\",\nkeywords = \"MUSTA flux\",\nkeywords = \"Two and three space dimensions \",\nabstract = \"In this paper we firstly carry out an extension of the\n                  finite-volume \\{WENO\\} schemes to three space\n                  dimensions and higher orders of accuracy. Secondly,\n                  we propose to use more accurate fluxes as the\n                  building block. These are the \\{HLLC\\} and \\{MUSTA\\}\n                  fluxes [Multi-stage predictor–corrector fluxes for\n                  hyperbolic equations, 2003; Restoration of the\n                  contact surface in the \\{HLL\\} Riemann solver,\n                  Report CoA 9204, June 1992; J. Shock Waves 4 (1994)\n                  25]. The numerical results suggest that the new\n                  WENO-HLLC and WENO-MUSTA schemes compare\n                  satisfactorily with the state-of-the-art\n                  finite-volume scheme of Shi et\n                  al. [J. Comput. Phys. 175 (2002) 108]. \"\n}\n\n@ARTICLE{sod:1978,\n   author = {{Sod}, G.~A.},\n    title = \"{A survey of several finite difference methods for systems of nonlinear hyperbolic conservation la\\\nws}\",\n  journal = JCP,\n     year = 1978,\n    month = apr,\n   volume = 27,\n    pages = {1-31},\n      doi = {10.1016/0021-9991(78)90023-2},\n   adsurl = {http://adsabs.harvard.edu/abs/1978JCoPh..27....1S},\n  adsnote = {Provided by the SAO/NASA Astrophysics Data System}\n}\n\n@ARTICLE{godunov:1959,\n  author = {S.~K.~Godunov},\n  title = {A~difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics},\n  journal = {Mat. Sb. (N.S.)},\n  year = 1959,\n  volume = {47(89)},\n  issue = {3},\n  pages = {271--306}\n}\n\n\n@ARTICLE{frenk:1999,\n   author = {{Frenk}, C.~S. and {White}, S.~D.~M. and {Bode}, P. and\n                  {Bond}, J.~R. and {Bryan}, G.~L. and {Cen}, R. and\n                  {Couchman}, H.~M.~P. and {Evrard}, A.~E. and\n                  {Gnedin}, N. and {Jenkins}, A. and {Khokhlov},\n                  A.~M. and {Klypin}, A. and {Navarro}, J.~F. and\n                  {Norman}, M.~L. and {Ostriker}, J.~P. and {Owen},\n                  J.~M. and {Pearce}, F.~R. and {Pen}, U.-L. and\n                  {Steinmetz}, M. and {Thomas}, P.~A. and {Villumsen},\n                  J.~V. and {Wadsley}, J.~W. and {Warren}, M.~S. and\n                  {Xu}, G. and {Yepes}, G.},\n    title = \"{The Santa Barbara Cluster Comparison Project: A Comparison of Cosmological Hydrodynamics Solutions}\",\n  journal = APJ,\n   eprint = {astro-ph/9906160},\n keywords = {COSMOLOGY: THEORY, COSMOLOGY: DARK MATTER, GALAXIES: CLUSTERS: GENERAL, COSMOLOGY: LARGE-SCALE STRUCTURE OF UNIVERSE, X-RAYS: GALAXIES, Cosmology: Theory, Cosmology: Dark Matter, Galaxies: Clusters: General, Cosmology: Large-Scale Structure of Universe, X-Rays: Galaxies},\n     year = 1999,\n    month = nov,\n   volume = 525,\n    pages = {554-582},\n      doi = {10.1086/307908},\n   adsurl = {http://adsabs.harvard.edu/abs/1999ApJ...525..554F},\n  adsnote = {Provided by the SAO/NASA Astrophysics Data System}\n}\n\n@ARTICLE{dimonte:2004,\n   author = {{Dimonte}, G. and {Youngs}, D.~L. and {Dimits}, A. and {Weber}, S. and\n\t{Marinak}, M. and {Wunsch}, S. and {Garasi}, C. and {Robinson}, A. and\n\t{Andrews}, M.~J. and {Ramaprabhu}, P. and {Calder}, A.~C. and\n\t{Fryxell}, B. and {Biello}, J. and {Dursi}, L. and {MacNeice}, P. and\n\t{Olson}, K. and {Ricker}, P. and {Rosner}, R. and {Timmes}, F. and\n\t{Tufo}, H. and {Young}, Y.-N. and {Zingale}, M.},\n    title = \"{A comparative study of the turbulent Rayleigh-Taylor instability using high-resolution three-dimensional numerical simulations: The Alpha-Group collaboration}\",\n  journal = {Physics of Fluids},\n keywords = {Interfacial instabilities},\n     year = 2004,\n    month = may,\n   volume = 16,\n    pages = {1668-1693},\n      doi = {10.1063/1.1688328},\n   adsurl = {http://adsabs.harvard.edu/abs/2004PhFl...16.1668D},\n  adsnote = {Provided by the SAO/NASA Astrophysics Data System}\n}\n\n@ARTICLE{devalborro:2006,\n   author = {{de Val-Borro}, M. and {Edgar}, R.~G. and {Artymowicz}, P. and\n\t{Ciecielag}, P. and {Cresswell}, P. and {D'Angelo}, G. and {Delgado-Donate}, E.~J. and\n\t{Dirksen}, G. and {Fromang}, S. and {Gawryszczak}, A. and {Klahr}, H. and\n\t{Kley}, W. and {Lyra}, W. and {Masset}, F. and {Mellema}, G. and\n\t{Nelson}, R.~P. and {Paardekooper}, S.-J. and {Peplinski}, A. and\n\t{Pierens}, A. and {Plewa}, T. and {Rice}, K. and {Sch{\\\"a}fer}, C. and\n\t{Speith}, R.},\n    title = \"{A comparative study of disc-planet interaction}\",\n  journal = MNRAS,\n   eprint = {astro-ph/0605237},\n keywords = {accretion, accretion discs: hydrodynamics: planets and satellites: general, accretion discs, hydrodynamics, planets and satellites: general},\n     year = 2006,\n    month = aug,\n   volume = 370,\n    pages = {529-558},\n      doi = {10.1111/j.1365-2966.2006.10488.x},\n   adsurl = {http://adsabs.harvard.edu/abs/2006MNRAS.370..529D},\n  adsnote = {Provided by the SAO/NASA Astrophysics Data System}\n}\n\n@ARTICLE{Zhang:2006,\n   author = {{Zhang}, W. and {MacFadyen}, A.~I.},\n    title = \"{RAM: A Relativistic Adaptive Mesh Refinement Hydrodynamics Code}\",\n  journal = {APJS},\n   eprint = {astro-ph/0505481},\n keywords = {Hydrodynamics, Methods: Numerical, Relativity},\n     year = 2006,\n    month = may,\n   volume = 164,\n    pages = {255-279},\n      doi = {10.1086/500792},\n   adsurl = {http://adsabs.harvard.edu/abs/2006ApJS..164..255Z},\n  adsnote = {Provided by the SAO/NASA Astrophysics Data System}\n}\n\n\n@BOOK{shu,\n   author = {{Shu}, F.~H.},\n    title = \"{Physics of Astrophysics, Vol. II}\",\n keywords = {GAS DYNAMICS, FLUID DYNAMICS, MAGNETOHYDRODYNAMICS, PLASMA PHYSICS, SHOCKS},\nbooktitle = {Physics of Astrophysics, Vol.~II, by Frank H.~Shu.~Published by University Science Books, ISBN 0-935702-65-2, 476pp, 1992.},\n     year = 1992,\npublisher = {University Science Books},\n   adsurl = {http://adsabs.harvard.edu/abs/1992phas.book.....S},\n  adsnote = {Provided by the SAO/NASA Astrophysics Data System}\n}\n\n\n@BOOK{choudhuri,\n   author = {{Choudhuri}, A.~R.},\n    title = \"{The physics of fluids and plasmas : an introduction for astrophysicists /}\",\nbooktitle = {Philosophy and Foundations of Physics},\n     year = 1998,\n    month = nov,\n   adsurl = {http://adsabs.harvard.edu/abs/1998pfp..book.....C},\n  adsnote = {Provided by the SAO/NASA Astrophysics Data System}\n}\n\n\n\n@ARTICLE{schmidt:2014,\n   author = {{Schmidt}, W.},\n    title = \"{Large Eddy Simulations in Astrophysics}\",\n  journal = {Living Reviews in Computational Astrophysics},\narchivePrefix = \"arXiv\",\n   eprint = {1404.2483},\n keywords = {Turbulence, Large eddy simulations (LES)},\n     year = 2015,\n    month = dec,\n   volume = 1,\n      eid = {2},\n    pages = {2},\n      doi = {10.1007/lrca-2015-2},\n   adsurl = {http://adsabs.harvard.edu/abs/2015LRCA....1....2S},\n  adsnote = {Provided by the SAO/NASA Astrophysics Data System}\n}\n\n@ARTICLE{aspden:2008,\n   author = {{Aspden}, A. and {Nikiforakis}, N. and {Dalziel}, S. and {Bell}, J.~B.},\n  title = \"{Analysis of implicit LES methods}\",\n  journal = {Communications in Applied Mathematics and Computational Science},\n  year = {2008},\n  volume = {3},\n  number = {1},\n  pages = {103--126}\n}\n\n\n\n@ARTICLE{timmeswoosley,\n   author = {{Timmes}, F.~X. and {Woosley}, S.~E.},\n    title = \"{The conductive propagation of nuclear flames. I - Degenerate C + O and O + NE + MG white dwarfs}\",\n  journal = APJ,\n keywords = {Degenerate Matter, Neutron Stars, Nuclear Fusion, Radiation Transport, Stellar Interiors, White Dwarf Stars, Computational Grids, Conductive Heat Transfer, Fractals, Gravitational Collapse, Reaction Kinetics, Supernovae},\n     year = 1992,\n    month = sep,\n   volume = 396,\n    pages = {649-667},\n      doi = {10.1086/171746},\n   adsurl = {http://adsabs.harvard.edu/abs/1992ApJ...396..649T},\n  adsnote = {Provided by the SAO/NASA Astrophysics Data System}\n}\n\n\n@ARTICLE{SNrt,\n   author = {{Bell}, J.~B. and {Day}, M.~S. and {Rendleman}, C.~A. and {Woosley}, S.~E. and\n\t{Zingale}, M.},\n    title = \"{Direct Numerical Simulations of Type Ia Supernovae Flames. II. The Rayleigh-Taylor Instability}\",\n  journal = APJ,\n   eprint = {astro-ph/0401247},\n keywords = {Conduction, Hydrodynamics, Methods: Numerical, Nuclear Reactions, Nucleosynthesis, Abundances, Stars: Supernovae: General, Stars: White Dwarfs},\n     year = 2004,\n    month = jun,\n   volume = 608,\n    pages = {883-906},\n      doi = {10.1086/420841},\n   adsurl = {http://adsabs.harvard.edu/abs/2004ApJ...608..883B},\n  adsnote = {Provided by the SAO/NASA Astrophysics Data System}\n}\n\n\n@ARTICLE{shuosher:1989b,\n   author = {{Shu}, C.-W. and {Osher}, S.},\n   title = \"{Efficient implementation of essentially non-oscillatory shock-capturing schemes II}\",\n   journal = JCP,\n   year = 1989,\n   volume = 83,\n   pages = {32--78}\n}\n\n@ARTICLE{flash,\n   author = {{Fryxell}, B. and {Olson}, K. and {Ricker}, P. and {Timmes}, F.~X. and\n\t{Zingale}, M. and {Lamb}, D.~Q. and {MacNeice}, P. and {Rosner}, R. and\n\t{Truran}, J.~W. and {Tufo}, H.},\n    title = \"{FLASH: An Adaptive Mesh Hydrodynamics Code for Modeling Astrophysical Thermonuclear Flashes}\",\n  journal = APJ:sup,\n keywords = {Equation of State, Hydrodynamics, Methods: Numerical, Nuclear Reactions, Nucleosynthesis, Abundances, Stars: General},\n     year = 2000,\n    month = nov,\n   volume = 131,\n    pages = {273-334},\n      doi = {10.1086/317361},\n   adsurl = {http://adsabs.harvard.edu/abs/2000ApJS..131..273F},\n  adsnote = {Provided by the SAO/NASA Astrophysics Data System}\n}\n\n@ARTICLE{zingalekatz,\n   author = {{Zingale}, M. and {Katz}, M.~P.},\n    title = \"{On the Piecewise Parabolic Method for Compressible Flow With Stellar Equations of State}\",\n  journal = APJ:sup,\narchivePrefix = \"arXiv\",\n   eprint = {1501.01923},\n primaryClass = \"astro-ph.IM\",\n keywords = {hydrodynamics, methods: numerical},\n     year = 2015,\n    month = feb,\n   volume = 216,\n      eid = {31},\n    pages = {31},\n      doi = {10.1088/0067-0049/216/2/31},\n   adsurl = {http://adsabs.harvard.edu/abs/2015ApJS..216...31Z},\n  adsnote = {Provided by the SAO/NASA Astrophysics Data System}\n}\n\n@BOOK{richtmyermorton,\n   author = {{Richtmyer}, R.~D. and {Morton}, K.~W.},\n   title = \"{Different Methods for Initial-Value Problems}\",\n   year = {1994},\n   publisher = {Krieger Publishing Company},\n   edition = {2nd}\n}\n\n\n@book{sedov:1959,\n      author=\"{Sedov}, L.~I.\",\n      title=\"Similarity and Dimensional Methods in Mechanics\",\n      publisher=\"Academic Press\",\n      year=\"1959\",\n      note=\"translated from the 4th Russian Ed.\"\n}\n\n\n@ARTICLE{gottliebshu:1996,\n   author = {{Gottlieb}, S. and {Shu}, C.-W.},\n   title=\"{Total Variation Diminishing Runge-Kutta Schemes}\",\n   journal = \"{ICASE Report No.\\ 96-50, NASA Langley Research Center}\",\n   year={1996},\n   month={July}\n}\n\n@ARTICLE{twodturbulence,\n   author = {{Boffetta}, G. and {Ecke}, R.~E.},\n    title = \"{Two-Dimensional Turbulence}\",\n  journal = {Annual Review of Fluid Mechanics},\n     year = 2012,\n    month = jan,\n   volume = 44,\n    pages = {427-451},\n      doi = {10.1146/annurev-fluid-120710-101240},\n   adsurl = {http://adsabs.harvard.edu/abs/2012AnRFM..44..427B},\n  adsnote = {Provided by the SAO/NASA Astrophysics Data System}\n}\n\n\n@article{timmes_sedov_code,\n      year=2007,\n      title = \"{On Efficient Generation of Numerically Robust Sedov Solutions}\",\n      author={{Kamm}, J.~R. and {Timmes}, F.~X.},\n      note=\"Los Alamos preprint la-ur-07-2849, \\url{http://cococubed.asu.edu/papers/la-ur-07-2849.pdf}\"\n}\n\n@ARTICLE{omang:2006,\n   author = {{Omang}, M. and {B{\\o}rve}, S. and {Trulsen}, J.},\n    title = \"{SPH in spherical and cylindrical coordinates}\",\n  journal = {Journal of Computational Physics},\n     year = 2006,\n    month = mar,\n   volume = 213,\n    pages = {391-412},\n      doi = {10.1016/j.jcp.2005.08.023},\n   adsurl = {http://adsabs.harvard.edu/abs/2006JCoPh.213..391O},\n  adsnote = {Provided by the SAO/NASA Astrophysics Data System}\n}\n\n\n@ARTICLE{timmes_nets,\n   author = {{Timmes}, F.~X.},\n    title = \"{Integration of Nuclear Reaction Networks for Stellar Hydrodynamics}\",\n  journal = {APJ:sup},\n keywords = {HYDRODYNAMICS, METHODS: NUMERICAL, NUCLEAR REACTIONS, NUCLEOSYNTHESIS, ABUNDANCES, STARS: INTERIORS, Hydrodynamics, Methods: Numerical, Nuclear Reactions, Nucleosynthesis, Abundances, Stars: Interiors},\n     year = 1999,\n    month = sep,\n   volume = 124,\n    pages = {241-263},\n      doi = {10.1086/313257},\n   adsurl = {http://adsabs.harvard.edu/abs/1999ApJS..124..241T},\n  adsnote = {Provided by the SAO/NASA Astrophysics Data System}\n}\n\n\n@ARTICLE{vode,\n  author={P. N. Brown and G. D. Byrne and A. C. Hindmarsh},\n  title={{VODE}: A Variable Coefficient ODE Solver},\n  journal={SIAM J. Sci. Stat. Comput.},\n  volume={10},\n  pages={1038-1051},\n  year={1989}\n}\n\n\n@article{hydro_test_quad,\nauthor = {Carsten W. Schulz-Rinne and James P. Collins and Harland M. Glaz},\ntitle = {Numerical Solution of the Riemann Problem for Two-Dimensional Gas Dynamics},\njournal = {SIAM Journal on Scientific Computing},\nvolume = {14},\nnumber = {6},\npages = {1394-1414},\nyear = {1993},\ndoi = {10.1137/0914082},\nURL = {http://dx.doi.org/10.1137/0914082},\neprint = {http://dx.doi.org/10.1137/0914082}\n}\n\n@ARTICLE{leveque:1997,\n   author = {{LeVeque}, R.~J.},\n    title = \"{Wave Propagation Algorithms for Multidimensional Hyperbolic Systems}\",\n  journal = {Journal of Computational Physics},\n     year = 1997,\n    month = mar,\n   volume = 131,\n    pages = {327-353},\n      doi = {10.1006/jcph.1996.5603},\n   adsurl = {http://adsabs.harvard.edu/abs/1997JCoPh.131..327L},\n  adsnote = {Provided by the SAO/NASA Astrophysics Data System}\n}\n\n@ARTICLE{heitmann:2008,\n   author = {{Heitmann}, K. and {Luki{\\'c}}, Z. and {Fasel}, P. and {Habib}, S. and\n\t{Warren}, M.~S. and {White}, M. and {Ahrens}, J. and {Ankeny}, L. and\n\t{Armstrong}, R. and {O'Shea}, B. and {Ricker}, P.~M. and {Springel}, V. and\n\t{Stadel}, J. and {Trac}, H.},\n    title = \"{The cosmic code comparison project}\",\n  journal = {Computational Science and Discovery},\narchivePrefix = \"arXiv\",\n   eprint = {0706.1270},\n     year = 2008,\n    month = oct,\n   volume = 1,\n   number = 1,\n      eid = {015003},\n    pages = {015003},\n      doi = {10.1088/1749-4699/1/1/015003},\n   adsurl = {http://adsabs.harvard.edu/abs/2008CS%26D....1a5003H},\n  adsnote = {Provided by the SAO/NASA Astrophysics Data System}\n}\n\n@article {MargolinRider:2002,\nauthor = {Margolin, Len G. and Rider, William J.},\ntitle = {A rationale for implicit turbulence modelling},\njournal = {International Journal for Numerical Methods in Fluids},\nvolume = {39},\nnumber = {9},\npublisher = {John Wiley & Sons, Ltd.},\nissn = {1097-0363},\nurl = {http://dx.doi.org/10.1002/fld.331},\ndoi = {10.1002/fld.331},\npages = {821--841},\nkeywords = {turbulence, large eddy simulation, Burgers' equation},\nyear = {2002},\n}\n\n@Book{peters:2000,\n  author={Norbert Peters},\n  title=\"{Turbulent Combustion}\",\n  publisher={Cambridge University Press},\n  year={2000}\n}\n\n@article{Gerolymos2009,\nauthor = {Gerolymos, G.A. and S{\\'{e}}n{\\'{e}}chal, D. and Vallet, I.},\ndoi = {10.1016/j.jcp.2009.07.039},\nissn = {00219991},\njournal = {Journal of Computational Physics},\nmonth = {dec},\nnumber = {23},\npages = {8481--8524},\npublisher = {Elsevier Inc.},\ntitle = {{Very-high-order {WENO} schemes}},\nurl = {http://linkinghub.elsevier.com/retrieve/pii/S0021999109003908},\nvolume = {228},\nyear = {2009}\n}\n\n@article{Gottlieb2008,\nauthor = {Gottlieb, Sigal and Ketcheson, David I. and Shu, Chi-Wang},\ndoi = {10.1007/s10915-008-9239-z},\nissn = {0885-7474},\njournal = {Journal of Scientific Computing},\nmonth = {sep},\nnumber = {3},\npages = {251--289},\npublisher = {Springer Netherlands},\ntitle = {{High Order Strong Stability Preserving Time Discretizations}},\nurl = {http://www.springerlink.com/content/61752031t10l2651/ http://www.springerlink.com/index/10.1007/s10915-008-9239-z},\nvolume = {38},\nyear = {2008}\n}\n\n@techreport{Shu1997,\nauthor = {Shu, Chi-wang},\nbooktitle = {Journal of Computational Physics},\nnumber = {97},\ntitle = {{Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws}},\nurl = {https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19980007543.pdf},\nyear = {1997}\n}\n"
  },
  {
    "path": "simulations/simulations.tex",
    "content": "\\section{How to setup a simulation?}\n\nWhen you perform a simulation, you are making a series of\napproximations.  The first comes with the choice of equation set.  As\ndiscussed earlier, for pure hydrodynamics, this often means solving\nthe Euler rather than the full Navier-Stokes equations.  The next\napproximation is made when discretizing the equations, using the \nmethods we described in the earlier chapters.  \n\nAstrophysical flows involve a range of length and timescales.  Often\nit is not possible to capture all these scales in a single simulation,\nso some comprimises must be made.  If you are interested in the\ndynamics on the full scale of the system, $L$, then you will resolve\ndown to some cutoff scale $l$, where $L/l \\sim O(10^3\\mbox{--}10^4)$.\nAt scales smaller than $l$, you neglect the flow features.  The\nsmallest scale of importance is the disipation scale $\\lambda \\ll L$.\n\nIf you are not resolving down to the dissipation scales in our\nproblem, then you are working in the realm of {\\em large eddy\n  simulations} (LES; see \\cite{schmidt:2014}).  Proper LES requires a\nsubgrid model to treat the unresolve scales and their feedback onto\nyour grid.  If you instead rely on numerical diffusion of your method\nto do the dissipation, you are doing {\\em implicit large eddy\n  simulation} or ILES---for many flows, this be sufficient to capture the\ndynamics (including turbulence) that are important~\\cite{aspden:2008,MargolinRider:2002}.\n\n\\section{Dimensionality and picking your resolution}\n\nNature is three-dimensional, so realistic simulaitons generally\nrequire full 3-d modeling.  Nevertheless, 2-d simulations can be\nuseful to scope out the problem space, and in particular, to get a\nfeel for the timescales and resolution needed to resolve smallscale\nfeatures.  The major place where dimensionality affects results is for\ninstabilities and turbulence.  The trend in 2-d is for small scale\nvortices to merge and build larger and larger structures (see, e.g.,\n\\cite{twodturbulence}), while in 3-d motions at the largest scale\ncascade down to smaller scales, driving turbulent flow down to the\nsmallest scales.\n\nTurbulence can be very important in astrophysical flows, and properly\ncapturing it requires modeling a large range of lengthscales\n(physically, the objective is to get a large {\\em Reynolds number}:\nthe ratio of the advective force to the dissipative force).\nGenerally, you'll just start to get a resolved inertial range over a\ndecade in wavenumber in your power spectrum at around 512 zones on a\nside of your domain (using the methods that we discussed here).  With\nfewer points than this, you are unlikely to capture turbulence.\n\nIf nuclear burning is modeled, the steepness of the reaction rate will\nset a lengthscale.  Burning can also wash out instabilities, a process\nsometimes called {\\em fire polishing} \\cite{timmeswoosley,SNrt}.\nHere, if the timescale for reactions is faster than the growth rate of\nan instability, the instability is suppressed.  This can set a small\nscale cutoff that is well above the viscous dissipation scale.\n\nHydrostatic equilibrium also puts demands on the resolution.  Recall that\nthe momentum equation is:\n\\begin{equation}\n\\rho \\DDt{\\Ub} + \\nabla p = \\rho \\gb\n\\end{equation}\nIf hydrostatic equilibrium, $\\nabla p = \\rho \\gb$, is not exactly\nsatisfied, then the residual will appear as a force that generates an\nacceleration and non-zero velocities will build up.  A good rule of\nthumb is that you need $10$ points per pressure scale height to get a\nreasonable hydrostatic balance (this is discussed at bit in\n\\cite{hse}).\n\nSpatial resolution is often in contention with improved physics---both\ncan add to the computational cost of a simulation, so for fixed cost,\nyou have often have to choose between more physics (e.g., larger\nreaction network) or more resolution.\n\nAn important part of the simulation process is to ensure that your\nresults are converged.  In a {\\em convergence study}, you run the same\nsimulation at various resolutions and look to see that the major\nresults (e.g., integral quantities) do not change significantly with\nresolution.  Ideally you would show that any results you are\ninterested are moving toward an asymptotic value.  Even if you are\nnot converged however, you could use the trends observed with resolution\nto help understand the possible error in your simulation.\n\n\n\n\n\\section{Boundary conditions}\n\nAnother approximation you make is how to treat the boundaries.\nUsually one chooses symmetry / reflecting, outflow / zero-gradient,\ninflow (specifying the value on the boundary), or periodic.  In the\ncase of symmetry, the fluid state is reflected and the normal velocity\nis reflected with a sign change.  The transverse velocity can be left\nas is (a slip wall) or set to zero at the boundary (no-slip wall).  \n\nGenerally speaking, you should put your boundaries as far away from\nthe place where the action is occuring as possible.  For example,\nif you are modeling a convective region bounded by stably stratified\nflow, then you should have a large buffer of stable atmosphere\nbounding it.\n\nPeriodic boundary conditions don't magically fix everything either.\nWith a triply-periodic box (periodic on all boundaries), then you are\nconfining the flow.  If energy is injected into the box, then there is\nno place for it to expand, and this can lead to artifical heating or\ncompression.\n\nSymmetry / reflecting boundaries provide a simple way to reduce a\nproblem size, for example, modeling only an octant of a star in 3-d\ninstead of the full star.  But they also introduce artifacts, since\nthey force the normal velocity to zero right on the boundary.  This\nmeans that flow through the center of a star (e.g., if we model only\nan octant) can be restricted, resulting in artifical flow / convective\nfeatures.  Wall heating can also be an issue---forcing the normal\nvelocity to zero at the symmetry boundary can lead to stagnation of\nthe flow there, allowing hot spots to develop and not be transported\naway.\n\n\n\\MarginPar{characteristic BCs?}\n\n\n\n\n\\section{Timestep considerations}\n\nThe timestep constraint we discussed (the CFL condition) is required\nfor stability, but that is different than accuracy.  In particular,\naccurately capturing some physical processes (that may appear as\nsource terms) might require more restrictive timesteps to ensure that they \nare properly coupled with the hydrodynamics.  \n\nAn example where this can commonly arise is reactive flows---combining\nhydrodynamics with nuclear or chemical reactions.  If the reactions\nare very vigorous, then you may run into a situation where you exaust\nyour fuel in a zone in a single step, without giving the hydrodynamics\na chance to advect new fuel in.  This represents a breakdown of the\ncoupling of the hydrodynamics and reactions.  A common way to improve\nthe accuracy here is to monitor the change in mass fractions of a fuel\nand set a timestep limit such that:\n\\begin{equation}\n\\Delta t_\\mathrm{react} = C_\\mathrm{react}\n\\frac{X_\\mathrm{fuel}}{|\\omegadot_\\mathrm{fuel}|}\n\\end{equation}\nwhere $C_\\mathrm{react} < 1$.  The species creation rate,\n$\\dot{X}_\\mathrm{react}$, is typically evaluated using the value from\nthe previous step.\n\n\n\\section{Convergence and multiphysics}\n\nFor a full simulation, with hydrodynamics coupled to other physics,\nyou should test your code to ensure that it is converging at the rate\nthat you expect.  Usually in this case there is no analytic solution,\nso you need to measure the convergence numerically.  A standard way to\ndo this is to run pairs of simulations that differ by a factor of two\nin resolution, we'll denote the coarser simulation with the subscript\n$c$ and the finer simulation with the subscript $f$.  The error is\ncomputed for a field $\\phi$ by coarsening the finer simulation by a\nfactor of two, and computing the norm of the difference:\n\\begin{equation}\n\\epsilon \\equiv \\| \\phi_c - \\mathcal{C}(\\phi_f) \\|\n\\end{equation}\nHere, we denote the coarsening operator as $\\mathcal{C}(.)$.  In this\ncase, two simulations give us a single error.  We then repeat this,\ntaking the fine simulation from this comparison as the coarse for the\nnext, and using an even finer simulation to generate a new error.  We\ncan then measure the convergence of a multiphysics simulation by\ncomparing the errors from the two pairs of the simulations.  This procedure\nwas used in, e.g., \\cite{ABNZ:III,mccorquodalecolella}.\n\nIdeally, a smooth problem should be used, since discontinuities by\ndefinition do not converge with resolution (and limiters, if used,\nwould kick in there as well).\n\n\n\\section{Computational cost}\n\nThe main limitation to your choice of resolution and timestep is\ncomputational cost.  For a three-dimensional simulation in a box with\n$N$ zones on a side, the work scales as $O(N^4)$.  The cost to advance\na single timestep is based on the volume, $N^3$, and the number of\ntimesteps to get you to a fixed simulation time, $t_\\mathrm{max}$, is\n$t_\\mathrm{max}/\\Delta t$, but $\\Delta t \\sim \\Delta x / s \\sim L/(N\ns)$, where $s$ is the fastest information speed.  So the number \nof timesteps increases with $N$, making the total cost $N^4$.\nThink about this for a minute---if you double the number of zones, then\nthe amount of work you need to do increases by 16$\\times$.\n\nA common conflict that arises is: Do you do one ``hero'' calculation\nor many smaller calculations to understand the sensitivity of your\nresults to input parameters?  Computing centers that grant allocations\nusually setup their queues to favor big jobs that use as much of the\nmachine as possible (as that's what their funders use as a metric for\ntheir success).\n\n\nIt is important to understand that no single algorithm will offer everything\nyou need.\n\n\n\\section{I/O}\n\nAnother challenge with simulations is data storage.  For a calculation\non a $1024^3$ grid, using double precision, each number you store\ntakes 8~bytes / zone, so a single variable on the grid will require\n8~GB.  For compressible hydro, you will have atleast 5 variables (in\n3-d), $\\rho$, $(\\rho {\\bf U})$, and $(\\rho E)$, so this is a minimum of 40~GB per\nfile.  Often you will want some derived quantities (like temperature)\nand composition quantities in your output, which will greatly increase\nyour storage requirements.\n\nFor this reason, simulation codes often have different types of\noutput.  Checkpoint files store all the variables that are needed to\nrestart the calculation from where you left off---these should be\nstored in a (portable) binary format at machine precision.  You don't\nneed to keep these around forever, perhaps only saving the last few as\nnecessary to restart your calculation as it works through a computing\ncenter's queue.  Plotfiles store a lot of variables---basically anything\nthat will be needed for the analysis of the simulation.  Since you don't\nneed these to restart, you can probably afford to store the variables\nat single precision.  Sometimes you might have multiple levels of plotfiles,\nstoring a few variables (like density) very frequently to allow for \nthe production of smooth movies, and storing a lot of variables and \nderived quantities at a much longer cadence.\n\nA perpetual challenge with the analysis and visualization of\nsimulations is that sometimes you don't know what you are looking for\nuntil the simulation is complete, so you cannot simply do runtime\nvisualization and not save the raw data as the simulation progresses.\nThis leads to collections of plotfiles that can easily top 100~TB per\nsimulation.  Runtime diagnostics can help a little---if you know any\nglobal quantities that you will be interested in over the course of\nthe simulation (like peak temperature, or total energy) you can\ncompute these on the fly and simply store a single number per step (or\nevery $N$ steps) into a file.\n"
  },
  {
    "path": "software-engineering/software-engineering.tex",
    "content": "\\section{Version control}\n\nVersion control gives you the ability to keep track of changes to your\nsource, interact with other developers on the same project, and to\nroll back changes when bugs are introduced.  It is easy to get started\nand is perhaps the single most important part of code development for\neveryone.  \n\nThere are several choices for version control, split loosely between\ncentralized systems and distributed version control systems.  In the\nformer category are CVS and subversion---these have been around for a long\ntime and are still in wide use, but not recommedned for starting a new project.\nMercurial and git are the two most popular distributed version control\nsystems, each actively used.  There are some subtle differences between\nthe two, but generally the workflow in each system follows the same ideas.\n\nDiagram showing a distributed workflow.\n\nWe generally want to pull\n\nbitbucket and github\n\n\n\n\n\\section{Testing}\n\nThere are several types of testing that are important in computational\nhydrodynamics.  When concerning source code quality, {\\em unit testing}\nand {\\em regression testing} are invaluable in making sure that your\ncode works as designed and that you don't change the results as you \ncontinue to develop.\n\n\\section{Reproducibility}\n\n"
  },
  {
    "path": "software-engineering/topics.txt",
    "content": "\n\n-- git & github\n\n-- diff\n\n-- gnuplot\n\n-- awk\n\n-- grep / ack\n\n-- basic scripting (for i in ...  [ seq, *.png ]\n\n   . basename\n\n\n-- screen\n"
  },
  {
    "path": "symbols/symbols.tex",
    "content": "\n\\renewcommand{\\arraystretch}{1.5}\n%\n\\begin{center}\n\\begin{longtable}{|l|p{3.25in}|l|}\n\\caption[Definition of symbols.]{Definition of symbols.} \\label{table:sym} \\\\\n%\n\\hline \\multicolumn{1}{|c|}{\\textbf{symbol}} & \n       \\multicolumn{1}{ c|}{\\textbf{meaning}} & \n       \\multicolumn{1}{ c|}{\\textbf{units}} \\\\ \\hline \n\\endfirsthead\n\n\\multicolumn{3}{c}%\n{{\\tablename\\ \\thetable{}---continued}} \\\\\n\\hline \\multicolumn{1}{|c|}{\\textbf{symbol}} & \n       \\multicolumn{1}{ c|}{\\textbf{meaning}} & \n       \\multicolumn{1}{ c|}{\\textbf{units}} \\\\ \\hline \n\\endhead\n\n\\multicolumn{3}{|r|}{{\\em continued on next page}} \\\\ \\hline\n\\endfoot\n\n\\hline \n\\endlastfoot\n$\\Ab$     & Jacobian matrix  & N/A \\\\\n%\n$C$       & Lagrangian sound speed, $C = \\sqrt{\\Gamma_1 p \\rho}$ & g~m$^{-2}$~s$^{-1}$\\\\\n%\n$\\cfl$    & CFL number & --\\\\\n%\n$c$     & sound speed, $c = \\sqrt{\\Gamma_1 p/\\rho}$  & m~s$^{-1}$ \\\\\n%\n$c_p$   & specific heat at constant pressure\n          ($c_p \\equiv \\left . \\partial h / \\partial T \\right |_{p,X_k}$)\n        & erg~g$^{-1}$~K$^{-1}$ \\\\\n%\n$c_v$   & specific heat at constant density\n          ($c_v \\equiv \\left . \\partial e / \\partial T \\right |_{\\rho,X_k}$)\n        & erg~g$^{-1}$~K$^{-1}$ \\\\\n%\n$E$     & specific total energy                      & erg~g$^{-1}$ \\\\\n%\n$e$     & specific internal energy                   & erg~g$^{-1}$ \\\\\n%\n$\\Fb$     & flux vector                                & N/A \\\\\n%\n$\\gb$     & gravitational acceleration                 & cm~s$^{-2}$ \\\\\n%\n$\\Gamma_1$ & first adiabatic exponent ($\\Gamma_1 \\equiv \\left . d \\log p/d \\log \\rho \\right |_s$) & -- \\\\\n%\n$\\gamma$ & ratio of specific heats, $\\gamma = c_p/c_v$ & -- \\\\\n%\n$\\gamma_e$ & the quantity $p/(\\rho e) + 1$ & -- \\\\\n%\n$H$     & heat sources                               & erg~g$^{-1}$~s$^{-1}$ \\\\\n%\n$\\Hnuc$     & nuclear energy source                               & erg~g$^{-1}$~s$^{-1}$ \\\\\n%\n$h$     & specific enthalpy                          & erg~g$^{-1}$ \\\\\n%\n$\\Ic$   & integral under a (piecewise) parabolic polynomial reconstruction & N/A \\\\\n%\n$\\kth$  & thermal conductivity                       & erg~cm$^{-1}$~s$^{-1}$~K$^-1$ \\\\\n%\n${\\bf L}$ & matrix of left eigenvectors              & -- \\\\\n%\n${\\boldsymbol{\\Lambda}}$ & diagonal matrix of eigenvalue & -- \\\\\n%\n$\\lb$     & left eigenvector                           & N/A \\\\\n%\n$\\lambda$ & eigenvalue                               & N/A \\\\\n%\n$M$       & Mach number, $M = |\\Ub|/c$               & -- \\\\\n%\n$\\omegadot_k$ & species creation rate                         & s$^{-1}$ \\\\\n% \n$p$     & pressure                                   & erg~cm$^{-3}$ \\\\\n%\n$\\qb$     & primitive variable vector                 & N/A \\\\\n%\n${\\bf R}$ & matrix of right eigenvectors              & -- \\\\\n%\n$\\riemann(q_L, q_R)$ & Riemann problem between states $q_L$ and $q_R$ & N/A \\\\\n%\n$\\rb$     & right eigenvector                          & N/A \\\\\n%\n$\\rho$  & mass density  & g~cm$^{-3}$ \\\\\n%\n$S$     & source term to the divergence constraint   & s$^{-1}$ \\\\\n%\n$s$     & specific entropy                           & erg~g$^{-1}$~K$^{-1}$ \\\\\n%\n$T$     & temperature                                & K \\\\\n%\n$t$     & time                                       & s \\\\\n%\n$\\tau$  & specific volume ($\\tau = 1/\\rho$)          & cm$^3$~g$^{-1}$ \\\\\n%\n$\\Ub$   & total velocity vector, $\\Ub = (u, v)^\\intercal$ in 2-d & cm~s$^{-1}$ \\\\\n%\n$\\Uc$   & conserved variable vector & N/A \\\\\n%\n$u$     & x-velocity   & cm~s$^{-1}$ \\\\\n%\n$v$     & y-velocity   & cm~s$^{-1}$ \\\\\n%\n$\\wb$     & characteristic variables vector                   & N/A \\\\\n%\n$X_k$   & mass fraction of the species ($\\sum_k X_k = 1$) & -- \\\\\n\\end{longtable}\n\\end{center}\n\\renewcommand{\\arraystretch}{1.0}\n\n\n"
  }
]