Repository: Open-Astrophysics-Bookshelf/numerical_exercises Branch: main Commit: 2d001455106e Files: 126 Total size: 3.7 MB Directory structure: gitextract_nerblykz/ ├── .github/ │ └── workflows/ │ └── build-book.yml ├── .gitignore ├── ADDITIONS ├── CompHydroTutorial.tex ├── Euler/ │ ├── Euler-methods.tex │ ├── Euler-theory.tex │ ├── README │ ├── axisymmetry.odg │ └── main.tex ├── GNUmakefile ├── LICENSE ├── NOTES ├── README.md ├── TODO ├── advection/ │ ├── advection-basics.tex │ ├── advection-higherorder.tex │ ├── main.tex │ └── old-scripts/ │ ├── fv.py │ ├── simplegrid.py │ └── simplegrid2.py ├── burgers/ │ └── burgers.tex ├── codes/ │ └── intro/ │ └── sin-converge.py ├── diffusion/ │ ├── GNUmakefile │ └── diffusion.tex ├── figures/ │ ├── Euler/ │ │ ├── multiple_interfaces.py │ │ ├── ppm-sequence.py │ │ ├── ppm-trace.py │ │ ├── ppm.py │ │ ├── rarefaction_cartoon.py │ │ ├── riemann-states-q.py │ │ ├── riemann-states.py │ │ ├── riemann-waves-jump.py │ │ ├── riemann-waves.py │ │ ├── riemann_cartoon.py │ │ ├── states.py │ │ └── update.sh │ ├── advection/ │ │ ├── 2dFD.py │ │ ├── 2dFV.py │ │ ├── 2dgrid.py │ │ ├── 2dgrid_hat.py │ │ ├── 2dgrid_transverse.py │ │ ├── characteristics.py │ │ ├── domains.py │ │ ├── fd_ghost.py │ │ ├── fv_ghost.py │ │ ├── generalgrid.py │ │ ├── rea-limitex.py │ │ ├── rea.py │ │ ├── riemann.py │ │ ├── riemann_bc.py │ │ ├── riemann_mol.py │ │ └── update.sh │ ├── burgers/ │ │ ├── characteristics.py │ │ ├── rh.py │ │ └── update.sh │ ├── finite-volume/ │ │ ├── ccfd.py │ │ ├── discretizations.odg │ │ ├── domain.py │ │ ├── fd.py │ │ ├── fv.py │ │ ├── simplegrid2.py │ │ ├── simplegrid_gc.py │ │ └── update.sh │ ├── gridFigs/ │ │ ├── fdrestrict.py │ │ ├── fvprolong.py │ │ ├── fvrestrict.py │ │ └── nested.py │ ├── incompressible/ │ │ ├── MAC-solve.py │ │ └── MAC.py │ ├── intro/ │ │ ├── fft_simple_examples.py │ │ ├── integrals.py │ │ ├── rk4_plot.py │ │ └── update.sh │ ├── multigrid/ │ │ ├── 2dgrid-mg.py │ │ ├── fv-fd_bnd.py │ │ ├── fvrestrict.py │ │ ├── laplacian.py │ │ ├── mgtower.py │ │ ├── red_black.py │ │ └── smooth-separate.py │ └── weno/ │ └── weno.py ├── finite-volume/ │ ├── discretizations.eps │ ├── finite-volume.tex │ ├── main.tex │ └── old-scripts/ │ ├── domain-1d.py │ └── simplegrid.py ├── gravity/ │ └── gravity.tex ├── higher-order/ │ ├── higher-order-burgers.tex │ ├── higher-order-euler.tex │ └── weno-coefficients.ipynb ├── hydro-test-problems/ │ └── hydro-test-problems.tex ├── hydro1d/ │ └── hydro1d.tex ├── hydro_examples/ │ └── hydro_examples.tex ├── incompressible/ │ ├── GNUmakefile │ ├── MAC.eps │ ├── MAC_solve.eps │ ├── incompressible.tex │ └── main.tex ├── instabilities/ │ └── instabilities.tex ├── intro/ │ └── intro.tex ├── low_mach/ │ ├── low_mach.tex │ ├── mg_vc_periodic_converge.eps │ └── mg_vc_periodic_test.eps ├── multigrid/ │ ├── 2dgrid-prolong.eps │ ├── GNUmakefile │ ├── main.tex │ ├── mg-converge.eps │ ├── mg_error_vs_cycle.eps │ ├── multigrid.tex │ └── rb.eps ├── multiphysics/ │ └── multiphysics.tex ├── mysymbols.tex ├── pde-classes/ │ └── pde-classes.tex ├── preface/ │ └── preface.tex ├── public-codes/ │ └── NOTES ├── pyro/ │ └── pyro.tex ├── radiation/ │ ├── mg_general_inhomogeneous_converge.eps │ ├── mg_general_inhomogeneous_test.eps │ └── radiation.tex ├── reactive_flow/ │ ├── notes.txt │ └── reactive_flow.tex ├── refs.bib ├── simulations/ │ └── simulations.tex ├── software-engineering/ │ ├── software-engineering.tex │ └── topics.txt └── symbols/ └── symbols.tex ================================================ FILE CONTENTS ================================================ ================================================ FILE: .github/workflows/build-book.yml ================================================ name: build book on: push: branches: - main pull_request: branches: - main jobs: build-book: runs-on: ubuntu-latest steps: - uses: actions/checkout@v3 with: fetch-depth: 0 - name: Install dependencies run: | sudo apt-get update -y -qq sudo apt-get -qq -y install texlive texlive-latex-extra texlive-fonts-extra texlive-font-utils build-essential - name: Build run: | make mkdir out mv CompHydroTutorial.pdf out - name: Archive uses: actions/upload-artifact@v3 with: name: book-pdf path: out/CompHydroTutorial.pdf - name: Deploy uses: peaceiris/actions-gh-pages@v3 with: github_token: ${{ secrets.GITHUB_TOKEN }} publish_dir: ./out keep_files: true ================================================ FILE: .gitignore ================================================ *.pyc *~ \#* .\#* *.log *.aux *.dvi *.bbl *.blg *-converted-to.pdf *.lof *.exc *.toc *.png *.bcf *.brf CompHydroTutorial.out CompHydroTutorial.pdf figures/*/*.eps figures/*/*.pdf ================================================ FILE: ADDITIONS ================================================ * convergence testing via Richardson extrapolation -- always test the convergence of the initial conditions * PPM vis * well-balancing ================================================ FILE: CompHydroTutorial.tex ================================================ \documentclass[11pt]{book} %\usepackage{microtype} \input conditional.tex %\def\debugmode{} % include figures \usepackage{epsfig,wrapfig} % rotating table in the appendix \usepackage{rotating} % prefer PDF to PNG \DeclareGraphicsExtensions{% .pdf, .png} % Bibliographic Package \usepackage[numbers,sort&compress]{natbib} % code listings \usepackage[procnames]{listings} \lstset{basicstyle=\ttfamily\small, keywordstyle=\color{keywords}, commentstyle=\color{comments}, stringstyle=\color{red}, showstringspaces=false, identifierstyle=\color{blue}, procnamekeys={def,class}} \graphicspath{{preface/}{intro/}{finite-volume/}{advection/}{burgers/}{diffusion/}{Euler/}{incompressible/}{multigrid/}{multiphysics/}{low_mach/}{radiation/}{pyro/}{higher-order/}} \usepackage{longtable} % AMS symbols \usepackage{amsmath,amssymb} % for forcing a figure to appear on a single page % see 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\usepackage{fancyhdr} %% \pagestyle{fancy} %% \fancyfoot[LO,LE]{\footnotesize \sffamily \color{mygray} M.\ Zingale---Notes on the Euler equations} %% \fancyfoot[RO,RE]{\footnotesize \sffamily \color{mygray} (\today)} %% \fancyfoot[CO,CE]{\thepage} %% \fancyhead{} %% \renewcommand{\headrulewidth}{0.0pt} %% \renewcommand{\footrulewidth}{0.0pt} % don't make the chapter/section headings uppercase. See the fancyhdr % documentation (section 9) \usepackage{fancyhdr} \renewcommand{\chaptermark}[1]{% \markboth{\chaptername \ \thechapter.\ #1}{}} \renewcommand{\sectionmark}[1]{\markright{\thesection---#1}} % don't put a header on blank pages, see % http://www.latex-community.org/forum/viewtopic.php?f=4&p=51559 % change ``plain'' to ``empty'' to eliminate the page number \makeatletter \renewcommand*\cleardoublepage{\clearpage\if@twoside \ifodd\c@page\else \hbox{} \thispagestyle{empty} \newpage \if@twocolumn\hbox{}\newpage\fi\fi\fi} \makeatother % no page number of the part page % http://latex.org/forum/viewtopic.php?f=47&t=4293&p=16738 \makeatletter \renewcommand\part{% \if@openright \cleardoublepage \else \clearpage \fi \thispagestyle{empty}% \if@twocolumn \onecolumn \@tempswatrue \else \@tempswafalse \fi \null\vfil \secdef\@part\@spart} \makeatother % skip a bit of space between paragraphs, to enhance readability \usepackage{parskip} % captions \usepackage{caption} \renewcommand{\captionfont}{\footnotesize} \renewcommand{\captionlabelfont}{\footnotesize} \setlength{\captionmargin}{3em} %----------------------------------------------------------------------------- % define a new environment for exercises %\newcounter{exercise} % simple way -- no TOC list % toclot allows us to make a list of exercises % the new environment stuff came from http://www.dickimaw-books.com/latex/novices/html/newenv.html \usepackage{tocloft} \newcommand{\listexercisename}{List of Exercises} % the [chapter] argumenet here means that we reset the numbers each chapter % exc is the short name we give to this new list \newlistof[chapter]{exercise}{exc}{\listexercisename} \usepackage{changepage} % used to adjust the margins within the environment % environment name, with optional argument for display in the list of % exercises \newenvironment{exercise}[1][] {% begin code % % this indents our text from the left and right \begin{adjustwidth}{1.0cm}{1.0cm}% % % this skips a line before %\par\vspace{\baselineskip}\noindent % % this increments our counter \refstepcounter{exercise}% \begin{tcolorbox}[enhanced, breakable, title={\sffamily Exercise \theexercise}] % % this displays the exercise number and starts italics \begin{itshape}% % % this adds an entry to our custom List of Exercises \addcontentsline{exc}{exercise}{\protect\numberline{\theexercise}~~ #1} }% {% end code -- this is done at the end of the environment % % stop the italics \end{itshape}% % \end{tcolorbox} % this stops our custom margins \end{adjustwidth}% % % this skips a line at the end %\vspace{\baselineskip}\ignorespacesafterend } % this makes the List of Exercises have a line break before each chapter, % just like in the main ToC. % % this has a bug -- it does not omit the space if the chapter has 0 exercises \usepackage{etoolbox} \preto\section{% \ifnum\value{chapter}=1{}\else % don't skip before Ch 1 \ifnum\value{section}=0\addtocontents{exc}{\vskip10pt}\fi \fi } % indent these the standard amount for a chaptered section \setlength{\cftexerciseindent}{1.5em} %----------------------------------------------------------------------------- % license \usepackage{ccicons} % computer keyboard symbol \usepackage{marvosym} % for dotted lines in the matrics/arrays \usepackage{arydshln} % fonts for TOC, list of figures, etc \renewcommand*\listfigurename{\bf\textsf{List of Figures}} \renewcommand*\listexercisename{\bf\textsf{List of Exercises}} \renewcommand*\contentsname{\bf\textsf{Table of Contents}} % short table of contents \usepackage{shorttoc} % pack more figures on a page \usepackage{float} \renewcommand\floatpagefraction{.9} \renewcommand\topfraction{.9} \renewcommand\bottomfraction{.9} \renewcommand\textfraction{.1} \newcommand{\hydroex}{{\sf hydro\_examples}} \newcommand{\hydroexdoit}[1]{{\color{red} \LARGE \Keyboard}\/ \hydroex: {\tt #1}} \newcommand{\hydroexsymb}{{\color{red} \LARGE \Keyboard}} \usepackage{multirow} % MarginPars \setlength{\marginparwidth}{0.75in} \newcommand{\aMarginPar}[1]{\marginpar{\vskip-\baselineskip\raggedright\tiny\sffamily\hrule\smallskip{\color{red}#1}\par\smallskip\hrule}} \newcommand{\MarginPar}[1]{\ifdefined\debugmode\aMarginPar{#1}\fi} \usepackage{eso-pic} \newcommand\BackgroundPic{% \put(-0,-0){% \parbox[b][\paperheight]{\paperwidth}{% \vfill \centering \includegraphics[width=\paperwidth,%height=\paperheight,% keepaspectratio]{images/crop_cover.png}% \vfill }}} \usepackage[many]{tcolorbox} \newtcolorbox{mybox}[1][]{ width=\textwidth, arc=0mm, auto outer arc, boxsep=0.05cm, toprule=2pt, leftrule=2pt, bottomrule=2pt, rightrule=2pt, colframe=black, colback=white, fontupper=\centering\fontsize{28pt}{28pt}\sffamily, breakable, nobeforeafter, enhanced jigsaw, opacityframe=0.7, opacityback=0.75, title={\hfill\sffamily part of the open astrophysics bookshelf}, } \pdfpageattr{/Group <>} \input mysymbols \begin{document} \AddToShipoutPicture*{\BackgroundPic} \frontmatter \begin{titlepage} \ \\[1.25in] %% \HRule\\[0.5em] %% {\Huge \textsf{{ %% Lecture Notes on\\[0.1em] %% Computational Hydrodynamics\\[0.3em] %% for Astrophysics}} %% } %% \HRule %% \\[2em] % %% {\Large \sf Michael Zingale} \\ {\sf Stony Brook University} % %% \begin{mybox}[] %% \vskip 3mm %% {Introduction to} \\%[-0.75em] %% {Computational Hydrodynamics \\%[-0.75em] %% for Astrophysics} %% \vspace{0.5em} %% \end{mybox} {\sffamily \color{white}% \begin{center} \fontsize{22}{18}\selectfont Introduction to \\ \fontsize{32}{26}\selectfont Computational Astrophysical Hydrodynamics \end{center} } \vspace{0.5em} \vfill {\color{red}\sffamily the Open Astrophysics Bookshelf \hfill\color{white} Michael Zingale} \end{titlepage} \pagestyle{plain} \null \vfill \noindent \ccCopy\ 2013, 2014, 2015, 2016, 2017 Michael Zingale \\ \noindent document git version: \input git_info.tex $\ldots$ \\ \noindent \today \noindent the source for these notes are available online (via \git):\\ \url{https://github.com/Open-Astrophysics-Bookshelf/numerical_exercises} \noindent \ccbyncsa \\ \noindent This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0) license. \clearpage \shorttoc{Chapter Listing}{0} \clearpage \setcounter{tocdepth}{2} \tableofcontents \clearpage \listoffigures \addcontentsline{toc}{chapter}{list of figures} \clearpage \listofexercise \addcontentsline{toc}{chapter}{list of exercises} \clearpage \chapter*{preface} \chaptermark{preface} \addcontentsline{toc}{chapter}{preface} \input preface/preface.tex \clearpage \pagestyle{headings} \renewcommand{\chaptermark}[1]{% \markboth{\chaptername \ \thechapter.\ #1}{}} \renewcommand{\sectionmark}[1]{\markright{\thesection---#1}} % put the git hash on the first page of each chapter -- see section % 7 of the fancyhdr docs to see how to override the plain % style \fancypagestyle{plain}{% \fancyhf{} % clear all header and footer fields \fancyfoot[C]{\thepage} % except the center \fancyfoot[L]{\scriptsize git version: \input git_info.tex $\ldots$} \renewcommand{\headrulewidth}{0pt} \renewcommand{\footrulewidth}{0pt}} \mainmatter \part{Basics} \chapter{Simulation Overview} \input intro/intro.tex \chapter{Classification of PDEs} \input pde-classes/pde-classes.tex %\addtocontents{exc}{\protect\addvspace{10pt}}% \chapter{Finite-Volume Grids} \input finite-volume/finite-volume.tex \part{Advection and Hydrodynamics} %\addtocontents{exc}{\protect\addvspace{10pt}}% \chapter{Advection Basics} \input advection/advection-basics.tex \chapter{Second- (and Higher-) Order Advection} \input advection/advection-higherorder.tex %\addtocontents{exc}{\protect\addvspace{10pt}}% \chapter{Burgers' Equation} \input burgers/burgers.tex \input higher-order/higher-order-burgers.tex %\addtocontents{exc}{\protect\addvspace{10pt}}% \chapter{Euler Equations: Theory} \input Euler/Euler-theory.tex \chapter{Euler Equations: Numerical Methods} \input Euler/Euler-methods.tex \input higher-order/higher-order-euler.tex \ifdefined\debugmode \chapter{Hydrodynamics Test Problems} \input hydro-test-problems/hydro-test-problems.tex \chapter{Instabilities and Turbulence} \input instabilities/instabilities.tex \fi \part{Elliptic and Parabolic Problems} %\addtocontents{exc}{\protect\addvspace{10pt}}% \chapter{Elliptic Equations and Multigrid} \input multigrid/multigrid.tex %\addtocontents{exc}{\protect\addvspace{10pt}}% \chapter{Diffusion} \input diffusion/diffusion.tex \part{Multiphysics applications} %\addtocontents{exc}{\protect\addvspace{10pt}}% \chapter{Model Multiphysics Problems} \input multiphysics/multiphysics.tex \chapter{Reactive Flow} \input reactive_flow/reactive_flow.tex \ifdefined\debugmode \chapter{Self-gravity} \input gravity/gravity.tex \fi \chapter{Planning a Simulation} \input simulations/simulations.tex \part{Low Speed Hydrodynamics} %\addtocontents{exc}{\protect\addvspace{10pt}}% \chapter{Incompressible Flow and Projection Methods} \input incompressible/incompressible.tex \chapter{Low Mach Number Methods} \input low_mach/low_mach.tex \ifdefined\debugmode \chapter{Radiation Hydrodynamics} \input radiation/radiation.tex \fi \part{Code Descriptions} \appendix \chapter{Using \hydroex} \input hydro_examples/hydro_examples.tex \chapter{Using \pyro} \input pyro/pyro.tex \chapter{Using \hydrooned} \input hydro1d/hydro1d.tex %% \chapter{Software Engineering Practices} %% \input software-engineering/software-engineering.tex \backmatter \addcontentsline{toc}{chapter}{References} \bibliographystyle{plain} \bibliography{refs} \end{document} ================================================ FILE: Euler/Euler-methods.tex ================================================ \label{ch:compressible} \section{Introduction} \label{sec:eulermeth:intro} The numerical methods we use for the Euler equations mirror those used with advection in Ch.~\ref{ch:advection}. The basic process is: \begin{itemize} \item Reconstruct the state variables to the interfaces. \item Solve the Riemann problem (as described in \S~\ref{euler:sec:riemann}) and construct the fluxes through the interfaces. \item Perform the conservative update on the state variables. \end{itemize} The Euler equations are much more complicated than linear advection---we already saw this with the Riemann problem. But we will also see that in different parts of the algorithm we will use conservative or primitive variables. Additional physics can enter as source terms, which we'll see how to add to both the interface states and conservative update here. %----------------------------------------------------------------------------- \section{Reconstruction of interface states} \label{sec:onedrecon} We will solve the Euler equations using a high-order {\em Godunov method}---a finite volume method whereby the fluxes through the interfaces are computed by solving the Riemann problem for our system. The finite-volume update for our system appears as: \begin{equation} \Uc^{n+1}_i = \Uc^n_i + \frac{\Delta t}{\Delta x} \left ( \Fb_{i-\myhalf}^{n+\myhalf} - \Fb_{i+\myhalf}^{n+\myhalf} \right ) \end{equation} This says that each of the conserved quantities in $\Uc$ change only due to the flux of that quantity through the boundary of the cell. Instead of approximating the flux itself on the interface, we find an approximation to the state on the interface, $\Uc_{i-\myhalf}^{n+\myhalf}$ and $\Uc_{i+\myhalf}^{n+\myhalf}$ and use this with the flux function to define the flux through the interface: \begin{align} \Fb_{i-\myhalf}^{n+\myhalf} &= \Fb(\Uc_{i-\myhalf}^{n+\myhalf}) \\ \Fb_{i+\myhalf}^{n+\myhalf} &= \Fb(\Uc_{i+\myhalf}^{n+\myhalf}) \end{align} To find this interface state, we predict left and right states at each interface (centered in time), which are the input to the Riemann solver. The Riemann solver will then look at the characteristic wave structure and determine the fluid state on the interface, which is then used to compute the flux. This is illustrated in Figure~\ref{fig:riemann}. Finally, although we use the conserved variables for the final update, in constructing the interface states it is often easier to work with the primitive variables. These have a simpler characteristic structure. The interface states in terms of the primitive variables can be converted into the interface states of the conserved variables through a simple algebraic transformation, \begin{equation} \Uc_{i+\myhalf,L}^{n+\myhalf} = \Uc(\qb_{i+\myhalf,L}^{n+\myhalf}) \end{equation} % figure made with figures/Euler/riemann-states.py \begin{figure}[t] \centering \includegraphics[width=\linewidth]{riemann_comp} \caption[The left and right states for the Riemann problem]{\label{fig:riemann} The left and right states at interface $i+\myhalf$. The arrow indicates the flux through the interface, as computed by the Riemann solver using these states as input.} \end{figure} As we saw with linear advection in \S~\ref{ch:adv:sndorder}, at the start of a timestep, each zones contains cell-averages. Consider a cell average quantity, $\qb_i$. Constructing the interface states requires reconstructing the cell-average data as a continuous function, $\qb(x)$, defined for each cell. This polynomial is only {\em piecewise} continuous, since each cell has its own $\qb(x)$. Standard choices are piecewise constant, \begin{equation} \qb(x) = \qb_i \end{equation} piecewise linear, \begin{equation} \qb(x) = \qb_i + \frac{\Delta \qb_i}{\Delta x}(x - x_i) \end{equation} or piecewise parabolic, \begin{equation} \qb(x) = \boldsymbol{\alpha} (x - x_i)^2 + \boldsymbol{\beta} (x - x_i) + \boldsymbol{\gamma} \end{equation} where in each case, the average of $\qb(x)$ over the cell gives $\qb_i$. Most of the complexity of the method is then figuring out the coefficients of the polynomial. Characteristic tracing is then done under this polynomial to see how much of each characteristic quantity comes to the interface over the timestep. The jump in the primitive variables is projected into the characteristic variables, and only jumps moving toward the interface are included in our reconstruction. We look at several methods below that build off of these ideas below. \subsection{Piecewise constant} The simplest possible reconstruction of the data is piecewise constant. This is what was done in the original Godunov method~\cite{godunov:1959}. For the interface marked by $i+\myhalf$, the left and right states on the interface are simply: \begin{align} \Uc_{i+\myhalf,L} &= \Uc_i \\ \Uc_{i+\myhalf,R} &= \Uc_{i+1} \end{align} This does not take into account in any way how the state $\Uc$ may be changing through the cell. As a result, it is first-order accurate in space, and since no attempt was made to center it in time, it is first-order accurate in time. \subsection{Piecewise linear} For higher-order reconstruction, we first convert from the conserved variables, $\Uc$, to the primitive variables, $\qb$. These have a simpler characteristic structure, making them easier to work with. Here we consider piecewise linear reconstruction---the cell average data is approximated by a line with non-zero slope within each cell. Figure~\ref{fig:plm} shows the piecewise linear reconstruction of some data. % figure made with figures/Euler/ppm.py \begin{figure}[t] \centering \includegraphics[width=\linewidth]{piecewise-linear} \caption[Piecewise linear reconstruction of cell average data]{\label{fig:plm} Piecewise linear reconstruction of the cell averages. The dotted line shows the unlimited center-difference slopes and the solid line shows the limited slopes.} \end{figure} Consider constructing the left state at the interface $i+\myhalf$ (see Figure~\ref{fig:riemann}). Just like for the advection equation, we do a Taylor expansion through $\Delta x/2$ to bring us to the interface, and $\Delta t/2$ to bring us to the midpoint in time. Starting with $\qb_i$, the cell-centered primitive variable, expanding to the right interface (to create the left state there) gives: \begin{align} \qb_{i+\myhalf,L}^{n+\myhalf} &= \qb_i^n + \left . \frac{\Delta x}{2} \frac{\partial \qb}{\partial x} \right |_i + \frac{\Delta t}{2} \underbrace{\left .\frac{\partial \qb}{\partial t} \right |_i}_{= -\Ab \partial \qb / \partial x} + \ldots \\ &= \qb_i^n + \frac{\Delta x}{2} \left . \frac{\partial \qb}{\partial x} \right |_i - \frac{\Delta t}{2} \left ( \Ab \frac{\partial \qb}{\partial x} \right )_i\\ &= \qb_i^n + \frac{1}{2} \left [ 1 - \frac{\Delta t}{\Delta x} \Ab_i \right ] \Delta \qb_i \label{eq:taylorstate} \end{align} where $\Delta \qb_i$ is the reconstructed slope of the primitive variable in that cell (similar to how we compute it for the advection equation). We note that the terms truncated in the first line are $O(\Delta x^2)$ and $O(\Delta t^2)$, so our method will be second-order accurate in space and time. As with the advection equation, we limit the slope such that no new minima or maxima are introduced. Any of the slope limiters used for linear advection apply here as well. We represent the limited slope as $\overline{\Delta \qb}_i$. We can decompose $\Ab \Delta \qb$ in terms of the left and right eigenvectors and sum over all the waves that move {\em toward} the interface. First, we recognize that $\Ab = R \Lambda L$ and recognizing that the `$1$' in Eq.~\ref{eq:taylorstate} is the identity, $I = LR$, we rewrite this expression as: \begin{equation} \qb_{i+\myhalf,L}^{n+\myhalf} = \qb_i^n + \frac{1}{2} \left [ \Rb\Lb - \frac{\Delta t}{\Delta x} \Rb\Lambdab \Lb \right ]_i \overline{\Delta \qb}_i \end{equation} We see the common factor of $\Lb \Delta \qb$. We now write this back in component form. Consider: \begin{equation} \renewcommand{\arraystretch}{1.5} \Rb\Lambdab \Lb \overline{\Delta \qb} = \left ( \begin{array}{c:c:c} r_1^\evm & r_1^\evz & r_1^\evp \\ r_2^\evm & r_2^\evz & r_2^\evp \\ r_3^\evm & r_3^\evz & r_3^\evp \end{array} \right ) \left ( \begin{array}{ccc} \lambda^\evm & & \\ & \lambda^\evz & \\ & & \lambda^\evp \end{array} \right ) \left ( \begin{array}{ccc} l_1^\evm & l_2^\evm & l_3^\evm \\ \hdashline l_1^\evz & l_2^\evz & l_3^\evz \\ \hdashline l_1^\evp & l_2^\evp & l_3^\evp \end{array} \right ) \left ( \begin{array}{c} \overline{\Delta \rho} \\ \overline{\Delta u} \\ \overline{\Delta p} \end{array} \right ) \renewcommand{\arraystretch}{1.0} \end{equation} Starting with $\Lb \overline{\Delta \qb}$, which is a vector with each component the dot-product of a left eigenvalue with $\overline{\Delta \qb}$, we have \begin{equation} \renewcommand{\arraystretch}{1.5} \Rb\Lambdab \Lb \overline{\Delta \qb} = \left ( \begin{array}{c:c:c} r_1^\evm & r_1^\evz & r_1^\evp \\ r_2^\evm & r_2^\evz & r_2^\evp \\ r_3^\evm & r_3^\evz & r_3^\evp \end{array} \right ) \left ( \begin{array}{ccc} \lambda^\evm & & \\ & \lambda^\evz & \\ & & \lambda^\evp \end{array} \right ) \left ( \begin{array}{c} \lb^\evm \cdot \overline{\Delta \qb} \\ \lb^\evz \cdot \overline{\Delta \qb} \\ \lb^\evp \cdot \overline{\Delta \qb} \end{array} \right ) \renewcommand{\arraystretch}{1.0} \end{equation} Next we see that multiplying this vector by $\Lambdab$ simply puts the eigenvalue with its respective eigenvector in the resulting column vector: \begin{equation} \renewcommand{\arraystretch}{1.5} \Rb\Lambdab \Lb \overline{\Delta \qb} = \left ( \begin{array}{c:c:c} r_1^\evm & r_1^\evz & r_1^\evp \\ r_2^\evm & r_2^\evz & r_2^\evp \\ r_3^\evm & r_3^\evz & r_3^\evp \end{array} \right ) \left ( \begin{array}{c} \lambda^\evm \, \lb^\evm \cdot \overline{\Delta \qb} \\ \lambda^\evz \, \lb^\evz \cdot \overline{\Delta \qb} \\ \lambda^\evp \, \lb^\evp \cdot \overline{\Delta \qb} \end{array} \right ) \renewcommand{\arraystretch}{1.0} \end{equation} Finally, the last multiply results in a column vector: \begin{equation} \renewcommand{\arraystretch}{1.5} R\Lambda L \overline{\Delta \qb} = \left ( \begin{array}{c} r_1^\evm \, \lambda^\evm \, \lb^\evm \cdot \overline{\Delta \qb} + r_1^\evz \, \lambda^\evz \, \lb^\evz \cdot \overline{\Delta \qb} + r_1^\evp \, \lambda^\evp \, \lb^\evp \cdot \overline{\Delta \qb} \\ % r_2^\evm \, \lambda^\evm \, \lb^\evm \cdot \overline{\Delta \qb} + r_2^\evz \, \lambda^\evz \, \lb^\evz \cdot \overline{\Delta \qb} + r_2^\evp \, \lambda^\evp \, \lb^\evp \cdot \overline{\Delta \qb} \\ % r_3^\evm \, \lambda^\evm \, \lb^\evm \cdot \overline{\Delta \qb} + r_3^\evz \, \lambda^\evz \, \lb^\evz \cdot \overline{\Delta \qb} + r_3^\evp \, \lambda^\evp \, \lb^\evp \cdot \overline{\Delta \qb} \\ \end{array} \right ) \renewcommand{\arraystretch}{1.0} \end{equation} We can rewrite this compactly as: \begin{equation} \sum_\nu \lambda^{(\nu)} (\lb^{(\nu)} \cdot \overline{\Delta \qb}) \rb^{(\nu)} \end{equation} where we use $\nu$ to indicate which wave we are summing over. A similar expansion is used for $\Rb\Lb \overline{\Delta \qb}$. In fact, any vector can be decomposed in this fashion: \begin{equation} {\boldsymbol{\chi}} = {\bf I} {\boldsymbol{\chi}} = \Rb \Lb {\boldsymbol{\chi}} = \sum_\nu (\lb^\enu \cdot {\boldsymbol{\chi}}) \rb^\enu \end{equation} And then it is easy to see that the above manipulations for $\Ab \Delta \qb$ can be expressed as: \begin{equation} \Ab \Delta \qb = \Ab \sum_\nu (\lb^\enu \cdot \overline{\Delta \qb}) \rb^\enu = \sum_\nu (\lb^\enu \cdot \overline{\Delta \qb}) \Ab \rb^\enu = \sum_\nu (\lb^\enu \cdot \overline{\Delta \qb}) \lambda^\enu \rb^\enu \end{equation} where we used $\Ab \rb^\enu = \lambda^\enu \rb^\enu$. The quantity $(\lb^\enu \cdot \overline{\Delta \qb})$ that shows up here is the projection of the vector $\overline{\Delta \qb}$ into the characteristic variable carried by wave $\nu$. This sum shows, as discussed earlier, that each wave carries a jump in the characteristic variable, with the jump in the primitive variables proportion to the right eigenvector, $\rb^\enu$. The resulting vector for the left state is: \begin{equation} \qb_{i+\myhalf,L}^{n+\myhalf} = \qb_i^n + \frac{1}{2} \sum_{\nu; \lambda^{(\nu)} \ge 0} \left [ 1 - \frac{\Delta t}{\Delta x} \lambda_i^{(\nu)} \right ] (\lb_i^{(\nu)} \cdot \overline{\Delta \qb}_i) \rb_i^{(\nu)} \end{equation} Note that we make a slight change here, and only include a term in the sum if its wave is moving toward the interface ($\lambda^\enu \ge 0$). The quantity $\Delta t \lambda^\enu/\Delta x$ inside the brackets is simply the CFL number for the wave $\nu$. Starting with the data in the $i+1$ zone and expanding to the left, we can find the right state on the $i+\myhalf$ interface: \begin{equation} \qb_{i+\myhalf,R}^{n+\myhalf} = \qb_{i+1}^n - \frac{1}{2} \sum_{\nu; \lambda^{(\nu)} \le 0} \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)} \end{equation} A good discussion of this is in Miller \& Colella \cite{millercolella:2002} (Eq.\ 85). This expression is saying that each wave carries a jump in $\rb^{(\nu)}$ and only those jumps moving toward the interface contribute to our interface state. This restriction of only summing up the waves moving toward the interface is sometimes called {\em characteristic tracing}. This decomposition in terms of the eigenvectors and eigenvalues is commonly called a {\em characteristic projection}. In terms of an operator, $P$, it can be expressed as: \begin{equation} P {\boldsymbol{\chi}} = \sum_\nu (\lb^{(\nu)} . {\boldsymbol{\chi}}) \rb^{(\nu)} \end{equation} \begin{exercise}[Characteristic projection] { Show that $P \qb = \qb$, using the eigenvectors corresponding to the primitive variable form of the Euler equations.} \end{exercise} In the literature, sometimes a `$>$' or `$<$' subscript on $P$ is used to indicate the characteristic tracing. We could stop here, but Colella \& Glaz~\cite{colellaglaz:1985} (p.\ 278) argue that the act of decomposing $\Ab$ in terms of the left and right eigenvectors is a linearization of the quasi-linear system, and we should minimize the size of the quantities that are subjected to this characteristic projection. To accomplish this, they suggest subtracting off a {\em reference state}. Saltzman (Eq.\ 8) further argues that since only jumps in the solution are used in constructing the interface state, and that the characteristic decomposition simply adds up all these jumps, we can subtract off the reference state and project the result. In other words, we can write: \begin{equation} \qb_{i+\myhalf,L}^{n+\myhalf} - \qb_\mathrm{ref} = \qb_i^n - \qb_\mathrm{ref} + \frac{1}{2} \left [ 1 - \frac{\Delta t}{\Delta x} \Ab_i \right ] \overline{\Delta \qb}_i \label{eq:lin_decomp} \end{equation} Then we subject the RHS to the characteristic projection---this tells us how much of the quantity $\qb_{i+\myhalf,L}^{n+\myhalf} - \qb_\mathrm{ref}$ reaches the interface. Colella \& Glaz (p.\ 278) and Colella (Eq.\ 2.11) suggest \begin{equation} \qb_\mathrm{ref} = \tilde{\qb}_{i,L} \equiv \qb_i^n + \frac{1}{2} \left [ 1 - \frac{\Delta t}{\Delta x} \max(\lambda_i^\evp, 0) \right ] \overline{\Delta \qb}_i \label{eq:qref} \end{equation} where $\lambda^\evp$ is the fastest eigenvalue, and thus will see the largest portion of the linear profiles. Physically, this reference state represents the jump carried by the fastest wave moving toward the interface. Then, \begin{equation} \qb_{i+\myhalf,L}^{n+\myhalf} - \tilde{\qb}_{i,L} = \frac{1}{2} \frac{\Delta t}{\Delta x} \left [ \max(\lambda_i^\evp,0) - \Ab_i \right ] \overline{\Delta \qb}_i \end{equation} and projecting this RHS (see Colella \& Glaz Eq. 43; Miller \& Colella Eq. 87), and isolating the interface state, we have \begin{align} \qb_{i+\myhalf,L}^{n+\myhalf} &= \tilde{\qb}_{i,L} + \frac{1}{2} \frac{\Delta t}{\Delta x} \sum_{\nu; \lambda^{(\nu)} \ge 0} \lb_i^{(\nu)} \cdot \left [ \max(\lambda_i^\evp,0) - \Ab_i \right ] \overline{\Delta \qb}_i \, \rb_i^{(\nu)} \\ &= \tilde{\qb}_{i,L} + \frac{1}{2} \frac{\Delta t}{\Delta x} \sum_{\nu; \lambda^{(\nu)} \ge 0} \left [ \max(\lambda_i^\evp,0) - \lambda_i^{(\nu)} \right ] (\lb_i^{(\nu)} \cdot \overline{\Delta \qb}_i) \, \rb_i^{(\nu)} \end{align} This is equivalent to the expression in Saltzman~\cite{saltzman:1994} (p.\ 161, first column, second-to-last equation) and Colella~\cite{colella:1990} (p.\ 191, the group of expressions at the end)\footnote{Since the combination $\lb^\enu \cdot \Delta \qb$ appears so frequently, some sources represent this as $\beta^\enu$ and then write the interface states as a linear combination of the $\beta$'s. Some variation in the literation, depending on whether $\rho$ or $\tau = 1/\rho$ is used as a primitive variable}. The corresponding state to the right of this interface is: \begin{equation} \qb_{i+\myhalf,R}^{n+\myhalf} = \tilde{\qb}_{i+1,R} + \frac{1}{2} \frac{\Delta t}{\Delta x} \sum_{\nu; \lambda^\enu \le 0} \left [ \min(\lambda_{i+1}^\evm,0) - \lambda_{i+1}^\enu \right ] (\lb_{i+1}^\enu \cdot \overline{\Delta \qb}_{i+1}) \, \rb_{i+1}^\enu \end{equation} where now the reference state captures the flow from the $i+1$ zone moving to the {\em left} to this interface (hence the appearance of $\lambda^\evm$, the leftmost eigenvalue): \begin{equation} \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} \end{equation} Side note: the data in zone $i$ will be used to construct the right state at $i-\myhalf$ (the left interface) and the left state at $i+\myhalf$ (the right interface) (see Figure~\ref{fig:states}). For this reason, codes usually compute the eigenvectors/eigenvalues for that zone and then compute $\qb_{i-\myhalf,R}^{n+\myhalf}$ together with $\qb_{i+\myhalf,L}^{n+\myhalf}$ in a loop over the zone centers. % figure made by figures/Euler/states.py \begin{figure} \centering \includegraphics[width=3.5in]{states} \caption[The two interface states derived from a cell-center quantity]{\label{fig:states} The two interface states that are constructed using $\qb_i$ as the starting point.} \end{figure} \subsection{Piecewise parabolic} \label{sec:hydro:ppm} The piecewise parabolic method uses a parabolic reconstruction in each cell. This is more accurate than the linear reconstruction. Figure~\ref{fig:ppm} shows the reconstructed parabolic profiles within a few cells. Since the original PPM paper~\cite{colellawoodward:1984}, there have been many discussions of the method, with many variations. Here we focus on the presentation by Miller \& Colella~\cite{millercolella:2002}, since that is the most straightforward. Note: even though a parabolic profile could be third-order accurate, the temporal discretization and prediction in this method is still only second-order. % % figure made by figures/Euler/ppm.py \begin{figure}[t] \centering \includegraphics[width=\linewidth]{piecewise-parabolic} \caption[Piecewise parabolic reconstruction of the cell averages]{\label{fig:ppm} Piecewise parabolic reconstruction of the cell averages. The dotted line shows the unlimited parabolas---note how they touch at each interface, since the interface values come from the same interpolant initially. The solid line shows the limited parabolas.} \end{figure} Miller \& Colella give an excellent description of how to take the results for piecewise linear reconstruction and generalize it to the case of PPM~\cite{colellawoodward:1984} (see Eqs.\ 88-90). Starting with Eq.~\ref{eq:lin_decomp}, we can write this (after the characteristic projection) as \begin{equation} \qb_{i+\myhalf,L}^{n+\myhalf} = \tilde{\qb}_+ - \sum_{\nu;\lambda^{(\nu)}\ge 0} \lb_i^{(\nu)} \cdot \left \{ \tilde{\qb}_+ - \left [ \qb_i^n + \frac{1}{2} \left ( 1 - \frac{\Delta t}{\Delta x} \lambda_i^{(\nu)} \right ) \overline{\Delta \qb}_i \right ] \right \} \rb_i^{(\nu)} \end{equation} Miller \& Colella rewrite the portion inside the $[\ldots]$ recognizing that (similar to M\&C Eq.\ 88, but for the $i+\myhalf,L$ interface): \begin{equation} \qb_i^n + \frac{1}{2} \left (1 - \frac{\Delta t}{\Delta x} \lambda_i^{(\nu)} \right ) \overline{\Delta \qb}_i \approx \frac{1}{\lambda \Delta t} \int_{x_{i+\myhalf} - \lambda \Delta t}^{x_{i+\myhalf}} \qb(x) dx \end{equation} where $\qb(x)$ is the reconstructed functional form of $\qb$ in the zone. \begin{exercise}[The average state reacting the interface] {Show that this is exactly true for a linear reconstruction of $\qb(x)$, i.e., $\qb(x) = \qb_i + (\partial \qb/\partial x) (x - x_i)$.} \end{exercise} \noindent The integral on the right represents the average of $\qb$ that can reach the right interface of the cell $i$ over timestep $\Delta t$, moving at the wavespeed $\lambda$. This suggests that we can replace the linear reconstruction of $\qb$ with a parabolic one, and keep our expressions for the interface states. % figure made by figures/Euler/ppm-trace.py \begin{figure} \centering \includegraphics[width=3.5in]{ppm-trace} \caption[Integration under the parabola profile for to an interface]{\label{fig:ppm_trace} Integration under the parabolic profile. For each of the waves, $\sigma^\enu$ is the fraction of the cell that they cross in a timestep, and $\sigma^\enu \Delta x = \lambda^\enu \Delta t$ is the distance they can travel. Here we are integrating under the parabola to the right interface of cell $i$ to define $\Ic_+^\enu$ (this is indicated by the shaded region). The $\Ic_+^\enu$ carried by this wave will be added to those carried by the other waves to form the left state at interface $i+\myhalf$.} \end{figure} In particular, we define \begin{equation} \Ic_+^{(\nu)}(\qb_i) = \frac{1}{\sigma^{(\nu)} \Delta x} \int _{x_{i+\myhalf} - \sigma^{(\nu)} \Delta x}^{x_{i+\myhalf}} \qb(x) dx \end{equation} with $\sigma^{(\nu)} = |\lambda^{(\nu)}|\Delta t / \Delta x$ (see Almgren et al. Eq. 31) (see Figure~\ref{fig:ppm_trace}). Then \begin{equation} \qb_{i+\myhalf,L}^{n+\myhalf} = \tilde{\qb}_+ - \sum_{\nu;\lambda^{(\nu)}\ge 0} \lb_i^{(\nu)} \cdot \left ( \tilde{\qb}_+ - \Ic_+^{(\nu)}(\qb_i) \right ) \rb_i^{(\nu)} \label{eq:ppmtrace} \end{equation} Miller \& Colella choose the reference state as \begin{equation} \tilde{\qb}_+ = \left \{ \begin{array}{cc} \Ic_+^{(+)}(\qb_i) & \mathrm{if~} u + c > 0 \\ \qb_i & \mathrm{otherwise} \end{array} \right . \end{equation} where the superscript $(+)$ on $\Ic$ indicates that the fastest eigenvalue ($\lambda^\evp = u + c$) is used. This is similar in spirit to Eq.~\ref{eq:qref}. Note: in the original PPM paper, if the wave is not approaching the interface, instead of using the cell-average, $\qb_i$, they use the limit of the quadratic interpolant. In contrast to the above, the Castro paper~\cite{almgren:2010} just uses $\qb_i$ for the reference state regardless of whether the wave is moving toward or away from the interface. Note that if the system were linear, then the choice of reference state would not matter. To finish the reconstruction, we need to know the parabolic form of $\qb(x)$. Here, we do the reconstruction from the original PPM paper: \begin{equation} \qb(x) = \qb_{-} + \xi(x) \left ( \Delta \qb + \qb_6 (1 - \xi(x) ) \right ) \end{equation} with $\Delta \qb = \qb_+ - \qb_-$, and $\qb_-$, $\qb_+$ the values of the polynomial on the left and right edges, respectively, of the current cell, and \begin{equation} \qb_6 \equiv 6 \left [ \qb_i - \frac{1}{2} (\qb_- + \qb_+) \right ] \end{equation} and \begin{equation} \xi(x) = \frac{x - x_{i-\myhalf}}{\Delta x} \end{equation} To complete the description, we need to determine the parameters of the parabola. The values of $\qb_-$ and $\qb_+$ are computed and limited as described in the original PPM paper. With this definition, we can do the integral $\Ic_+$: \begin{equation} \Ic_+^{(\nu)}(\qb_i) = \qb_{+,i} - \frac{\sigma_i^{(\nu)}}{2} \left [ \Delta \qb_i - \qb_{6,i} \left ( 1 - \frac{2}{3} \sigma_i^{(\nu)} \right ) \right ] \end{equation} Figure~\ref{fig:ppm_trace} illustrates the process of integrating under the parabolic profile. \begin{exercise}[Conservative interpolation] {Show that $\qb(x)$ is a conservative interpolant. That is \begin{equation} \frac{1}{\Delta x} \int_{x_{i-\myhalf}}^{x_{i+\myhalf}} \qb(x) dx = \qb_i \end{equation} You can also see that the average over the left half of the zone is $\qb_i -\frac{1}{4}\Delta \qb$ and the average over the right half of the zone is $\qb_i + \frac{1}{4}\Delta \qb$. This means that there are equal areas between the integral and zone average on the left and right sides of the zone. This can be seen by looking at Figure~\ref{fig:ppm}. } \end{exercise} \begin{quote} \noindent\makebox[\linewidth]{\rule{0.9\textwidth}{1pt}} {\em Aside}: Note that this characteristic projection of $\tilde{\qb}_+ - \Ic_+^{(\nu)}$ is discussed in the original PPM paper in the paragraph following their Eq.~3.5. They do not keep things in this form however, and instead explicitly multiply out the $l\cdot [\ldots] r$ terms to arrive at their Eq.~3.6. For example, starting with Eq.~\ref{eq:ppmtrace}, we can write the left velocity state as (leaving off the $i$ subscripts on the vectors): \begin{equation} u_{i+\myhalf,L}^{n+\myhalf} = \tilde{u}_+ - \sum_\nu \lb^{(\nu)} \cdot ( \tilde{\qb}_+ - \Ic_+^{(\nu)}(\qb) ) \underbrace{\rb^{(\nu)}}_{\begin{smallmatrix}\mathrm{only~the}\\ u~\mathrm{`slot'} \end{smallmatrix}} \end{equation} (where, as above, the $\sim$ indicates the reference state). Here the $r$ eigenvector on the end is representative---we only pick the row corresponding to $u$ in the $\qb$ vector (in our case, the second row). Putting in the eigenvectors and writing out the sum, we have: \begin{align} u_{i+\myhalf,L}^{n+\myhalf} = \tilde{u}_+ &- \left ( \begin{array}{ccc} 0 & -\frac{\rho}{2c} & \frac{1}{2c^2} \end{array} \right ) \left ( \begin{array}{c} \tilde{\rho}_+ - \Ic_+^\evm(\rho) \\ \tilde{u}_+ - \Ic_+^\evm(u) \\ \tilde{p}_+ - \Ic_+^\evm(p) \end{array} \right ) {\color{mygray} \left ( \begin{array}{c} 1 \\ {\color{black} -c/\rho} \\ c^2 \end{array} \right ) } \nonumber \\ % &-\left ( \begin{array}{ccc} 1 & 0 & -\frac{1}{c^2} \end{array} \right ) \left ( \begin{array}{c} \tilde{\rho}_+ - \Ic_+^\evz(\rho) \\ \tilde{u}_+ - \Ic_+^\evz(u) \\ \tilde{p}_+ - \Ic_+^\evz(p) \end{array} \right ) {\color{mygray} \left ( \begin{array}{c} 1 \\ {\color{black} 0} \\ 0 \end{array} \right ) } \nonumber \\ % &-\left ( \begin{array}{ccc} 0 & \frac{\rho}{2c} & \frac{1}{2c^2} \end{array} \right ) \left ( \begin{array}{c} \tilde{\rho}_+ - \Ic_+^\evp(\rho) \\ \tilde{u}_+ - \Ic_+^\evp(u) \\ \tilde{p}_+ - \Ic_+^\evp(p) \end{array} \right ) {\color{mygray} \left ( \begin{array}{c} 1 \\ {\color{black} c/\rho} \\ c^2 \end{array} \right ) } % \end{align} Here again we show the entire right eigenvector for illustration, but only the element that comes into play is drawn in black. This shows that the second term is $0$---the contact wave does not carry a jump in velocity. Multiplying out $\lb^{(\nu)} \cdot (\tilde{\qb}_+ - \Ic_+^{(\nu)})$ we have: \begin{align} u_{i+\myhalf,L}^{n+\myhalf} = \tilde{u}_+ &- \frac{1}{2} \left [ (\tilde{u}_+ - \Ic_+^\evm(u) ) - \frac{\tilde{p}_+ - \Ic_+^\evm(p)}{C} \right ] \nonumber \\ &- \frac{1}{2} \left [ (\tilde{u}_+ - \Ic_+^\evp(u) ) + \frac{\tilde{p}_+ - \Ic_+^\evp(p)}{C} \right ] \label{eq:ufull} \end{align} where $C$ is the Lagrangian sound speed ($C = \sqrt{\gamma p \rho}$). Defining \begin{align} \beta^\evp &= - \frac{1}{2C} \left [ (\tilde{u}_+ - \Ic_+^\evp(u) ) + \frac{\tilde{p}_+ - \Ic_+^\evp(p)}{C} \right ] \\ % \beta^\evm &= + \frac{1}{2C} \left [ (\tilde{u}_+ - \Ic_+^\evm(u) ) - \frac{\tilde{p}_+ - \Ic_+^\evm(p)}{C} \right ] \end{align} we can write our left state as: \begin{equation} u_{i+\myhalf,L}^{n+\myhalf} = \tilde{u}_+ + C ( \beta^\evp - \beta^\evm) \end{equation} This is Eqs.~3.6 and 3.7 in the PPM paper. Note that in their construction appears to use the reference state in defining the Lagrangian sound speed (in their $\beta$ expressions is written as $\tilde{C}$). This may follow from the comment before Eq.~3.6, ``modified slightly for the present application''. Similarly, the expressions for $\rho_L$ and $p_L$ can be written out. \\ \noindent\makebox[\linewidth]{\rule{0.9\textwidth}{1pt}} \end{quote} Similar expressions can be derived for the right state at the left interface of the zone ($\qb_{i-\myhalf,R}^{n+\myhalf}$). Here, the integral under the parabolic reconstruction is done over the region of each wave that can reach the left interface over our timestep: \begin{equation} \Ic_-^{(\nu)}(\qb) = \frac{1}{\sigma^{(\nu)} \Delta x} \int_{x_{i-\myhalf}}^{x_{i-\myhalf} + \sigma^{(\nu)} \Delta x} \qb(x) dx \end{equation} The right state at $i-\myhalf$ using zone $i$ data is: \begin{equation} \qb_{i-\myhalf,R}^{n+\myhalf} = \tilde{\qb}_- - \sum_{\nu; \lambda_\nu \le 0} \lb_i^{(\nu)} \cdot \left ( \tilde{\qb}_- - \Ic_-^{(\nu)}(\qb_i) \right ) \rb_i^{(\nu)} \end{equation} where the reference state is now: \begin{equation} \tilde{\qb}_- = \left \{ \begin{array}{cc} \Ic_-^{(-)}(\qb_i) & \mathrm{if~} u - c < 0 \\ \qb_i & \mathrm{otherwise} \end{array} \right . \end{equation} where the $(-)$ superscript on $\Ic$ indicates that the most negative eigenvalue $(\lambda^- = u - c)$ is used. The integral $\Ic_-^{(\nu)}(\qb)$ can be computed analytically by substituting in the parabolic interpolant, giving: \begin{equation} \Ic_-^{(\nu)}(\qb_i) = \qb_{-,i} + \frac{\sigma_i^{(\nu)}}{2} \left [ \Delta \qb_i + \qb_{6,i} \left ( 1 - \frac{2}{3} \sigma_i^{(\nu)} \right ) \right ] \end{equation} This is equivalent to Eq.~31b in the Castro paper. \subsubsection{New PPM limiters} Recent work \cite{colellasekora} has formulated improved limiters for PPM that do not clip the profiles at extrema. This only changes the limiting process in defining $\qb_+$ and $\qb_-$, and does not affect the subsequent parts of the algorithm. \subsection{Flattening and contact steepening} Shocks are self-steepening (this is how we detect them in the Riemann solver---we look for converging characteristics). This can cause trouble with the methods here, because the shocks may become too steep. Flattening is a procedure to add additional dissipation at shocks, to ensure that they are smeared out over $\sim 2$ zones. The flattening procedure is a multi-dimensional operation that looks at the pressure and velocity profiles and returns a coefficient, $\chi \in [0,1]$ that multiplies the limited slopes. The convention most sources use is that $\chi = 1$ means no flattening (the slopes are unaltered), while $\chi = 0$ means complete flattening---the slopes are zeroed, dropping us to a first-order method. See for example in Saltzman~\cite{saltzman:1994}. Once the flattening coefficient is determined, the interface state is blended with the cell-centered value via: \begin{equation} \qb_{i+\myhalf,\{L,R\}}^{n+\myhalf} \leftarrow (1 - \chi) \qb_i + \chi \qb_{i+\myhalf,\{L,R\}}^{n+\myhalf} \end{equation} Note that the flattening algorithm increases the stencil size of piecewise-linear and piecewise-parabolic reconstruction to 4 ghost cells on each side. This is because the flattening procedure itself looks at the pressure 2 zones away, and we need to construct the flattening coefficient in both the first ghost cell (since we need the interface values there) and the second ghost cell (since the flattening procedure looks at the coefficients in its immediate upwinded neighbor). In contrast to shocks, contact waves do not steepen (they are associated with the middle characteristic wave, and the velocity does not change across that, meaning there cannot be any convergence). The original PPM paper advocates a contact steepening method to artificially steepen contact waves. While it shows good results in 1-d, it can be problematic in multi-dimensions. Overall, the community seems split over whether this term should be used. Many people advocate that if you reach a situation where you think contact steepening may be necessary, it is more likely that the issue is that you do not have enough resolution. \subsection{Limiting on characteristic variables} Some authors (see for example, \cite{athena} Eqs.~37, 38) advocate limiting on the characteristic variables rather than the primitive variables. The characteristic slopes for the quantity carried by the wave $\nu$ can be found from the primitive variables as: % \begin{equation} \Delta w^{(\nu)} = \lb^{(\nu)} \cdot \Delta \qb \end{equation} % any limiting would then be done to $\Delta w^{(\nu)}$ and the limited primitive variables would be recovered as: \begin{equation} \overline{\Delta \qb} = \sum_\nu \overline{\Delta w}^{(\nu)} \rb^{(\nu)} \end{equation} (here we use an overline to indicate limiting). This is attractive because it is more in the spirit of the linear advection equation and the formalism that was developed there. A potential downside is that when you limit on the characteristic variables and convert back to the primitive, the primitive variables may now fall outside of valid physical ranges (for example, negative density). %----------------------------------------------------------------------------- \section{Riemann solvers} Once the interface states are created, the Riemann solver is called. This returns the solution at the interface: \begin{equation} \qb_{i+\myhalf}^{n+\myhalf} = \mathcal{R}(\qb_{i+\myhalf,L}^{n+\myhalf}, \qb_{i+\myhalf,R}^{n+\myhalf}) \end{equation} This is done for every interface, as illustrated in Figure~\ref{fig:euler:multiple_interfaces}. % figure drawn with figures/Euler/multiple_interfaces.py \begin{figure} \centering \includegraphics[width=\linewidth]{multiple_interfaces} \caption[Riemann wave structure at each interface]{\label{fig:euler:multiple_interfaces} The Riemann wave structure resulting from the separate Riemann problems at each interface. For each interface, we find the state on the interface and use this to evaluate the flux through the interface.} \end{figure} As discussed in \S~\ref{Euler:riemann:solution}, we need to determine which state is on our interface to compute the fluxes through the interface. The full solution of the Riemann problem can be quite expensive, so in practice, approximate Riemann solvers are used to speed the computation. Different Riemann solvers will have different approximations for finding the speeds of the left, center, and right wave, and evaluating the star state. In the `star' region, only $\rho$ jumps across the middle (contact) wave, the pressure and velocity are constant across that wave (see $\rb^\evz$). We determine the state in the star region ($\rho_l^*, \rho_r^*, u^*, p^*$) by using the jump conditions for the Euler equations or the Riemann invariants, as shown in \S~\ref{Euler:riemann:starstate}. Some approximate Riemann solvers assume that both waves are shocks or both are rarefactions, aimplying the form of the equation that must be solved to find $p_\star$. Two-shock solvers, like the one in \cite{colellaglaz:1985} are quite common in astrophyiscs. Figure~\ref{fig:euler:riemann-2shock-curve} shows the Hugoniot curves for the Sod problem under the 2-shock assumption---they are quite close to those for the true solution. To further save on cost, approximate Riemann solvers for general equations of state often include additional thermodynamic information at the interfaces that overconstrains the system, but can remove the need to call an expensive equation of state routine in solving for the star state. \begin{figure}[t] \centering \includegraphics[width=0.9\linewidth]{riemann-2shock-sod-phase} \caption[The approximate (2-shock) Hugoniot curves corresponding to the Sod problem]{\label{fig:euler:riemann-2shock-curve} The approximate Hugoniot curves under the 2-shock approximation corresponding to the Sod problem. Also shown as the gray dotted line is the exact Hugoniot curves (compare to Figure~\ref{fig:euler:riemann-curve}). We see that the 2-shock approximation does a reasonable job near the intersection and only diverges significantly for small $p$ (which is where the solution should really be a rarefaction).\\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/compressible/riemann-2shock.py}{riemann-2shock.py}}} \end{figure} An additional approximation concerns rarefactions. Recall that a rarefaction involves diverging flow---it spreads out with time. Special consideration needs to be taken if the rarefaction wave spans the interface (a {\em transonic rarefaction}, \S~\ref{Euler:riemann:solution}). In this case, most approximate Riemann solvers simply linearly interpolate between the left or right state and the appropriate star state instead of solving for the structure inside the rarefaction. With this approximate state, the fluxes are computed as: \begin{equation} \renewcommand{\arraystretch}{1.5} \Fb_{i+\myhalf}^{n+\myhalf} = \left ( \begin{array}{c} \rho_{i+\myhalf}^{n+\myhalf} u_{i+\myhalf}^{n+\myhalf} \\ \rho_{i+\myhalf}^{n+\myhalf} (u_{i+\myhalf}^{n+\myhalf})^2 + p_{i+\myhalf}^{n+\myhalf} \\ u_{i+\myhalf}^{n+\myhalf} p_{i+\myhalf}^{n+\myhalf} / (\gamma - 1) + \frac{1}{2} \rho_{i+\myhalf}^{n+\myhalf} (u_{i+\myhalf}^{n+\myhalf})^3 + u_{i+\myhalf}^{n+\myhalf} p_{i+\myhalf}^{n+\myhalf} \end{array} \right ) \renewcommand{\arraystretch}{1.0} \end{equation} A different class of approximate Riemann solvers (the HLL family) approximate the fluxes directly instead of approximating the state first. These require estimates of the wave speeds, and care must be taken to ensure those estimates are valid for a general equation of state. These wave speeds are then used together with the Rankine-Hugoniot jump conditions to give the fluxes. %----------------------------------------------------------------------------- \section{Conservative update} Once we have the fluxes, the conservative update is done as \begin{equation} \Uc^{n+1}_i = \Uc^n_i + \frac{\Delta t}{\Delta x} \left ( \Fb_{i-\myhalf}^{n+\myhalf} - \Fb_{i+\myhalf}^{n+\myhalf} \right ) \end{equation} The timestep, $\Delta t$ is determined by the time it takes for the fastest wave to cross a single zone: \begin{equation} \Delta t < \min_d \left \{ \frac{\Delta x}{|\Ub \cdot {\bf e}_d| + c}\right \} \end{equation} (This is the limit for the CTU unsplit scheme here, but see the discussion in \S~\ref{sec:adv:timestep}). Often simulation codes will further restrict the timestep. Commonly used restrictions for pure hydrodynamics include a limit on the factor by which a timestep can grow from one step to the next (a typical value is $1.2$), and an initial scaling, say of 1/10, for the first timestep. Together, these will force the code to take a few initial steps before working up to the CFL limit. \subsection{Artificial viscosity} Colella and Woodward argue that behind slow-moving shocks these methods can have oscillations. The fix they propose is to use some artificial viscosity---this is additional dissipation that kicks in at shocks. (They argue that flattening alone is not enough). We use a multidimensional analog of their artificial viscosity (\cite{colellawoodward:1984}, Eq.\ 4.5) which modifies the fluxes. By design, it only kicks in for converging flows, such that you would find around a shock. %----------------------------------------------------------------------------- \section{Boundary conditions} Boundary conditions are implemented through ghost cells. The following are the most commonly used boundary conditions. For the expressions below, we use the subscript $\mathrm{lo}$ to denote the spatial index of the first valid zone in the domain (just inside the left boundary). \begin{itemize} \item {\em Outflow}: the idea here is that the flow should gracefully leave the domain. The simplest form is to simply give all variables a zero-gradient: \begin{equation} \left ( \begin{array}{c} \rho_{\mathrm{lo-1},j} \\ (\rho u)_{\mathrm{lo-1},j} \\ (\rho v)_{\mathrm{lo-1},j} \\ (\rho E)_{\mathrm{lo-1},j} \end{array} \right ) = \left ( \begin{array}{c} \rho_{\mathrm{lo},j} \\ (\rho u)_{\mathrm{lo},j} \\ (\rho v)_{\mathrm{lo},j} \\ (\rho E)_{\mathrm{lo},j} \end{array} \right ) \end{equation} Note that these boundaries are not perfect. At the boundary, one (or more) of the waves from the Riemann problem can still enter the domain. Only for supersonic flow, do all waves point outward. \item {\em Reflect}: this is appropriate at a solid wall or symmetry plane. All variables are reflected across the boundary, with the normal velocity given the opposite sign. At the $x$-boundary, the first ghost cell is: \begin{equation} \left ( \begin{array}{c} \rho_{\mathrm{lo-1},j} \\ (\rho u)_{\mathrm{lo-1},j} \\ (\rho v)_{\mathrm{lo-1},j} \\ (\rho E)_{\mathrm{lo-1},j} \end{array} \right ) = \left ( \begin{array}{c} \rho_{\mathrm{lo},j} \\ -(\rho u)_{\mathrm{lo},j} \\ (\rho v)_{\mathrm{lo},j} \\ (\rho E)_{\mathrm{lo},j} \end{array} \right ) \end{equation} The next is: \begin{equation} \left ( \begin{array}{c} \rho_{\mathrm{lo-2},j} \\ (\rho u)_{\mathrm{lo-2},j} \\ (\rho v)_{\mathrm{lo-2},j} \\ (\rho E)_{\mathrm{lo-2},j} \end{array} \right ) = \left ( \begin{array}{c} \rho_{\mathrm{lo+1},j} \\ -(\rho u)_{\mathrm{lo+1},j} \\ (\rho v)_{\mathrm{lo+1},j} \\ (\rho E)_{\mathrm{lo+1},j} \end{array} \right ) \end{equation} and so on $\ldots$ \item {\em Inflow}: inflow boundary conditions specify the state directly on the boundary. Technically, this state is on the boundary itself, not the cell-center. This can be accounted for by modifying the stencils used in the reconstruction near inflow boundaries. \item {\em Hydrostatic}: a hydrostatic boundary can be used at the base of an atmosphere to provide the pressure support necessary to hold up the atmosphere against gravity while still letting acoustic waves pass through. An example of this is described in \cite{hse}. \end{itemize} %----------------------------------------------------------------------------- \section{Multidimensional problems} The multidimensional case is very similar to the multidimensional advection problem. Our system of equations is now: \begin{equation} \Uc_t + [\Fb^{(x)}(\Uc)]_x + [\Fb^{(y)}(\Uc)]_y = 0 \end{equation} with \begin{equation} \Uc = \left ( \begin{array}{c} \rho \\ \rho u \\ \rho v \\ \rho E \end{array} \right ) % \qquad % \Fb^{(x)}(\Uc) = \left ( \begin{array}{c} \rho u \\ \rho uu + p \\ \rho v u \\ \rho u E + up \end{array} \right ) % \qquad \Fb^{(y)}(\Uc) = \left ( \begin{array}{c} \rho v \\ \rho vu \\ \rho v v + p \\ \rho v E + vp \end{array} \right ) \end{equation} We note that there is no transformation that can convert the multidimensional system into characteristic variables, since we cannot simultaneously diagonalize the Jacobians corresponding to $\Fb^{(x)}$ and $\Fb^{(y)}$. Related to this is that when limiting, we limit one-dimensional slopes instead of doing a full multidimensional reconstruction and limiting (see \cite{BDS} for a multidimensional limiting procedure for linear advection. For the Euler equations, since we cannot write the multidimensional system in a characteristic form, we cannot use this type of method). For a directionally-unsplit discretization, we predict the cell-centered quantities to the edges by Taylor expanding the conservative state, $\Uc$, in space and time. Now, when replacing the time derivative ($\partial \Uc/\partial t$) with the divergence of the fluxes, we gain a transverse flux derivative term. For example, predicting to the upper $x$ edge of zone $i,j$, we have: \begin{align} \Uc_{i+\myhalf,j,L}^{n+\myhalf} &= \Uc_{i,j}^n + \frac{\Delta x}{2} \frac{\partial \Uc}{\partial x} + \frac{\Delta t}{2} \frac{\partial \Uc}{\partial t} + \ldots \\ &= \Uc_{i,j}^n + \frac{\Delta x}{2} \frac{\partial \Uc}{\partial x} - \frac{\Delta t}{2} \frac{\partial \Fb^{(x)}}{\partial x} - \frac{\Delta t}{2} \frac{\partial \Fb^{(y)}}{\partial y} \\ &= \Uc_{i,j}^n + \frac{1}{2} \left [ 1 - \frac{\Delta t}{\Delta x} \Ab^{(x)}(\Uc) \right ] \Delta \Uc - \frac{\Delta t}{2} \frac{\partial \Fb^{(y)}}{\partial y} \label{eq:Utaylorstate} \end{align} where $\Ab^{(x)}(\Uc) \equiv \partial \Fb^{(x)} / \partial \Uc$. We decompose this into a {\em normal state} and a {\em transverse flux difference}. Adopting the notation from Colella (1990), we use $\hat{\Uc}$ to denote the normal state: \begin{align} \hat{\Uc}_{i+\myhalf,j,L}^{n+\myhalf} &\equiv \Uc_{i,j}^n + \frac{1}{2} \left [ 1 - \frac{\Delta t}{\Delta x} \Ab^{(x)}(\Uc) \right ] \Delta \Uc \\ \Uc_{i+\myhalf,j,L}^{n+\myhalf} &= \hat{\Uc}_{i+\myhalf,j,L}^{n+\myhalf} - \frac{\Delta t}{2} \frac{\partial \Fb^{(y)}}{\partial y} \label{eq:fullleftstate} \end{align} The primitive variable form for this system is \begin{equation} \qb_t + \Ab^{(x)}(\qb) \qb_x + \Ab^{(y)}(\qb) \qb_y = 0 \end{equation} where \begin{equation} \qb = \left ( \begin{array}{c} \rho \\ u \\ v \\ p \end{array} \right ) % \qquad \Ab^{(x)}(\qb) = \left ( \begin{array}{cccc} u & \rho & 0 & 0 \\ 0 & u & 0 & 1/\rho \\ 0 & 0 & u & 0 \\ 0 & \gamma p & 0 & u \end{array} \right ) \qquad \Ab^{(y)}(\qb) = \left ( \begin{array}{cccc} v & 0 & \rho & 0 \\ 0 & v & 0 & 0 \\ 0 & 0 & v & 1/\rho \\ 0 & 0 & \gamma p & v \end{array} \right ) \end{equation} There are now 4 eigenvalues. For $\Ab^{(x)}(\qb)$, they are $u-c$, $u$, $u$, $u+c$. If we just look at the system for the $x$ evolution, we see that the transverse velocity (in this case, $v$) just advects with velocity $u$, corresponding to the additional eigenvalue. \begin{exercise}[Eigenvectors for the 2-d Euler equations] { Derive the form of $\Ab^{(x)}(\qb)$ and $\Ab^{(y)}(\qb)$ and find their left and right eigenvectors.} \end{exercise} We note here that $\hat{\Uc}_{i+\myhalf,j,L}^{n+\myhalf}$ is essentially one-dimensional, since only the $x$-fluxes are involved (through $\Ab^{(x)}(\Uc)$). This means that we can compute this term using the one-dimensional techniques developed in \S~\ref{sec:onedrecon}. In particular, Colella (1990) suggest that we switch to primitive variables and compute this as: \begin{equation} \hat{\Uc}_{i+\myhalf,j,L}^{n+\myhalf} = \Uc(\hat{\qb}_{i+\myhalf,j,L}^{n+\myhalf}) \end{equation} Similarly, we consider the system projected along the $y$-direction to define the normal states on the $y$-edges, again using the one-dimensional reconstruction on the primitive variables from \S~\ref{sec:onedrecon}: \begin{equation} \hat{\Uc}_{i,j+\myhalf,L}^{n+\myhalf} = \Uc(\hat{\qb}_{i,j+\myhalf,L}^{n+\myhalf}) \end{equation} To compute the full interface state (Eq.~\ref{eq:fullleftstate}), we need to include the transverse term. Colella (1990) gives two different procedures for evaluating the transverse fluxes. The first is to simply use the cell-centered $\Uc_{i,j}$ (Colella 1990, Eq. 2.13); the second is to use the reconstructed normal states (the $\hat{\Uc}$'s) (Eq. 2.15). In both cases, we need to solve a {\em transverse Riemann problem} to find the true state on the transverse interface. This latter approach is what we prefer. In particular, for computing the full $x$-interface left state, $\Uc_{i+\myhalf,j,L}^{n+\myhalf}$, we need the transverse ($y$) states, which we define as \begin{align} \Uc^T_{i,j+\myhalf} &= \mathcal{R}(\hat{\Uc}^{n+\myhalf}_{i,j+\myhalf,L}, \hat{\Uc}^{n+\myhalf}_{i,j+\myhalf,R}) \\ \Uc^T_{i,j-\myhalf} &= \mathcal{R}(\hat{\Uc}^{n+\myhalf}_{i,j-\myhalf,L}, \hat{\Uc}^{n+\myhalf}_{i,j-\myhalf,R}) \end{align} Taken together, the full interface state is now: \begin{equation} \Uc_{i+\myhalf,j,L}^{n+\myhalf} = \Uc(\hat{\qb}_{i+\myhalf,j,L}^{n+\myhalf}) - \frac{\Delta t}{2} \frac{\Fb^{(y)}(\Uc^T_{i,j+\myhalf}) - \Fb^{(y)}(\Uc^T_{i,j-\myhalf})}{\Delta y} \end{equation} The right state at the $i+\myhalf$ interface can be similarly computed (starting with the data in zone $i+1,j$ and expanding to the left) as: \begin{equation} \Uc_{i+\myhalf,j,R}^{n+\myhalf} = \Uc(\hat{\qb}_{i+\myhalf,j,R}^{n+\myhalf}) - \frac{\Delta t}{2} \frac{\Fb^{(y)}(\Uc^T_{i+1,j+\myhalf}) - \Fb^{(y)}(\Uc^T_{i+1,j-\myhalf})}{\Delta y} \end{equation} Note the indices on the transverse states---they are now to the right of the interface (since we are dealing with the right state). We then find the $x$-interface state by solving the Riemann problem normal to our interface: \begin{equation} \Uc_{i+\myhalf,j}^{n+\myhalf} = \mathcal{R}(\Uc_{i+\myhalf,j,L}^{n+\myhalf}, \Uc_{i+\myhalf,j,R}^{n+\myhalf}) \end{equation} Therefore, construction of the interface states now requires two Riemann solves: a transverse and normal one. The fluxes are then evaluated as: \begin{equation} \Fb^{(x),n+\myhalf}_{i+\myhalf,j} = \Fb^{(x)}(\Uc_{i+\myhalf,j}^{n+\myhalf}) \end{equation} Note, for multi-dimensional problems, in the Riemann solver, the transverse velocities are simply selected based on the speed of the contact, giving either the left or right state. The final conservative update is done as: \begin{equation} \Uc^{n+1}_{i,j} = \Uc^n_{i,j} + \frac{\Delta t}{\Delta x} \left ( \Fb^{(x),n+\myhalf}_{i-\myhalf,j} - \Fb^{(x),n+\myhalf}_{i+\myhalf,j} \right ) + \frac{\Delta t}{\Delta y} \left ( \Fb^{(y),n+\myhalf}_{i,j-\myhalf} - \Fb^{(y),n+\myhalf}_{i,j+\myhalf} \right ) \end{equation} \subsection{3-d unsplit} The extension of the unsplit methodology to 3-d is described by Saltzman~\cite{saltzman:1994}. The basic idea is the same as in 2-d, except now additional transverse Riemann solve are needed to fully couple in the corners. \section{Source terms} \label{euler:sec:sourceterms} Adding source terms is straightforward. For a system described by \begin{equation} \Uc_t + [\Fb^{(x)}(\Uc)]_x + [\Fb^{(y)}(\Uc)]_y = \Hb \end{equation} we predict to the edges in the same fashion as described above, but now when we replace $\partial \Uc/\partial t$ with the divergence of the fluxes, we also pick up the source term. This appears as: \begin{align} \Uc_{i+\myhalf,j,L}^{n+\myhalf} &= \Uc_{i,j}^n + \frac{\Delta x}{2} \frac{\partial \Uc}{\partial x} + \frac{\Delta t}{2} \frac{\partial \Uc}{\partial t} + \ldots \\ &= \Uc_{i,j}^n + \frac{\Delta x}{2} \frac{\partial \Uc}{\partial x} - \frac{\Delta t}{2} \frac{\partial \Fb^{(x)}}{\partial x} - \frac{\Delta t}{2} \frac{\partial \Fb^{(y)}}{\partial y} + \frac{\Delta t}{2} \Hb_{i,j} \\ &= \Uc_{i,j}^n + \frac{1}{2} \left [1 -\frac{\Delta t}{\Delta x} \Ab^{(x)}(\Uc)\right ] \Delta \Uc - \frac{\Delta t}{2} \frac{\partial \Fb^{(y)}}{\partial y} + \frac{\Delta t}{2} \Hb_{i,j} \label{eq:Utaylorstatesource} \end{align} We can compute things as above, but simply add the source term to the $\hat{\Uc}$'s and carry it through. Note that the source here is cell-centered. This expansion is second-order accurate. This is the approach outlined in Miller \& Colella \cite{millercolella:2002}. Also notice the similarity of this source term to a second-order Euler method for integrating ODEs. Alternately, we can include the source terms in the characteristic tracing of the interface states. This is the approach taken in, e.g., the original PPM paper. To make things more concrete, let's consider just gravity as the source. Our primitive variable equations in this case are: \begin{equation} \qb_t + \Ab(\qb) \qb_x = \Gb \end{equation} where $\Gb = (0, g, 0)^T$---i.e. the gravitational source only affects $u$, not $\rho$ or $p$\footnote{Note that in the PPM paper, they put $\Gb$ on the lefthand side of the primitive variable equation, so our signs are opposite.}. First we construct a parabolic profile of $g$ in each zone and integrate under that profile to determine the average $g$ carried by each wave to the interface, we'll denote these as $\Ic^\enu_\pm(g)$. Then we include the gravitational source term in the characteristic projection itself. Our projections are now: \begin{equation} \sum_{\nu; \lambda^\enu \ge 0}\lb^\enu \cdot (\tilde{\qb} - \Ic^\enu_+(\qb) - \tfrac{\Delta t}{2} \Gb) \rb^\enu \end{equation} for the left state, and \begin{equation} \sum_{\nu; \lambda^\enu \le 0} \lb^\enu \cdot (\tilde{\qb} - \Ic^\enu_-(\qb) - \tfrac{\Delta t}{2} \Gb) \rb^\enu \end{equation} for the right state. Since $\Gb$ is only non-zero for velocity, only the velocity changes. Writing out the sum (and performing the vector products), we get: \begin{align} u_{i+\myhalf,L}^{n+\myhalf} = \tilde{u}_+ &- \frac{1}{2} \left [ \left (\tilde{u}_+ - \Ic_+^\evm(u) - \frac{\Delta t}{2} \Ic^\evm_+(g) \right ) - \frac{\tilde{p}_+ - \Ic_+^\evm(p)}{C} \right ] \nonumber \\ &- \frac{1}{2} \left [ \left (\tilde{u}_+ - \Ic_+^\evp(u) - \frac{\Delta t}{2} \Ic^\evp_+(g) \right ) + \frac{\tilde{p}_+ - \Ic_+^\evp(p)}{C} \right ] \end{align} where the only change from Eq.~\ref{eq:ufull} are the $\Ic^\evm_+(g)$ and $\Ic^\evp_+(g)$ terms. \footnote{These differ from the expression in the PPM paper, where $\Delta t \Gb$, not $(\Delta t/2) \Gb$ is used in the projection, however this appears to be a typo. To see this, notice that if both waves are moving toward the interface, then the source term that is added to the interface state is $(\Delta t/4) (\Ic_+^\evm(g) + \Ic_+^\evp(g))$ for the left state, which reduces to $(\Delta t/2) g$ for constant g---this matches the result from the Taylor expansion above (Eq.~\ref{eq:Utaylorstatesource}).} Regardless of how the source term information is included in the interface states, we also need to include it in the conservative update. To second-order, we need it to be time-centered, which usually means averaging the time-level $n$ and $n+1$ sources: \begin{align} \Uc^{n+1}_{i,j} = \Uc^n_{i,j} &+ \frac{\Delta t}{\Delta x} \left ( \Fb^{(x),n+\myhalf}_{i-\myhalf,j} - \Fb^{(x),n+\myhalf}_{i+\myhalf,j} \right ) \nonumber \\ &+ \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 ) \end{align} As written, this appears to be an implicit update (since $\Uc^{n+1}$ depends on $\Hb^{n+1}$), but often, the form of the source terms allows you to update the equations in sequence explicitly. Again, using a constant gravitational acceleration as an example, and looking in 1-d for simplicity, $\Uc = (\rho, \rho u, \rho E)^\intercal$ and $\Hb = (0, \rho g, \rho u g)^\intercal$, so our update sequence is: \begin{align} \rho^{n+1}_i = \rho^n_i &+ \frac{\Delta t}{\Delta x} \left [\rho_{i-\myhalf}^{n+\myhalf} u_{i-\myhalf}^{n+\myhalf} - \rho_{i+\myhalf}^{n+\myhalf} u_{i+\myhalf}^{n+\myhalf} \right ] \\ (\rho u)^{n+1}_i = (\rho u)^n_i &+ \frac{\Delta t}{\Delta x} \left [ \rho_{i-\myhalf}^{n+\myhalf} (u_{i-\myhalf}^{n+\myhalf})^2 - \rho_{i+\myhalf}^{n+\myhalf} (u_{i+\myhalf}^{n+\myhalf})^2 \right ] + \frac{\Delta t}{\Delta x} \left ( p_{i-\myhalf}^{n+\myhalf} - p_{i+\myhalf}^{n+\myhalf} \right ) \nonumber \\ &+ \frac{\Delta t}{2}(\rho^n_i + \rho^{n+1}_i) g \\ (\rho E)^{n+1}_i = (\rho E)^n_i &+ \frac{\Delta t}{\Delta x} \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 \\ &\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 ] + \frac{\Delta t}{2} \left [ (\rho u)^n + (\rho u)^{n+1} \right] g \end{align} These updates can be done one after another without any implicit coupling. Other sources, like the Coriolis force, will involve an implicit update, but even then, it is local to a single zone and can be solved analytically. %----------------------------------------------------------------------------- \section{Simple geometries} \label{ch:euler:geom} So far we have been working only with Cartesian geometries, but it is easy to extend these methods to simple non-Cartesian geometies, like spherical and cylindrical. These geometies allow us to capture 3-d volume expansion effects in lower dimensions. For a 1-d solver, a spherical geometry means that our coordinate is the radius in the sphere, and as we move outward from the origin, the volume of a zone (which is actually now a spherical shell) grows. There are two places where we need to take the geometry into account: the reconstruction of the interface states and the conservative update. Our 1-d system in spherical coordinates appears as: \begin{equation} \ddt{\Uc} + \frac{1}{r^2} \ddr{r^2 \Fb} = 0 \end{equation} The geometry factors that appear are from the spherical form of the divergence. Expanding out the radial derivative, we have: \begin{equation} \ddt{\Uc} + \ddr{\Fb} = -\frac{2\Fb}{r} \end{equation} Likewise, the primitive variable equations now have source terms: \begin{align} \ddt{\rho} + u \ddx{\rho} + \rho \ddx{u} &= -\frac{2\rho u}{r} \\ \ddt{u} + u \ddx{u} + \frac{1}{\rho} \ddx{p} &= 0 \\ \ddt{p} + u \ddx{p} + \Gamma_1 p \ddx{u} &= -\frac{2\Gamma_1 p u}{r} \end{align} \begin{exercise}[Spherical form of primitive variable equations] {Derive the above system of 1-d spherical primitive variable equations starting from the conservative equations. Be sure to include the geometry factors everywhere there is a divergence.} \end{exercise} For the prediction of the interface states, we now include these geometric source terms as sources to the interface state, following the same ideas as in \S~\ref{euler:sec:sourceterms}. The conservative update now needs to include the geometry factors. However, it is complicated by the fact that in the momentum equation, the pressure term is a gradient, not a divergence, and therefore has different geometic factors. The update of the system appears as: \begin{align} \rho^{n+1}_i = \rho^n_i &+ \frac{\Delta t}{V_i} \left [A_{i-\myhalf} \rho_{i-\myhalf}^{n+\myhalf} u_{i-\myhalf}^{n+\myhalf} - A_{i+\myhalf} \rho_{i+\myhalf}^{n+\myhalf} u_{i+\myhalf}^{n+\myhalf} \right ] \\ (\rho u)^{n+1}_i = (\rho u)^n_i &+ \frac{\Delta t}{V_i} \left [ A_{i-\myhalf} \rho_{i-\myhalf}^{n+\myhalf} (u_{i-\myhalf}^{n+\myhalf})^2 - A_{i+\myhalf} \rho_{i+\myhalf}^{n+\myhalf} (u_{i+\myhalf}^{n+\myhalf})^2 \right ]\nonumber \\ & + \frac{\Delta t}{\Delta r} \left ( p_{i-\myhalf}^{n+\myhalf} - p_{i+\myhalf}^{n+\myhalf} \right ) \\ (\rho E)^{n+1}_i = (\rho E)^n_i &+ \frac{\Delta t}{V_i} \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 \\ &\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 ] \end{align} Here, the geometry factors are: \begin{align} A_{i-\myhalf} &= (r_{i-\myhalf})^2\\ V_i &= (r_i)^2 \Delta r \end{align} \begin{figure} \centering \includegraphics[width=0.8\linewidth]{axisymmetry} \caption[The axisymmetric computational domain] {\label{fig:axisymmetri} The axisymmetric computational domain. Here we imagine that the 2-d plane rotates through the cylindrical $\theta$ angle to create a volume.} \end{figure} It is also common to do 2-d axisymmetric models---here the $r$ and $z$ coordinates from a cylindrical geometry are modeled. Again, this appears Cartesian, except there is a volume factor implicit in the divergence that must be accounted for. Our system in axisymmetric coordinates\footnote{Some sources will call this cylindrical coordinates, but note that the grid is not a polar grid---it is still Cartesian} is: \begin{equation} \frac{\partial \Uc}{\partial t} + \frac{1}{r}\frac{\partial r \Fb^{(r)}}{\partial r} + \frac{\partial \Fb^{(z)}}{\partial z} = 0 \end{equation} Expanding out the $r$ derivative, we can write this as: \begin{equation} \frac{\partial \Uc}{\partial t} + \frac{\partial \Fb^{(r)}}{\partial r} + \frac{\partial \Fb^{(z)}}{\partial z} = -\frac{\Fb^{(r)}}{r} \end{equation} Again, the primitive variable version of this expanded form is used for the interface state prediction. The conservative update follows the same idea as the 1-d spherical version. The area and volume factors only differ from their Cartesian counterparts in the radial direction, and take the form: \begin{align} A_{i-\myhalf,j} &= r_{i-\myhalf} \Delta z \\ A_{i,j-\myhalf} &= r_i \Delta z\\ V_{i,j} &= r_i \Delta r \Delta z \end{align} These choices of geometric factors reproduce a discretized form of the cylindrical divergence: \begin{equation} \nabla \cdot {\boldsymbol{\phi}} = \frac{1}{r} \ddr{(r\phi^{(r)})} + \ddz{\phi^{(z)}} \end{equation} Just as with the 1-d spherical case, the pressure term in the momentum equation needs to be treated separately from the flux, since it enters as a gradient and not a divergence.\footnote{It is common to see the divergence term expressed as \begin{equation} \nabla \cdot {\boldsymbol{\phi}} = \frac{1}{r^\alpha} \ddr{(r^\alpha \phi^{(r)})} + \ldots \end{equation} where $\alpha = 1$ for axisymmetric and $\alpha = 2$ for 1-d spherical. This allows for the geometry correction to be expressed more concisely, as the source term takes the form $-\alpha F/r$. } %----------------------------------------------------------------------------- \section{Some Test problems} There are a large number of standard test problems that are used to test out our methods. Like we say with advection, it is best to have a problem with an analytic solution. Here we show just a few of the most popular test problems. \subsection{Shock tubes} \label{sec:euler-methods:shocktubes} Shock tubes are Riemann problems---consider the evolution of an initial discontinuity in the domain. The evolution with time will just be the solution to the Riemann problem that we described \S~\ref{euler:sec:riemann}. The initial conditions can be varied to produce any combination of shocks and rarefactions as the left and right waves. A popular initial condition is the Sod problem~\cite{sod:1978} which results is a right moving shock and contact and a left moving rarefaction. Note: it is a good test of your code's ability to preserve symmetry to flip the initial conditions left/right and rerun. The results should be the same to machine precision, but atleast to roundoff error. A common reason for breaking symmetry is using inequalities in your code that are biases in a direction, e.g., \begin{lstlisting} if u > 0: # positive velocity test case else: # negative or zero velocity test case \end{lstlisting} This has a left-right bias, since we don't handle the case where $u = 0$ separately. A better construction would test on $u < 0$ alone, and then have a final {\em else} clause to catch $u = 0$. Since these tests start out with a discontinuity, they are not the best tests to use for convergence testing. Wherever there is an initiali discontinuity, the limiters will kick in and drop your method to first-order accurate. For a general equation of state, you can still solve the Riemann problem exactly and define analogous test problems to those commonly used for a gamma-law gas. Some shock tube test problems for a stellar equation of state are shown in \cite{zingalekatz}. For the tests shown below, we use the \hydrooned\ code described in Appendix~\ref{app:hydro1d}. \subsubsection{Sod problem} The initial conditions for the Sod problem \cite{sod:1978} are: \begin{align} \rho_l &= 1 & \rho_r &= 1/8 \nonumber \\ u_l &= 0 & u_r &= 0 \\ p_l &= 1 & p_r &= 1/10 \nonumber \end{align} usually with $\gamma = 1.4$\MarginPar{double check} These result in a left wave moving contact and rightward moving contact and shock. The shock is not particularly strong, but this problem is a nice demonstration of the types of hydrodynamic waves. \begin{figure}[t] \centering \includegraphics[width=0.7\linewidth]{hydro1d_god_sod} \caption[Piecewise constant reconstruction Sod problem]{\label{fig:Euler:sod:god} Piecewise constant reconstruction with the Sod problem, using 128 zones, $\cfl = 0.8$, and the CGF Riemann solver. This was run with \hydrooned\ using the {\tt sod} problem setup, setting {\tt godunov\_type=0} and visualized with the {\tt sod\_compare.py} script there.} \end{figure} \begin{figure}[t] \centering \includegraphics[width=0.7\linewidth]{hydro1d_ppm_sod} \caption[Piecewise parabolic reconstruction Sod problem]{\label{fig:Euler:sod:ppm} Piecewise parabolic reconstruction with the Sod problem, using 128 zones, $\cfl = 0.8$, and the CGF Riemann solver. This was run with \hydrooned\ using the {\tt sod} problem setup, setting {\tt godunov\_type=2} and visualized with the {\tt sod\_compare.py} script there.} \end{figure} \subsubsection{Double rarefaction} The double rarefaction problem starts with initially diverging flow that creates a vacuum state in the middle. The initial conditions (as given in \cite{toro:1997} are \begin{align} \rho_l &= 1 & \rho_r &= 1 \nonumber \\ u_l &= -2.0 & u_r &= 2.0 \\ p_l &= 0.4 & p_r &= 0.4 \nonumber \end{align} \begin{figure}[t] \centering \includegraphics[width=0.7\linewidth]{hydro1d_god_double_rare} \caption[Piecewise constant reconstruction double rarefaction problem]{\label{fig:Euler:doublerare:god} Piecewise constant reconstruction with the double rarefaction problem, using 128 zones, $\cfl = 0.8$, and the CGF Riemann solver. This was run with \hydrooned\ using the {\tt sod} problem setup, setting {\tt godunov\_type=0} and visualized with the {\tt sod\_compare.py} script there.} \end{figure} \begin{figure}[t] \centering \includegraphics[width=0.7\linewidth]{hydro1d_ppm_double_rare} \caption[Piecewise constant reconstruction double rarefaction problem]{\label{fig:Euler:doublerare:ppm} Piecewise parabolic reconstruction with the double rarefaction problem, using 128 zones, $\cfl = 0.8$, and the CGF Riemann solver. This was run with \hydrooned\ using the {\tt sod} problem setup, setting {\tt godunov\_type=2} and visualized with the {\tt sod\_compare.py} script there.} \end{figure} \subsection{Sedov blast wave} The Sedov (or Sedov-Taylor) blast wave~\cite{sedov:1959} is a point explosion---energy, $\mathcal{E}_\mathrm{expl}$, is placed at a point in a uniform domain. A spherical shockwave propagates outward, evacuating the region in the center. The Sedov problem also has an analytic solution\footnote{See \cite{timmes_sedov_code} for a nice code to general the solutions}, and this problem is a good way of testing out the geometric factors for 1-d spherical and 2-d axisymmetric geometries. \begin{figure}[t] \centering \includegraphics[width=0.7\linewidth]{hydro1d_sedov_1dsph} \caption[1-d spherical Sedov problem]{\label{fig:Euler:sedov1d} 1-d Sedov explosion with piecewise parabolic reconstruction, using 128 zones (on $r \in [0,1]$), $\cfl = 0.8$, and the CGF Riemann solver. This was run with \hydrooned\ using the {\tt sedov} problem setup and visualized with the {\tt sedov\_compare.py} script there. The 1-d spherical geometry used makes this act as if it were a sphere.} \end{figure} The major difficulty with initializing the Sedov problem is representing a point, where all the energy desposited, on the grid. If you just initialize a singe zone, then the size of the point changes as you change resolution. Additionally, in 2- or 3-d Cartesian coordinates, the point will be squared off. A standard way of initializing this (see, e.g., \cite{omang:2006}) is to imagine the energy deposition inflating a region of radius $r_\mathrm{init}$ like a balloon, resulting in an energy density, $E = \mathcal{E}_\mathrm{expl}/V_\mathrm{init}$, where $V_\mathrm{init} = 4\pi r_\mathrm{init}^3/3$ for a spherical blast wave and $V_\mathrm{init} = \pi r_\mathrm{init}^2$ for a cylindrical blast wave. Then the pressure is \begin{equation} p = \frac{\mathcal{E}_\mathrm{expl}}{V_\mathrm{init}} (\gamma - 1) \end{equation} Figure~\ref{fig:Euler:sedov1d} shows the solution to the Sedov problem in 1-d in a spherical geometry. This is compared to the spherical Sedov solution (solid line). There is good agreement in the position of the shock and density and velocity profiles. The pressure behind the shock is a little low---this is likely an artifact of the initialization process. In 2-d, we run the problem in Cartesian coordinates---this produces a cylindrical blast wave. Figure~\ref{fig:Euler:sedov2d} shows the solution in 2-d. \begin{figure}[t] \centering \includegraphics[width=0.85\linewidth]{sedov_pyro} \caption[2-d cylindrical Sedov problem]{\label{fig:Euler:sedov2d} 2-d Sedov explosion with piecewise linear reconstruction, using 128$^2$ zones (on $r \in [0,1]\times[0,1]$), $\cfl = 0.8$, and the HLLC Riemann solver. This was run with \pyro\ as {\tt ./pyro.py compressible sedov inputs.sedov}. An initial perturbation size of $r_\mathrm{init} = 0.01$ was used.} \end{figure} The density and pressure look very symmetric. In the velocity magnitude plot, we see an imprint of the grid along coordinate axes, and likewise in the internal energy. The bump in internal energy near the origin arises because of the error in defining $e$ from $E$ and $U$. We can produce an angle-averaged profile of this and compare to the analytic solution, shown in Figure~\ref{fig:Euler:sedov2d_compare}. \begin{figure}[t] \centering \includegraphics[width=0.75\linewidth]{sedov_compare} \caption[2-d cylindrical Sedov problem]{\label{fig:Euler:sedov2d_compare} Angle-average profile for the 2-d Sedov explosion from Figure~\ref{fig:Euler:sedov2d} shown with the analytic solution. This was constructed using the {\tt sedov\_compare.py} script in \pyro.} \end{figure} \subsection{Advection} We can run a simple advection test analogous to the tests we used in Ch.~\ref{ch:advection}. However, because we are now doing hydrodynamics, we need to suppress the dynamics. This is accomplished by putting the profile we want to advect in the density field and then put it in pressure equilibrium by adjusting the internal energy. For example, consider a Gaussian profile. We initialize the density as: \begin{equation} \rho = (\rho_1 - \rho_0) e^{-(x - x_c)^2/\sigma^2} + \rho_0 \end{equation} To advect the profile to the right, we choose a constant velocity, \begin{equation} u = u_0 \end{equation} and to suppress dynamics, we make the pressure constant, \begin{equation} p = p_0 \end{equation} Finally, we compute the total energy as \begin{equation} \rho E = \frac{p}{\gamma - 1} + \frac{1}{2}\rho u^2 \end{equation} \begin{figure}[t] \centering \includegraphics[width=0.7\linewidth]{advect_hydro1d} \caption[Simple advection test]{\label{fig:Euler:advect:1d} Piecewise parabolic reconstruction with an advection test, using 128 zones, $\cfl = 0.8$, and the CGF Riemann solver. A Gaussian profile was advected for 10 periods. This was run with \hydrooned\ using the {\tt advect} problem setup and visualized with the {\tt advect\_compare.py} script there.} \end{figure} Figure~\ref{fig:Euler:advect:1d} shows an example of a Gaussian profile advected for 10 periods. The initial conditions used $\rho_0 = 10^{-3}$, $\rho_1 = 1$, $p_0 = 10^{-6}$, $u_0 = 1$, and $\sigma = 0.1$ (these coincide with the choices used in \cite{flash}). The non-zero value for the ambient density, $\rho_0$, ensures that any quantities that are derived by dividing by density remain well-defined. The result is similar to what we saw when we considered pure advection---the shape is mostly preserved, but the peak of the Gaussian is clipped. \subsection{Slow moving shock} Slow moving (or stationary) shocks can be difficult to model, as oscillations can be setup behind the shock (this is discussed a little in \cite{colellawoodward:1984,leveque:2002}). We can produce a slow moving shock as a shock tube, and we can use the jump conditions across a shock that were derived for the Riemann problem to find the conditions to setup a stationary (or slow-moving) shock. The speed of a right-moving shock was found (see Eq.~\ref{eq:euler:shockspeedjump}) as: \begin{equation} S = u_r + c_r \left [ \left ( \frac{p_\star}{p_r} \right ) \frac{\gamma+1}{2\gamma} + \frac{\gamma-1}{2\gamma} \right ]^{\myhalf} \end{equation} We want $S = 0$, which allows us to express the pre-shock velocity, $u_r$, as: \begin{equation} u_r = -c_r \left [ \left ( \frac{p_\star}{p_r} \right ) \frac{\gamma+1}{2\gamma} + \frac{\gamma-1}{2\gamma} \right ]^{\myhalf} \end{equation} We have the freedom to choose the density and pressure ahead of the shock, $\rho_r$ and $p_r$, which in turn gives us $c_r$. Next, we can pick the strength of the shock by choosing the jump in pressure, $p_\star/p_r$. Together, this allows us to compute $u_r$, and thus we know the entire pre-shock state. We can compute the post-shock state (which was the star state when we discussed the Riemann problem) using the jump conditions, Eqs.~\ref{eq:euler:shockrhojump} and \ref{eq:euler:shockujump}.\footnote{The script \href{https://github.com/zingale/hydro_examples/blob/master/compressible/slow_shock.py}{\tt slow\_shock.py} will find the initial conditions to generate a stationary shock.} For a pressure jump of 100 across a shock, the following conditions will generate a stationary right-facing shock (with $\gamma = 1.4$): \begin{align} \rho_l &= 5.6698 & \rho_r &= 1 \nonumber \\ u_l &= -1.9336 & u_r &= -10.9636 \\ p_l &= 100 & p_r &= 1 \nonumber \end{align} By adjusting the velocity of both the left and right state, we can produce a strong shock that moves slowly across the grid. For instance, a shock with $S = 0.4$ results from \begin{align} \rho_l &= 5.6698 & \rho_r &= 1 \nonumber \\ u_l &= -1.5336 & u_r &= -10.5636 \\ p_l &= 100 & p_r &= 1 \nonumber \end{align} \begin{figure}[t] \centering \includegraphics[width=0.7\linewidth]{slowshock} \caption[1-d spherical Sedov problem]{\label{fig:Euler:slowshock:ppm} 1-d slow moving shock problem with piecewise parabolic reconstruction, using 128 zones (on $r \in [0,1]$), $\cfl = 0.8$, and the CGF Riemann solver. This was run with \hydrooned\ using the {\tt sod} problem setup and visualized with the {\tt sod\_compare.py} script there.} \end{figure} \subsection{Two-dimensional Riemann problems} Several different 2-d Riemann problems were introduced in \cite{hydro_test_quad}, to explore the multi-dimensional interactive of the different hydrodynamic waves. These problems initialize the 4 quadrants of the domain with different states, and watch the ensuing evolution. There are some analytic estimates that can be compared to, but also these tests can provide a means of assessing the symmetry of a code in the presence of complex flows. We use the setup corresponding to {\em configuration 3} in that paper (this same setup is used in \cite{leveque:1997}). \begin{figure}[t] \centering \includegraphics[width=0.85\linewidth]{quad} \caption{\label{fig:Euler:quad} Two-dimensional Riemann problem from \cite{hydro_test_quad}.} \end{figure} \ifdefined \debugmode \subsection{Odd-even decoupling} \subsection{Gresho vortex} The Gresho vortex is a vortex in with a stabilizing pressure gradient designed such that the overall structure is unchanging in time. It has a nice feature in that it allows you to set the Mach number of the flow as one of the parameters. \subsection{Hydrostatic equilibrium} Hydrodynamics codes can have a difficult time keeping an atmosphere in hydrostatic equilibrium. This is because any departure from $\nabla p = \rho {\bf g}$ due to truncation error in the method will be seen as an acceleration by the momentum equation: \begin{equation} \rho \DDt{\Ub} = \rho {\bf g} - \nabla p \end{equation} The result will be growing velocities in the atmosphere, which can overwhelm any signal in the flow that you are interested in studying. Since this acceleration is driven by truncation error, higher resolution will result in lower velocities. % top of the star % scale height resolved -- do uniform T atmosphere %\subsection{Rayleigh-Taylor instability} \fi %----------------------------------------------------------------------------- \section{Method of lines integration and higher order} \label{sec:comp:mol} Just like we explored with linear advection (\S~\ref{adv:sec:mol_2d}), instead of doing the characteristic tracing, we could rely on the integrator to do the work for us. Discretizing our system in space leads to the following system: \begin{equation} \frac{d\Uc_{i,j}}{dt} = -\frac{\Fb^{(x)}(\Uc_{i+\myhalf,j}) - \Fb^{(x)}(\Uc_{i-\myhalf,j})}{\Delta x} -\frac{\Fb^{(y)}(\Uc_{i,j+\myhalf}) - \Fb^{(y)}(\Uc_{i,j-\myhalf})}{\Delta y} \end{equation} Note that there is no time superscript in the $\Uc$ used to evaluate the fluxes on the righthand side---we have not done any time discretization yet. Now we can use an ODE integrator to solve this system. Consider second-order Runge-Kutta. We evaluate two slopes, \begin{alignat}{3} {\bf k}_1 &= \Delta t \bigg [ &-&\frac{\Fb^{(x)}(\Uc^n_{i+\myhalf,j}) - \Fb^{(x)}(\Uc^n_{i-\myhalf,j})}{\Delta x} \nonumber \\ &\, &-&\frac{\Fb^{(y)}(\Uc^n_{i,j+\myhalf}) - \Fb^{(y)}(\Uc^n_{i,j-\myhalf})}{\Delta y} \bigg ] \\ {\bf k}_2 &= \Delta t \bigg [ &-&\frac{\Fb^{(x)}([\Uc^n + {\bf k}_\myhalf]_{i+\myhalf,j}) - \Fb^{(x)}([\Uc^n + {\bf k}_\myhalf]_{i-\myhalf,j})}{\Delta x} \nonumber \\ &\, &-&\frac{\Fb^{(y)}([\Uc^n + {\bf k}_\myhalf]_{i,j+\myhalf}) - \Fb^{(y)}([\Uc^n + {\bf k}_\myhalf]_{i,j-\myhalf})}{\Delta y} \bigg ] \end{alignat} and then \begin{equation} \Uc^{n+1}_{i,j} = \Uc^n_{i,j} + {\bf k}_2 \end{equation} In the construction of the interface states, $\Uc^n_{i+\myhalf,j}$ or $[\Uc^n + {\bf k}_\myhalf]_{i+\myhalf,j}$, there is no explicit transverse term, since that arose from Taylor expanding $\Uc^n_{i,j}$ in time through $\Delta t/2$. Instead, we simply construct these interface states using a one-dimensional reconstruction and solve a Riemann problem at each interface. The evaluation of the second slope, ${\bf k}_2$, implicitly includes the transverse information since we add ${\bf k}_\myhalf$ to $\Uc^n_{i,j}$ before doing the prediction to the interfaces. Also note, however, that in this construction of the interface states, there is no characteristic projection, since that arises from predicting the interface states forward in time. Again, these interface states are at a constant time, not predicted into the future. Generally speaking we want the order of accuracy in time to match that in space. The fourth-order Runge-Kutta method is a popular method for integrating ODEs, so it makes sense to couple this with a fourth-order-in-space method. However, going higher-order than second-order is more challenging. The key issue is that we can no longer simply approximate the cell average as the cell-center value, i.e., $\langle \phi\rangle_i \ne \phi_i$. This comes into play, for instance, in translating between the conserved and primitive variables. A fully fourth-order method is presented in ~\cite{mccorquodalecolella} Note that when using a Runge-Kutta method-of-lines integrator for the time-discretization of a multidimensional system, the timestep is actually more restrictive than the cases presented above that predicted the interface states to the half-time and performed characteristic tracing. Titarev \& Toro \cite{titarevtoro} claim that you need $0 < C < \myhalf$ for 2-d flows and $0 < C < 1/3$ for 3-d flows. An additional complexity arises when doing multiphysics. Often we split the different physical processes up and treat them in turn. There are standard methods to do this with second-order accuracy in time, but higher-order is more tricky. \section{Thermodynamic issues} \subsection{Defining temperature} Although not needed for the pure Euler equations, it is sometimes desirable to define the temperature for source terms (like reactions) or complex equations of state. The temperature can typically be found from the equation of state given the internal energy: \begin{eqnarray} e &=& E - \frac{1}{2} u^2 \\ T &=& T(e, \rho) \end{eqnarray} Trouble can arise when you are in a region of flow where the kinetic energy dominates (high Mach number flow). In this case, the $e$ defined via subtraction can become negative due to truncation error in the evolution of $u$ compared to $E$. In this instance, one must either impose a floor value for $e$ or find an alternate method of deriving it. In~\cite{bryan:1995}, an alternate formulation of the Euler equations is proposed. Both the total energy equation {\em and} the internal energy equation are evolved in each zone. When the flow is dominated by kinetic energy, then the internal energy from the internal energy evolution equation is used. The cost of this is conservation---the internal energy is not a conserved quantity, and switching to it introduces conservation of energy errors. \subsection{General equation of state} The above methods were formulated with a constant gamma equation of state. A general equation of state (such as degenerate electrons) requires a more complex method. Most methods are designed to reduce the need to call a complex equation of state frequently, and work by augmenting the vector of primitive variables with additional thermodynamic information. There are two parts of the adaption to a general equation of state: the interface states and the Riemann problem. \subsubsection{Carrying $\gamma_e$} The classic prescription for extending this methodology is presented by Colella and Glaz~\cite{colellaglaz:1985}. They construct a thermodynamic index, \begin{equation} \gamma_e = \frac{p}{\rho e} + 1 \end{equation} and derive an evolution equation for $\gamma_e$ (C\&G, Eq.\ 26). We can derive a similar expression as \begin{align} \frac{D\gamma_e}{Dt} &= \frac{D}{Dt} \left ( \frac{p}{\rho e} + 1 \right ) = - \frac{p}{(\rho e)^2} \frac{D(\rho e)}{Dt} + \frac{1}{\rho e} \frac{Dp}{Dt} \nonumber \\ &= (\gamma_e - 1) (\gamma_e - \Gamma_1) \nabla \cdot \Ub \label{eq:gammae} \end{align} where we used Eqs.~\ref{eq:euler:pgeneral} and \ref{eq:euler:econs}, and the definition of the sound speed. This evolution equation is used to predict $\gamma_e$ to interfaces, and these interface values of $\gamma_e$ are used in the Riemann solver presented there to find the fluxes through the interface. A different adiabatic index (they call $\Gamma$, we call $\Gamma_1$) appears in the definition of the sound speed. They argue that this can be brought to interfaces in a piecewise constant fashion while still making the overall method second order, since $\Gamma_1$ does not explicitly appear in the fluxes (see the discussion at the top of page 277). We can derive the characteristic structure of this system for use in the tracing in the construction of interface states% \footnote{A {\sf Jupyter} notebook using {\sf SymPy} that derives these eigenvectors is available here: \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/compressible/euler-generaleos.ipynb}{euler-generaleos.ipynb}}}. If we write our system as \begin{equation} \qb = \left ( \begin{array}{c} \tau \\ u \\ p \\ \gamma_e \end{array} \right ) \end{equation} (we use $\tau = 1/\rho$ here for consistency with CG), we have \begin{equation} \Ab = \left ( \begin{array}{cccc} u & -\tau & 0 & 0 \\ 0 & u & \tau & 0 \\ 0 & c^2/\tau & u & 0 \\ 0 & -\alpha & 0 & u \end{array} \right ) \end{equation} where we define $\alpha = (\gamma_e - 1)(\gamma_e - \Gamma_1)$ for convenience. The right eigenvectors are: \begin{equation} \rb^\evm = \left( \begin{array}{c} 1\\ {c}/{\tau}\\ -{c^{2}}/{\tau^{2}}\\ {\alpha}/{\tau}\end{array}\right) % \qquad \rb^\evz = \left( \begin{array}{c} 1\\0\\0\\0 \end{array}\right) % \qquad \rb^\evzs{\gamma_e} = \left( \begin{array}{c} 0\\0\\0\\1\end{array}\right) % \qquad \rb^\evp = \left( \begin{array}{c} 1\\ -{c}/{\tau}\\ -{c^{2}}/{\tau^{2}}\\ {\alpha}/{\tau} \end{array}\right ) \end{equation} and corresponding left eigenvectors are: \begin{align} \lb^\evm &= \left( \begin{array}{cccc} 0 & \frac{\tau}{2 c} & - \frac{\tau^{2}}{2 c^{2}} & 0\end{array}\right) \\ % \lb^\evz &= \left( \begin{array}{cccc} 0 & \frac{\tau}{2 c} & - \frac{\tau^{2}}{2 c^{2}} & 0\end{array}\right) \\ % \lb^\evzs{\gamma_e} &= \left( \begin{array}{cccc} 0 & 0 & \frac{\alpha \tau}{c^{2}} & 1\end{array}\right) \\ % \lb^\evp &= \left ( \begin{array}{cccc} 0 & - \frac{\tau}{2 c} & - \frac{\tau^{2}}{2 c^{2}} & 0\end{array}\right) \end{align} \subsubsection{Carrying $(\rho e)$} Alternately, the Castro paper~\cite{almgren:2010} relies on an idea from an unpublished manuscript by Colella, Glaz, and Ferguson that predicts $\rho e$ to edges in addition to $\rho$, $u$, and $p$. Since $\rho e$ comes from a conservation-like equation (Eq.~\ref{eq:euler:econs}), predicting it to the interface in the unsplit formulation is straightforward. This over-specifies the thermodynamics, but eliminates the need for $\gamma_e$. With the addition of $\rho e$, our system becomes: \begin{equation} \qb = \left ( \begin{array}{c} \rho \\ u \\ p \\ \rho e \end{array} \right ) % \qquad % \Ab = \left ( \begin{array}{cccc} u & \rho & 0 & 0 \\ 0 & u & 1/\rho & 0 \\ 0 & \rho c^2 & u & 0 \\ 0 & \rho h & 0 & u \end{array} \right ) \end{equation} where $h = e + p/\rho$ is the specific enthalpy. The eigenvalues of this system are: \begin{equation} \lambda^\evm = u - c \qquad \lambda^\evz = u \qquad \lambda^\evzs{\rho e} = u \qquad \lambda^\evp = u + c \end{equation} and the eigenvectors are: \begin{equation} \rb^\evm = \left ( \begin{array}{c} 1 \\ -c/\rho \\ c^2 \\ h \end{array} \right ) \qquad \rb^\evz = \left ( \begin{array}{c} 1 \\ 0 \\ 0 \\ 0 \end{array} \right ) \qquad \rb^\evzs{\rho e} = \left ( \begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \end{array} \right ) \qquad \rb^\evp = \left ( \begin{array}{c} 1 \\ c/\rho \\ c^2 \\ h \end{array} \right ) \end{equation} and \begin{eqnarray} && \lb^\evm = ( \begin{array}{cccc} 0 & -\frac{\rho}{2c} & \frac{1}{2c^2} & 0 \end{array} ) \nonumber \\ && \lb^\evz = ( \begin{array}{cccc} 1 & 0 & -\frac{1}{c^2} & 0 \end{array} ) \nonumber \\ && \lb^\evzs{\rho e} = ( \begin{array}{cccc} 0 & 0 & -\frac{h}{c^2} & 1 \end{array} ) \nonumber \\ && \lb^\evp = ( \begin{array}{cccc} 0 & \frac{\rho}{2c} & \frac{1}{2c^2} & 0 \end{array} ) \end{eqnarray} Remember that the state variables in the $\qb$ vector are mixed into the other states by $\lb \cdot \qb$. Since all $\lb^{(\nu)}$'s have $0$ in the $\rho e$ `slot' (the last position) except for $\lb^\evzs{\rho e}$, and the corresponding $\rb^\evzs{\rho e}$ is only non-zero in the $\rho e$ slot, this means that $\rho e$ is not mixed into the other state variables. This is as expected, since $\rho e$ is not needed in the system. Also recall that the jump carried by the wave $\nu$ is proportional to $\rb^{(\nu)}$---since $\rb^\evm$, $\rb^\evzs{\rho e}$, and $\rb^\evp$ have non-zero $\rho e$ elements, this means that $\rho e$ jumps across these three waves. Working through the sum for the $(\rho e)$ state, and using a $\sim$ to denote the reference states, we arrive at: \begin{align} (\rho e)_{i+\myhalf,L}^{n+\myhalf} = \widetilde{(\rho e)} &- \frac{1}{2} \left [ -\frac{\rho}{c} \left (\tilde{u} - \Ic^{(1)}_+(u) \right ) +\frac{1}{c^2} \left (\tilde{p} - \Ic^{(1)}_+(p) \right ) \right ] h \nonumber \\ &- \left [ -\frac{h}{c^2} \left (\tilde{p} - \Ic^{(3)}_+(p) \right ) + \left (\widetilde{(\rho e)} - \Ic^{(3)}_+(\rho e) \right ) \right ] \nonumber \\ &- \frac{1}{2} \left [ \frac{\rho}{c} \left (\tilde{u} - \Ic^{(4)}_+(u) \right ) +\frac{1}{c^2} \left (\tilde{p} - \Ic^{(4)}_+(p) \right ) \right ] h \end{align} This is the expression that is found in the Castro code. If you are dealing with a simple geometry, then the divergences have metric terms. In 1-d, we then have: \begin{equation} \ddt{(\rho e)} + \frac{1}{r^\alpha} \ddr{(r^\alpha \rho u e)} + p \frac{1}{r^\alpha} \ddr{(r^\alpha u)} \end{equation} Expanding out the derivatives gives use the equation in Cartesian form with a geometric source term: \begin{equation} \ddt{(\rho e)} + \ddr{(\rho u e)} + p \ddr{u} = -\frac{\alpha \rho hu}{r} \end{equation} where $h$ is the specific enthalpy. This can be accommodated using the procedure described in \S~\ref{ch:euler:geom}. All of these methods are designed to avoid EOS calls where possible, since general equations of state can be expensive. Extending these to an unsplit formulation requires carrying an additional auxiliary variable from the primitive state back to the conserved state and adding the transverse gradients to its interface state. Castro deals with a conserved state of $\Uc = (\rho, \rho \Ub, \rho E, p)$, and explicitly adds the transverse terms found in the multi-dimensional form of Eq.~\ref{eq:euler:pgeneral} to the normal states of $p$. ================================================ FILE: Euler/Euler-theory.tex ================================================ \label{ch:compressible-theory} %----------------------------------------------------------------------------- \section{Euler equation properties} The Euler equations\footnote{ We focus on the Euler equations, which are the most commonly modeled set of fluid equations in astrophysics. The more general equation set, the Navier-Stokes equations, includes dissipative terms. However, for astrophysical flows, the scales on which these dissipative terms operate are usually much smaller than the system of interest (equivalently, Reynolds numbers of astrophysical flows are very large).} describe conservation of mass, momentum, and energy in the fluid approximation. Their general form, without any source terms, is: \MarginPar{need to add a discussion of the N-S equations and dimensionless numbers} \begin{align} \ddt{\rho} + \nabla \cdot (\rho \Ub) &= 0 \\ \ddt{(\rho \Ub)} + \nabla \cdot (\rho \Ub \Ub) + \nabla p &= 0 \\ \ddt{(\rho E)} + \nabla \cdot (\rho E \Ub + p \Ub ) &= 0 \end{align} Here $\rho$ is the density, $\Ub$ is the velocity vector, $\Ub = u\hat{x} + v\hat{y}$, $p$ is the pressure, and $E$ is the total energy / mass, and can be expressed in terms of the specific internal energy, $e$, and kinetic energy as: \begin{equation} E = e + \frac{1}{2} |\Ub|^2 \end{equation} The equations are closed with the addition of an equation of state. A common choice is the gamma-law EOS: \begin{equation} p = \rho e(\gamma - 1) \end{equation} where $\gamma$ is the ratio of specific heats for the gas/fluid (for an ideal, monatomic gas, $\gamma = 5/3$), but any relation of the form: \begin{equation} p = p(\rho, e) \end{equation} will work. For many astrophysical environments, we may not be able to express this relation analytically, but instead will solve it via numerical integration or by interpolating from tabulated results. For a derivation of the equations of hydrodynamics using moments of the Boltzmann equation see \cite{shu,choudhuri}. For a physically-motivated derivation from conservation, see \cite{choudhuri,leveque:2002}. In one dimension, they appear as\footnote{assuming Cartesian coordinates}: \begin{align} \frac{\partial \rho}{\partial t} + \frac{\partial (\rho u)}{\partial x} &= 0 \\ % \frac{\partial(\rho u)}{\partial t} + \frac{\partial (\rho uu + p)}{\partial x} &= 0 \\ % \frac{\partial(\rho E)}{\partial t} + \frac{\partial(\rho u E + u p)}{\partial x} &= 0 \end{align} One thing that we can notice immediately is that there is no need for temperature in this equation set, although often, when source terms are present, we will need to obtain temperature from the equation of state. In this form, the equations are said to be in {\em conservative form}, i.e.\ they can be written as: \begin{equation} \Uc_t + \left [\Fb(\Uc) \right ]_x = 0 \end{equation} with \begin{equation} \Uc = \left ( \begin{array}{c} \rho \\ \rho u \\ \rho E \end{array} \right ) % \qquad % \Fb(\Uc) = \left ( \begin{array}{c} \rho u \\ \rho uu + p \\ \rho u E + up \end{array} \right ) \end{equation} % We can write this in {\em quasi-linear} form by first expressing the flux vector in terms of the conserved variables directly. Taking $m \equiv \rho u$, $\mathcal{E} \equiv \rho E$, and assuming a gamma-law EOS% \footnote{we can relax this assumption by writing $p = p(\rho, e)$, and then taking the derivatives of this as needed: $\partial p/\partial \rho$, $\partial p/\partial m = \partial p /\partial e|_\rho \partial e/\partial m$, and $\partial p/\partial \mathcal{E} = \partial p/\partial e|_\rho \partial e/\partial \mathcal{E}$, with $e = (\mathcal{E} - \myhalf m^2/\rho)/\rho$. But as we'll see, there are simpler systems to work with.}, \begin{equation} p = \rho e (\gamma-1) = \left (\mathcal{E} - \frac{1}{2} \frac{m^2}{\rho}\right )(\gamma - 1) \end{equation} we have \begin{equation} \Fb(\Uc) = \left ( \begin{array}{c} m \\ \frac{1}{2}\frac{m^2}{\rho} \left (3 - \gamma \right ) + \mathcal{E} (\gamma - 1) \\ \frac{m\mathcal{E}}{\rho} \gamma -\frac{1}{2} \frac{m^3}{\rho^2} (\gamma -1) \end{array} \right ) \end{equation} The Jacobian% \footnote{ The Jacobian, ${\bf J}$ of a vector $\Fb(\Uc)$ with $\Fb = (f_1, f_2, \ldots, f_n)^\intercal$ and $\Uc = (u_1, u_2, \ldots, u_n)^\intercal$ is \begin{equation*} {\bf J} \equiv \frac{\partial \Fb}{\partial \Uc} = \left ( \begin{array}{cccc} \partial f_1/\partial u_1 & \partial f_1/\partial u_2 & \ldots & \partial f_1/\partial u_n \\ \partial f_2/\partial u_1 & \partial f_2/\partial u_2 & \ldots & \partial f_2/\partial u_n \\ \vdots & \vdots & \ddots & \vdots \\ \partial f_n/\partial u_1 & \partial f_n/\partial u_2 & \ldots & \partial f_n/\partial u_n \end{array} \right ) \end{equation*} } of this flux vector can now be computed as $\Ab = \partial \Fb/\partial \Uc$: \begin{equation} \Ab(\Uc) = \left ( \begin{array}{ccc} 0 & 1 & 0 \\ -\frac{1}{2}u^2(3 -\gamma) & u (3 -\gamma) & \gamma - 1 \\ \frac{1}{2}(\gamma -2)u^3 - \frac{uc^2}{\gamma -1} & \frac{3-2\gamma}{2} u^2 + \frac{c^2}{\gamma -1} & u \gamma \end{array} \right ) \end{equation}\MarginPar{SymPy notebook?} where the speed of sound is $c = \sqrt{\gamma p/\rho}$. With this, our system can be written as: \begin{equation} \Uc_t + \Ab(\Uc) \Uc_x = 0 \end{equation} This matrix is quite complex and difficult to work with. The eigenvectors of this matrix can be found in a variety of sources (e.g. \cite{toro:1997,athena}). An alternate way to express these equations is using the {\em primitive variables}: $\rho, u, p$. \begin{exercise}[Primitive variable form of the Euler equations] {Show that the Euler equations in primitive form can be written as \begin{align} \ddt{\rho} + u \ddx{\rho} + \rho \ddx{u} &= 0 \\ \ddt{u} + u \ddx{u} + \frac{1}{\rho} \ddx{p} &= 0 \\ \ddt{p} + u \ddx{p} + \gamma p \ddx{u} &= 0 \label{eq:prim:p} \end{align} } \end{exercise} Notice that the velocity equation looks like Burgers' equation, and is nonlinear. This nonlinearity will admit shock and rarefaction solutions like we saw with Burgers' equation. % The primitive variable system can be written compactly as: \begin{equation} \qb_t + \Ab(\qb) \qb_x = 0 \end{equation} where \begin{equation} \qb = \left ( \begin{array}{c} \rho \\ u \\ p \end{array} \right ) % \qquad \Ab(\qb) = \left ( \begin{array}{ccc} u & \rho & 0 \\ 0 & u & 1/\rho \\ 0 & \gamma p & u \end{array} \right ) \label{eq:primitivesystem} \end{equation} The eigenvalues of $\Ab$ can be found via $| \Ab - \lambda {\bf I} | = 0$, where $|\ldots|$ indicates the determinant and $\lambda$ are the eigenvalues. \begin{exercise}[The eigenvalues of the Euler system] { Show that the eigenvalues of $\Ab$ are $\lambda^\evm = u -c$, $\lambda^\evz = u$, $\lambda^\evp = u+c$. } \end{exercise} Note that both the conserved Jacobian matrix, $\Ab(\Uc)$, and the primitive variable matrix, $\Ab(\qb)$, have the same eigenvalues, since they represent the same physics. In Eq.~\ref{eq:prim:p}, we used the algebraic gamma-law equation of state to replace $e$ with $p$, however, for a general equation of state, we can get the appropriate expression by writing $p = p(\rho, s)$: \begin{equation} \label{eq:euler:presnosource} \frac{Dp}{Dt} = \left . \frac{\partial p}{\partial \rho} \right |_s \frac{D\rho}{Dt} + \left . \frac{\partial p}{\partial s} \right |_\rho \cancelto{0}{\frac{Ds}{Dt}} \end{equation} where $Ds/Dt = 0$ when no entropy sources are present (we will relax this assumption in \S~\ref{ch:lm:constraints}). Recognizing that $\Gamma_1 \equiv \partial \log p/\partial \log \rho |_s$, we have: \begin{equation} \frac{\partial p}{\partial t} + u \frac{\partial p}{\partial x} + \Gamma_1 p \frac{\partial u}{\partial x} = 0 \label{eq:euler:pgeneral} \end{equation} as the generalization of the pressure equation\footnote{$\Gamma_1$ is one of the many adiabatic indices that describe the coupling between different thermodynamic quantities under compression/expansion. Any stellar structure book should give a good background, e.g., \cite{HKT}. Note that for an ideal gas, $\Gamma_1 = \gamma$}. The sound speed is then $c^2 = \Gamma_1 p /\rho$. These eigenvalues are the speeds at which information propagates through the fluid. Since the eigenvalues are real, this system (the Euler equations) is said to be {\em hyperbolic}. Additionally, since $\Ab = \Ab(\qb)$, the system is said to be {\em quasi-linear}. Note, physically, these eigenvalues are telling us that information is communicated at the speed of the fluid ($u$), as well as the speed of sound moving with respect to the fluid ($u \pm c$). We'll use the symbols $\{-,\circ,+\}$ to denote the eigenvalues and their corresponding eigenvectors throughout these notes. % The right and left eigenvectors can be found via: \begin{equation} \Ab \, \rb^\enu = \lambda^\enu \rb^\enu \; ; \qquad \lb^\enu \, \Ab = \lambda^\enu \lb^\enu \end{equation} where $\nu = \{-,\circ,+\}$ corresponding to the three waves, and there is one right and one left eigenvector for each of the eigenvalues. % \begin{exercise}[Eigenvectors of the Euler system] { Show that the right eigenvectors are: \begin{equation} \label{eq:euler:primRevs} \rb^\evm = \left ( \begin{array}{c} 1 \\ -c/\rho \\ c^2 \end{array} \right ) % \qquad \rb^\evz = \left ( \begin{array}{c} 1 \\ 0 \\ 0 \end{array} \right ) % \qquad \rb^\evp = \left ( \begin{array}{c} 1 \\ c/\rho \\ c^2 \end{array} \right ) \end{equation} and the left eigenvectors are: \begin{align} \lb^\evm &= \left ( \begin{array}{ccc} 0 & -\frac{\rho}{2c} & \frac{1}{2c^2} \end{array} \right ) \\ % \qquad \lb^\evz &= \left ( \begin{array}{ccc} 1 & 0 & -\frac{1}{c^2} \end{array} \right ) \\ % \qquad \lb^\evp &= \left ( \begin{array}{ccc} 0 & \phantom{+}\frac{\rho}{2c} & \frac{1}{2c^2} \end{array} \right ) \end{align} Note that in general, there can be an arbitrary constant in front of each eigenvector. Here they are normalized such that $\lb^{(i)} \cdot \rb^{(j)} = \delta_{ij}$. } \end{exercise} A {\sf Jupyter} notebook using {\sf SymPy} that derives these eigenvectors is available: \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/compressible/euler.ipynb}{euler.ipynb}} \footnote{if you don't have {\sf Jupyter}, you can view this rendered online via the github link.} A final form of the equations is called the {\em characteristic form}. Here, we wish to diagonalize the matrix $\Ab$. We take the matrix $\Rb$ to be the matrix of right eigenvectors, \begin{equation} \Rb = (\rb^\evm | \rb^\evz | \rb^\evp ) \end{equation} (i.e., $\Rb$ is a square matrix whose columns are the eigenvectors), and $\Lb$ is the corresponding matrix of left eigenvectors: \begin{equation} \Lb = \left ( \begin{array}{c} \lb^\evm \\ \hdashline \lb^\evz \\ \hdashline \lb^\evp \end{array} \right ) \end{equation} Note that $\Lb\, \Rb = {\bf I} = \Rb\, \Lb$, and $\Lb = \Rb^{-1}$. \begin{exercise}[Characteristic form of the Euler equations] { Show that $\Lambdab = \Lb \Ab \Rb$ is a diagonal matrix with the diagonal elements simply the 3 eigenvalues we found above: \begin{equation} \Lambdab = \left ( \begin{array}{ccc} \lambda^\evm & & \\ & \lambda^\evz & \\ & & \lambda^\evp \end{array} \right ) \end{equation} } \end{exercise} To transform our primitive variable system into the characteristic form, we start by multiplying by $\Lb$: \begin{equation} \Lb\qb_t + \Lb \Ab \qb_x = 0 \end{equation} Next, recalling that $\Rb\Lb = {\bf I}$ (we can insert the identity matrix where we please), we can rewrite our system as: \begin{equation} \Lb\qb_t + \Lb \Ab \Rb \Lb \qb_x = 0 \end{equation} Finally, defining $d\wb = \Lb d\qb$, \begin{equation} \wb_t + \Lambda \wb_x = 0 \end{equation} Here, the $\wb$ are the {\em characteristic variables}. Note that we cannot in general integrate $d\wb = \Lb d\qb$ to write down the characteristic quantities. Since $\Lambdab$ is diagonal, this system is a set of decoupled advection-like equations: \begin{align} w_t^\evm + \lambda^\evm w_x^\evm &= 0 \\ w_t^\evz + \lambda^\evz w_x^\evz &= 0 \\ w_t^\evp + \lambda^\evp w_x^\evp &= 0 \end{align} If the system were linear, then the solution to each would simply be to advect the quantity $w^\enu$ at the wave speed $\lambda^\enu$. We could then get back the primitive variables as $d\qb = \Lb^{-1} d\wb = \Rb d\wb$. The basic idea of hyperbolic systems of PDEs is that there is one wave for each eigenvalue, and these define the characteristic curves, $dx/dt = \lambda^\enu$. As we discussed with linear advection and Burgers' equation, along the characteristics the solution is constant. Across them, the solution jumps. For a linear system, the characteristics are straight lines. Imagine initial conditions consisting of a jump in the primitive variables, $\Delta \qb$. The corresponding jumps in characteristic variables are $\Delta \wb \sim \Lb \Delta \qb$ (where the $\sim$ accounts for the fact that in a nonlinear system, $\Lb = \Lb(\qb)$). The characteristic form of the equations says that each of the waves will carry its respective jump, $\Delta \wb$. Since $d\qb = \Lb^{-1}d\wb = \Rb d\wb$, the jump in the primitive variable across each wave is proportional to the right-eigenvector associated with that wave. So, for example, since $\rb^\evz$ is only non-zero for the density element (see Eq.~\ref{eq:euler:primRevs}), this then means that only density jumps across the $\lambda^\evz = u$ wave---pressure and velocity are constant across this wave (see for example, Toro~\cite{toro:1997}, Ch.\ 2, 3 or LeVeque~\cite{leveque:2002} for a thorough discussion). Figure~\ref{fig:sod} shows the three waves emanating from an initial discontinuity. This is a classic problem called the Sod problem~\cite{sod:1978}. The initial conditions are \begin{align} \rho_l &= 1 & \rho_r &= 1/8 \nonumber \\ u_l &= 0 & u_r &= 0 \\ p_l &= 1 & p_r &= 1/10 \nonumber \end{align} Here we see the three waves propagating away from the initial discontinuity. The left ($u-c$) wave is a rarefaction, the middle ($u$) is the contact discontinuity, and the right ($u+c$) is a shock. Note that all 3 primitive variables jump across the left and right waves, but only the density jumps across the middle wave. This reflects the right eigenvectors. Also note that no waves have reached the far left and far right yet, the conditions there are the same as the initial conditions. \MarginPar{we need a python version of the Riemann exact script} \begin{figure} \centering \includegraphics[width=0.8\linewidth]{riemann-sod} \caption[The Sod problem]{\label{fig:sod} Evolution following from an initial discontinuity at $x = 0.5$. These particular conditions are called the {\em Sod problem}, and in general, a setup with two states separated by a discontinuity is called a shock-tube problem. We see three waves carrying changes in the solution. \\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/compressible/riemann-sod.py}{riemann-sod.py}}} \end{figure} %----------------------------------------------------------------------------- \section{The Riemann problem} \label{euler:sec:riemann} Just like with advection and Burgers' equation, we will need to solve a Riemann problem. However, as our system is nonlinear, and has 3 waves, the solution is considerably more complex, but the general ideas are straightforward. Here we review the basic outline of operations, and refer to Toro~\cite{toro:1997} for full details on a variety of methods for solving the Riemann problem. The Riemann problem consists of a left and right state separated by an interface. For the Euler equations, there are three eigenvalues, which are the speeds at which information propagates. Each of these correspond to a wave that will move out from the interface with time, and each wave will carry with it a jump in the characteristic variables. Figure~\ref{fig:riemann:waves} shows the three waves moving out from the interface, separating space into 4 regions, marked: $L$, $L_*$, $R_*$, and $R$. \begin{figure}[h] \centering \includegraphics[width=\linewidth]{riemann-waves} \caption[The Riemann problem wave structure for the Euler equations]{\label{fig:riemann:waves} The wave structure and 4 distinct regions for the Riemann problem. Time can be thought of as the vertical axis here, so we see the waves moving outward from the interface.} \end{figure} We typically work in terms of primitive variables. The states in the $L$ and $R$ regions are simply the left and right input states---the waves have not had time to reach here, so they are unmodified. Looking at the middle right eigenvector, $\rb^\evz$, we identify the middle wave as a {\em contact discontinuity}. There is neither compression nor expansion across this wave (since the velocity does not jump across it) and it simply separates two states. We also know from the eigenvectors that the pressure is constant across the contact. The left and right waves can be either a rarefaction or a shock. It will be a shock if the flow is compressing and a rarefaction otherwise. We already see that the only unknowns are $u_\star$, $p_\star$, $\rho_{\star,L}$, and $\rho_{\star,R}$. \subsection{Rarefactions} The conditions across a rarefaction can be understood by considering a system where entropy replaces pressure, $\qb_s = (\rho, u, s)^\intercal$. The entropy equation is simply $Ds/Dt = 0$. We need to express the pressure gradient in the velocity equation in terms of $\qb_s$. \begin{equation} \frac{\partial p(\rho, s)}{\partial x} = \left . \frac{\partial p}{\partial s} \right |_\rho \frac{\partial s}{\partial x} + \left . \frac{\partial p}{\partial \rho} \right |_s \frac{\partial \rho}{\partial x} = \left . \frac{\partial p}{\partial s} \right |_\rho \frac{\partial s}{\partial x} + \frac{p\Gamma_1}{\rho} \frac{\partial \rho}{\partial x} \end{equation} giving \begin{equation} \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + \frac{1}{\rho} \left [ \left . \frac{\partial p}{\partial s} \right |_\rho \frac{\partial s}{\partial x} + \frac{p\Gamma_1}{\rho} \frac{\partial \rho}{\partial x} \right ] = 0 \end{equation} This gives a system: \begin{equation} {\qb_s}_t + \Ab_s(\qb_s) {\qb_s}_x = 0 \end{equation} where \begin{equation} \Ab_s = \left ( \begin{array}{ccc} u & \rho & 0 \\ c^2/\rho & u & \frac{1}{\rho} \left . \frac{\partial p}{\partial s}\right |_\rho \\ 0 & 0 & u \end{array} \right ) \end{equation} The eigenvalues of $\Ab_s$ are again $u$, $u-c$, and $u+c$. The right eigenvectors are\footnote{Again, see the {\sf SymPy} {\sf Jupyter} notebook: \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/compressible/euler.ipynb}{euler.ipynb}}} \begin{align} \rb_s^\evm = \left ( \begin{array}{c} 1 \\ -c/\rho \\ 0 \end{array} \right ) % \qquad \rb_s^\evz = \left ( \begin{array}{c} 1 \\ 0 \\ -c^2/p_s \end{array} \right ) % \qquad \rb_s^\evp = \left ( \begin{array}{c} 1 \\ c/\rho \\ 0 \end{array} \right ) \end{align} Here, we define $p_s \equiv \partial p/\partial s |_\rho$. Since the jump in $\qb_s$ is proportional to these eigenvectors, we see that entropy does not change across the left, $(-)$, and right, $(+)$, waves. Consider the $\evp$ wave, which moves at a speed $\lambda^\evp = u +c$. Across this wave, the characteristic variable $\wb^\evp$ will jump, but the others will not. Similarly, $\wb^\evp$ will be constant across the $\evm$ wave. So we can use this constancy to tell us how the evolution behaves across that wave. The constancy of the middle and right characteristic variables across the left wave gives us the conditions (using our original primitive variable system): \begin{align} \lb^\evp \cdot d\qb &= 0 \\ \lb^\evz \cdot d\qb &= 0 \end{align} or \begin{equation} \left ( \begin{array}{ccc} 0 & \frac{\rho}{2c} & \frac{1}{2c^2} \end{array} \right) \left ( \begin{array}{c} d\rho \\ du \\ dp \end{array} \right ) = 0 \qquad % \left ( \begin{array}{ccc} 1 & 0 & -\frac{1}{c^2} \end{array} \right) \left ( \begin{array}{c} d\rho \\ du \\ dp \end{array} \right ) = 0 \end{equation} Expanding these, we have the system: \begin{align} du + \frac{1}{\rho c} dp &= 0 \\ d\rho - \frac{1}{c^2} dp &= 0 \end{align} Defining the {\em Lagrangian sound speed}, $C \equiv \rho c$, and the specific volume, $\tau = 1/\rho$, we can rewrite this system as: \begin{equation} du = -\frac{dp}{C} , \,\,\, d\tau = -\frac{dp}{C^2} \quad \mbox{across the left wave} \end{equation} Similar arguments would give the condition across the right wave as: \begin{equation} du = \frac{dp}{C} , \,\,\, d\tau = -\frac{dp}{C^2} \quad \mbox{across the right wave} \end{equation} These are completely general at this point. Note that the condition of the second wave, $d\tau/dp$ is essentially the definition of the adiabatic index $\Gamma_1$, $dp/p - \Gamma_1 d\rho/\rho = 0$ for constant entropy. Finding the solution across the rarefaction then involves integrating the system from $p_{l,r}$ to $p_\star$, where $l, r$ is the original left or right state, depending on which wave you are considering. Coupled with this, we need our equation of state to provide $C = C(\tau, p)$. These solutions are called the {\em Riemann invariants}. Considerable simplification can be made if we assume a gamma-law gas. The eigensystem with entropy showed us that entropy is constant across the solution. This allows us to write our equation of state as: \begin{equation} p = K \rho^\gamma \end{equation} where $K$ is a constant that depends on the entropy, and do the integrals analytically. \begin{exercise}[Riemann invariants for gamma-law gas] Assume a gamma-law gas, then you only need to integrate a single equation to find the solution across the left rarefaction: \begin{equation} u = - \int \frac{dp}{\rho c} \end{equation} where $c = \sqrt{\gamma p/\rho}$. Using $p = K\rho^\gamma$, show that \begin{equation} u + \frac{2c}{\gamma - 1} = \mbox{constant} \quad \mbox{across the left rarefaction} \end{equation} This allows you to link the state to the left of the rarefaction to the star state: \begin{equation} u_L + \frac{2c_L}{\gamma - 1} = u_\star + \frac{2c_\star}{\gamma -1} \end{equation} and \begin{equation} \frac{p_L}{\rho_L^\gamma} = \frac{p_\star}{\rho_\star^\gamma} \end{equation} from our constant-entropy equation of state. Together, show that this gives: \begin{equation} u_{\star,L} = u_L + \frac{2c_L}{\gamma - 1} \left [ 1 - \left ( \frac{p_\star}{p_L}\right )^{(\gamma - 1)/2\gamma} \right ] \end{equation} A similar construction can be done for the right rarefaction, yielding: \begin{equation} u_{\star,R} = u_R - \frac{2c_R}{\gamma - 1} \left [ 1 - \left ( \frac{p_\star}{p_R}\right )^{(\gamma - 1)/2\gamma} \right ] \end{equation} \end{exercise} For a general equation of state, there is not a closed form, but we would need to integrate our system to get $u_\star = u(p_\star)$ and $\rho_{\star,s} = 1/\tau(p_\star)$. We'll denote the solution across the left rarefaction as $u_\star = u_L^{rare}(p_\star)$ and across the right rarefaction as $u_\star = u_R^{rare}(p_\star)$. \subsection{Shocks} Entropy is not constant across a shock---there is dissipation, so we need a different way to connect the states across the shock. As with Burgers' equation, we can understand the shock by looking at the Rankine-Hugoniot jump conditions. There will be one condition for each of our conservation laws, and together they tell us the speed of the shock and how density and pressure jump across it. The Rankine-Hugoniot conditions are: \begin{equation} \frac{\Fb(\Uc_\star) - \Fb(\Uc_s)}{\Uc_\star - \Uc_s} = S \end{equation} where $S$ is the shock speed. It is easiest to work in the frame of the shock. If the shock speed is $S$, then we define the velocity in the shock frame as: \begin{align} \hat{u}_s &= u_s - S \\ \hat{u}_\star &= u_\star - S \end{align} and the Rankine-Hugoniot conditions become: \begin{equation} \frac{\Fb(\hat{\Uc}_\star) - \Fb(\hat{\Uc}_s)}{\hat{\Uc}_\star - \hat{\Uc}_s} = 0 \end{equation} For the one-dimensional system of Euler equations, we have the conditions: \begin{align} \rho_\star \hat{u}_\star &= \rho_s \hat{u}_s \\ \rho_\star \hat{u}_\star^2 + p_\star &= \rho_s \hat{u}_s^2 + p_s \\ \rho_\star \hat{u}_\star e_\star + \frac{1}{2} \rho_\star \hat{u}_\star^3 + \hat{u}_\star p_\star &= \rho_s \hat{u}_s e_s + \frac{1}{2} \rho_s \hat{u}_s^3 + \hat{u}_s p_s \end{align} Our goal is to find how each variable jumps across the shock. We'll work this out for the general EOS case, following the ideas in \cite{colellaglaz:1985}. Starting with the mass flux, we can express: \begin{equation} \hat{u}_\star = \frac{\rho_s}{\rho_\star} \hat{u}_s \end{equation} and then insert this into the momentum equation to get: \begin{equation} \rho_s^2 \left ( \frac{1}{\rho_s} - \frac{1}{\rho_\star} \right ) \hat{u}_s^2 = p_\star - p_s \end{equation} Introducing compact notation for the jump across the shock: \begin{equation} [q] \equiv q_\star - q \end{equation} we have: \begin{equation} -[\tau] = \frac{[p]}{\rho_s^2 \hat{u}_s^2} \end{equation} We now introduce the mass flux, $W_s$. For a shock separating $L$/$L\star$ (the 1-shock), mass will be moving through the shock to the right, so, in the frame of the shock, the mass flux is \begin{equation} W_L \equiv \rho_L \hat{u}_L = \rho_\star \hat{u}_\star \end{equation} Likewise, for a shock separating $R$/$R\star$ (the 3-shock), mass will move to the left passing through the shock, and in the frame of the shock, the mass flux is \begin{equation} W_R \equiv -\rho_R \hat{u}_R = -\rho_\star \hat{u}_\star \end{equation} Thus, our first jump condition becomes \begin{equation} -[\tau] = \frac{[p]}{W_s^2} \end{equation} Next, going back to the momentum equation, we can substitute in the mass flux. For the left (1-shock), we have: \begin{equation} W_L \hat{u}_\star + p_\star = W_L \hat{u}_L + p_L \end{equation} giving \begin{equation} [u] = -\frac{[p]}{W_L} \end{equation} where we recognized that $\hat{u}_\star - \hat{u}_L = u_\star - u_L = [u]$, since the shock speed is the same for $u_\star$ and $u$. Likewise, for the right (3-shock), we have: \begin{equation} -W_R \hat{u}_\star + p_\star = -W_R \hat{u}_R + p_R \end{equation} giving \begin{equation} [u] = \frac{[p]}{W_R} \end{equation} The last jump condition is for energy. Since all the terms in the energy equation are proportional to velocity, there will be no sign difference between the left and right shock jump conditions. We start by introducing the mass flux: \begin{equation} W_s e_\star + \frac{1}{2} W_s \hat{u}_\star^2 + \frac{W_s}{\rho_\star} p_\star = W_s e_s + \frac{1}{2} W_s \hat{u}_s^2 + \frac{W_s}{\rho_s} p_s \end{equation} The mass flux cancels, leaving \begin{equation} [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 \end{equation} getting rid of the velocities using $\hat{u}_\star^2 = W_s^2/\rho_\star^2$ and $\hat{u}_s^2 = W_s^2/\rho_s^2$, we have: \begin{equation} [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 \end{equation} Then introducing $W_s^2 = -[p]/[\tau]$, and after a lot of algebra, we arrive at: \begin{equation} [e] = -\frac{p_\star + p_s}{2} [\tau] \end{equation} Some sources write $\bar{p} \equiv (p_\star + p_s)/2$. To summarize, our jump conditions across the shock are: \begin{align} [\tau] &= -\frac{[p]}{W_s^2} \\ [u] &= \mp \frac{[p]}{W_s} \quad\mbox{`$-$' for left, `+' for right}\\ [e] &= - \frac{p_\star + p_s}{2} [\tau] \label{eq:euler:shock:ejump} \end{align} As with the rarefaction, the goal is to express this as a function, $u_\star = u_s^\mathrm{shock}(p_\star)$. The general solution procedure is starts with a proposed value for $p_\star$, then: \begin{enumerate} \item root find to find $\rho_\star$ corresponding to the $p_\star$: \begin{enumerate} \item guess a value for $\rho_\star$ \item using the equation of state, express $e_\star = e(p_\star, \rho_\star)$ \item use Newton's method (or another technique) with $[e] = -\bar{p} [\tau]$ to find a correction to $\rho_\star$ \end{enumerate} \item compute \begin{equation} \frac{1}{W_s^2} = - \frac{[\tau]}{[p]} \end{equation} \item find the star velocity: \begin{equation} u_\star = u_s \mp \frac{[p]}{W_s} \end{equation} where we use `$-$' for the left shock, and `+' for the right shock \end{enumerate} The one other piece of information we need is the shock speed. We can get this from the mass flux, $W_s$, definition, e.g., $W_L = \rho_L \hat{u}_L = \rho_L (u_L - S)$,: \begin{equation} \label{eq:euler-theory:shockgeneral} S = u_s \mp \frac{W_s}{\rho_s} \quad\mbox{`$-$' for left, `+' for right}\\ \end{equation} This procedure gives us $\rho_{\star,s}$, $p_\star$, and $S$, the shock speed. Just as with the rarefaction, considerable simplification can be made if we assume a gamma-law gas. \begin{exercise}[Shock jump conditions for $\gamma$-law EOS] { Introducing \begin{equation} e = \frac{p}{\rho} \frac{1}{\gamma -1} \end{equation} into the jump condition for energy, Eq.~\ref{eq:euler:shock:ejump}, show that we can express the jump in density in terms of the ratio of pressure, $p_\star/p_s$, as: \begin{equation} \label{eq:euler:shockrhojump} \rho_{\star,s} = \rho_s \left [ \frac{ \frac{p_\star}{p_s} (\gamma + 1) + (\gamma - 1)} {(\gamma + 1) + \frac{p_\star}{p_s} (\gamma -1)} \right ] \end{equation} } Now, compute the mass flux, $W_s$ starting with \begin{equation} W_s^2 = -\frac{[p]}{[\tau]} \end{equation} use the $\gamma$-law equation of state and show that \begin{equation} W_s^2 = \frac{1}{2} p_s \rho_s \left [ \left(\frac{p_\star}{p_s}\right) (\gamma + 1) + (\gamma -1) \right ] \end{equation} With this, show that the star velocity is: \begin{equation} \label{eq:euler:shockujump} u_\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}} \end{equation} with `+` for the left shock and `$-$` for the right shock. Also show that the shock speed is: \begin{equation} \label{eq:euler:shockspeedjump} S = 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} \end{equation} with the `$-$' or the left shock and the `+' for the right shock. \end{exercise} \subsection{Finding the Star State} \label{Euler:riemann:starstate} The left and right states are connected to the state in the star region by a Hugoniot curve---this is a curve in the $u$-$p$ plane that shows all of the possible states one can reach from the current state through either a shock or rarefaction. There are two such curves, one corresponding to the left and one to the right state: \begin{equation} u_{\star,L}(p) = \begin{cases} u_{\star,L}^\mathrm{shock}(p) & p > p_L \\ u_{\star,L}^\mathrm{rare}(p) & p \le p_L \end{cases} \qquad u_{\star,r}(p) = \begin{cases} u_{\star,R}^\mathrm{shock}(p) & p > p_R \\ u_{\star,R}^\mathrm{rare}(p) & p \le p_R \end{cases} \end{equation} The solution to the Riemann problem is the point in the $u$-$p$ plane where these two curves intersect, e.g., we solve for $p_\star$ in: \begin{equation} u_{\star,l}(p_\star) - u_{\star,r}(p_\star) = 0 \end{equation} This is equivalent to saying that pressure and velocity do not jump across the contact wave. For a gamma-law equation of state, using the results from the exercises, we have: \begin{equation} \renewcommand{\arraystretch}{1.5} u_{\star,s}(p_\star) = \begin{cases} 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 \\ \\ % 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 \end{cases} \renewcommand{\arraystretch}{1.0} \end{equation} Figure~\ref{fig:euler:riemann-curve} shows the Hugoniot curves for the Sod problem. Comparing to Figure~\ref{fig:sod}, we see that the right state is linked to the star state with a shock while the left state is linked to the star state with a rarefaction. \begin{exercise}[Hugoniot curves] Download the code demonstrated in Figure~\ref{fig:euler:riemann-curve} and experiment with different initial conditions to see how the solution changes based on the states. Try to create initial states that give rise to two rarefactions and a separate set of states that give rise to two shocks. \end{exercise} \begin{figure}[t] \centering \includegraphics[width=0.9\linewidth]{riemann-phase} \caption[The Hugoniot curves corresponding to the Sod problem]{\label{fig:euler:riemann-curve} The Hugoniot curves corresponding to the Sod problem. The shock and rarefaction curves are shown. The solution to the Riemann problem is the point where the curves intersect. The line style of the curve indicates where we are a shock or rarefaction. Where $p > p_s$, where $s \in {L,R}$, we have a shock.\\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/compressible/riemann-phase.py}{riemann-phase.py}}} \end{figure} \subsection{Complete Solution} \label{Euler:riemann:solution} To complete the solution, we need to find which of the 4 regions, $L, L\star, R\star, R$ we fall in. For hydrodynamics, we are usually interested only in the solution on the interface, but we can look along any ray in the $x$-$t$ plane by defining $\xi = (x - x_\mathrm{interface})/t$. If the initial discontinuity is on the interface, then $\xi = 0$. % figure from figures/Euler/riemann_cartoon.py \begin{figure} \centering \includegraphics[width=0.49\linewidth]{riemann_waves_ifc_R} \includegraphics[width=0.49\linewidth]{riemann_waves_ifc_Rstar} \\ \includegraphics[width=0.49\linewidth]{riemann_waves_ifc_Lstar} \includegraphics[width=0.49\linewidth]{riemann_waves_ifc_L} \caption[Wave configuration for the Riemann problem] {\label{fig:euler:riemann_sample} An illustration of the 3 waves emanating from an initial discontinuity (the origin of the axes). The 4 cases show all possible cases of waves being to the left or right of the interface (corresponding to the choice $\xi = 0$). In all cases, the middle wave is a contact and the left (1) and right (3) waves are either a shock or a rarefaction. The state on the interface is: $R$ (case a); $R\star$ (case b); $L\star$ (case c); $L$ (case d).} \end{figure} Figure~\ref{fig:euler:riemann_sample} shows the possible configurations of the waves. Note that the middle wave is always a contact but the left (1) and right (3) waves can be either a shock or rarefaction. To find out which region the interface falls in, we simply look at the speeds. The first speed to consider is the contact wave, that has a speed of simply $S_c = u_\star$. If $S_c < \xi$, then we are choosing between $R$ and $R_\star$ states (cases a and b). If $S_c > \xi$, then we are choosing between the $L$ and $L_\star$ states (cases c and d in Figure~\ref{fig:euler:riemann_sample})\footnote{for the special case where $S_c = 0$, we usually take the solution to be the average of the $L_\star$ and $R_\star$ states. Testing for this explicitly rather than using $\ge$ in a check is important for maintaining symmetry (see \S~\ref{sec:euler-methods:shocktubes}).}. The direction of the contact reduces the problem down to choosing between just two regions, either $L$-$L_\star$ or $R$-$R_\star$. This leaves just a single wave to consider: the right wave for cases a and b; and the left wave for cases c and d. We need the wave speed for this---it will depend on whether it is a shock or a rarefaction. For a shock, the wave speed is given by Eq.~\ref{eq:euler-theory:shockgeneral}. We do the same procedure as before. For example, for cases a and b, we look at $S_R$, the right-moving shock speed for the 3-wave. If $S_R > 0$, then the interface is in the $R_\star$ state (case b), while if $S_R < 0$, then the interface is in the $R$ state (case a). A similar process holds for the left-moving shock and cases c and d. For a rarefaction, the relevant speeds are the characteristic speeds on either side of the rarefaction. We usually talk about the leading part of the rarefaction (the {\em head}) and trailing part (the {\em tail}). The corresponding wave speeds are: \begin{itemize} \item left (1) rarefaction: \begin{itemize} \item $\lambda_\mathrm{head} = u_L - c_L$ \item $\lambda_\mathrm{tail} = u_\star - c_\star$ \end{itemize} \item right (3) rarefaction: \begin{itemize} \item $\lambda_\mathrm{head} = u_R + c_R$ \item $\lambda_\mathrm{tail} = u_\star + c_\star$ \end{itemize} \end{itemize} The nature of the rarefaction is such that it spreads out, $|\lambda_\mathrm{head}| > |\lambda_\mathrm{tail}|$. To determine which state we are in, we look to see where both the head and tail are with respect to the interface. Figure~\ref{fig:euler:rarefaction_sample} shows the possibilities for a left (1) rarefaction. In case (a), $\lambda_\mathrm{head}, \lambda_\mathrm{tail} < 0$, so the $L_\star$ state is on the interface. In case (c), $\lambda_\mathrm{head}, \lambda_\mathrm{tail} > 0$, so the $L$ state is on the interface. Case (b) is more complicated---we have $\lambda_\mathrm{head} <0, \lambda_\mathrm{tail} > 0$, so the rarefaction spans the interface. We need to therefore solve for the state in the rarefaction itself. To find the structure inside of a rarefaction, consider some point, $(x, t)$, in the $x$-$t$ plane. For an initial discontinuity at the origin, we can imagine a line connecting this point and the origin, which has the form: \begin{equation} \frac{x}{t} = u - c \end{equation} % figure from figures/Euler/rarefaction_cartoon.py \begin{figure} \centering \includegraphics[width=0.6\linewidth]{rarefaction_left} \\ \includegraphics[width=0.6\linewidth]{rarefaction_center} \\ \includegraphics[width=0.6\linewidth]{rarefaction_right} \caption[Rarefaction configuration for the Riemann problem] {\label{fig:euler:rarefaction_sample} An illustration of the structure of the left rarefaction wave, for the case where the contact and right wave are to the right of the interface. Here, we are choosing between states $L$ and $L_\star$. In (a), both the head and tail of the rarefaction are to the left of the interface, so state $L_\star$ is on the interface. In (c), both the head and tail of the rarefaction are to the right of the interface, so state $L$ is on the interface. Inbetween, case (b), shows a rarefaction that spans the interface. For this case, we need to integrate the Riemann invariants to find the state on the interface.} \end{figure} %----------------------------------------------------------------------------- \section{Other thermodynamic equations} \label{sec:euler:otherthermo} At times we will want to use alternate forms of the energy equation. The internal energy is governed by the first law of thermodynamics. In the absence of any heat sources, we have: \begin{equation} dq = 0 = de + pd(1/\rho) \end{equation} where $e$ is the specific internal energy. Applying this to a Lagrangian fluid element, we have: \begin{align} \DDt{e} + p \DDt{(1/\rho)} &= 0 \\ \DDt{e} - \frac{p}{\rho^2} \frac{D\rho}{Dt} &= 0 \\ \rho \frac{De}{Dt} + p \nabla \cdot \Ub &= 0 \end{align} where we used the continuity equation in the last step to eliminate $D\rho/Dt$. This can be rewritten by adding $e \times$ the continuity equation to give: \begin{equation} \ddt{(\rho e)} + \nabla \cdot (\rho \Ub e) + p \nabla \cdot \Ub = 0 \label{eq:euler:econs} \end{equation} Notice that internal energy, $(\rho e)$ is not a conserved quantity (in particular, the $p\nabla \cdot \Ub$ term is not in conservative form). Another energy-like quantity that we can consider is specific enthalpy, \begin{equation} h = e + \frac{p}{\rho} \end{equation} Differentiating this, and using the internal energy equation, \begin{align} \DDt{h} &= \DDt{e} - \frac{p}{\rho^2} \DDt{\rho} + \frac{1}{\rho} \DDt{p} \\ &= \frac{p}{\rho^2}\DDt{\rho} - \frac{p}{\rho^2} \DDt{\rho} + \frac{1}{\rho} \DDt{p} \end{align} so \begin{equation} \rho \DDt{h} = \DDt{p} \end{equation} This form is useful for very subsonic flows (see, e.g.~\cite{SNpaper}), where $Dp/Dt = 0$, which then shows that enthalpy is conserved: \begin{equation} \ddt{(\rho h)} + \nabla \cdot (\rho h \Ub) = 0 \end{equation} (here we added $h \times$ the continuity equation to transform from the Lagrangian derivative to a conservation equation). We can also look at the temperature evolution. It is interesting to approach this from both the internal energy and enthalpy equations. Starting with internal energy, writing it as $e(\rho, T)$, we have: \begin{equation} \DDt{e} = \left . \frac{\partial e}{\partial \rho} \right |_T \DDt{\rho} + \left . \frac{\partial e}{\partial T} \right |_\rho \DDt{T} = -\frac{p}{\rho} \nabla \cdot \Ub \end{equation} The specific heat at constant volume is defined as $c_v = \partial e/\partial T|_\rho$, allowing us to write: \begin{equation} c_v \DDt{T} = \left ( \left . \frac{\partial e}{\partial \rho} \right |_T - \frac{p}{\rho} \right ) \nabla \cdot \Ub \end{equation} If we alternately start with enthalpy, expressing it as $h = h(p, T)$, we have: \begin{equation} \DDt{h} = \left . \frac{\partial h}{\partial p} \right |_T \DDt{p} + \left . \frac{\partial h}{\partial T} \right |_p \DDt{T} = \frac{1}{\rho} \DDt{p} \end{equation} The specific heat at constant pressure is defined as $c_p = \partial h/\partial T|_p$, letting us write: \begin{equation} c_p \DDt{T} = \left ( \frac{1}{\rho} - \left . \frac{\partial h}{\partial p} \right |_T \right ) \DDt{p} \end{equation} These two temperature evolution equations are equivalent, but one describes the evolution in terms of density and the other in terms of pressure. Later, we'll see how these can be useful when we approximate the evolution of reactive flow under constant density or constant pressure behaviors. \if debug \subsection{Eigensystem with temperature} Here we illustrate how the eigensystem changes when we replace pressure with temperature in our primitive variable system. We write this set of variables as $\hat{\qb} = (\tau, u, T)^\intercal$---note that we keep $\tau$ instead of $\rho$ here. The motivation for this comes from the fact that with temperature in the mix, the temperature will jump across the contact wave. Since pressure should be constant across the contact, and, even with the general EOS, a temperature jump needs a density drop for pressure to remain constant, $\tau$ should counteract the behavior of $T$ across the contact. The temperature evolution equation appears as: \begin{equation} \frac{\partial T}{\partial t} = -u\ddx{T} + \frac{1}{\rho c_p} \left [ (1 - \rho h_p) \frac{Dp}{Dt} \right ] \end{equation} (see, e.g.\ \cite{ABRZ:I}) where $c_p$ is the specific heat at constant pressure, $c_p = \partial h/\partial T|_p$ and $h_p \equiv \partial h / \partial p |_T$, with $h = e + p/\rho$ the specific enthalpy. We can use the standard pressure evolution equation (Eq.~\ref{eq:euler:pgeneral}) to eliminate the Lagrangian pressure term, resulting in: \begin{equation} \frac{\partial T}{\partial t} = -u\ddx{T} - \eta \ddx{u} \end{equation} where we defined \begin{equation} \eta \equiv \frac{1 - \rho h_p}{c_p} c^2 \end{equation} The density equation remains unchanged from the traditional primitive variable formulation, but for the velocity, we need to write the $\partial p/\partial x$ term in terms of $\hat{\qb}$. We do this via the chain rule. We define $p_\rho \equiv {\partial p}/{\partial \rho} |_T$ and $p_T \equiv {\partial p}/{\partial T} |_\rho$, then our velocity equation is: \begin{equation} \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 \end{equation} Note that here we neglected any composition dependence in the EOS. Our primitive variable system in matrix form is then: \begin{equation} \hat{\qb}_t + \hat{\Ab} \hat{\qb}_x = 0 \end{equation} with \begin{equation} \renewcommand{\arraystretch}{1.5} \hat{\Ab}(\hat{\qb}) = \left ( \begin{array}{ccc} u & -\tau & 0 \\ -\dfrac{p_\rho}{\tau} & u & \tau p_T \\ 0 & \eta & u \end{array} \right ) \end{equation} The eigenvalues can be found through the characteristic polynomial, $|\hat{\Ab} - \lambda I| = 0$: \begin{equation} (u - \lambda)^3 - (u -\lambda) \left ( {p_T \eta\tau} + p_\rho \right ) = 0 \end{equation} We can simplify the last term in parenthesis: \begin{equation} {p_T \eta\tau} + p_\rho = \frac{p_T (1 - \rho h_p) c^2}{\rho c_p} + p_\rho = \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} \end{equation} where we substituted in \begin{equation} h_p = \frac{1}{\rho} \left ( 1 - \frac{p}{\rho p_\rho} \right ) + \frac{e_\rho}{p_\rho} \end{equation} (see \cite{ABRZ:I}, appendix A) where $e_\rho = \partial e/\partial \rho |_T$. The term in parenthesis in Eq.~\ref{eq:step1} is simply $c_p - c_v$ (\cite{cg}, Eq.~9.81) where $c_v$ is the specific heat at constant volume, and Eq.~\ref{eq:step1} further reduces to: \begin{equation} {p_T \eta\tau} + p_\rho = \frac{c^2}{c_p} (c_p - c_v) + p_\rho = c^2 - c^2 \frac{\chi_p}{\Gamma_1} + p_\rho = c^2 \end{equation} where we used $c_v/c_p = \chi_p/\Gamma_1$ and $\chi_p = \rho p_\rho / p$ (\cite{cg}, Eqs.~9.87, 9.82). Putting this into our characteristic polynomial, we see that the eigenvalues are, as expected, $\lambda = u, u \pm c$. It is also useful to note that \begin{equation} \eta = \frac{c^2 - p_\rho}{\tau p_T} \end{equation} We construct the left and right eigenvectors such that they are orthonormal, and find: \begin{equation} \renewcommand{\arraystretch}{1.3} \hat{R} = \left ( \begin{array}{ccc} 1 & 1 & 1 \\ c/\tau & 0 & -c/\tau \\ -(c^2 - p_\rho)/\tau^2 p_T & p_\rho/\tau^2 p_T & -(c^2 - p_\rho)/\tau^2 p_T \end{array} \right ) \end{equation} and \begin{equation} \hat{L} = \left ( \begin{array}{ccc} {p_\rho}/{2 c^2} & {\tau}/{2c} & -{\tau^2 p_T}/{2 c^2} \\ 1 - {p_\rho}/{c^2} &0 & {\tau^2 p_T}/{c^2} \\ {p_\rho}/{2 c^2} & -{\tau}/{2c} & -{\tau^2 p_T}/{2 c^2} \end{array} \right ) \end{equation} We note that all thermodynamic derivatives are expressed in terms of $\rho$ or $T$ with the other quantity held constant. This is in the form we expect a $(T, \rho)$-based EOS to return derivatives. Notice also that the temperature jumps across the $\evz$ wave (the contact discontinuity; this is seen from the non-zero value in $\hat{r}^\evz$ for the temperature). We write: \begin{eqnarray} \hat{\beta}^\evm_s &\equiv& (\hat{l}^\evm \cdot \Delta \hat{\qb}^\evm ) = \frac{1}{2C} \left (\frac{\rho^2 p_\rho}{C} \Delta \tau^\evm + \Delta u^\evm - \frac{p_T}{C}\Delta T^\evm \right ) \\ % \hat{\beta}^\evz_s &\equiv& (\hat{l}^\evz \cdot \Delta \hat{\qb}^\evz ) = \Delta \tau^\evz + \frac{1}{C^2} \left (-\rho^2 p_\rho \Delta \tau^\evz + {p_T}\Delta T^\evz \right ) \\ % \hat{\beta}^\evp_s &\equiv& (\hat{l}^\evp \cdot \Delta \hat{\qb}^\evp ) = \frac{1}{2C} \left (\frac{\rho^2 p_\rho}{C} \Delta \tau^\evp - \Delta u^\evp - \frac{p_T}{C}\Delta T^\evp \right ) \end{eqnarray} Note that since $p_\tau = -\rho^2 p_\rho$, we can form a $\Delta p^\enu$ as \begin{equation} \Delta p^\enu = p_\tau \Delta \tau^\enu + p_T \Delta T^\enu \end{equation} and then we see that the $\hat{\beta}^\enu$'s above have the same functional form as the $\mathring{\beta}^\enu$ from the $\mathring{\qb} = (\tau, u, p, e)^\intercal$ eigensystem. This is no surprise, since the $l\cdot \Delta \qb$ are the characteristic variables of the Euler equations. The numerical values will differ though, because the $\hat{\beta}^\enu$ and $\mathring{\beta}^\enu$ use different reference states and reconstructed variables. Finally, we can write out the interface states: \begin{eqnarray} \tau_s &=& \tilde{\tau} - (\hat{\beta}^\evm + \hat{\beta}^\evz + \hat{\beta}^\evp) \\ % u_s &=& \tilde{u} - (C \hat{\beta}^\evm - C \hat{\beta}^\evp) \\ % T_s &=& \tilde{T} - \frac{1}{p_T} \left \{ \left [ -C^2 \hat{\beta}^\evm - C^2 \hat{\beta}^\evp \right ] + \rho^2 p_\rho \left [ \hat{\beta}^\evm + \hat{\beta}^\evz + \hat{\beta}^\evp \right ] \right \} \end{eqnarray} For $T_s$, we recognize the first quantity in the square brackets as being the same as $-(p_s - \tilde{p})$ in the $\mathring{\qb}$ system, and the second term in the square brackets being the same as $-(\tau_s - \tilde{\tau})$ in the $\mathring{\qb}$ system, then we see \begin{equation} p_T (T_s - \tilde{T} ) \approx (p_s - \tilde{p} ) - p_\tau (\tau_s - \tilde{\tau}) \end{equation} which is what we would expect for jumps in $p$ when applying the chain rule. Note that this is not a strict equality, since the reference states and interpolation between the two eigensystems are different. Nevertheless, this demonstrates the connection between the two methods. The above did not consider variations in the composition of the fluid. Multiple species complicate things---now the replacement of the pressure gradient picks up a composition gradient term. The eigensystem will change with this addition. We don't explore this here at this time. \fi ================================================ FILE: Euler/README ================================================ sod.eps: made by hydro1d in the exact/ subdirector ================================================ FILE: Euler/main.tex ================================================ \documentclass[11pt]{article} % margins \usepackage[margin=0.75in]{geometry} % \usepackage{amsmath} % figures \usepackage{graphicx} % font \usepackage{mathpazo} \usepackage{helvet} % coloring \usepackage{color} \definecolor{mygray}{gray}{0.5} % footer \usepackage{fancyhdr} \pagestyle{fancy} \fancyfoot[LO,LE]{\footnotesize \sffamily \color{mygray} M.\ Zingale---Notes on the Euler equations} \fancyfoot[RO,RE]{\footnotesize \sffamily \color{mygray} (\today)} \fancyfoot[CO,CE]{\thepage} \fancyhead{} \renewcommand{\headrulewidth}{0.0pt} \renewcommand{\footrulewidth}{0.0pt} % captions \usepackage{caption} \renewcommand{\captionfont}{\footnotesize} \renewcommand{\captionlabelfont}{\footnotesize} \setlength{\captionmargin}{5em} \newcommand{\evm}{{(-)}} \newcommand{\evz}{{(\circ)}} \newcommand{\evp}{{(+)}} \newcommand{\enu}{{(\nu)}} % for dotted lines in the matrics/arrays \usepackage{arydshln} \usepackage{sectsty} \allsectionsfont{\sffamily} \begin{document} \begin{center} {\LARGE \textsf{\textbf{ Notes on the Euler Equations}} } \end{center} \input Euler \bibliographystyle{plain} \bibliography{../refs} \end{document} ================================================ FILE: GNUmakefile ================================================ # by default, typing make will build the `production' version of the # Computational Hydrodynamics notes. # # if you instead for 'make DEBUG=t', then it will make the draft version. # this will enable margin comments and build in any chapters that are # not yet complete. ALL: CompHydroTutorial.pdf DEBUG := DIRS := preface \ intro \ pde-classes \ finite-volume \ advection \ burgers \ Euler \ hydro-test-problems \ multigrid \ diffusion \ multiphysics \ incompressible \ low_mach \ pyro \ hydro_examples \ software-engineering \ symbols \ radiation # dependencies TEXS := $(foreach dir, $(DIRS), $(wildcard $(dir)/*.tex)) EPSS := $(foreach dir, $(DIRS), $(wildcard $(dir)/*.eps)) # for PDFs, pdflatex will automagically convert file.eps to # file-converted-to.pdf at build time. These lines create # a variable, PDFS, that just contains the original PDF files # as dependencies tPDFS := $(foreach dir, $(DIRS), $(wildcard $(dir)/*.pdf)) cPDFS := $(foreach dir, $(DIRS), $(wildcard $(dir)/*converted-to.pdf)) PDFS := $(filter-out $(cPDFS), $(tPDFS)) conditional.tex: ifdef DEBUG echo "\def\debugmode{}" > conditional.tex else echo "" > conditional.tex endif CompHydroTutorial.pdf: CompHydroTutorial.tex conditional.tex $(TEXS) $(EPSS) $(PDFS) refs.bib git rev-parse --short=12 HEAD > git_info.tex pdflatex CompHydroTutorial < /dev/null bibtex CompHydroTutorial.aux pdflatex CompHydroTutorial < /dev/null pdflatex CompHydroTutorial < /dev/null #pdflatex CompHydroTutorial < /dev/null $(RM) git_info.tex $(RM) conditional.tex clean: $(RM) *.aux *.log *.dvi *.bbl *.blg *.lof *.toc *.exc *.out *.brf $(RM) *~ conditional.tex realclean: clean $(RM) CompHydroTutorial.pdf find . -name "*-converted-to.pdf" -exec $(RM) {} \; .PHONY: clean print-%: ; @echo $* is $($*) ================================================ FILE: LICENSE ================================================ Attribution-NonCommercial-ShareAlike 4.0 International ======================================================================= Creative Commons Corporation ("Creative Commons") is not a law firm and does not provide legal services or legal advice. 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Except for the limited purpose of indicating that material is shared under a Creative Commons public license or as otherwise permitted by the Creative Commons policies published at creativecommons.org/policies, Creative Commons does not authorize the use of the trademark "Creative Commons" or any other trademark or logo of Creative Commons without its prior written consent including, without limitation, in connection with any unauthorized modifications to any of its public licenses or any other arrangements, understandings, or agreements concerning use of licensed material. For the avoidance of doubt, this paragraph does not form part of the public licenses. Creative Commons may be contacted at creativecommons.org. ================================================ FILE: NOTES ================================================ -- add to Ch 1 a review of numerical methods: . truncation vs. roundoff error . numerical differentiation and integration . ODEs . linear algebra -- add section to each chapter: "Numerical Explorations" . create a separate, non-pyro git repo that has basic examples for all of the methods in 1-d also add discussions from pyro. . start with simple grid class and BC discussion . 1-d advection example -- preface: "Each of the main chapters has 2 sections at the end: "Numerical Explorations" and "Going further". The first guides the reader through some exercises using the codes in the git repo associated with these notes. The latter briefly discusses some of the popular extensions done to the core algorithms discussed here, with pointers to the relevant papers in the literature. -- burgers: http://onlinelibrary.wiley.com/doi/10.1002/fld.1650030302/abstract Generating exact solutions of the two-dimensional Burgers' equations Clive A. J. Fletcher ================================================ FILE: README.md ================================================ # Computational Hydrodynamics for Astrophysics *part of the Open Astrophysics Bookshelf* Notes on numerical methods for computational astrophysical hydrodynamics. These notes describe the way I think about the numerical methods commonly used with grid-based codes in astrophysical hydrodynamics. The notes are written in LaTeX, and should build by typing 'make' in the main directory. Working implementations for all of the solvers are contained either in the main pyro code or in the set of hydro examples, both referenced below. ## Chapters The following chapters are mostly written: - Simulation Overview - Finite-Volume Grids - Advection - Burgers' Equation - Euler Equations: Theory - Euler Equations: Numerical Methods - Elliptic Equations and Multigrid - Diffusion - Model Multiphysics Problems - Reactive Flow - Planning a Simulation - Incompressible Flow and Projection Methods - Low Mach Number Methods The following are things I'd like to add in the next 1-2 years: - Fluid Instabilities - Rotation and Self-gravity - Radiation Hydrodynamics - MHD - AMR - Mapped Grids The following are things hopefully will eventually get written: - Relativisitc Flows - Higher-Order Methods - Implicit Hydrodynamics ## Getting PDFs and Source A PDF version of these notes is available at: http://open-astrophysics-bookshelf.github.io/numerical_exercises/ (this PDF is automatically generated by a Github action each time changes are pushed to `main`.) There are two sets of companion codes that go along with these notes: - *hydro_examples*: https://github.com/zingale/hydro_examples simple, standalone, 1-d solvers that illustrate the basic ideas. - *pyro*: https://github.com/python-hydro/pyro2 pyro is a 2-d full simulation code designed with simplicity in mind that implements the core solvers described in these notes along with various test problems. ================================================ FILE: TODO ================================================ Add implementation details throughout, explaining how to translate stuff into code -- maybe in the margins with the Tufte class? https://en.wikipedia.org/wiki/Shallow_water_equations#/media/File:Characteristics_saint-venant.svg new chapters: -- self-gravity -- radiation hydro -- AMR -- mapped grids * general quadralateral grids * moving grids - 1-d moving grid showing -- MHD -- relativisitc -- WENO -- higher-order o method of lines integration with RK3/4 and high-order reconstruction -- implicit hydro -- shallow water -- cosmological flows -- understanding simulations * convergence tests * sanity checks * parameter studies vs. hero calculations -- volume of fluid sample of public codes -- Athena -- Castro -- Enzo -- Flash -- Maestro -- Pluto -- Zeus ======================= Chapter 2: -- in "what is a simulation", add a discussion of DNS, LES, and ILES -- add BC and difference on a grid examples perhaps through a ipython notebook. ======================= Chapter 4: -- add a notebook showing FTCS and upwind on a simple FV grid pyro exercises: -- try out different limiters ======================= Chapter 5: -- exercise: notebook showing burgers' solution ======================= Chapter 6/7: -- add a figure for section 5.2.1, showing piecewise constant reconstruction -- add a discussion (interlude) motivating how limiters are derived -- bulletted list on different Riemann solvers -- derive the jump conditions and the entropy conditions for the Riemann problem and show the function we zero. -- implementation detail: for Riemann solvers, usually the godunov velocity is set to 0 at symmetry boundaries -- it is important to odd-reflect the gravitational acceleration at reflecting boundaries examples: -- 1-d Riemann solver -- 1-d Godunov pyro... ======================= Multigrid chapter: examples: -- 1-d relaxation -- 1-d MG ======================= Diffusion chapter: examples: -- 1-d diffusion solver pyro ---- reference for the choice of epsilon in numerical derivatives as \sqrt{machine epsilon} -- this gives some suggestions: http://mathoverflow.net/questions/28463/optimum-small-number-for-numerical-differentiation ---- tcolorbox for exercises: http://tex.stackexchange.com/questions/172475/how-can-i-define-a-custom-tcolorbox-environment-with-color-as-a-parameter http://www.scriptscoop2.com/t/e6de0140d668/counters-for-chapter-and-section-environments-when-using-tcolorbox-for.html http://tex.stackexchange.com/questions/254401/use-the-exercise-environment-from-legrand-book-template http://www.latextemplates.com/template/the-legrand-orange-book ================================================ FILE: advection/advection-basics.tex ================================================ \label{ch:advection} \section{The linear advection equation} The linear advection equation is simply: \begin{equation} \label{eq:advect} a_t + u a_x = 0 \end{equation} where $a(x,t)$ is some scalar quantity and $u$ is the velocity at which it is advected ($u > 0$ advects to the right). The solution to Eq.~\ref{eq:advect} is to simply take the initial data, $a(x,t=0)$, and displace it to the right at a speed $u$. The shape of the initial data is preserved in the advection. Many hyperbolic systems of PDEs, e.g.\ the equations of hydrodynamics, can be written in a form that looks like a system of (nonlinear) advection equations, so the advection equation provides important insight into the methods used for these systems. % \begin{exercise}[Linear advection analytic solution] {Show via substitution that $a(x - ut)$ is a solution to Eq.~\ref{eq:advect} for any choice of a. This means that the solution is constant along the lines $x = u t$ (the curves along which the solution is constant are called the characteristics).} \end{exercise} Figure~\ref{fig:advection_char} shows an initial profile, $a(x)$, and the corresponding characteristics in the $t$-$x$ plane. With time, since the solution is constant along these characteristics, it simply each point simply follows the characteristic curve, resulting in a shift of the profile to the right. An important concept that we will discuss shortly is {\em stability}. Not every discretization that we write down will be well behaved. For some, our initial state will begin to ``blow-up'', and take on obscenely large and unphysical values after just a few steps. This is the hallmark of a method that is unstable. Some methods have restrictions on the size of the timestep that result in a stable methods as well. % this figure is produced by figures/advection/characteristics.py \begin{figure}[t] \centering \includegraphics[width=0.6\linewidth]{advection-characteristics} \caption[Characteristics for linear advection] {\label{fig:advection_char} (top) Initially sinusoidal distribution (bottom) Characteristic structure for the linear advection equation $u = 1$. Note that for this linear equation the characteristics are all parallel (they have the same slope in the $t$-$x$ plane.} \end{figure} \section{First-order advection in 1-d and finite-differences} To get a flavor of the methods for advection, we will use a simple finite-difference discretization---here, the domain is divided into a sequence of points where we store the solution. We will solve Eq.~\ref{eq:advect} numerically by discretizing the solution at these points. The index $i$ denotes the point's location, and $a_i$ denotes the discrete value of $a(x)$ in zone $i$. The data in each zone can be initialized as $a_i = a(x_i)$. Figure~\ref{fig:fdgrid} shows the grid. We also need to discretize in time. We denote the time-level of the solution with a superscript, so $a_i^n = a(x_i,t^n)$. For a fixed $\Delta t$, time level $n$ corresponds to a time of $t = n\Delta t$. A simple first-order accurate discretization is: \begin{equation} \frac{a_i^{n+1} - a_i^n}{\Delta t} = - u \frac{a_i^n - a_{i-1}^n}{\Delta x} \label{eq:fo} \end{equation} This is an {\em explicit} method, since the new solution, $a_i^{n+1}$, depends only on information at the old time level, $n$. Finally, we also need to specify a boundary condition for this. Our choice of spatial derivative is one-sided---it uses information from the zone to the left of the zone we are updating. This is because information is flowing from left to right in this problem ($u > 0$). This choice of the derivative is called {\em upwinding}---this choice of derivative results in a stable method. Notice that if we use Eq.~\ref{eq:fo} to update the data in the first zone inside the boundary, we need data to the left of this zone---outside of the domain. This means that we need a single {\em ghost point} to implement the boundary conditions for our method. The presence of the ghost points allow us to use the same update equation (e.g.\ Eq.~\ref{eq:fo}) for all zones in the domain. % figure created by figures/advection/fd-ghost.py \begin{figure}[t] \centering \includegraphics[width=\linewidth]{fd_ghost} \caption[A simple finite-difference grid]{\label{fig:fdgrid} A simple finite-difference grid. The solution is stored at each of the labeled points. The dotted lines show the ghost points used to extend our grid past the physical boundaries to accommodate boundary conditions. Note that if we are periodic, then points $0$ and $N-1$ are at the same physical point in space, so we would only need to update one of them.} \end{figure} The last piece of information needed to update the solution is the timestep, $\Delta t$. It can be shown that for the solution to be {\em stable}, the timestep must be less than the time it takes information to propagate across a single zone. That is: \begin{equation} \Delta t \le \frac{\Delta x}{u} \enskip . \end{equation} This is called the {\em Courant-Friedrichs-Lewy} or {\em CFL} condition. A dimensionless quantity called the {\em CFL number} is defined as \begin{equation} \cfl = \frac{\Delta t u }{\Delta x} \end{equation} Stability requires $\cfl \le 1$. % We traditionally write the timestep as \begin{equation} \label{eq:timestep} \Delta t = \cfl \frac{\Delta x}{u} \end{equation} and specify $\cfl$ as part of the problem (a typical value may be $\cfl = 0.7$). \begin{exercise}[Perfect advection with a Courant number of 1] {Show analytically that when you use $\cfl=1$ in the first-order differenced advection equation (Eq.~\ref{eq:fo}) that you advect the profile exactly, without any numerical error.} \end{exercise} Keep in mind that, in general, we will be solving a non-linear system of equations, so it is not possible to run with $\cfl=1$, since $u$ (and therefore $\cfl$) will change from zone to zone. Instead, one looks at the most restrictive timestep over all the zones and uses that for the entire system. \begin{exercise}[A 1-d finite-difference solver for linear advection] {Write a code to solve the 1-d linear advection equation using the discretization of Eq.~\ref{eq:fo} on the domain $[0,1]$ with $u=1$ and periodic boundary conditions. For initial conditions, try both a Gaussian profile and a top-hat:} \begin{equation} a = \left \{ \begin{array}{lllll}0 & \mathit{~if~~} & &x& < 1/3 \\ 1 & \mathit{~if~~} & 1/3 \le &x& < 2/3 \\ 0 & \mathit{~if~~} & 2/3 \le &x& \end{array} \right . \end{equation} \noindent Note: For a general treatment of boundary conditions, you would initialize the ghost points to their corresponding periodic data and apply the difference equations to zones $0, \ldots, N-1$. However, for periodic BCs on this grid, points $0$ and $N-1$ are identical, so you could do the update in this special case on points $1, \ldots, N-1$ without the need for ghost points and then set $a_0 = a_{N-1}$ after the update. \\ \noindent Run you program for one or more periods (one period is $T=1/u$) with several different CFL numbers and notice that there is substantial numerical dissipation (see Figure~\ref{fig:fdadvect}). \end{exercise} \begin{figure}[t] \centering % figure generated by hydro_examples/advection/fdadvect.py \includegraphics[width=0.8\linewidth]{fdadvect-upwind} \caption[First-order finite-difference solution to linear advection] {\label{fig:fdadvect} Finite-difference solution to the first-order finite-difference upwind method for advection, using 65 points and a variety of CFL numbers. \\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/advection/fdadvect.py}{fdadvect.py}}} \end{figure} % This method is first-order accurate. Ultimately we will want higher-order accurate methods. The most obvious change from our initial discretization is to try a higher-order spatial derivative. \begin{exercise}[FTCS and stability] {You may think that using a centered-difference for the spatial derivative, $a_x \sim (a_{i+1} - a_{i-1})/(2 \Delta x)$ would be more accurate. This method is called FTCS (forward-time, centered-space). Try this on the same test problems.} \end{exercise} \begin{figure}[t] \centering % figure generated by hydro_examples/advection/fdadvect.py \includegraphics[width=0.48\linewidth]{fdadvect-FTCS-C0_1} \includegraphics[width=0.48\linewidth]{fdadvect-FTCS-C0_5} \caption[FTCS finite-difference solution to linear advection] {\label{fig:fdadvect-ftcs} Finite-difference solution using the FTCS finite-difference method for advection using 65 points, modeling for only 1/10$^\mathrm{th}$ of a period. The panel of the left is with $\cfl = 0.1$ and the panel on the right is $\cfl = 0.5$.\\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/advection/fdadvect.py}{fdadvect.py}}} \end{figure} You will find that no matter what value of $\cfl$ you choose with the FTCS method, the solution is unconditionally {\em unstable} (see Figure~\ref{fig:fdadvect-ftcs}). If you continue to evolve the equation with this method, you would find that the amplitude grows without bound. There is something about that discretization that simply gets the physics wrong. \section{Stability} \MarginPar{add a discussion of domain of dependence} The classic method for understanding stability is to consider the growth of a single Fourier mode in our discretization. That is, substitute in $a_i^n = A^n e^{Ii\theta}$, 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 phase. A method is stable if $|A^{n+1}/A^n| \le 1$. FTCS appears as: \begin{equation} a_i^{n +1} = a_i^n - \frac{\cfl}{2} (a_{i+1}^n - a_{i-1}^n) \end{equation} Examining a Fourier mode shows that: \begin{align} A^{n+1}e^{Ii\theta} &= A^n e^{Ii\theta} - \frac{\cfl}{2} \left ( A^n e^{I(i+1)\theta} - A^n e^{I(i-1)\theta} \right ) \\ A^{n+1} &= A^n - \frac{\cfl}{2} A^n \left ( e^{I\theta} - e^{-I\theta}\right ) \\ A^{n+1} &= A^n \left ( 1 - I \cfl \sin\theta \right ) \end{align} so the magnitude of the amplification is \begin{equation} \left | \frac{A^{n+1}}{A^n}\right |^2 = 1 + \cfl^2\sin^2\theta \end{equation} We see that there is no value of $\cfl$ that can make the method stable ($|A^{n+1}/A^n| > 1$ always). Doing the same analysis for Eq.~\ref{eq:fo} would show that the upwind method is stable for $0\le \cfl \le 1$. \begin{exercise}[Stability of the upwind method] {Using the above stability analysis, considering the amplitude of a single Fourier mode, show that the growth of a mode for the upwind method (Eq.~\ref{eq:fo}) is: \begin{equation} \left | \frac{A^{n+1}}{A^n} \right |^2 = 1 - 2\cfl(1-\cfl)(1-\cos\theta) \end{equation} and stability requires $2\cfl(1-\cfl) \ge 0$ or $0 \le \cfl \le 1$. } \end{exercise} It is important to note that this stability analysis only works for linear equations, where the different Fourier modes are decoupled, nevertheless, we use its ideas for nonlinear advection problems as well. Truncation analysis can also help us understand stability. The idea here is to keep the higher order terms in the Taylor series to understand how they modify the actual equation you are trying to solve. \begin{exercise}[Stability analysis] {To get an alternate feel for stability, we can ask what the terms left out by truncation look like. That is, we can begin with the discretized equation: \begin{equation} a_i^{n+1} - a_i^n = -\frac{u \Delta t}{\Delta x} ( a_i^n - a_{i-1}^n ) \end{equation} and replace $a_i^{n+1}$ with a Taylor expansion in time, and replace $a_{i-1}^n$ with a Taylor expansion in space, keeping terms to $O(\Delta t^3)$ and $O(\Delta x^3)$. Replacing $\partial a/\partial t$ with $-u \partial a/ \partial x$ in the higher-order terms, show that our difference equation more closely corresponds to \begin{eqnarray} \label{eq:advect_trunc_analysis} a_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} \\ &=& \frac{u \Delta x}{2} (1 - \cfl) \frac{\partial^2 a}{\partial x^2} \end{eqnarray} } \end{exercise} Notice that the righthand side of Eq.~\ref{eq:advect_trunc_analysis} looks like a diffusion term, however, if $\cfl > 1$, then the coefficient of the diffusion is negative---this is unphysical. This means that the diffusion would act to take smooth features and make them more strongly peaked---the opposite of physical diffusion. For FTCS, a similar truncation analysis would show that the diffusion term is always negative. \subsection{Domain of dependence} Another important view of our numerical difference scheme is to look at the {\em domain of dependence}. Figure~\ref{fig:adv:domains} illustrates this for the updated point $a_i^{n+1}$ (shown at the top of center of the space-time diagram). The numerical domain of dependence shows the points that can influence the updated value of $a_i$ using our difference method. For the upwind scheme, we see that this is a triangle that includes $a_{i-1}^n$ and $a_i^n$. The physical domain of dependence is shown as the orange triangle---this is formed by tracing backwards in time from $a_i^{n+1}$ along a characteristic, reaching out to the point $x_i - C\Delta x = x_i - u\Delta t$ over $\Delta t$. Any stable numerical method must have a numerical domain of dependence that includes the physical domain of dependence. If it does not, then the update to the solution simply does not see the points that contribute to the solution over the timestep $\Delta x$. Notice that this is a necessary, but not sufficient condition for stability. FTCS has a domain of dependence that includes the physical domain of dependence, but it is not stable. \begin{figure}[t] \centering \includegraphics[width=0.7\linewidth]{domains_upwind}\\ \includegraphics[width=0.7\linewidth]{domains_downwind}\\ \includegraphics[width=0.7\linewidth]{domains_FTCS} \caption[Domain of dependence space-time diagram]{\label{fig:adv:domains} Space-time diagrams showing the numerical domain of dependence (blue region) for three different difference methods. Also show is the physical domain of dependence---that formed by tracing backwards from $a_i^{n+1}$ along a characteristic.} \end{figure} \section{Implicit-in-time} An alternate approach to time-discretization is to do an implicit discretization. Here our upwind method would appear as: \begin{equation} \frac{a^{n+1}_i - a^n_i}{\Delta t} = -u \frac{a^{n+1}_i - a^{n+1}_{i-1}}{\Delta x} \end{equation} The only change here is that the righthand side is evaluated at the new timelevel, $n+1$. We can write this as a linear system with coupled equations: \begin{equation} -\cfl a^{n+1}_{i-1} + (1 + \cfl) a^{n+1}_i = a_i^n \end{equation} If we use periodic boundary conditions, then point $0$ and $N-1$ are identical, 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: \begin{equation} \renewcommand{\arraystretch}{1.5} \left ( \begin{array}{ccccccc} 1+\cfl & & & & & & -\cfl \\ -\cfl & 1+\cfl & \\ & -\cfl & 1+\cfl & \\ & & -\cfl & 1+\cfl & \\ &&&\ddots&\ddots &\\ &&&&-\cfl & 1+\cfl & \\ &&&&& -\cfl &1+\cfl \end{array} \right ) % \left ( \begin{array}{c} a_1^{n+1} \\ a_2^{n+1} \\ a_3^{n+1} \\ a_4^{n+1} \\ \vdots \\ a_{N-2}^{n+1} \\ a_{N-1}^{n+1} \end{array} \right ) = \left ( \begin{array}{c} a_1^{n} \\ a_2^{n} \\ a_3^{n} \\ a_4^{n} \\ \vdots \\ a_{N-2}^{n} \\ a_{N-1}^{n} \end{array} \right ) \end{equation} This requires a matrix solve---this makes implicit methods generally more expensive than explicit methods. However, stability analysis would show that this implicit discretization is stable for any choice of $\cfl$. (But one must not confuse stability with accuracy---the most accurate solutions with this method will still have a small $\cfl$). Also note that the form of the matrix will change depending on the choice of boundary conditions. Figure~\ref{fig:fdadvect-implicit} shows the result of solving this implicit system. \begin{figure}[t] \centering % figure generated by hydro_examples/advection/fdadvect_implicit.py \includegraphics[width=0.8\linewidth]{fdadvect-implicit} \caption[First-order implicit finite-difference solution to linear advection] {\label{fig:fdadvect-implicit} Finite-difference solution to the implicit first-order finite-difference upwind method for advection, using 65 points and a variety of CFL numbers. \\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/advection/fdadvect_implicit.py}{fdadvect\_implicit.py}}} \end{figure} \begin{exercise}[Implicit advection] {Code up the implicit advection scheme, but using outflow instead of periodic boundary conditions. This will change the form of the matrix. You can use the code from Figure~\ref{fig:fdadvect-implicit} as a starting point.} \end{exercise} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Eulerian vs.\ Lagrangian frames} It is useful to think about how our advected quantity, $a(x,t)$, changes in time. The full time derivative is: \begin{equation} \frac{d a(x,t)}{dt} = \frac{\partial a}{\partial t} + \frac{\partial a}{\partial x} \frac{dx}{dt} \end{equation} So the value of this derivative depends on the path, $x(t)$, that we choose to follow. Consider an observer who is stationary. They will watch the flow move past them, so $dx/dt = 0$, and $da/dt = \partial a/\partial t$. This fixed frame is called {\em Eulerian frame}. Instead imagine an observer who moves with the flow, at the velocity $u$. This way they keep pace with an individual feature in the flow and track the changes it experiences. In this case, $dx/dt = u$, and our derivative, commonly written as $D/Dt$ is: \begin{equation} \frac{D}{Dt} = \frac{\partial}{\partial t} + u \frac{\partial}{\partial x} \end{equation} This is the {\em Lagrangian frame}, and the derivative, $D/Dt$ is called the {\em Lagrangian derivative}, {\em material derivative}, {\em convective derivative}, or {\em advective derivative}\footnote{and there are actually many more names...}. Our linear advection equation can be written simply as $Da/Dt = 0$. We've been solving the equations in the Eulerian frame---our grid is fixed and the fluid moves through it. For hydrodynamics, it will be useful conceptually to consider the Lagrangian frame to understand how the fluid properties change in a particular fluid element over time. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Errors and convergence rate} For the advection problem (with $u>0$), the analytic solution is to simply propagate the initial profile to the right. This means that with periodic boundary conditions, after advecting for one period, our numerical solution should be identical to the initial conditions. Any differences are our numerical error. We can quantify the error by taking the norm of error\footnote{see \S~\ref{intro:sec:norm} for the definition of the norms} as: \begin{equation} \epsilon^\mathrm{abs} = \| a^\mathrm{final} - a^\mathrm{init} \|_2 \equiv \left [ \frac{1}{N} \sum_{i=1}^N ( a_i^\mathrm{final} - a_i^\mathrm{init} )^2 \right ]^{\myhalf} \end{equation} It is sometimes useful to compare to the norm of the original solution to get a measure of the relative error: \begin{equation} \epsilon^\mathrm{rel} \equiv \frac{\| a^\mathrm{final} - a^\mathrm{init} \|_2} {\| a^\mathrm{init} \|_2} \end{equation} Note that for the absolute norm, it is important in these definitions to normalize by the number of zones, $N$, otherwise our error will be resolution-dependent. For the relative norm, since we scale by a norm on the same grid, this normalization will cancel. ================================================ FILE: advection/advection-higherorder.tex ================================================ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Advection and the finite-volume method} \label{ch:adv:sndorder} In these notes, we will typically use a {\em finite-volume} discretization. Here we explore this method for the advection equation. First we rewrite the advection equation in {\em conservation form}: \begin{equation} a_t + \left[f(a)\right]_x = 0 \label{eq:advect-cons} \end{equation} where $f(a) = ua$ is the flux of the quantity $a$. In conservation form, the time derivative of a quantity is related to the divergence of its flux. % figure created with figures/advection/fv-ghost.py \begin{figure}[t] \centering \includegraphics[width=\linewidth]{fv_ghost} \caption[A finite-volume grid with valid cells labeled]{\label{fig:fvghost} A finite-volume grid running from $\mathrm{lo}, \ldots, \mathrm{hi}$, with two ghost cells at each end.} \end{figure} Recall that in the finite-volume discretization, $\langle a\rangle_i$ represents the average of $a(x,t)$ over the interval $x_{i-\myhalf}$ to $x_{i+\myhalf}$, where the half-integer indexes denote the zone edges (i.e.\ $x_{i-\myhalf} = x_i - \Delta x/2$). Figure~\ref{fig:fvghost} shows an example of such a grid with 2 ghost cells at each end. (For simplicity of notation, we drop the $\langle \rangle$ going forward). To discretize Eq.~\ref{eq:advect-cons}, we integrate it over a zone, from ${x_{i-\myhalf}}$ to ${x_{i+\myhalf}}$, normalizing by the zone width, $\Delta x$: \begin{align} \frac{1}{\Delta x} \int_{x_{i-\myhalf}}^{x_{i+\myhalf}} a_t \, dx &= - \frac{1}{\Delta x} \int_{x_{i-\myhalf}}^{x_{i+\myhalf}} \frac{\partial}{\partial x} f(a) \, dx \\ \frac{\partial}{\partial t} a_i &= - \frac{1}{\Delta x} \left \{ \left [f(a) \right ]_{i+\myhalf} - \left [f(a) \right ]_{i-\myhalf} \right \} \label{adv:eq:fvadv} \end{align} This is an evolution equation for the zone-average of $a$, and shows that it updates in time based on the fluxes through the boundary of the zone. We now have a choice on how to proceed with the time-discretization: \begin{enumerate} \item We can discretize ${\partial a_i}/{\partial t}$ directly as $(a_i^{n+1} - a_i^n)/\Delta t$. Then to achieve second-order accuracy, we need to evaluate the righthand side of Eq.~\ref{adv:eq:fvadv} at the midpoint in time ($n+1/2$). This gives rise to a predictor-corrector method (as described in \cite{colella:1990}. Sometimes this is called a characteristic tracing method (a name which will be more clear when we discuss the Euler equations). \item We can recognize that with the spatial discretization done, our PDE has now become an ODE, and we can use standard ODE methods (like Runge-Kutta) to integrate the system in time. This is a method-of-lines integration. \end{enumerate} We'll look at both of these in the next sections. \section{Second-order predictor-corrector scheme} We discretize Eq.~\ref{adv:eq:fvadv} in time by evaluating the righthand side at the midpoint in time---this gives a centered-difference in time, which is second-order accurate: \begin{equation} \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} \label{eq:consupdate1d} \end{equation} To evaluate the fluxes at the half-time, we need the state at the half-time, that is, we do : \begin{equation} \label{adv:eq:fluxeval} \left [f(a) \right ]_{i+\myhalf}^{n+\myhalf} = f(a_{i+\myhalf}^{n+\myhalf}) \enskip . \end{equation} We construct a second-order accurate approximation to $a_{i+\myhalf}^{n+\myhalf}$ by Taylor expanding the data in the cell to the interface. The construction of the interface state at the midpoint in time is the prediction and the conservative update in the correction here. Notice that for each interface, there are two possible interface states we can construct---one using the data to the left of the interface (which we will denote with a ``L'' subscript) and the other using the data to the right of the interface (denoted with an ``R'' subscript)---see Figure~\ref{fig:riemann_adv}. These states are: \begin{eqnarray} a_{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 \\ &=& 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 \\ &=& 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}\\ %% a_{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 \\ &=& 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 \\ &=& 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} \end{eqnarray} % figure created with figures/advection/riemann.py \begin{figure}[t] \centering \includegraphics[width=5.0in]{riemann-adv} \caption[The input state to the Riemann problem]{\label{fig:riemann_adv} The left and right interface state at the $i+\myhalf$ interface. Here, the left state, $a_{i+\myhalf,L}^{n+\myhalf}$, was predicted to the interface from the zone to the left of the interface, using $a_i$, and the right state, $a_{i+\myhalf,R}^{n+\myhalf}$, was predicted to the interface from the zone to the right, using $a_{i+1}$.} \end{figure} A suitable estimate is needed for the slope of $a$ that appears in these expressions (as $\partial a/\partial x$). We can approximate this simply as \begin{equation} \left . \frac{\partial a}{\partial x}\right |_i = \frac{a_{i+1} - a_{i-1}}{2 \Delta x} \label{eq:slopecentered} \end{equation} We can think of this method as reconstructing the function form of the data from the cell-average data in each cell using a piecewise linear polynomial. Don't be worried that this looks like FTCS---we'll do upwinding next. We now have two states, $a_{i+\myhalf,L}^{n+\myhalf}$ and $a_{i+\myhalf,R}^{n+\myhalf}$ separated by an interface---this is called the {\em Riemann problem}. % The solution to this will depend on the equation being solved, and results in a single state at the interface: \begin{equation} a_{i+\myhalf}^{n+\myhalf} = \mathcal{R}(a_{i+\myhalf,L}^{n+\myhalf},a_{i+\myhalf,R}^{n+\myhalf}) \end{equation} In our case, the advection equation simply propagates the state to the right (for $u > 0$), so the solution to the Riemann problem is to take the left state (this is another example of upwinding). That is we do: \begin{equation} \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 . \label{eq:riemannsolve} \end{equation} To complete the update, we use this interface state to evaluate the flux and update the advected quantity via Eq.~\ref{eq:consupdate1d} and \ref{adv:eq:fluxeval}. Boundary conditions are implemented by filling the ghost cells outside each end of the domain based on data in the interior. Note that at the very left edge of the domain, the state $a^{n+\myhalf}_{\mathrm{lo}-\myhalf}$ requires the construction of states on the left and right. The left state at that interface, $a^{n+\myhalf}_{\mathrm{lo}-\myhalf,L}$ depends on the slope reconstructed in the $\mathrm{lo}-1$ ghost cell, $\partial a/\partial x |_{\mathrm{lo}-1}$. This in turn is constructed using a limited center-difference that will consider the value in the cell to the left, $\mathrm{lo-2}$. Therefore, we need two ghost cells at each end of the domain for this method---figure~\ref{fig:advect_ghost} illustrates this. Higher-order limiters may require even more ghost cells. % figure from figures/advection/riemann_bc.py \begin{figure}[t] \centering \includegraphics[width=\linewidth]{riemann-bc} \caption[Reconstruction at the domain boundary] {\label{fig:advect_ghost} Reconstruction near the boundary, showing the need for two ghostcells. Here we see the left and right state at the left physical boundary of the domain (marked as $\mathrm{lo}-\myhalf$). The gray dotted lines are the piecewise constant cell averages and the red lines are the reconstructed slopes. Note that we need the slope in $\mathrm{lo}-1$ to get the left interface state at $\mathrm{lo}-\myhalf$, and that slope in turn needed the data in zone $\mathrm{lo}-2$ to construct a centered-difference.} \end{figure} \begin{exercise}[A second-order finite-volume solver for linear advection] {\label{adv:ex:fv} Write a second-order solver for the linear advection equation. To mimic a real hydrodynamics code, your code should have routines for finding initializing the state, filling boundary conditions, computing the timestep, constructing the interface states, solving the Riemann problem, and doing the update. The problem flow should look like: \begin{itemize} \item set initial conditions \item main evolution loop---loop until final time reached \begin{itemize} \item fill boundary conditions \item get timestep (Eq.~\ref{eq:timestep}) \item compute interface states (Eqs.~\ref{eq:statel} and \ref{eq:stater}) \item solve Riemann problem at all interfaces (Eq.~\ref{eq:riemannsolve}) \item do conservative update (Eqs.~\ref{eq:consupdate1d} and \ref{adv:eq:fluxeval}) \end{itemize} \end{itemize} Use both the top-hat and Gaussian initial conditions and periodic boundary conditions and compare to the first-order method. See Figure~\ref{fig:fvadvect}. } \end{exercise} \begin{figure} \centering % figure from codes/advection/fv_advection.py \includegraphics[width=0.8\linewidth]{fv-advect} \caption[Second-order finite-volume advection] {\label{fig:fvadvect} Second-order finite volume advection showing the result of advecting a tophat profile through five periods with both unlimited and limited slopes. This calculation used 64 zones and $\cfl=0.7$. \\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/advection/advection.py}{advection.py}}} \end{figure} \subsection{Limiting} The second-order method likely showed some oscillations in the solution, especially for the top-hat initial conditions. {\em Godunov's theorem} says that any monotonic linear method for advection is first-order accurate (see, e.g.,~\cite{laney}). In this context, monotonic means that no new minima or maxima are introduced. The converse is true too, which suggests that in order to have a second-order accurate method for this linear equation, the algorithm itself must be nonlinear. \begin{exercise}[Limiting and overshoots] {To remove the oscillations in practice, we limit the slopes to ensure that no new minima or maxima are introduced during the advection process. There are many choices for limited slopes. A popular one is the {\em minmod} limiter. Here, we construct the slopes in the interface states as: \begin{equation} \left . \frac{\partial a}{\partial x} \right |_i = \mathtt{minmod} \left ( \frac{a_i - a_{i-1}}{\Delta x}, \frac{a_{i+1} - a_i}{\Delta x} \right ) \end{equation} instead of Eq.~\ref{eq:slopecentered}. with \begin{equation} \mathtt{minmod}(a,b) = \left \{ \begin{array}{ll} a & \mathit{if~} |a| < |b| \mathrm{~and~} a\cdot b > 0 \\ b & \mathit{if~} |b| < |a| \mathrm{~and~} a\cdot b > 0 \\ 0 & \mathit{otherwise} \end{array} \right . \end{equation} Use this slope in your second-order advection code and notice that the oscillations go away---see Figure~\ref{fig:fvadvect}.} \end{exercise} We can get a feel for what happens with and without limiting pictorially. \MarginPar{add a discussion on how a limiter is designed/constructed} Figures~\ref{fig:limitingex} and \ref{fig:limitingexb} show the evolution of an initial discontinuity with and without limiting. Without limiting, we see an overshoot behind the discontinuity and an undershoot ahead of it---these develop after only a single step. With each additional step, the overshoots and undershoots move further away from the discontinuity. In contrast, the case with limiting shows no over- or undershoots around the initial discontinuity. By the end of the evolution, we see that the profile is much narrower in the limiting case than in the case without limiting. See the text by LeVeque~\cite{leveque:2002} for alternate choices of limiters. Note: most limiters will have some sort of test on the product of a left-sided and right-sided difference ($a\cdot b$ above)---this is $< 0$ at an extremum, which is precisely where we want to limit. % these two sets of figures can be created with % figures/advection/rea-limitex.py \begin{figure}[h] \centering \includegraphics[width=0.75\linewidth]{rea-nolimit-start_001} \\ \includegraphics[width=0.75\linewidth]{rea-nolimit-start_002} \\ \includegraphics[width=0.75\linewidth]{rea-nolimit-start_003} \\ \includegraphics[width=0.75\linewidth]{rea-nolimit-start_004} \\ \includegraphics[width=0.75\linewidth]{rea-nolimit-start_005} \\ \includegraphics[width=0.75\linewidth]{rea-nolimit-start_006} %\includegraphics[width=0.55\linewidth]{rea-nolimit-start_007} \\ %\includegraphics[width=0.55\linewidth]{rea-nolimit-start_008} \\ \caption[The effect of no limiting on initially discontinuous data]{\label{fig:limitingex}Initially discontinuous data evolved for several steps with no limiting. Notice that there are overshoots and undershoots surrounding the discontinuity.} \end{figure} \begin{figure}[h] \centering \includegraphics[width=0.75\linewidth]{rea-start_001} \\ \includegraphics[width=0.75\linewidth]{rea-start_002} \\ \includegraphics[width=0.75\linewidth]{rea-start_003} \\ \includegraphics[width=0.75\linewidth]{rea-start_004} \\ \includegraphics[width=0.75\linewidth]{rea-start_005} \\ \includegraphics[width=0.75\linewidth]{rea-start_006} %\includegraphics[width=0.55\linewidth]{rea-start_007} %\includegraphics[width=0.55\linewidth]{rea-start_008} \\ \caption[The effect of limiters on initially discontinuous data]{\label{fig:limitingexb}Initially discontinuous data evolved for several steps with limiting. Note that unlike the sequence without limiting (Figure~\ref{fig:limitingex}), the discontinuity remains sharper with limiting and there are no over- or undershoots.} \end{figure} A slightly more complex limiter is the MC limiter (monotonized central difference). First we define an extrema test, \begin{equation} \xi = (a_{i+1} - a_i) \cdot (a_i - a_{i-1}) \end{equation} Then the limited difference is \begin{equation} \left . \frac{\partial a}{\partial x} \right |_i = \left \{ \begin{array}{ll} \min \left [ \frac{| a_{i+1} - a_{i-1} |}{2 \Delta x}, 2 \frac{| a_{i+1} - a_i |}{\Delta x}, 2 \frac{| a_{i} - a_{i-1} |}{\Delta x} \right ] \mathrm{sign}(a_{i+1} - a_{i-1}) & \xi > 0 \\ 0 & \mathit{otherwise} \end{array} \right . \end{equation} % Note that a slightly different form of this limiter is presented in \cite{leveque:2002}, where all quantities are in a {\tt minmod}, which appears to limit a bit less. This is second-order accurate for smooth flows. The main goal of a limiter is to reduce the slope near extrema. Figure~\ref{fig:limit} shows a finite-volume grid with the original data, cell-centered slopes, and MC limited slopes. Note that near the strong gradients is where the limiting kicks in. The different limiters are all constructed by enforcing a condition requiring the method to be {\em total variation diminishing}, or TVD. More details on TVD limiters can be found in \cite{toro:1997,leveque:2002}. % this figure can be create with figures/advection/generalgrid.py \begin{figure} \centering \includegraphics[width=\linewidth]{generalgrid} \caption[Piecewise linear slopes with an without limiting]{\label{fig:limit} A finite-volume grid showing the cell averages (gray, dotted, horizontal lines), unlimited center-difference slopes (gray, solid) and MC limited slopes (red). Note that in zones $i$ and $i+1$, the slopes are limited slightly, so as not to overshoot or undershoot the neighboring cell value. Cell $i-1$ is not limited at all, whereas cells $i-2$, and $i+2$ are fully limited---the slope is set to $0$---these are extrema.} \end{figure} A popular extension of the MC limiter is the $4^\mathrm{th}$-order MC limiter, which is more accurate in smooth flows (this is shown in \cite{colella:1985}, Eqs. 2.5 and 2.6; and \cite{colella:1990}, Eq. 191). \begin{exercise}[Limiting and reduction in order-of-accuracy] {Show analytically that if you fully limit the slopes (i.e.\ set $\partial a/\partial x |_i = 0$),then the second-order method reduces to precisely our first-order finite-difference discretization, Eq.~\ref{eq:fo}. } \end{exercise} It is common to express the slope simply as the change in the state variable: \begin{equation} \Delta a_i = \left . \frac{\partial a}{\partial x} \right |_i \Delta x \end{equation} and to indicate the limited slope as $\overline{\Delta a}_i$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Reconstruct-evolve-average} Another way to think about these methods is as a reconstruction, evolve, and average (R-E-A) process (see Figure~\ref{fig:rea}). We can write the conservative update as: \begin{align} a_i^{n+1} &= a_i^n + \frac{\Delta t}{\Delta x} (u a^{n+\myhalf}_{i-\myhalf} - u a^{n+\myhalf}_{i+\myhalf} ) \\ &= a_i^n + \cfl (a^{n+\myhalf}_{i-\myhalf} - a^{n+\myhalf}_{i+\myhalf} ) \end{align} If we take $u > 0$, then the Riemann problem will always choose the left state, so we can write this as: \begin{equation} a_i^{n+1} = a_i^n + \cfl \biggl [\underbrace{\left (a_{i-1}^n + \frac{1}{2} (1-\cfl) \Delta a_{i-1} \right )}_{a_{i-\myhalf,L}} - \underbrace{\left (a_{i}^n + \frac{1}{2} (1-\cfl) \Delta a_{i} \right )}_{a_{i+\myhalf,L}} \biggr ] \label{eq:rea_orig} \end{equation} If we instead look at this via the R-E-A procedure, we write the reconstructed $a$ in each zone in the form of a piecewise linear polynomial \begin{equation} a_i(x) = a_i + \frac{\Delta a_i}{\Delta x} (x - x_i) \end{equation} Consider zone $i$. If we are advecting with a CFL number $\cfl$, then that means that the fraction $\cfl$ of the zone immediately to the left of the $i-\myhalf$ interface will advect into zone $i$ over the timestep. And only the fraction $1-\cfl$ in zone $i$ immediately to the right of the interface will stay in that zone. This is indicated by the shaded regions in Figure~\ref{fig:rea}. The average of the quantity $a$ from zone $i-1$ that will advect into zone $i$ is \begin{eqnarray} \mathcal{I}_< &=& \frac{1}{\cfl \Delta x} \int_{x_{i-\myhalf} - \cfl\Delta x}^{x_{i-\myhalf}} a_{i-1}(x) dx \\ % &=& \frac{1}{\cfl \Delta x} \int_{x_{i-\myhalf} - \cfl\Delta x}^{x_{i-\myhalf}} \left [ a_{i-1} + \frac{\Delta a_{i-1}}{\Delta x} (x - x_{i-1} ) \right ] dx \\ &=& a_{i-1} + \frac{1}{2} \Delta a_{i-1} (1-\cfl) \end{eqnarray} And the average of the quantity $a$ in zone $i$ that will remain in zone $i$ is \begin{eqnarray} \mathcal{I}_> &=& \frac{1}{(1-\cfl) \Delta x} \int_{x_{i-\myhalf}}^{x_{i-\myhalf} + (1-\cfl) \Delta x} a_{i}(x) dx \\ % &=& \frac{1}{(1-\cfl) \Delta x} \int_{x_{i-\myhalf}}^{x_{i-\myhalf} + (1-\cfl)\Delta x} \left [ a_{i} + \frac{\Delta a_{i}}{\Delta x} (x - x_{i} ) \right ] dx \\ &=& a_{i} - \frac{1}{2} \Delta a_{i} \cfl \end{eqnarray} The final part of the R-E-A procedure is to average over the advected profiles in the new cell. The weighted average of the amount brought in from the left of the interface and that remains in the cell is \begin{align} a_i^{n+1} &= \cfl \mathcal{I}_< + (1 - \cfl) \mathcal{I}_> \\ &= \cfl \left [ a^n_{i-1} + \frac{1}{2} \Delta a_{i-1} (1 - \cfl) \right ] + (1-\cfl) \left [ a^n_i - \frac{1}{2} \Delta a_i \cfl \right ] \\ &= a^n_i + \cfl \left [a^n_{i-1} + \frac{1}{2}(1 - \cfl) \Delta a_{i-1} \right ] - \cfl \left [ a^n_i + \frac{1}{2} (1-\cfl) \Delta a_i \right ] \end{align} This is identical to Eq.~\ref{eq:rea_orig}. This demonstrates that the R-E-A procedure is equivalent to our reconstruction, prediction of the interface states, solving the Riemann problem, and doing the conservative flux update. % this figure sequence is created by figures/advection/rea.py \begin{figure} \centering \includegraphics[width=\linewidth]{rea-start} \\ \includegraphics[width=\linewidth]{rea-reconstruction} \\ \includegraphics[width=\linewidth]{rea-trace} \\ \includegraphics[width=\linewidth]{rea-evolve} \\ \includegraphics[width=\linewidth]{rea-final} \caption[The Reconstruct-Evolve-Average procedure]{\label{fig:rea} Reconstruct-Evolve-Average. The top panel shows the original cell-average data. The second panel shows the (limited) piecewise linear reconstruction of the data. Assuming a CFL number of 0.6 and advection to the right, the shaded regions in the third panel show the data that will wind up in cell $i$ after advecting for a single step. The fourth panel shows the piecewise-linear data advected to the right by 0.6 of a cell-width (corresponding to a CFL of 0.6). The final panel shows the new averages of the data, constructed by averaging the advected piecewise linear data in each cell.} \end{figure} \begin{exercise}[Convergence testing] { Run the first-order solver for several different values of $\Delta x$, each a factor of 2 smaller than the previous. Compute $\epsilon$ for each resolution and observe that it converges in a first-order fashion (i.e.\ $\epsilon$ decreases by 2 when we decrease $\Delta x$ by a factor of 2). \noindent Do the same with the second-order solver and observe that it converges as second-order. However: you may find less than second-order if your initial conditions have discontinuities and you are limiting. Figure~\ref{fig:advnorm} shows the convergence of the method with no limiting, MC, and minmod limiting, $\cfl = 0.8$, and a Gaussian initial condition for 5 periods.} \end{exercise} % figure from hydro_examples: advection/advection.py \begin{figure}[t] \centering \includegraphics[width=0.8\linewidth]{plm-converge} \caption[Convergence of second-order finite-volume advection] {\label{fig:advnorm} Convergence for the second-order finite-volume method with no limiting, MC, and minmod limiting advecting a Gaussian initial profile with $\cfl = 0.8$ for 5 periods. \\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/advection/advection.py}{advection.py}}} \end{figure} \begin{figure}[ht] \centering \includegraphics[width=0.8\linewidth]{fv-gaussian-limiters} \\[1em] \includegraphics[width=0.8\linewidth]{fv-tophat-limiters} \caption[Effect of different limiters on evolution] {\label{fig:limiter_panel} The effect of different limiters on the evolution of a Gaussian initial profile (top) and a tophat initial profile (bottom), using $\cfl = 0.8$ and 5 periods. \\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/advection/advection.py}{advection.py}}} \end{figure} As seen in figure~\ref{fig:advnorm}, the choice of limiters can have a great effect on the accuracy. Figure~\ref{fig:limiter_panel} shows the Gaussian and tophat profiles with several different limiters. \section{Method of lines approach} \label{adv:sec:mol_2d} In the above constructions, we computed a time-centered flux by predicting the interface state to the half-time (by Taylor-expanding in time through $\Delta t /2$). Instead, we can recognize that with the spatial discretization done in Eq.~\ref{adv:eq:fvadv}, we are left with an ordinary differential equation in time that can then be solved using standard ODE techniques. Substituting in the flux as $f(a) = ua$, we have the ODE: \begin{equation} \ddt{a_i} = - \frac{u a_{i+1/2} - u a_{i-1/2}}{\Delta x} \end{equation} Note that there is no time-level indicated on the righthand side. We find the interface values of $a$ by interpolating in space only. For a second-order accurate method, we can use a piecewise linear reconstruction (see Figure~\ref{fig:advection:riemann-mol}), and evaluate the interface states as simply the point on the reconstructed line on the interface, e.g., \begin{align} a_{i+\myhalf,L} &= a_{i} + \frac{1}{2} \Delta a_{i} \\ a_{i+\myhalf,R} &= a_{i+1} - \frac{1}{2} \Delta a_{i+1} \end{align} % \begin{figure}[t] \centering \includegraphics[width=\linewidth]{riemann-mol} \caption[Method-of-lines spatial reconstruction]{\label{fig:advection:riemann-mol} Piecewise linear reconstruction of the zones showing the interface states used with the method of lines approach.} \end{figure} % The same limiting ideas discussed above apply here. As usual, we resolve the left-right degeneracy on the interface by solving the Riemann problem: \begin{equation} a_{i+\myhalf,j} = \mathcal{R}(a_{i+\myhalf,j,L}, a_{i+\myhalf,j,R}) \end{equation} Once we have this interface state, we can integrate the ODE. A Runge-Kutta method works fine here (it should be atleast second-order to match the spatial accuracy). \section{Multi-dimensional advection} The two-dimensional linear advection equation is: \begin{equation} a_t + u a_x + v a_y = 0 \label{eq:advect2d} \end{equation} where $u$ is the velocity in the $x$-direction and $v$ is the velocity in the $y$-direction. We denote the average of $a(x,y,t)$ in a zone $i,j$ as $a_{i,j}$. Here, $i$ is the index in the $x$-direction and $j$ is the index in the $y$-direction. A 2-d grid is shown in Figure~\ref{fig:2dgrid}. Just as in the one-dimensional case, we will extend the domain with a perimeter of ghost cells to set the boundary conditions. % this is create by figures/advection/2dgrid.py \begin{figure}[h] \centering \includegraphics[width=4.0in]{2dgrid} \caption[A 2-d grid with zone-centered indexes]{\label{fig:2dgrid} A simple 2-d grid with the zone-centered indexes. The $\times$s mark the left and right interface states at the upper edge of the $i,j$ zone in each coordinate direction. For a finite-volume discretization, $a_{i,j}$ represents the average of $a(x,y)$ over the zone.} \end{figure} To derive the finite-volume form of the update, we start by writing this in conservative form. Since $u$ and $v$ are constant, we can move them inside the divergences: \begin{equation} a_t + (u a)_x + (v a)_y = 0 \label{eq:advect2d-cons} \end{equation} This is the form we will integrate over zones. As before, we will define the average of $a$ in a zone by integrating it over the volume: \begin{equation} a_{i,j} = \frac{1}{\Delta x \Delta y} \int_{x_{i-\myhalf}}^{x_{i+\myhalf}} \int_{y_{j-\myhalf}}^{y_{j+\myhalf}} a(x,y,t) \, dx \, dy \end{equation} Integrating Eq.~\ref{eq:advect2d-cons} over $x$ and $y$, we have: \begin{align} \frac{1}{\Delta x \Delta y} \int_{x_{i-\myhalf}}^{x_{i+\myhalf}} \int_{y_{j-\myhalf}}^{y_{j+\myhalf}} a_t \, dx \, dy = &- \frac{1}{\Delta x \Delta y} \int_{x_{i-\myhalf}}^{x_{i+\myhalf}} \int_{y_{j-\myhalf}}^{y_{j+\myhalf}} (u a)_x \, dx \, dy \nonumber \\ &- \frac{1}{\Delta x \Delta y} \int_{x_{i-\myhalf}}^{x_{i+\myhalf}} \int_{y_{j-\myhalf}}^{y_{j+\myhalf}} (v a)_y \, dx \, dy \end{align} or using the divergence theorem, \begin{align} \label{eq:adv:multiddivergence} \frac{\partial a_{i,j}}{\partial t} = &- \frac{1}{\Delta x\Delta y} \int_{y_{j-\myhalf}}^{y_{j+\myhalf}} \left \{ (u a)_{i+\myhalf,j} - (u a)_{i-\myhalf,j} \right \} dy \nonumber \\ &- \frac{1}{\Delta x\Delta y} \int_{x_{i-\myhalf}}^{x_{i+\myhalf}} \left \{ (v a)_{i,j+\myhalf} - (v a)_{i,j-\myhalf} \right \} dx \end{align} Now we integrate over time---the left side of our expression becomes just the difference between the new and old state. \begin{align} a_{i,j}^{n+1} - a_{i,j}^n = &- \frac{1}{\Delta x\Delta y} \int_{t^n}^{t^{n+1}} \int_{y_{j-\myhalf}}^{y_{j+\myhalf}} \left \{ (u a)_{i+\myhalf,j} - (u a)_{i-\myhalf,j} \right \} dy dt \nonumber \\ &- \frac{1}{\Delta x\Delta y} \int_{t^n}^{t^{n+1}} \int_{x_{i-\myhalf}}^{x_{i+\myhalf}} \left \{ (v a)_{i,j+\myhalf} - (v a)_{i,j-\myhalf} \right \} dx dt \label{eq:update2du} \end{align} We define the flux through the interface as the average over the face of that interface and time: \MarginPar{is there a way to illustrate this, as some space-time diagram?} \begin{itemize} \item x-face: \begin{equation} \langle (ua)_{i+\myhalf,j}\rangle_{(t)} = \frac{1}{\Delta y \Delta t} \int_{t^n}^{t^{n+1}} \int_{y_{j-\myhalf}}^{y_{j+\myhalf}} (ua)_{i+\myhalf,j}\, dy dt \end{equation} \item y-face \begin{equation} \langle (va)_{i,j+\myhalf}\rangle_{(t)} = \frac{1}{\Delta x \Delta t} \int_{t^n}^{t^{n+1}} \int_{x_{i-\myhalf}}^{x_{i+\myhalf}} (va)_{i,j+\myhalf}\, dx dt \end{equation} \end{itemize} where $\langle . \rangle_{(t)}$ denotes the time-average over the face. For a second-order accurate method in time, we replace the average in time with the flux at the midpoint in time and the average over the face with the flux at the center of the face: \begin{equation} \langle (ua)_{i+\myhalf,j} \rangle_{(t)} \approx (ua)_{i+\myhalf,j}^{n+\myhalf} \end{equation} Then we have: \begin{equation} a_{i,j}^{n+1} = a_{i,j}^n - \Delta t \left [ \frac{(ua)_{i+\myhalf,j}^{n+\myhalf} - (ua)_{i-\myhalf,j}^{n+\myhalf}}{\Delta x} + \frac{(va)_{i,j+\myhalf}^{n+\myhalf} - (va)_{i,j-\myhalf}^{n+\myhalf}}{\Delta y} \right ] \end{equation} For the advection problem, since $u$ and $v$ are constant, we need to find the interface states of $a$ alone. There are two methods for computing these interface states, $a_{i\pm\myhalf,j}^{n+\myhalf}$ on $x$-interfaces and $a_{i,j\pm\myhalf}^{n+\myhalf}$ on $y$-interfaces: dimensionally split and unsplit. Dimensionally split methods are easier to code, since each dimension is operated on independent of the others, so you can simply call a one-dimensional method for each direction. Unsplit methods, however, are more accurate and less susceptible to grid effects. \MarginPar{show an example + refs? Zalesak test?} \subsection{Dimensionally split} \label{adv:sec:dimensionalsplit} In a split method, we update the state in each coordinate direction independently. This is simple and a straightforward way to use one-dimensional methods in multi-d. To be second-order accurate in time, we do {\em Strang splitting}~\cite{strang}, where we alternate the order of the dimensional updates each timestep. An update through $\Delta t$ consists of $x$ and $y$ sweeps and appears as: \begin{eqnarray} \frac{a_{i,j}^\star - a_{i,j}^n}{\Delta t} &=& - \frac{ u a_{i+\myhalf,j}^{n+\myhalf} - u a_{i-\myhalf,j}^{n+\myhalf} }{\Delta x} \label{eq:splitx}\\ \frac{a_{i,j}^{n+1} - a_{i,j}^\star}{\Delta t} &=& - \frac{ v a_{i,j+\myhalf}^{\star,n+\myhalf} - v a_{i,j-\myhalf}^{\star,n+\myhalf} }{\Delta y} \label{eq:splity} \end{eqnarray} Here, Eq.~\ref{eq:splitx} is the update in the $x$-direction. In constructing the interface states, $a_{i+\myhalf,j}^{n+\myhalf}$ and $a_{i-\myhalf,j}^{n+\myhalf}$, we do the exact same procedure as the one-dimensional case, constructing the left and right states at each interface and then solving the same Riemann problem to find the unique state on the interface. Each dimensional sweep is done without knowledge of the other dimensions. For example, in the $x$-update, we are solving: \begin{equation} a_t + u a_x = 0 \label{eq:advect1dx} \end{equation} and in the $y$-update, we are solving: \begin{equation} a_t + v a_y = 0 \end{equation} The construction of the interface states largely mimics the one-dimensional case (Eq.~\ref{eq:statel} and \ref{eq:stater}). For example, the $a_{i+\myhalf,j,L}^{n+\myhalf}$ state is: \begin{eqnarray} a_{i+\myhalf,j,L}^{n+\myhalf} &=& a_{i,j}^n + \frac{\Delta x}{2} \left .\frac{\partial a}{\partial x} \right |_{i,j} + \frac{\Delta t}{2} \left .\frac{\partial a}{\partial t} \right |_{i,j} + \mathcal{O}(\Delta x^2) + \mathcal{O}(\Delta t^2) \nonumber \\ &=& a_{i,j}^n + \frac{\Delta x}{2} \left .\frac{\partial a}{\partial x} \right |_{i,j} + \frac{\Delta t}{2} \left ( - u \left .\frac{\partial a}{\partial x} \right |_{i,j} \right ) + \ldots \nonumber \\ &=& a_{i,j}^n + \frac{\Delta x}{2} \left ( 1 - \frac{\Delta t}{\Delta x} u \right ) \left .\frac{\partial a}{\partial x} \right |_{i,j} + \ldots \label{eq:statels} \end{eqnarray} Notice that when substituting for $\partial a / \partial t$, we use the one-dimensional split version of the advection equation (Eq.~\ref{eq:advect1dx}) instead of the full multi-dimensional equation. There are no $y$-direction terms that come into play in the split method when considering the $x$-direction. The $x$-update (Eq.~\ref{eq:splitx}) updates the state only accounting for the $x$-fluxes---we denote this intermediate state with the `$\star$' superscript. For the $y$-update, we construct our interface states in the analogous way as in the $x$-direction, but begin with the `$\star$' state instead of the old-time state. In this fashion, the $y$-update `sees' the result of the $x$-update and couples things together. To achieve second-order accuracy in time, it is necessary to alternate the directions of the sweeps each timestep, i.e., $x$-$y$ then $y$-$x$. Furthermore, this pair of sweeps should use the same timestep, $\Delta t$. \subsection{Unsplit multi-dimensional advection} \label{adv:sec:unsplit_2d} The unsplit case differs from the dimensionally split case in two ways: (1) in predicting the interface states, we use knowledge of the flow in the transverse direction, and (2), only a single conservative update is done per timestep, with all directions updating simultaneously. See \cite{colella:1990} for more details. This idea is sometimes called the ``corner transport upwind'' or CTU method. The construction of the $a_{i+\myhalf,j,L}^{n+\myhalf}$ interface state appears as \begin{eqnarray} a_{i+\myhalf,j,L}^{n+\myhalf} &=& a_{i,j}^n + \frac{\Delta x}{2} \left .\frac{\partial a}{\partial x} \right |_{i,j} + \frac{\Delta t}{2} \left .\frac{\partial a}{\partial t} \right |_{i,j} + \mathcal{O}(\Delta x^2) + \mathcal{O}(\Delta t^2) \nonumber \\ &=& a_{i,j}^n + \frac{\Delta x}{2} \left .\frac{\partial a}{\partial x} \right |_{i,j} + \frac{\Delta t}{2} \left ( - u \left .\frac{\partial a}{\partial x} \right |_{i,j} - v \left .\frac{\partial a}{\partial y} \right |_{i,j} \right ) + \ldots \nonumber \\ &=& \underbrace{a_{i,j}^n + \frac{\Delta x}{2} \left ( 1 - \frac{\Delta t}{\Delta x} u \right ) \left .\frac{\partial a}{\partial x} \right |_{i,j}}_{\hat{a}_{i+\myhalf,j,L}^{n+\myhalf}} \underbrace{- \frac{\Delta t}{2} v \left .\frac{\partial a}{\partial y} \right |_{i,j}}_{\mathrm{``transverse~flux~difference''}} + \ldots \label{eq:statelu} \end{eqnarray} The main difference between the split and unsplit interface states is the explicitly appearance of the ``transverse flux difference'' in the unsplit interface state. We rewrite this as: \begin{equation} a_{i+\myhalf,j,L}^{n+\myhalf} = \hat{a}_{i+\myhalf,j,L}^{n+\myhalf} - \frac{\Delta t}{2} v \left .\frac{\partial a}{\partial y} \right |_{i,j} \end{equation} Here, the $\hat{a}_{i+\myhalf,j,L}^{n+\myhalf}$ term is just the normal prediction without considering the transverse direction (e.g., Eq.~\ref{eq:statels}). The basic update procedure is: \begin{itemize} \item Construct the normal predictor term, $\hat{a}_{i+\myhalf,j,L}^{n+\myhalf}$, in a fashion identical to the one-dimensional and split methods. We compute these one-dimensional $\hat{a}$'s at the left and right every interface in both coordinate directions. Note that these states are still one-dimensional, since we have not used any information from the transverse direction in their computation. \item Solve a Riemann problem at each of these interfaces: \begin{eqnarray} a^T_{i+\myhalf,j} &=& \mathcal{R}(\hat{a}_{i+\myhalf,j,L}^{n+\myhalf}, \hat{a}_{i+\myhalf,j,R}^{n+\myhalf}) \\ a^T_{i,j+\myhalf} &=& \mathcal{R}(\hat{a}_{i,j+\myhalf,L}^{n+\myhalf}, \hat{a}_{i,j+\myhalf,R}^{n+\myhalf}) \\ \end{eqnarray} These states are given the `$^T$' superscript since they are the states that are used in computing the transverse flux difference. \item Correct the previously computed normal interface states (the $\hat{a}$'s) with the transverse flux difference: \begin{equation} a_{i+\myhalf,j,L}^{n+\myhalf} = \hat{a}_{i+\myhalf,j,L}^{n+\myhalf} - \frac{\Delta t}{2} v \frac{a^T_{i,j+\myhalf} - a^T_{i,j-\myhalf}}{\Delta y} \end{equation} Notice that the fluxes that are differenced for the left state are those that are transverse, but to the left of the interface. Similarly, for the right state, it would be those that are transverse, but to the right of the interface: \begin{equation} a_{i+\myhalf,j,R}^{n+\myhalf} = \hat{a}_{i+\myhalf,j,R}^{n+\myhalf} - \frac{\Delta t}{2} v \frac{a^T_{i+1,j+\myhalf} - a^T_{i+1,j-\myhalf}}{\Delta y} \end{equation} \end{itemize} A similar procedure happens at the $y$-interfaces. Figure~\ref{fig:unsplitstates} illustrates the steps involved in the construction of the $a_{i+\myhalf,j,L}^{n+\myhalf}$ state. \MarginPar{more panels illustrating the sequence?} % % this is created by figures/advection/2dgrid-hat.py % and 2dgrid-transverse.py \begin{figure} \centering \includegraphics[width=0.7\linewidth]{2dgrid-hat} \\ \includegraphics[width=0.7\linewidth]{2dgrid-transverse} \caption[The construction of an interface state with the transverse component]{\label{fig:unsplitstates} The construction of the $a_{i+\myhalf,j,L}^{n+\myhalf}$ state. Top: first we compute the $\hat{a}$'s---here we show all of the $\hat{a}$'s that will be used in computing the full left interface state at $(i+\myhalf,j)$. Bottom: after the transverse Riemann solves, we have the two transverse states ($a^T_{i,j+\myhalf}$ and $a^T_{i,j-\myhalf}$) that will be differenced and used to correct $\hat{a}_{i+\myhalf,j,L}^{n+\myhalf}$ (illustrated by the dotted lines) to make $a_{i+\myhalf,j,L}^{n+\myhalf}$.} \end{figure} Once all of the full states (normal prediction $+$ transverse flux difference) are computed to the left and right of all the interfaces ($x$ and $y$), we solve another Riemann problem to find the final state on each interface. \begin{equation} a_{i+\myhalf,j}^{n+\myhalf} = \mathcal{R}(a_{i+\myhalf,j,L}^{n+\myhalf}, a_{i+\myhalf,j,R}^{n+\myhalf}) \\ \end{equation} The final conservative update is then done via Eq.~\ref{eq:update2du}. See \cite{colella:1990} for more details on this unsplit method, and \cite{saltzman:1994} for details of the extension to 3-d. Figures~\ref{fig:adv:gaussian-2d} and \ref{fig:adv:tophat-2d} show the advection of a smooth Gaussian and a discontinuous tophat diagonally on a coarse $32\times 32$ zone domain. Each show a diffusion of the initial function, similar to what we saw in 1-d. In the tophat, we also see a slight undershoot on the trailing side of the tophat after one period. % run as: ./pyro.py advection smooth inputs.smooth % visualized as: ./plot.py -W 7.7 -H 6 -o smooth_init.pdf advection smooth_0000 \begin{figure} \centering \includegraphics[width=0.48\linewidth]{smooth_init}\hspace{1em} \includegraphics[width=0.48\linewidth]{smooth_final} \caption[Advection of Gaussian profile in 2-d] {\label{fig:adv:gaussian-2d} Advection of a Gaussian profile with $u = v = 1$ for one period on a $32\times 32$ grid, with $\cfl = 0.8$. This was run with \pyro\ as {\tt ./pyro.py advection smooth inputs.smooth}} \end{figure} \begin{figure} \centering \includegraphics[width=0.48\linewidth]{tophat_init}\hspace{1em} \includegraphics[width=0.48\linewidth]{tophat_final} \caption[Advection of tophat profile in 2-d]{\label{fig:adv:tophat-2d} Advection of a tophat function with $u = v = 1$ for one period on a $32\times 32$ grid, with $\cfl = 0.8$. This was run as {\tt ./pyro.py advection tophat inputs.tophat}} \end{figure} \subsection{Timestep limiter for multi-dimensions} \label{sec:adv:timestep} The timestep criterion we used in one-dimension (Eq.~\ref{eq:timestep}) needs to be generalized for multi-dimensions. For the dimensionally split case, since we are basically piecing together two one-dimensional sweeps, the timestep limit is usually taken as: \begin{equation} \label{eq:adv:timestep:multid} \Delta t = \cfl \min \left \{ \frac{\Delta x}{|u|} , \frac{\Delta y}{|v|} \right \} \end{equation} This is, for example, the construction that is used with the dimensionally-split hydrodynamics solver in the original Flash code~\cite{flash}\footnote{although, the paper does not explicitly write this out}. \MarginPar{does the prometheus MPA 449 write it out?} Stability requires picking a CFL number, $\cfl \le 1$. A general extension of the timestep restriction for a fully-multi-dimensional algorithm is often written in the form (see, e.g., \cite{shuosher:1989b}) \begin{equation} \label{eq:adv:timstep:mol} \Delta t = \cfl \left ( \sum_{i=1}^d \frac{|\Ub \cdot \mathbf{e}_d|}{\Delta x_d} \right )^{-1} \end{equation} where $d$ is the coordinate direction, $\Ub = (u, v)$ is the velocity vector, and $\mathbf{e}_d$ is the unit vector in direction $d$. For these methods $\cfl \lesssim 1$. This is usually more restrictive than Eq.~\ref{eq:adv:timestep:multid}. For the CTU method described above, \cite{colella:1990} argues that the inclusion of the transverse information removes some of the instability inherent in simpler schemes, allowing for a larger timestep, restricted by Eq.~\ref{eq:adv:timestep:multid}. Note, because of the different form of the timestep limiters here, it is not enough to simply cite a CFL number when describing results, you must also explain which form of the timestep calculation (Eq.~\ref{eq:adv:timestep:multid} or Eq.~\ref{eq:adv:timstep:mol}) is being used. \subsection{Method-of-lines in multi-dimensions} The multidimensional version of method-of-lines integration follows the ideas from the one-dimensional case. We can start with Eq.~\ref{eq:adv:multiddivergence}, and define the fluxes over though a face as: \begin{itemize} \item x-face: \begin{equation} \langle (ua)_{i+\myhalf,j} \rangle = \frac{1}{\Delta y} \int_{y_{j-\myhalf}}^{y_{j+\myhalf}} (ua)_{i+\myhalf,j}\, dy \end{equation} \item y-face \begin{equation} \langle (va)_{i,j+\myhalf} \rangle = \frac{1}{\Delta x} \int_{x_{i-\myhalf}}^{x_{i+\myhalf}} (va)_{i,j+\myhalf}\, dx \end{equation} \end{itemize} These are similar to the expressions above, except there is no integral over time. Again, to second order in space, we can just use the flux evaluated at the midpoint in the face: \begin{equation} \langle (ua)_{i+\myhalf,j} \rangle \approx (ua)_{i+\myhalf,j} \end{equation} The result is the ODE: \begin{equation} \frac{d a_{i,j}}{d t} = - \frac{(ua)_{i+\myhalf,j} - (ua)_{i-\myhalf,j}}{\Delta x} - \frac{(va)_{i,j+\myhalf} - (va)_{i,j-\myhalf}}{\Delta y} \end{equation} For the interface states, we separately construct slopes in each dimension: $\Delta^{(x)}a_{i,j}$ is the slope in the $x$-direction, and $\Delta^{(y)}a_{i,j}$ is the slope in the $y$-direction. Then the interface states are: \begin{align} a_{i+\myhalf,j,L} &= a_{i,j} + \frac{1}{2} \Delta a^{(x)}_{i,j} \\ a_{i+\myhalf,j,R} &= a_{i+1,j} - \frac{1}{2} \Delta a^{(x)}_{i+1,j} \end{align} and \begin{align} a_{i,j+\myhalf,L} &= a_{i,j} + \frac{1}{2} \Delta a^{(y)}_{i,j} \\ a_{i,j+\myhalf,R} &= a_{i,j+1} - \frac{1}{2} \Delta a^{(y)}_{i,j+1} \end{align} With these interface states, we have the method for evaluating the righthand side of our ODE. We can then evolve the ODE using a Runge-Kutta integration method. At each stage of the RK integration, we do the same construction of the interface states, solve the Riemann problem, etc. Notice that unlike the directionally-split method, we are doing the update in both directions at the same time---we are relying on the ODE integrator to do the coupling in the different directions for us. This construction is a lot simpler than the unsplit method described previously. The cost of this simplicity is a reduced range of stability. Performaing the stability analysis here is much more complicated. For simplicity, consider a simple first-order upwind scheme in two-dimensions. Higher order time integrators (like RK4) will have separate stages that look like a first-order-in-time update, so this stability analysis can be thought of as applying to a single stage. Our difference equation is: \begin{equation} \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} - v \frac{a^n_{i,j} - a^n_{i,j-1}}{\Delta y} \end{equation} We introduce a single-mode Fourier function, as before, but this time with a mode in each direction: \begin{equation} a^n_{i,j} = A^n e^{Ii\theta}e^{Ij\phi} \end{equation} Next, to simplify things, we'll assume that $u = v$, and $\Delta x = \Delta y$, then $\cfl = u \Delta t/ \Delta x = v \Delta t / \Delta y$. Inserting this into our difference equation gives: \begin{equation} \frac{A^{n+1}}{A^n} = 1 - \cfl \left (1 - e^{-I\theta} \right ) - \cfl \left (1 - e^{-I\phi} \right ) \end{equation} This form looks very close to the one-dimensional case, but it is difficult to analytically find the values of $\cfl$, but we can numerically find the maximum $ | {A^{n+1}}/{A^n} |^2$ for $\theta \in [0, 2\pi]$ and $\phi \in [0, 2\pi]$ as a function of $\cfl$. In doing this, we find that this discretization is unstable ($| {A^{n+1}}/{A^n} |^2 > 1$) for $\cfl > 1/2$\footnote{See the script \href{https://github.com/zingale/hydro_examples/blob/master/compressible/cfl.py}{cfl.py}}. This restriction is $1/d$, where $d$ is the dimensionality, and it is analogous to the difference between the timestep limits for stability from the more generous Eq.~\ref{eq:adv:timestep:multid} to the more restrictive Eq.~\ref{eq:adv:timstep:mol}. This reduced $\cfl$ for stability is discussed in \cite{shuosher:1989b} (see section 4), and is also discussed in \cite{titarevtoro,mccorquodalecolella}. Different reconstruction and integration methods can vary this limit some, for instance, the fourth-order method 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 for this application~\cite{gottliebshu:1996}. % run as: % ./pyro.py advection_rk tophat inputs.tophat % ./plot.py -W 7.7 -H 6 -o tophat_rk4_cfl08.pdf advection_rk tophat_0040 % % ./pyro.py advection_rk tophat inputs.tophat driver.cfl=0.4 % ./plot.py -W 7.7 -H 6 -o tophat_rk4_cfl04.pdf advection_rk tophat_0080 \begin{figure} \centering \includegraphics[width=0.48\linewidth]{tophat_rk4_cfl08}\hspace{1em} \includegraphics[width=0.48\linewidth]{tophat_rk4_cfl04} \caption[Advection of tophat function with method-of-lines integration] {\label{fig:adv:tophat-2d-rk4} Advection of a tophat function using the method-of-lines approach with 4th-order Runge-Kutta time integration. We use $u = v = 1$ for one period on a $32\times 32$ grid. On the left is $\cfl = 0.8$---this is clearly unstable. On the right is $\cfl = 0.4$, which is stable. This was run as {\tt ./pyro.py advection\_rk tophat inputs.tophat}} \end{figure} \section{High-Order Finite difference methods} \label{sec:ho-fd} Chapter~\ref{ch:advection} introduced numerical methods for the linear advection equation \begin{equation} \label{eq:ho-advect} a_t + u a_x = 0 \end{equation} that were first or second order in space. The advantages of using numerical methods that are higher order can be large, especially when in regions where the flow is smooth. However, these must be balanced against the additional computational cost and complexity. In addition, many higher order methods do no better than second order methods in cases with discontinuities. \subsection{The problem with higher-order finite volume methods} The finite volume approach outlined in chapter~\ref{ch:fv} %\todo{label missing in fv tex file?} matches the physics of hyperbolic balance laws perfectly, by maintaining the conservation of appropriate quantities using the fluxes through the surfaces of each control volume or computational cell. However, this leads to significant computational costs and complexities when going beyond second order methods. The inter-cell fluxes are computed by integrals over the face of the computational cell. In one dimension the face is a single point and the integral needs the flux evaluated at a single point, as in equation~\ref{eq:consup}. In higher dimensions the face is one dimensional (e.g., a line) or two dimensional (e.g., a square) and so the integral must be approximated by evaluating the flux at \emph{multiple} points. If we use Gauss quadrature, this means that a third order method would require at least two flux evaluations per face in two dimensions, and typically four evaluations per face in three dimensions. % % This comment about the timestep doesn't hold in general, as pointed out by % Mike. I think it *is* relevant for directionally split codes where the % transverse Riemann Problem isn't solved. That's probably too technical a % distinction to make this early. % %As well as the additional computational cost, there is also typically an %additional restriction on the timestep. Intuitively this can be seen by %thinking about the flux evaluations again. We fix the values either side of the %cell boundary in order to compute the flux at one point. The timestep %restriction is given by the time it take for information to propagate from a %different flux evaluation point, which would change the information available. %As we need to compute the flux at more points within the cell faces, the %distance between points where the flux is evaluated goes down, reducing the %time it takes for information to propagate from one point to another. \subsection{Finite differences} The conservative finite difference method starts from the endpoint for finite volume methods, by considering the update formula~\eqref{eq:consup} %\todo{is this the right equation to ref? Don't think it's labelled?} \begin{equation} \tag{\ref{eq:consup}} \frac{\partial \langle \Uc\rangle_{i}}{\partial t} = - \frac{1}{\Delta x} \left \{ \Fb_{i+\myhalf} - \Fb_{i-\myhalf} \right \}. \end{equation} In the finite difference method the interpretation used is different. We are now thinking of \begin{equation} \label{eq:fd-deriv} \frac{1}{\Delta x} \left \{ \Fb_{i+\myhalf} - \Fb_{i-\myhalf} \right \} \end{equation} as representing a direct approximation to the derivative of the flux at $x_i$. In particular, we are \emph{not} thinking of $\Fb_{i+\myhalf}$ as being directly linked to the flux. Crucially the update still ensures global conservation, as the ``flux'' term is re-used for neighbouring points in the same way as in the finite volume case. When we combine this viewpoint with the Method of Lines approach in section~\ref{adv:sec:mol_2d}, the implementation of a high-order finite boils down to \begin{itemize} \item using a high-order time integration scheme (such as a high order Runge-Kutta method); \item finding a way that the finite difference in equation~\eqref{eq:fd-deriv} approximates the derivative of the flux both stably, and to a sufficiently high order. \end{itemize} \subsection{WENO reconstruction} \label{sec:WENO} \begin{figure} \includegraphics[width=0.96\linewidth]{weno-convergence} \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.} \label{fig:weno-convergence} \end{figure} For the advection equation with constant advection speed $u$, our higher-order method requires us to find a high-order approximation to $a_{i+\myhalf}$. If we considered the five points $a_{j}, j \in [i-2, \dots, i+2]$ then we could use the information from all five of these points to approximate $a_{i+\myhalf}$ as \begin{equation} \label{eq:weno-advection-5pt} 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). \end{equation} Alternatively we could use only three of the points, in three different ways: \begin{subequations} \label{eq:weno-advection-3pt} \begin{align} 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), \\ 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), \\ 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). \end{align} \end{subequations} Which is better? The five point stencil gives higher order accuracy, but includes information from both sides of the point. This ignores information from the characteristics and, near discontinuities, will lead to Gibbs oscillations. The three point stencils give lower accuracy. However, they give us the flexibility to choose a stencil to avoid discontinuities, possibly using the characteristic 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}. \begin{exercise}[ENO stencils] {Check that equations~\eqref{eq:weno-advection-5pt} and \eqref{eq:weno-advection-3pt} give the claimed order of accuracy results.} \end{exercise} The method of choosing the ``best'' stencil (to avoid discontinuities as far as possible) is called \emph{Essentially Non-Oscillatory} (ENO) reconstruction. It is not as accurate as it could be given the size of the stencil it uses, and the logical branches required to find the ``best'' stencil can reduce computational performance. An alternative that avoids the problems of ENO schemes are the \emph{Weighted} ENO, or WENO, schemes. These use a combination of the possible ENO stencils to reconstruct $a_{i+\myhalf}$. This relies on the observation that there are constants $C_k$ such that, for example, \begin{equation} \label{eq:WENO-C-weights} a^{(\text{Fifth})}_{i+\myhalf} = \sum_{k=0}^{2} C_k a^{(\text{Third}, r)}_{i+\myhalf}. \end{equation} \begin{exercise}[WENO weights] {Check that the constants $C_0 = \tfrac{3}{10}, C_1 = \tfrac{3}{5}, C_2 = \tfrac{1}{10}$ make equation~\eqref{eq:WENO-C-weights} consistent with equations~\eqref{eq:weno-advection-5pt} and \eqref{eq:weno-advection-3pt}.} \end{exercise} The WENO methods work by retaining the sum over all stencils as in equation~\eqref{eq:WENO-C-weights}, but adjusting the weights to avoid oscillations. Ideally, in regions where the $a^{(0,1)}$ stencils include a shock but the $a^{(2)}$ stencil does not the weights should be $\tilde{C}_0 = 0 = \tilde{C}_1$ and $\tilde{C}_2 = 1$. In order for this sum to make sense we will need the weights to add to $1$. \subsubsection{WENO algorithm} \label{sec:WENO-algorithm} \begin{figure} \includegraphics[width=0.96\linewidth]{weno-weights} \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.} \label{fig:weno-weights} \end{figure} We can now write out the full WENO algorithm of order $(2 r - 1)$. The constant $r$ sets the size of the individual stencils as well as the (optimal) order of the 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 $a_i$). First compute the individual stencils \begin{equation} \label{eq:weno-stencils} a_{r, k, i+\myhalf} = \sum_{\ell=0}^{r-1} A_{r, k, \ell} a_{i+k-\ell}, \quad k = 0, \dots, r-1. \end{equation} Here the $A_{r, k, \ell}$ terms are constants which, for given scheme accuracy $r$ are needed to compute the $k^{\text{th}}$ stencil. We then want to combine these individual stencils to compute the WENO result. For this we write \begin{equation} \label{eq:weno-reconstruction} a_{r, WENO, i+\myhalf} = \sum_{k=0}^{r-1} \omega_{r, k, i+\myhalf} a_{r, k, i+\myhalf}. \end{equation} Here the $\omega_{r, k, i+\myhalf}$ terms are the weights that vary from point to point. They must sum to one, so that we have a convex combination of the stencils: \begin{equation} \label{eq:weno-convex} \sum_{k=1}^{r-1} \omega_{r, k, i+\myhalf} = 1. \end{equation} We also want the weights to match the \emph{optimal} weights $C_{r, k}$ in smooth regions. The optimal weights are defined such that \begin{equation} \label{eq:weno-optimal} a_{r, WENO, i+\myhalf} = \sum_{k=0}^{r-1} C_{r, k} a_{r, k, i+\myhalf} \end{equation} is accurate to order ${\cal O}(\Delta x^{(2 r - 1)})$ when $q$ is smooth. The \emph{choice} of how to get from the optimal weights $C_{r, k}$ to the nonlinear weights $\omega_{r, k, i+\myhalf}$ defines the WENO scheme. The standard choice of Jiang and Shu is to introduce a measure of the smoothness of the $k^{\text{th}}$ stencil by computing the sum of the integral averages of its derivatives from order $1$ up to order $2 r - 1$. For practical implementation purposes this means computing the \emph{smoothness indicators} $\beta_{r, k, i+\myhalf}$ as \begin{equation} \label{eq:weno-smoothness} \beta_{r, k, i+\myhalf} = \sigma_{r, k, \ell, m} a_{i+k-\ell} a_{i+k-m}. \end{equation} The terms $\sigma_{r, k, \ell, m}$ are pre-computed constants which give the coefficients in the quadratic form for the $k^{\text{th}}$ stencil of width $r$ in the reconstruction. These smoothness indicators are non-negative, and will be large when the derivatives are large in magnitude. The Jiang and Shu weights are then set by \begin{subequations} \label{eq:weno-nonlinear-weights} \begin{align} \label{eq:weno-nonlinear-weights-omega} \omega_{r, k, i+\myhalf} &= \frac{\alpha_{r, k, i+\myhalf}}{\sum_{k=0}^{r-1} \alpha_{r, k, i+\myhalf}}, \\ \label{eq:weno-nonlinear-weights-alpha} \alpha_{r, k, i+\myhalf} &= \frac{C_{r, k}}{\epsilon + \beta_{r, k, i+\myhalf}^2}. \end{align} \end{subequations} Here $\epsilon$ is a small number introduced to avoid division-by-zero problems. The form of $\omega_{r, k, i+\myhalf}$ in equation~\eqref{eq:weno-nonlinear-weights-omega} guarantees that the convex sum condition in equation~\eqref{eq:weno-convex} holds. The form of $\alpha_{r, k, i+\myhalf}$ ensures that when all the smoothness indicators $\beta_{r, k, i+\myhalf}$ are the same magnitude the weights $\omega_{r, k, i+\myhalf}$ will match the optimal weights $C_{r, k}$, but when a smoothness indicator is large (i.e., when the associated stencil has large derivatives, typically associated with discontinuities), the contribution of its associated stencil will be small. An example of this is shown in figure~\ref{fig:weno-weights}. Finally, we note that this method has reconstructed $a_{i+\myhalf}$ from the left. For the advection equation case where the advection speed $u$ is positive this is all we need: the characteristic information tells us that we should use this reconstruction. In the case where the advection speed is negative we should reconstruct from the right. In the implementation it is easiest to implement only one reconstruction direction and pass the data in in reverse order. \begin{exercise}[WENO reconstruction] {Taking the values for the $C$, $A$ and $\sigma$ constants from Gerolymos et al.\ or Shu's review~\cite{Shu1997}, construct a WENO reconstruction method for $r=2$. Check your results on smooth and non-smooth functions.} \end{exercise} \subsubsection{Implementation issues with WENO schemes} \label{sec:weno-implementation} A convergence test on the WENO schemes with $r=3$ and $r=5$ is shown in figure~\ref{fig:weno-converge-gaussian-rk4}. This should be compared to figure~\ref{fig:advnorm}, as in both a Gaussian profile is advected five times around a periodic domain. The WENO schemes show higher absolute accuracy for moderate size grids ($N \gtrsim 100$) and show faster convergence. However, it is clear that whilst the $r=3$ method converges at fifth order as expected, the $r=5$ WENO method does not converge at ninth order but at fourth order. This is explained by the time integrator. We remember that any solver has an error both from the spatial and the time discretization. In figure~\ref{fig:weno-converge-gaussian-rk4} the time integrator is the classic fourth order Runge-Kutta, and it is the error from this integrator that is dominating. To confirm this, figure~\ref{fig:weno-converge-gaussian} uses the eigth order Dormand-Price Runge-Kutta method with adaptive step size control (using the \texttt{scipy.integrate.ode} routine). Here we see high order convergence for all $r$, and in each case the convergence rate is $2 r - 2$ for \textbf{no reason I can understand}. \begin{figure}[t] \centering % figure generated by hydro_examples/advection/weno.py \includegraphics[width=0.8\linewidth]{weno-converge-gaussian-rk4} \caption[High order WENO convergence rates for linear advection] {\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. \\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/advection/weno.py}{weno.py}}} \end{figure} % \begin{figure}[t] \centering % figure generated by hydro_examples/advection/weno.py \includegraphics[width=0.8\linewidth]{weno-converge-gaussian} \caption[Very high order WENO convergence rates for linear advection] {\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. \\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/advection/weno.py}{weno.py}}} \end{figure} % \section{Going further} \begin{itemize} \item {\em Slope limiting}: there are a wide variety of slope limiters. All of them are designed to reduce oscillations in the presence of discontinuities or extrema, but some are higher-order and can be less restrictive when dealing with smooth flows. Most hydro texts (e.g.\ \cite{leveque:2002,toro:1997}) provide an introduction to the design of such limiters. \item {\em Multi-dimensional limiting}: the procedure described above still does the limiting in each dimension independent of the other when doing the unsplit reconstruction. This can lead to overshoots/ undershoots. An example of a method that considers the limiting in multi-dimensions is \cite{BDS,quadBDS}. \item {\em Spatially-varying velocity field}: if we consider a spatially varying velocity field, $u(x,y)$ and $v(x,y)$ that is specified externally, then we can describe the advection of a quantity $\phi$ as: \begin{equation} \phi_t + (\phi u)_x + (\phi v)_y = 0 \end{equation} The solution procedure is largely the same as described above. We write: \begin{align} \phi_{i+\myhalf,j,L}^{n+\myhalf} &= \phi_{i,j}^n + \frac{\Delta x}{2} \frac{\partial \phi}{\partial x} + \frac{\Delta t}{2} \frac{\partial \phi}{\partial t} + \ldots \nonumber \\ % &= \phi_{i,j}^n + \frac{\Delta x}{2} \frac{\partial \phi}{\partial x} + \frac{\Delta t}{2} \left [ -(\phi u)_x -(\phi v)_y \right ]_{i,j} \nonumber\\ % &= \underbrace{\phi_{i,j}^n + \frac{\Delta x}{2} \left ( 1 - \frac{\Delta t}{\Delta x} u_{i,j} \right ) \frac{\partial \phi}{\partial x}}_{\hat{\phi}_{i+\myhalf,j,L}^{n+\myhalf}} - \frac{\Delta t}{2} \left [\phi u_x \right ]_{i,j} - \frac{\Delta t}{2} \left [ (\phi v)_y \right ]_{i,j} \end{align} and upwinding is used to resolve the Riemann problem for both the transverse and normal interface states. This type of construction comes up in low Mach number flows, where the density can be advected according to the velocity field in much the fashion shown here, and is described in \S~\ref{sec:lm:density}. For compressible hydrodynamics, we often have density-weighted quantities that we advect. This extension is described in \S~\ref{sec:euler:further}. \end{itemize} \section{\pyro\ experimentation} To gain some experiences with these ideas, we can use the advection solver in \pyro\ (see Appendix~\ref{app:pyro} to get started). The \pyro\ advection solver implements the second-order unsplit advection algorithm described in the previous sections. To run this solver on the Gaussian advection problem, do: \begin{verbatim} ./pyro.py advection smooth inputs.smooth \end{verbatim} By default, this will advect a Gaussian profile diagonally across the domain for a single period. To get a feel for the advection algorithm, here are some suggested exercises: \begin{exercise}[Role of limiters] {Implement a tophat initial profile and run with and without limiters (this is controlled by the {\tt advection.limiter} runtime parameter).} \end{exercise} \begin{exercise}[Grid effects] {Look at the solution when you advect purely in the $x$- or $y$-direction and compare to the diagonal case---notice how the direction affects the error in the solution. This is the imprint of the grid we discretize on.} \end{exercise} \begin{exercise}[Split vs.\ unsplit] {Implement a dimensionally-split version of the advection algorithm and compare the results to the unsplit version. Pick a suitable norm and measure the convergence of the methods.} \end{exercise} ================================================ FILE: advection/main.tex ================================================ \documentclass[11pt]{article} % margins \usepackage[margin=0.75in]{geometry} % figures \usepackage{graphicx} % font \usepackage{mathpazo} \usepackage{helvet} % coloring \usepackage{color} \definecolor{mygray}{gray}{0.5} % footer \usepackage{fancyhdr} \pagestyle{fancy} \fancyfoot[LO,LE]{\footnotesize \sffamily \color{mygray} M.\ Zingale---Notes on Advection} \fancyfoot[RO,RE]{\footnotesize \sffamily \color{mygray} (\today)} \fancyfoot[CO,CE]{\thepage} \fancyhead{} \renewcommand{\headrulewidth}{0.0pt} \renewcommand{\footrulewidth}{0.0pt} % captions \usepackage{caption} \renewcommand{\captionfont}{\footnotesize} \renewcommand{\captionlabelfont}{\footnotesize} \setlength{\captionmargin}{5em} \usepackage{sectsty} \allsectionsfont{\sffamily} \usepackage{amsmath} \newcounter{exercise} \usepackage{changepage} \newenvironment{exercise}% environment name {% begin code \begin{adjustwidth}{1.0cm}{0.5cm} \par\vspace{\baselineskip}\noindent \refstepcounter{exercise}% \textbf{Exercise \theexercise\ }\begin{itshape}% %\par\vspace{\baselineskip}\noindent\ignorespaces }% {% end code \end{itshape}\end{adjustwidth}\vspace{\baselineskip}\ignorespacesafterend } \begin{document} \begin{center} {\LARGE \textsf{\textbf{ Notes on Advection}} } \end{center} \input advection \bibliographystyle{plain} \bibliography{../refs} \end{document} ================================================ FILE: advection/old-scripts/fv.py ================================================ import math import numpy import pylab import grid_plot_util as gpu nzones = 5 # data that lives on the grid a = numpy.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.15, 0.5]) gr = gpu.grid(nzones) gpu.drawGrid(gr) gpu.labelCenter(gr, nzones/2, r"$x_i$") gpu.labelAvg(gr, nzones/2, a[nzones/2], r"$\langle f\rangle_i$") gpu.labelCenter(gr, nzones/2-1, r"$x_{i-1}$") gpu.labelCenter(gr, nzones/2+1, r"$x_{i+1}$") gpu.labelCenter(gr, nzones/2-2, r"$x_{i-2}$") gpu.labelCenter(gr, nzones/2+2, r"$x_{i+2}$") gpu.labelEdge(gr, nzones/2, r"$x_{i-1/2}$") gpu.labelEdge(gr, nzones/2+1, r"$x_{i+1/2}$") n = 0 while (n < nzones): gpu.drawCellAvg(gr, n, a[n]) n += 1 pylab.axis([gr.xmin-0.5*gr.dx,gr.xmax+0.5*gr.dx, -0.25, 1.2]) pylab.axis("off") pylab.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95) f = pylab.gcf() f.set_size_inches(8.0,2.0) pylab.savefig("fv.png", dpi=200) pylab.savefig("fv.eps") ================================================ FILE: advection/old-scripts/simplegrid.py ================================================ import math import numpy import pylab def simplegrid(): # grid info xmin = 0.0 xmax = 1.0 nzones = 7 ng = 1 dx = (xmax - xmin)/float(nzones) xl = (numpy.arange(2*ng+nzones) - ng)*dx xr = (numpy.arange(2*ng+nzones)+1 - ng)*dx xc = 0.5*(xl + xr) #------------------------------------------------------------------------ # plot a domain without ghostcells pylab.plot([xmin,xmax], [0,0], color="k", lw=2) # domain left edge pylab.plot([xl[ng], xl[ng]], [0, 0.5], color="k", lw=4) n = ng while (n < ng+nzones): # draw right edge pylab.plot([xr[n], xr[n]], [0, 0.5], color="k", lw=2) # draw center marker pylab.plot([xc[n], xc[n]], [-0.05, 0], color="k") n += 1 # domain right edge pylab.plot([xl[ng+nzones], xl[ng+nzones]], [0, 0.5], color="k", lw=4) # label a few pylab.text(xc[ng+nzones/2], -0.1, r"$i$", horizontalalignment='center', verticalalignment='top') pylab.text(xc[ng+nzones/2-1], -0.1, r"$i-1$", horizontalalignment='center', verticalalignment='top') pylab.text(xc[ng+nzones/2+1], -0.1, r"$i+1$", horizontalalignment='center', verticalalignment='top') pylab.xlim(xl[0]-0.5*dx,xr[2*ng+nzones-1]+0.5*dx) pylab.ylim(-0.25, 0.6) pylab.axis("off") pylab.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95) f = pylab.gcf() f.set_size_inches(8.0,2.0) pylab.savefig("simplegrid.png") pylab.savefig("simplegrid.eps") #------------------------------------------------------------------------ # now draw one with ghostcells pylab.plot([xmin-ng*dx,xmax+ng*dx], [0,0], color="k", lw=2) # domain (w/ ghostcells) left edge pylab.plot([xl[0], xl[0]], [0, 0.5], color="k", lw=2) n = 0 while (n < 2*ng+nzones): # draw right edge pylab.plot([xr[n], xr[n]], [0, 0.5], color="k", lw=2) # draw center marker pylab.plot([xc[n], xc[n]], [-0.05, 0], color="k") n += 1 # domain left edge pylab.plot([xl[ng], xl[ng]], [0, 0.5], color="k", lw=4) # domain right edge pylab.plot([xl[ng+nzones], xl[ng+nzones]], [0, 0.5], color="k", lw=4) # label a few pylab.text(xc[ng+nzones/2], -0.1, r"$i$", horizontalalignment='center', verticalalignment='top') pylab.text(xc[ng+nzones/2-1], -0.1, r"$i-1$", horizontalalignment='center', verticalalignment='top') pylab.text(xc[ng+nzones/2+1], -0.1, r"$i+1$", horizontalalignment='center', verticalalignment='top') pylab.text(xc[ng-1], -0.1, r"$-1$", horizontalalignment='center', verticalalignment='top') pylab.text(xc[ng], -0.1, r"$0$", horizontalalignment='center', verticalalignment='top') pylab.text(xc[ng+nzones-1], -0.1, r"$N-1$", horizontalalignment='center', verticalalignment='top') pylab.text(xc[ng+nzones], -0.1, r"$N$", horizontalalignment='center', verticalalignment='top') pylab.xlim(xl[0]-0.5*dx,xr[2*ng+nzones-1]+0.5*dx) pylab.ylim(-0.25, 0.6) pylab.axis("off") pylab.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95) f = pylab.gcf() f.set_size_inches(8.0,2.0) pylab.savefig("simplegrid_gc.png") pylab.savefig("simplegrid_gc.eps") if __name__== "__main__": simplegrid() ================================================ FILE: advection/old-scripts/simplegrid2.py ================================================ import math import numpy import pylab def simplegrid(): xmin = 0.0 xmax = 1.0 nzones = 7 dx = (xmax - xmin)/float(nzones) xl = numpy.arange(nzones)*dx xr = (numpy.arange(nzones)+1)*dx xc = 0.5*(xl + xr) pylab.plot([xmin-0.5*dx,xmax+0.5*dx], [0,0], color="k", lw=2) pylab.plot([xl[0], xl[0]], [0, 0.5], color="k", lw=2) n = 0 while (n < nzones): # draw right edge pylab.plot([xr[n], xr[n]], [0, 0.5], color="k", lw=2) # draw center marker pylab.plot([xc[n], xc[n]], [-0.05, 0], color="k") # draw edge marker if (n == 0): pylab.plot([xl[0], xl[0]], [-0.05, 0], color="k") pylab.plot([xr[n], xr[n]], [-0.05, 0], color="k") n += 1 # label a few cell-centers pylab.text(xc[nzones/2], -0.1, r"$i$", horizontalalignment='center', verticalalignment='top', fontsize="small") pylab.text(xc[nzones/2-1], -0.1, r"$i-1$", horizontalalignment='center', verticalalignment='top', fontsize="small") pylab.text(xc[nzones/2+1], -0.1, r"$i+1$", horizontalalignment='center', verticalalignment='top', fontsize="small") # label a few edges pylab.text(xl[nzones/2], -0.075, r"$i-1/2$", horizontalalignment='center', verticalalignment='top', fontsize="small") pylab.text(xr[nzones/2], -0.075, r"$i+1/2$", horizontalalignment='center', verticalalignment='top', fontsize="small") pylab.axis([xmin-0.5*dx,xmax+0.5*dx, -0.25, 0.6]) pylab.axis("off") pylab.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95) f = pylab.gcf() f.set_size_inches(8.0,2.0) pylab.savefig("simplegrid2.png") pylab.savefig("simplegrid2.eps") if __name__== "__main__": simplegrid() ================================================ FILE: burgers/burgers.tex ================================================ \label{ch:burgers} \section{Burgers' equation} The inviscid Burgers' equation is the simplest {\em nonlinear} hyperbolic equation: \begin{equation} u_t + u u_x = 0 \end{equation} Here $u$ is both the quantity being advected and the speed at which it is moving. Recall that for the linear advection equation, we saw that the solution was constant along lines $x = ut + x_0$, which are parallel, since $u$ is spatially constant. For Burgers' equation, this is no longer the case, and the characteristic lines are now given by $dx/dt = u$, with $x(0) = x_0$. Since $u = u(t)$, we cannot integrate this directly. \begin{exercise}[Burgers' characteristics] {To find the lines along which the solution to Burgers' equation is constant, consider the full change in $u$: \begin{equation} du = \left . \frac{\partial u}{\partial t} \right |_x dt + \left . \frac{\partial u}{\partial x} \right |_t dx = 0 \end{equation} Here, we seek $u = \mathrm{constant}$, so we set $du = 0$. Now, using the Burgers' equation itself, show that this implies that the lines along which the solution is constant are: \begin{equation} \frac{dx}{dt} = u(x,t) \end{equation} } \end{exercise} If we take $u_0 = u(t=0)$, then we can look at how the characteristic behave over a small time interval (before $u(x,t)$ changes significantly). Figure~\ref{fig:burgers_char} shows the behavior for an initial velocity with sinusoidal profile. We see that after a short period of time, the characteristics intersect. At the point, $(x_s, t_s)$ where they intersect, there is no way to trace backwards along the characteristics to find a unique initial state. This merging of the characteristics in the $x$-$t$ plane is a {\em shock}, and represents just one way that nonlinear problems can differ from linear ones. Another type of wave not present in a linear system is a {\em rarefaction}. Figure~\ref{fig:burgers_char_rare} shows initial conditions of slower velocity to the left of faster velocity. We see that the characteristics diverge in this case, and we will be left with having to fill in the solution inbetween as some intermediate state. % figure created with figures/burgers/characteristics.py \begin{figure}[t] \centering \includegraphics[width=0.6\linewidth]{burgers-characteristics} \caption[Characteristics for shock initial conditions] {\label{fig:burgers_char} (top) Initially sinusoidal velocity distribution (bottom) Approximate characteristic structure for Burgers' equation, using $u_0 = u(t)$. Note that after a short period of time, the characteristics intersect, signaling the formation of a shock.} \end{figure} % figure created with figures/burgers/characteristics.py \begin{figure}[t] \centering \includegraphics[width=0.6\linewidth]{burgers-characteristics-rare} \caption[Characteristics for rarefaction initial conditions] {\label{fig:burgers_char_rare} (top) Initially discontinuous velocity distribution with low velocity left of high velocity (bottom) Approximate characteristic structure for Burgers' equation, using $u_0 = u(t)$. Note that after a short period of time, the characteristics diverge---this is a rarefaction.} \end{figure} In conservative form, Burgers' equation appears as: \begin{equation} u_t + \left [\tfrac{1}{2} u^2 \right ]_x = 0 \end{equation} The solution of this follows the same methodology as described in Chapter~\ref{ch:advection}. The interface states are predicted as to second order by Taylor expanding in space and time: \begin{eqnarray} u^{n+1}_{i+\myhalf,L} &=& u^n_i + \frac{\Delta x}{2} \frac{\partial u}{\partial x} + \frac{\Delta t}{2} \left . \frac{\partial u}{\partial t} \right |_i + \ldots \\ &=& u^n_i + \frac{\Delta x}{2} \frac{\partial u}{\partial x} + \frac{\Delta t}{2} \left . \left (-u_i \frac{\partial u}{\partial x} \right ) \right |_i + \ldots \\ &=& u^n_i + \frac{\Delta x}{2} \left ( 1 - \frac{\Delta t}{\Delta x} u_i \right ) \left . \frac{\partial u}{\partial x} \right |_i + \ldots \end{eqnarray} The only difference with the linear advection equation is that now $u_i \Delta t/\Delta x$ varies from zone to zone, whereas with linear advection, it is the constant $C$. The slopes are computed using the same limiters as with linear advection. The Riemann problem differs from linear advection. As we saw above, the characteristic curves can intersect in the $x$-$t$ plane, and it is not possible to trace backward from time to learn where the flow originated. This is the condition for a {\em shock}. % figure made by figures/burgers/rh.py \begin{figure}[t] \centering \includegraphics[width=4in]{rh} \caption[Rankine-Hugoniot conditions]{\label{fig:rh} A rightward moving shock in the $x$-$t$ plane separating two states: $u_l$ and $u_r$.} \end{figure} The shock speed is computed through the {\em Rankine-Hugoniot} jump conditions. For a scalar equation, these are easy to construct. We'll follow the method of \cite{leveque:2002}. Figure~\ref{fig:rh} shows two states separated by a rightward moving shock in the $x$-$t$ plane. At time $t^n$, the state in our interval ($x \in [x_l,x_r]$) is entirely $u_r$. As time evolves, we imagine our interval $[x_l,x_r]$ moving vertically 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 $u_l$. The shock moves with a speed $S = \Delta x/\Delta t$ in this figure. To determine the speed, we integrate our conservation law over both space and time (and normalize by $\Delta x = x_r - x_l$): \begin{equation} \frac{1}{\Delta x} \int_{x_l}^{x_r} dx \int_{t^n}^{t^{n+1}} dt\, u_t = - \frac{1}{\Delta x} \int_{x_l}^{x_r} dx \int_{t^n}^{t^{n+1}} dt \left [ f(u) \right ]_x \end{equation} Doing the $t$ integral on the left and $x$ integral on the right, we have \begin{equation} \frac{1}{\Delta x} \int_{x_l}^{x_r}\left \{ u(t^{n+1}) - u(t^n) \right \} dx = - \frac{1}{\Delta x} \int_{t^n}^{t^{n+1}} \left \{ f(u) |_{x=x_r} - f(u) |_{x=x_l} \right \} dt \end{equation} Recognizing that at $t=t^n$, $u = u_r$ and at $t=t^{n+1}$, $u = u_l$, in the left side becomes \begin{equation} \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 . \end{equation} For the right side, we see that all along $x=x_l$ the flux is $f = f(u_l)$ for $t\in [t^n, t^{n+1}]$. Likewise, all along $x=x_r$, the flux is $f = f(u_r)$ in the same time interval (see the figure). Therefore, our expression becomes: \begin{equation} (u_l - u_r) = -\frac{\Delta t}{\Delta x} \left [ f(u_r) - f(u_l)\right ] \end{equation} and using $S = \Delta x/\Delta t$, we see \begin{equation} S = \frac{f(u_r) - f(u_l)}{u_r - u_l} \end{equation} For Burgers' equation, substituting in $f(u) = u^2/2$, we get \begin{equation} S = \frac{1}{2}(u_l + u_r) \end{equation} With the shock speed known, the Riemann problem is straightforward. If there is a shock (compression, so $u_l > u_r$) then we compute the shock speed and check whether the shock is moving to the left or right, and then use the appropriate state. If there is no shock, then we can simply use upwinding, as there is no ambiguity as to how to trace backwards in time to the correct state. Putting this together, we have: \begin{eqnarray} \mathrm{if~} \underset{\text{(shock)}}{u_l > u_r}:&& u_s = \left \{ \begin{array}{cl} u_l & \mathrm{if~} S > 0 \\ u_r & \mathrm{if~} S < 0 \\ 0 & \mathrm{if~} S = 0 \end{array} \right . \\[1em] % \mathrm{otherwise:}&& u_s = \left \{ \begin{array}{clc} u_l & \mathrm{if~} u_l > 0 \\ u_r & \mathrm{if~} u_r < 0 \\ 0 & \mathrm{otherwise} \end{array} \right . \end{eqnarray} \MarginPar{other forms of this from incompressible?} Once the interface states are constructed, the flux is calculated as: \begin{equation} F^{n+\myhalf}_{i+\myhalf} = \frac{1}{2} \left (u_{i+\myhalf}^{n+\myhalf} \right )^2 \end{equation} and the conservative update is \begin{equation} u_i^{n+1} = u_i^n + \frac{\Delta t}{\Delta x} \left ( F_{i-\myhalf}^{n+\myhalf} - F_{i+\myhalf}^{n+\myhalf} \right ) \end{equation} The timestep constraint now must consider the most restrictive Courant condition over all the zones: \begin{equation} \Delta t = \min_i \left \{ \Delta x / u_i \right \} \end{equation} \begin{figure}[t] \centering \includegraphics[width=0.8\linewidth]{fv-burger-rarefaction} \caption[Rarefaction solution to the inviscid Burgers' equation]{\label{fig:burgers-rarefaction} Solution to the inviscid Burgers' equation with 256 zones and a Courant number, $C = 0.8$ for initial conditions that generate a rarefaction: the left half of the domain was initialized with $u = 1$ and the right half with $u = 2$. This initial velocity state creates a divergent flow. The curves are shown 0.02~s apart, with the darker grayscale representing later in time. \\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/burgers/burgers.py}{burgers.py}}} \end{figure} Figure~\ref{fig:burgers-rarefaction} shows the solution to Burgers' equation using the 2nd-order piecewise linear method described here, with the MC limiter. The initial conditions chosen are all positive velocity, with a lower velocity to the left of the higher velocity. As the solution evolves, the state on the right will rush away from the state on the left, and spread out like a fan. This is called a {\em rarefaction wave} or simply a {\em rarefaction}. \begin{figure}[t] \centering \includegraphics[width=0.8\linewidth]{fv-burger-sine} \caption[Shock solutions to the inviscid Burgers' equation]{\label{fig:burgers-shock} Solution to the inviscid Burgers' equation with 256 zones and a Courant number, $C = 0.8$. The initial conditions here are sinusoidal and the solution quickly steepens into a shock.The curves are shown 0.02~s apart, with the darker grayscale representing later in time. \\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/burgers/burgers.py}{burgers.py}}} \end{figure} Figure~\ref{fig:burgers-shock} shows the solution to Burgers' equation with initially sinusoidal data: \begin{equation} \label{eq:burgers:shockic} \renewcommand{\arraystretch}{1.75} u(x, t=0) = \left \{ \begin{array}{cc} 1 & x < 1/3 \\ 1 + \frac{1}{2} \sin\left (\frac{2\pi (x - 1/3)}{1/3}\right ) & 1/3 \le x \le 2/3 \\ 1 & x > 2/3 \end{array} \right . \renewcommand{\arraystretch}{1.0} \end{equation} This is analogous to the case shown in Figure~\ref{fig:burgers_char}---we see the solution steepen and form a shock which propagates to the right. The shock is practically infinitesimally thin here, since there is no explicitly viscosity to smear it out\footnote{Numerical diffusion, in the form of truncation error of our method, will smear things a little.}. \begin{exercise}[Simple Burgers' solver] {Extend your 1-d finite-volume solver for advection (from Exercise~\ref{adv:ex:fv}) to solve Burgers' equation. You will need to change the Riemann solver and use the local velocity in the construction of the interface states. Run the examples shown in Figures~\ref{fig:burgers-rarefaction} and \ref{fig:burgers-shock}}. \end{exercise} As we'll see shortly, these two types of waves can also appear in the Euler equations for hydrodynamics. A final thing to note is that we solved Burgers' equation in conservative form. For shock solutions, this is essential, since as we noted earlier, that finite-volume method relates to the integral form of the PDEs, the discontinuity is nicely handled by the Riemann solver. We could also imagine differencing Burgers' equation in non-conservative form, i.e., starting with: \begin{equation} u_t + u u_x = 0 \end{equation} If you did this (for instance, using a finite difference scheme and an upwind difference for $u_x$), you would find that you get the wrong shock speed. \begin{exercise}[Conservative form of Burgers' equation] { Using a simple first-order finite-difference method like we described in Ch.~\ref{ch:advection} for linear advection, difference the conservative and non-conservation formulations of Burgers' equation as: \begin{equation} \frac{u^{n+1}_i - u^n_i}{\Delta t} = -\frac{1}{2} \frac{(u^n_i)^2 - (u^n_{i-1})^2}{\Delta x} \end{equation} and \begin{equation} \frac{u_{i}^{n+1} - u_i^n}{\Delta t} = - \frac{u_i^n (u_i^n - u_{i-1}^n)}{\Delta x} \end{equation} (Note: these discretizations are upwind so long as $u > 0$). Run these with the shock initial Riemann conditions: \begin{equation} \renewcommand{\arraystretch}{1.75} u(x, t=0) = \left \{ \begin{array}{cc} 2 & x < 1/2 \\ 1 & x > 1/2 \end{array} \right . \renewcommand{\arraystretch}{1.0} \end{equation} and measure the shock speed from your solution by simply differencing the location of the discontinuity at two different times. Compare to the analytic solution for a shock for the Riemann problem.} \end{exercise} \section{Characteristic tracing} A concept that will be useful in the next section is {\em characteristic tracing}. The idea is that we only include information in the interface states if the characteristic that carries that information is moving toward the interface. For Burgers equation, this is simple, since there is only a single characteristic---the velocity. So for the left state on an interface, we'd only add the change if the velocity $u_i > 0$ (moving toward the interface $i+\myhalf$): \begin{equation} u^{n+1}_{i+\myhalf,L} = u^n_i + \frac{\Delta x}{2} \left ( 1 - \frac{\Delta t}{\Delta x} \max(0, u_i) \right ) \left . \frac{\partial u}{\partial x} \right |_i + \ldots \end{equation} Notice that the effect of this is to set the interface state simply to the value given by the piecewise linear reconstruction on the interface if the wave isn't moving to the interface. \section{Going further} \begin{itemize} \item The equation we've been dealing with here is the {\em inviscid} Burgers' equation. The full Burgers' equation includes viscosity (a velocity diffusion): \begin{equation} u_t + u u_x = \epsilon u_{xx} \end{equation} To solve this, we need to first learn about techniques for diffusion, and then how to solve equations that span multiple PDE types. This will be described in \S \ref{ch:multiphysics:sec:adburgers}. \item Aside from pedagogical interest, Burgers' equation can be used as a simple model of traffic flow (where shocks can arise from people slamming on the brakes). Many sources discuss this application, including the text by~\cite{leveque:2002}. \MarginPar{more refs?} \end{itemize} ================================================ FILE: codes/intro/sin-converge.py ================================================ import math X = [0.5, 0.25, 0.125, 0.0625] for x in X: print x, math.sin(x) - (x - x**3/6.0) ================================================ FILE: diffusion/GNUmakefile ================================================ EPStoPDF = epstopdf ALL: diffusion.pdf diffusion.pdf: main.tex diffusion.tex pdflatex -jobname=diffusion main.tex < /dev/null bibtex diffusion.aux pdflatex -jobname=diffusion main.tex < /dev/null pdflatex -jobname=diffusion main.tex < /dev/null clean: $(RM) *.aux *.log *.dvi *.bbl *.blg $(RM) *~ .PHONY: clean ================================================ FILE: diffusion/diffusion.tex ================================================ \label{ch:diffusion} \section{Diffusion} \label{sec:diffusion} Physically, a diffusive process obeys Fick's law---the quantity that is diffusing, $\phi$, moves from higher to lower concentration at a rate proportional to the gradient, \begin{equation} F = - k \nabla \phi \end{equation} If we think of this as a diffusive flux, then we can write the time-rate-of-change of $\phi$ as a conservation law: \begin{equation} \phi_t + \nabla\cdot F(\phi) = 0 \end{equation} This gives rise to the diffusion equation: \begin{equation} \phi_t = \nabla \cdot (k \nabla \phi) \end{equation} Diffusion has a time-dependence like the advection equations we already saw, but also has a global character like an elliptic problem. Unlike advection, there is no sense of upwinding in diffusion. As we will see, we can cast our differenced equations in a form reminiscent of an elliptic problem. In one-dimension, and assuming that the diffusion coefficient, $k$, is constant, we have \begin{equation} \frac{\partial \phi}{\partial t} = \frac{\partial }{\partial x} \left ( k \frac{\partial \phi}{\partial x} \right ) \end{equation} The diffusion equation can describe thermal diffusion (for example, as part of the energy equation in compressible flow), species/mass diffusion for multi-species flows, or the viscous terms in incompressible flows. In this form, the diffusion coefficient (or conductivity), $k$, can be a function of $x$, or even $\phi$. We will consider a constant diffusion coefficient as our model problem: \begin{equation} \frac{\partial \phi}{\partial t} = k \frac{\partial^2 \phi}{\partial x^2} \end{equation} The diffusion equation is the prototypical parabolic PDE. The basic behavior of the diffusion equation is to take strongly peaked concentrations of $\phi$ and smooth them out with time. \section{Explicit differencing} A nice overview of the discretizations for diffusion and their stability is given in \cite{richtmyermorton}. The simplest way to difference this equation is {\em explicit} in time (i.e., the righthand side depends only on the old state): \begin{equation} \label{eq:diff:explicitdiff} \frac{\phi_i^{n+1} - \phi_i^n}{\Delta t} = k \frac{\phi_{i+1}^n - 2\phi_i^n + \phi_{i-1}^n}{\Delta x^2} \end{equation} This is second-order accurate in space, but only first order accurate in time (since the righthand side is not centered in time). As with the advection equation, when differenced explicitly, there is a constraint on the timestep required for stability. Looking at the growth of a single Fourier mode, $\phi^n_i = A^n e^{i\imag\theta}$ with $\imag = \sqrt{-1}$, we find: \begin{equation} \label{eq:diff:amplitude} \frac{A^{n+1}}{A^n} = 1 + 2 \frac{k \Delta t}{\Delta x^2} ( \cos\theta - 1) \end{equation} Stability requires that $|A^{n+1}/A^n| \le 1$, which can only be true if $2k\Delta t/\Delta x^2 \le 1$. Therefore, our timestep constraint in this case is \begin{equation} \label{eq:diff:dt} \Delta t < \frac{1}{2} \frac{\Delta x^2}{k} \end{equation} As with advection, we often write the timestep as \begin{equation} \Delta t = \frac{C}{2} \frac{\Delta x^2}{k} \end{equation} where $C$ is a constant, $C < 1$. Note the $\Delta x^2$ dependence---this constraint can become really restrictive. \begin{exercise} [Explicit diffusion stability condition] Derive Eq.~\ref{eq:diff:amplitude} by inserting $\phi^n_i = A^n e^{ij\theta}$ into Eq.~\ref{eq:diff:explicitdiff} \end{exercise} To complete the solution, we need boundary conditions at the left ($x_l$) and right ($x_r$) boundaries. These are typically either Dirichlet: \begin{eqnarray} \phi |_{x=x_l} &=& \phi_l\\ \phi |_{x=x_r} &=& \phi_r \end{eqnarray} or Neumann: \begin{eqnarray} \phi_x |_{x=x_l} &=& \phi_x |_l\\ \phi_x |_{x=x_r} &=& \phi_x |_r \end{eqnarray} Physically, a Dirichlet BC means fixing the value of $\phi$ (e.g., temperature) on the boundary. A Neumann BC means fixing the flux on the boundary. Like with advection, it helps to have a good test problem to evaluate our methods for diffusion. We can use the fact that the diffusion of a Gaussian is a Gaussian with a smaller amplitude and greaterwidth. \begin{exercise}[1-d explicit diffusion] {Write a one-dimensional explicit diffusion solver for the domain $[0,1]$ with Neumann boundary conditions at each end and $k = 1$, using the discretization of Eq.~\ref{eq:diff:explicitdiff}. If we begin with a Gaussian, the resulting solution is also a Gaussian, giving a solution\footnote{Note: the 2- and 3-d solutions are slightly different than this 1-d solution}: \begin{equation} \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 \end{equation} Initialize our problem with $t = 0$, and take $t_0 = 0.001$, $\phi_1 = 1$, and $\phi_2 = 2$, and $x_c$ is the coordinate of the center of the domain. Run until $t = 0.01$ and compare to the analytic solution above.} \end{exercise} Figure~\ref{fig:diff_explicit} shows the solution for 64 zones and a value of $C = 0.8$. We see good agreement with the analytic solution. Note that if the initial conditions are not well resolve initially, then the solution will be bad (see Figure~\ref{fig:diff_explicit_res}). \begin{figure} \centering \includegraphics[width=\linewidth]{diff-explicit-64} \caption[Explicit diffusion of a Gaussian]{\label{fig:diff_explicit} Diffusion of a Gaussian using the explicit differencing of Eq.~\ref{eq:diff:explicitdiff} with 64 zones and C = 0.8, shown at several times. The dotted line is the analytic solution. \\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/diffusion/diffusion_explicit.py}{diffusion\_explicit.py}}} \end{figure} \begin{figure} \centering \includegraphics[width=\linewidth]{diffexplicit-res} \caption[Underresolved explicit diffusion of a Gaussian]{\label{fig:diff_explicit_res} Diffusion of a Gaussian using the explicit differencing of Eq.~\ref{eq:diff:explicitdiff} with different resolutions. several times. The dotted line is the analytic solution. \\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/diffusion/diffusion_explicit.py}{diffusion\_explicit.py}}} \end{figure} As with advection, if we exceed the timestep limit ($C > 1$), then the solution is unstable. This is shown in Figure~\ref{fig:diff_explicit_bad}. \begin{figure} \centering \includegraphics[width=\linewidth]{diff-explicit-64-bad} \caption[Unstable explicit diffusion]{\label{fig:diff_explicit_bad} Diffusion of a Gaussian using the explicit differencing of Eq.~\ref{eq:diff:explicitdiff} with 64 zones, but a timestep with $C > 1$, showing that the solution is unstable. \\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/diffusion/diffusion_explicit.py}{diffusion\_explicit.py}}} \end{figure} Our spatial order-of-accuracy is second-order. Figure~\ref{fig:diffexplicit_converge} shows the error as a function of number of zones, using the $L_2$ norm of the solution with respect to the analytic solution. Notice that at the coarsest resolution the error is very high---we are not resolving the initial conditions well enough to have a meaningful solution. At higher resolutions, we converge as $O(\Delta x^2)$. Recall though that our method has a truncation error that is $O(\Delta t) + O(\Delta x^2)$, but we don't see the first-order in time scaling. This is because of the timestep restriction. Since $\Delta t \sim \Delta x^2/k$, as we cut $\Delta x$ by 2, $\Delta t$ drops by 4, so the timestep choice makes our $O(\Delta t)$ truncation error go as $O(\Delta x^2)$. Note, this is not the case for advection, where $\Delta t \sim \Delta x$, so for our advection discretizations we will want the same order of accuracy in our spatial and temporal discretizations. \begin{figure} \centering \includegraphics[width=\linewidth]{diffexplicit-converge-0_8} \caption[Error convergence of explicit diffusion]{\label{fig:diffexplicit_converge} Error convergence with resolution of the explicit diffusion using the differencing of Eq.~\ref{eq:diff:explicitdiff} \\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/diffusion/diffusion_explicit.py}{diffusion\_explicit.py}}} \end{figure} \section{Implicit with direct solve} Recall that an implicit discretization of advection did not have a timestep restriction for stability. The same holds true for diffusion. A backward-Euler implicit discretization would be: \begin{equation} \label{eq:diff:foimplicit} \frac{\phi_i^{n+1} - \phi_i^n}{\Delta t} = k \frac{\phi_{i+1}^{n+1} - 2\phi_i^{n+1} + \phi_{i-1}^{n+1}}{\Delta x^2} \end{equation} The only difference with Eq.~\ref{eq:diff:explicitdiff} is the time-level of $\phi$ on the righthand side. Defining: \begin{equation} \alpha \equiv k \frac{\Delta t}{\Delta x^2} \end{equation} This is still first-order in time. We can do the stability analysis to see the growth of a mode, giving, \begin{equation} \frac{A^{n+1}}{A^n} = \frac{1}{1 - 2\alpha (\cos\theta - 1)} \end{equation} We see that $|A^{n+1}/A^n| \le 1$ for all $\theta$, $\alpha$, so this discretization is unconditionally stable. However, the timestep will still determine the accuracy. We can write Eq.~\ref{eq:diff:foimplicit} as: \begin{equation} \label{eq:diff:implicit} -\alpha \phi_{i+1}^{n+1} + (1 + 2\alpha) \phi_{i}^{n+1} - \alpha \phi_{i-1}^{n+1} = \phi_i^n \end{equation} This is a set of coupled algebraic equations. We can write this in matrix form. Using a cell-centered grid, we will solve for the values $[\mathrm{lo},\mathrm{hi}]$. The implicit method can use any $C > 0$. We specify boundary conditions by modifying the stencil (Eq.~\ref{eq:diff:implicit}) for the updates to $\mathrm{lo}$ and $\mathrm{hi}$\footnote{Here, we use the $\mathrm{lo}$, $\mathrm{hi}$ notation for grid indices from \S~\ref{sec:fv:bcs}}. For example, homogeneous Neumann BCs on the left mean: \begin{equation} \phi_\mathrm{lo-1} = \phi_\mathrm{lo} \end{equation} and substituting this into Eq~\ref{eq:diff:implicit}, the update for the leftmost cell is: \begin{equation} (1 + \alpha) \phi_\mathrm{lo}^{n+1} -\alpha \phi_\mathrm{lo+1}^{n+1} = \phi_\mathrm{lo}^n \end{equation} If we choose Dirichlet BCs on the right ($\phi |_{x=x_l} = A$), then: \begin{equation} \phi_\mathrm{hi+1} = 2 A - \phi_\mathrm{hi} \end{equation} Substituting this into Eq~\ref{eq:diff:implicit} the update for the rightmost cell is: \begin{equation} - \alpha \phi_\mathrm{hi-1}^{n+1} + (1 + 3\alpha) \phi_\mathrm{hi}^{n+1} = \phi_\mathrm{hi}^n + \alpha 2 A \end{equation} For all other interior cells, the stencil is unchanged. The resulting system appears as a {\em tridiagonal} matrix. \begin{equation} \renewcommand\arraystretch{1.5} \left ( \begin{array}{ccccccc} 1+\alpha & -\alpha & & & & & \\ -\alpha & 1+2\alpha & -\alpha & & & & \\ & -\alpha & 1+2\alpha & -\alpha& & & \\ & & \ddots & \ddots & \ddots & & \\ & & & \ddots & \ddots & \ddots & \\ & & & & -\alpha & 1+2\alpha &-\alpha \\ & & & & & -\alpha &1+3\alpha \\ \end{array} \right ) \left ( \begin{array}{c} \phi_\mathrm{lo}^{n+1} \\ \phi_\mathrm{lo+1}^{n+1} \\ \phi_\mathrm{lo+2}^{n+1} \\ \vdots \\ \vdots \\ \phi_\mathrm{hi-1}^{n+1} \\ \phi_\mathrm{hi}^{n+1} \\ \end{array} \right ) = \left ( \begin{array}{c} \phi_\mathrm{lo}^{n} \\ \phi_\mathrm{lo+1}^{n} \\ \phi_\mathrm{lo+2}^{n} \\ \vdots \\ \vdots \\ \phi_\mathrm{hi-1}^{n} \\ \phi_\mathrm{hi}^{n} + \alpha 2 A\\ \end{array} \right ) \end{equation} This can be solved by standard matrix operations, using a tridiagonal solvers (for example). Notice that the ghost cells do not appear in this linear system---we are only updating the interior points. \subsubsection{Crank-Nicolson time discretization} A second-order in time discretization requires us to center the righthand side in time. We do this as: \begin{equation} \frac{\phi_i^{n+1} - \phi_i^n}{\Delta t} = \frac{k}{2} \left ( \frac{\phi_{i+1}^{n} - 2\phi_i^{n} + \phi_{i-1}^{n}}{\Delta x^2} + \frac{\phi_{i+1}^{n+1} - 2\phi_i^{n+1} + \phi_{i-1}^{n+1}}{\Delta x^2} \right ) \end{equation} This looks like the average of the explicit and implicit systems we just saw. This time-discretization is called {\em Crank-Nicolson}. Again, using $\alpha \equiv k\Delta t / \Delta x^2$, and grouping all the $n+1$ terms on the left we have: \begin{equation} \phi^{n+1}_i - \frac{\alpha}{2} \left ( \phi^{n+1}_{i+1} - 2\phi^{n+1}_i + \phi^{n+1}_{i-1} \right ) = \phi^n_i + \frac{\alpha}{2} \left ( \phi^{n}_{i+1} - 2\phi^{n}_i + \phi^{n}_{i-1} \right ) \end{equation} and grouping together the the $n+1$ terms by zone, we have: \begin{equation} -\frac{\alpha}{2} \phi^{n+1}_{i+1} + (1 + \alpha)\phi^{n+1}_i - \frac{\alpha}{2} \phi^{n+1}_{i-1} = \phi^n_i + \frac{\alpha}{2} \left ( \phi^n_{i+1} - 2 \phi^n_i + \phi^n_{i-1} \right ) \end{equation} Considering homogeneous Neumann boundary conditions on the left and right, we again have $\phi^{n+1}_\mathrm{lo-1} = \phi^{n+1}_\mathrm{lo}$ and $\phi^{n+1}_\mathrm{hi+1} = \phi^{n+1}_\mathrm{hi}$, and our stencil at the boundary becomes \begin{equation} -\frac{\alpha}{2} \phi^{n+1}_\mathrm{lo+1} + \left (1 + \frac{\alpha}{2} \right ) \phi^{n+1}_\mathrm{lo} = \phi^n_\mathrm{lo} + \frac{\alpha}{2} \left ( \phi^{n}_\mathrm{lo+1} - 2\phi^{n}_\mathrm{lo} + \phi^{n}_\mathrm{lo-1} \right ) \end{equation} The matrix form of this system is: \newcommand{\atwo}{\frac{\alpha}{2}} \begin{equation} \label{eq:diff:cnmatrix} \renewcommand\arraystretch{1.5} \left ( \begin{array}{ccccccc} 1+\atwo & -\atwo & & & & & \\ -\atwo & 1+\alpha & -\atwo & & & & \\ & -\atwo & 1+\alpha & -\atwo & & & \\ & & \ddots & \ddots & \ddots & & \\ & & & \ddots & \ddots & \ddots & \\ & & & & -\atwo & 1+\alpha &-\atwo \\ & & & & & -\atwo &1+\atwo \\ \end{array} \right ) \left ( \begin{array}{c} \phi_\mathrm{lo}^{n+1} \\ \phi_\mathrm{lo+1}^{n+1} \\ \phi_\mathrm{lo+2}^{n+1} \\ \vdots \\ \vdots \\ \phi_\mathrm{hi-1}^{n+1} \\ \phi_\mathrm{hi}^{n+1} \\ \end{array} \right ) = \left ( \begin{array}{c} \phi_\mathrm{lo}^{n} + \frac{k\Delta t}{2} [\nabla^2 \phi]^n_\mathrm{lo}\\ \phi_\mathrm{lo+1}^{n} + \frac{k\Delta t}{2} [\nabla^2 \phi]^n_\mathrm{lo+1} \\ \phi_\mathrm{lo+2}^{n} + \frac{k\Delta t}{2} [\nabla^2 \phi]^n_\mathrm{lo+2}\\ \vdots \\ \vdots \\ \phi_\mathrm{hi-1}^{n} + \frac{k\Delta t}{2} [\nabla^2 \phi]^n_\mathrm{hi-1}\\ \phi_\mathrm{hi}^{n} + \frac{k\Delta t}{2} [\nabla^2 \phi]^n_\mathrm{hi}\\ \end{array} \right ) \end{equation} Figure~\ref{fig:diffuse} shows the result of using $\alpha = 0.8$ and $\alpha = 8.0$. We see that they are both stable, but that the smaller timestep is closer to the analytic solution (especially at early times). \begin{figure}[t] \centering \includegraphics[width=0.75\linewidth]{diff-implicit-128-CFL_0_8}\\ \includegraphics[width=0.75\linewidth]{diff-implicit-128-CFL_8_0} \caption[Implicit diffusion of a Gaussian]{\label{fig:diffuse} Implicit diffusion of a Gaussian (with Crank-Nicolson discretization) with $C = 0.8$ and $C = 8.0$. The exact solution at each time is shown as the dotted line. \\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/diffusion/diffusion_implicit.py}{diffusion\_implicit.py}}} \end{figure} \begin{exercise}[1-d implicit diffusion] {Write a one-dimensional implicit diffusion solver for the domain $[0,1]$ with Neumann boundary conditions at each end and $k = 1$. Your solver should use a tridiagonal solver and initialize a matrix like that above. Use a timestep close to the explicit step, a grid with N = 128 zones. Use a Gaussian for your initial conditions (as you did for the explicit problem.} \end{exercise} \section{Implicit multi-dimensional diffusion via multigrid} \label{diff:sec:implicit_mg} Instead of doing a direct solve of the matrix form of the system, we can use multigrid techniques. Consider the Crank-Nicolson system we just looked at: \begin{equation} \frac{\phi^{n+1}_i - \phi^n_i}{\Delta t} = \frac{1}{2} \left ( k \nabla^2 \phi^n_i + k \nabla^2 \phi^{n+1}_i \right ) \end{equation} Grouping all the $n+1$ terms on the left, we find: \begin{equation} \phi^{n+1}_i - \frac{\Delta t}{2} k \nabla^2 \phi^{n+1}_i = \phi^n_i + \frac{\Delta t}{2} k \nabla^2 \phi^n_i \end{equation} This is in the form of a constant-coefficient Helmholtz equation, \begin{equation} \label{eq:diff:helmholtz} (\alpha - \beta \nabla^2) \phi^{n+1} = f \end{equation} with \begin{eqnarray} \alpha &=& 1 \\ \beta &=& \frac{\Delta t}{2} k \\ f &=& \phi^n + \frac{\Delta t}{2} k \nabla^2 \phi^n \end{eqnarray} This can be solved using multigrid techniques with a Helmholtz operator. The same boundary conditions discussed in Chapter~\ref{ch:multigrid} apply here. The main difference between the multigrid technique for the Poisson problem and the Helmholtz problem is the form of the smoother. In 1-d, we discretize Eq.~\ref{eq:diff:helmholtz} with a second-order difference expression for the second derivative, and isolate $\phi_i$, giving a smoothing operation of the form: \begin{equation} \label{eq:diff:discrete1d} \phi_{i} \leftarrow \left . \left ( f_{i} + \frac{\beta}{\Delta x^2} [\phi_{i+1} + \phi_{i-1}] \right ) \middle / \left ( \alpha + \frac{2 \beta}{\Delta x^2} \right ) \right . \end{equation} Note that when you take $\alpha = 0$ and $\beta = -1$ in Eq.~\ref{eq:diff:discrete1d}, the smoothing operation reduces to the form that we saw in Chapter~\ref{ch:multigrid} for just the Poisson equation. Recall, when using multigrid, you do not need to actually construct the matrix. This is usually the most efficient way to implement diffusion in a multi-dimensional simulation code, especially when distributing the grid across parallel processors. Since the discretization is the same as the direct matrix solve, Eq.~\ref{eq:diff:cnmatrix}, the result will be exactly the same (to the tolerance of the multigrid solver). \begin{figure}[t] \centering \includegraphics[width=0.48\linewidth]{gauss_diff_start} \includegraphics[width=0.48\linewidth]{gauss_diff_end} \caption[2-d diffusion of a Gaussian]{\label{fig:diff:twodcompare} Diffusion of a Gaussian in 2-d with $128^2$ zones using $k = 1.0$ and $\cfl = 2.0$. This used Crank-Nicolson time-discretization and multigrid. This can be run in \pyro\ as {\tt ./pyro diffusion gaussian inputs.diffusion}.} \end{figure} \begin{figure}[t] \centering \includegraphics[width=0.8\linewidth]{gauss_diffusion_compare} \caption[Comparison of 2-d implicit diffusion with analytic solution]{\label{fig:diffusion:multid} A comparison of 2-d implicit diffusion from Figure~\ref{fig:diff:twodcompare} with the analytic solution at several times. The 2-d solution was averaged over angles to yield a profile as a function of radius, using the {\tt gauss\_diffusion\_compare.py} in \pyro.} \end{figure} Figure~\ref{fig:diffusion:multid} shows the result of diffusing a Gaussian profile on a 2-d grid using Crank-Nicolson time-discretization, a constant coefficient, and multigrid to solve the discretized system, using \pyro. There is a good agreement between the analytic and numerical solution, showing that this scheme models the diffusion of an initially resolved Gaussian well. \begin{exercise}[Implicit multi-dimensional diffusion] {The diffusion solver in \pyro\ uses Crank-Nicolson differencing in time. Modify the solver to do first-order backward Euler. This will change the form of the linear system (coefficients and righthand side), but should not require any changes to the multigrid solver itself. Compare the solution with the Crank-Nicolson solution for a very coarsely-resolved Gaussian and a finely-resolved Gaussian. } \end{exercise} \subsection{Convergence} One needs to be careful with the Crank-Nicolson discretization. If the initial data is under-resolved and you are taking a big timestep ($C \gg 1$), then the method can be unstable. Figure~\ref{fig:diff:cnunstable} shows such a case for the Gaussian diffusion with Crank-Nicolson discretization and 64 zones with $C = 10$. For this reason, simulation codes often drop down to the simple backwards difference time-discretization for implicit diffusion of under-resolved flows. A good discussion of the different types of stability can be found in \cite{leveque:fd}. \begin{figure} \centering \includegraphics[width=\linewidth]{diff-implicit-64-CFL_10_0} \caption[Under-resolved Crank-Nicolson diffusion] {\label{fig:diff:cnunstable} Crank-Nicolson diffusion of a Gaussian on under-resolved initial data with a large timestep. Here we use $64$ zones and $C= 10$. \\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/diffusion/diffusion_implicit.py}{diffusion\_implicit.py}}} \end{figure} \begin{figure} \centering \includegraphics[width=0.75\linewidth]{diffimplicit-converge-0_8} \\ \includegraphics[width=0.75\linewidth]{diffimplicit-converge-8_0} \caption[Convergence of diffusion methods] {\label{fig:diff:convergence} Convergence of the explicit, backward-difference, and Crank-Nicolson diffusion methods for $C = 0.8$ (top) and $C = 8.0$ (bottom). For the latter case, the explicit method is not valid and is not shown. We see that the Crank-Nicolson method has the lowest error, and when resolving the data, has second-order convergence.\\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/diffusion/diff_converge.py}{diff\_converge.py}}} \end{figure} Figure~\ref{fig:diff:convergence} shows the convergence of several of the methods discussed here on the diffusion of a Gaussian, for two different Courant numbers, 0.8 and 8.0. For the lower case, we can do the diffusion explicitly as well as implicitly, and we see that all methods show the same trends, with the Crank-Nicolson method being the most accurate (since it is the only second-order-in-time method shown). For the larger timestep, we only run the implicit methods (the explicit case is unstable). We see that at coarse resolution, the errors in the methods are similar---this is because the solution is unresolved. At higher resolution, the error for the Crank-Nicolson method is an order of magnitude lower. \section{Non-constant Conductivity} For a non-constant conductivity, our equation has the form: \begin{equation} \frac{\partial \phi}{\partial t} = \frac{\partial}{\partial x} \left (k \frac{\partial \phi}{\partial x} \right ) \end{equation} For the case where $k = k(x)$, we discretize as: \begin{equation} \frac{\phi_i^{n+1} - \phi_i^n}{\Delta t} = \frac{ \{ k \nabla \phi \}_{i+\myhalf} - \{ k \nabla \phi \}_{i-\myhalf}}{\Delta x} \end{equation} Here we need the values of $k$ at the interfaces, $k_{i-\myhalf}$ and $k_{i+\myhalf}$. We can get these from the cell-centered values in a variety of ways including straight-averaging: \begin{equation} k_{i+\myhalf} = \frac{1}{2} (k_i + k_{i+1}) \end{equation} or averaging the inverses: \begin{equation} \frac{1}{k_{i+\myhalf}} = \frac{1}{2} \left (\frac{1}{k_i} + \frac{1}{k_{i+1}} \right ) \end{equation} The actual form should be motivated by the physics Slightly more complicated are state-dependent transport coefficients---the transport coefficients themselves depend on the quantity being diffused: \begin{equation} \frac{\phi_i^{n+1} - \phi_i^n}{\Delta t} = \frac{1}{2} \left \{ \nabla \cdot [ k(\phi^n) \nabla \phi^n ]_i + \nabla \cdot [ k(\phi^{n+1}) \nabla \phi^{n+1} ]_i \right \} \end{equation} (for example, with thermal diffusion, the conductivity can be temperature dependent). In this case, we can achieve second-order accuracy by doing a predictor-corrector. First we diffuse with the transport coefficients evaluated at the old time, giving a provisional state, $\phi^\star$: \begin{equation} \frac{\phi_i^\star - \phi_i^n}{\Delta t} = \frac{1}{2} \left \{ \nabla \cdot [ k(\phi^n) \nabla \phi^n ] + \nabla \cdot [ k(\phi^n) \nabla \phi^\star ] \right \} \end{equation} Then we redo the diffusion, evaluating $k$ with $\phi^\star$ to center the righthand side in time, giving the new state, $\phi^{n+1}$: \begin{equation} \frac{\phi_i^{n+1} - \phi_i^n}{\Delta t} = \frac{1}{2} \left \{ \nabla \cdot [ k(\phi^n) \nabla \phi^n ] + \nabla \cdot [ k(\phi^\star) \nabla \phi^{n+1} ] \right \} \end{equation} This is the approach used, for example, in \cite{SNpaper}. \section{Diffusion in Hydrodynamics} Often we find diffusion represented as one of many physical processes in a single equation. For example, consider the internal energy equation with both reactions and diffusion: \begin{equation} \rho\frac{\partial e}{\partial t} + \rho U \cdot \nabla e + p \nabla \cdot U = \nabla \cdot k \nabla T + \rho S \end{equation} This can be solved via an explicit-implicit discretization. First the advection terms are computed as: \begin{equation} A = \rho U \cdot \nabla e + p \nabla \cdot U \end{equation} Then the advective-diffusive part is solved implicitly. Expressing $e = e(\rho, T)$, and using the chain rule, \begin{equation} \nabla e = e_T \nabla T + e_\rho \nabla \rho \end{equation} where $e_T = \partial e / \partial T |_\rho \equiv c_v$ is the specific heat at constant volume and $e_\rho = \partial e / \partial \rho |_T$. Rewriting, we have: \begin{equation} \nabla T = (\nabla e - e_\rho \nabla \rho)/c_v \end{equation} and then \begin{equation} \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 \end{equation} This is now a diffusion equation for $e$, which can be solved by the techniques described above. Note: if the transport coefficients (including $c_v$, $e_\rho$) are dependent on $e$, then we still need to do a predictor-corrector method here. This is discussed, for example, in \cite{SNpaper,malone:2011}. A simple case for this type of advection-diffusion is also shown in \S~\ref{sec:multiphys:diffreact}. ================================================ FILE: figures/Euler/multiple_interfaces.py ================================================ import numpy as np import matplotlib.pyplot as plt import matplotlib as mpl import random import grid_plot as gp mpl.rcParams['mathtext.fontset'] = 'cm' mpl.rcParams['mathtext.rm'] = 'serif' # font sizes mpl.rcParams['font.size'] = 12 mpl.rcParams['legend.fontsize'] = 'large' mpl.rcParams['figure.titlesize'] = 'medium' class RiemannWaves(object): """draw a diagram with a the three waves """ def __init__(self, x0, dx): """ create a random collection of Riemann waves """ states = ["shock", "rarefaction"] self.x0 = x0 self.dx = dx # first pick if the angle of the contact, from a normalize distribution centered on 0 _a = np.degrees(0.4*np.random.randn()) while _a > 45: _a = np.degrees(0.4*np.random.randn()) self.contact_pos = _a print(_a) # now the left state -- first set how far it is from the contact _a = np.degrees(0.4*np.random.randn()) while _a > 45: _a = np.degrees(0.4*np.random.randn()) print(_a) self.left_pos = self.contact_pos - np.abs(_a) self.left_type = random.choice(states) # and the right _a = np.degrees(0.4*np.random.randn()) while _a > 45: _a = np.degrees(0.4*np.random.randn()) print(_a) self.right_pos = self.contact_pos + np.abs(_a) self.right_type = random.choice(states) def draw(self, output="plot.png", label=None): """ draw the wave structure """ print(self.left_pos, self.contact_pos, self.right_pos) # each wave has a point on the origin. The maximum height we # want to draw to is self.dx -- we'll do the center and right x1 = self.x0 + self.dx * np.tan(np.radians(self.left_pos)) xc = self.x0 + self.dx * np.tan(np.radians(self.contact_pos)) x3 = self.x0 + self.dx * np.tan(np.radians(self.right_pos)) # draw the waves if self.left_type == "shock": plt.plot([self.x0, x1], [0, self.dx], color="C0", lw=3) else: for n in range(5): x1 = self.x0 + self.dx * np.tan(np.radians(self.left_pos - 2*n)) plt.plot([self.x0, x1], [0, self.dx], color="C0", lw=1) plt.plot([self.x0, xc], [0, self.dx], color="C1") if self.right_type == "shock": plt.plot([self.x0, x3], [0, self.dx], color="C0", lw=3) else: for n in range(5): x3 = self.x0 + self.dx * np.tan(np.radians(self.right_pos + 2*n)) plt.plot([self.x0, x3], [0, self.dx], color="C0", lw=1) if __name__ == "__main__": xmin = 0.0 xmax = 8.0 nzones = 4 ng = 0 gr = gp.FVGrid(nzones, xmin=xmin, xmax=xmax) gr.draw_grid() gr.label_edge(0, r"$i-\mythreehalf$", fontsize="medium") gr.label_edge(1, r"$i-\myhalf$", fontsize="medium") gr.label_edge(2, r"$i+\myhalf$", fontsize="medium") gr.label_edge(3, r"$i+\mythreehalf$", fontsize="medium") gr.label_edge(3, r"$i+\myfivehalf$", fontsize="medium", right_edge=True) gr.label_center(0, r"$i-1$", fontsize="medium") gr.label_center(1, r"$i$", fontsize="medium") gr.label_center(2, r"$i+1$", fontsize="medium") gr.label_center(3, r"$i+2$", fontsize="medium") riemann = [] for n in range(nzones): riemann.append(RiemannWaves(gr.xl[n], 0.5*gr.dx)) riemann.append(RiemannWaves(gr.xr[nzones-1], 0.5*gr.dx)) # draw for r in riemann: r.draw() gr.clean_axes() f = plt.gcf() f.set_size_inches(8.0, 2.5) plt.tight_layout() plt.savefig("multiple_interfaces.pdf", dpi=150) ================================================ FILE: figures/Euler/ppm-sequence.py ================================================ import math import numpy as np import matplotlib.pyplot as plt import grid_plot as gp import riemann class RiemannProblem(object): def __init__(self, q_l, q_r, gamma=5./3., N=3): self.q_l = q_l self.q_r = q_r self.nx = 2*N self.gr = gp. FVGrid(self.nx) # allocate space -- we just care about N zones to the left and # right of the interface self.vars = {} rho = self.gr.scratch_array() rho[0:N] = q_l.rho rho[N:2**N] = q_r.rho u = self.gr.scratch_array() u[0:N] = q_l.u u[N:2*N] = q_r.u p = self.gr.scratch_array() p[0:N] = q_l.p p[N:2*N] = q_r.p rscale = max(q_l.rho, q_r.rho) uscale = max(q_l.u, q_r.u) pscale = max(q_l.p, q_r.p) # do the ppm reconstruction on all of these variables self.vars["rho"] = gp.PiecewiseParabolic(self.gr, rho, scale=rscale) self.vars["u"] = gp.PiecewiseParabolic(self.gr, u, scale=uscale) self.vars["p"] = gp.PiecewiseParabolic(self.gr, p, scale=pscale) def draw_cubic_points(self, var, color="r"): # these are defined on the i+1/2 interface # note that we require 2 zones on either side of the interface, so # we cannot draw points for the first 2 or last 2 zones aint = self.vars[var].aint plt.scatter(self.gr.xr[1:-2], aint[1:-2], marker="x", s=40, color=color, zorder=10) def draw_parabola(self, var): ap, am = self.vars[var].ap, self.vars[var].am for n in range(2,self.nx-2): self.vars[var].draw_parabola(n) def draw_grid(self): self.gr.draw_grid() self.gr.label_center(self.nx/2, r"$i$") self.gr.label_center(self.nx/2-1, r"$i-1$") self.gr.label_center(self.nx/2+1, r"$i+1$") self.gr.label_center(self.nx/2-2, r"$i-2$") self.gr.label_center(self.nx/2+2, r"$i+2$") def draw_var_avg(self, var, scale=1.0): for n in range(self.gr.nx): self.vars[var].draw_cell_avg(n, color="r") #----------------------------------------------------------------------------- # multipage PDF # http://matplotlib.org/examples/pylab_examples/multipage_pdf.html # left state q_l = riemann.State(rho=1.0, u=0.0, p=1.0) # right state q_r = riemann.State(rho=0.125, u=0.0, p=0.1) r = RiemannProblem(q_l, q_r) subidx = [311, 312, 313] states = ["rho", "u", "p"] state_labels = [r"$\rho$", r"$u$", r"$p$"] #----------------------------------------------------------------------------- # plot 1: initial state plt.clf() for n, s in enumerate(states): plt.subplot(subidx[n]) r.draw_grid() r.draw_var_avg(s) r.gr.clean_axes() plt.title(state_labels[n]) f = plt.gcf() f.set_size_inches(8.0,7.0) plt.tight_layout() plt.savefig("ppm-seq-1.png") #----------------------------------------------------------------------------- # plot 2: cubic points (three vertical) plt.clf() for n, s in enumerate(states): plt.subplot(subidx[n]) r.draw_grid() r.draw_var_avg(s) r.draw_cubic_points(s) r.gr.clean_axes() plt.title(state_labels[n]) f = plt.gcf() f.set_size_inches(8.0,7.0) plt.tight_layout() plt.savefig("ppm-seq-2.png") #----------------------------------------------------------------------------- # plot 3: parabola (three vertical) plt.clf() for n, s in enumerate(states): plt.subplot(subidx[n]) r.draw_grid() r.draw_var_avg(s) r.draw_cubic_points(s) r.draw_parabola(s) r.gr.clean_axes() plt.title(state_labels[n]) f = plt.gcf() f.set_size_inches(8.0,7.0) plt.tight_layout() plt.savefig("ppm-seq-3.png") #----------------------------------------------------------------------------- # plot 4: cell-center Riemann waves to show what hits the interface #----------------------------------------------------------------------------- # plot 5: tracing (zoom in, 3 horizontal) #----------------------------------------------------------------------------- # plot 6: final interface state (zoom in, 3 horizontal) #----------------------------------------------------------------------------- # plot 7: Riemann phase #----------------------------------------------------------------------------- # plot 8: Riemann solution ================================================ FILE: figures/Euler/ppm-trace.py ================================================ import numpy as np import matplotlib.pyplot as plt import grid_plot as gp #----------------------------------------------------------------------------- nzones = 8 # data that lives on the grid #a = np.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.15, 0.5, 0.55]) a = np.array([0.3, 1.0, 1.0, 0.8, 0.2, 0.15, 0.5, 0.55]) gr = gp.FVGrid(nzones) plt.clf() gr.draw_grid(center_only=1) gr.label_center(nzones//2-1, r"$i$") gr.label_center(nzones//2, r"$i+1$") ppm = gp.PiecewiseParabolic(gr, a) for n in range(gr.nx//2-1, gr.nx//2+1): ppm.draw_parabola(n, color="r") nn = gr.nx//2-1 sigma = 0.6 ppm.ppm_trace_left(nn, sigma, color="0.75") plt.plot([gr.xr[gr.nx//2-1]-sigma*gr.dx, gr.xr[gr.nx//2-1]], [1.2, 1.2], color="k") plt.plot([gr.xr[gr.nx//2-1]-sigma*gr.dx, gr.xr[gr.nx//2-1]-sigma*gr.dx], [1.15, 1.25], color="k") plt.plot([gr.xr[gr.nx//2-1], gr.xr[gr.nx//2-1]], [1.15, 1.25], color="k") plt.text(gr.xr[gr.nx//2-1]-0.5*sigma*gr.dx, 1.3, r"$\sigma_{i}^{(\nu)} \Delta x$", horizontalalignment="center") plt.text(gr.xr[gr.nx//2-1]-0.5*sigma*gr.dx, 0.4, r"$\mathcal{I}_+^{(\nu)}$", color="r", horizontalalignment="center") plt.axis([gr.xl[gr.nx//2-1]-0.5*gr.dx,gr.xr[gr.nx//2]+0.5*gr.dx, -0.25, 1.4]) plt.axis("off") plt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95) f = plt.gcf() f.set_size_inches(3.0,2.0) plt.savefig("ppm-trace.pdf") plt.savefig("ppm-trace.png") ================================================ FILE: figures/Euler/ppm.py ================================================ import math import numpy as np import matplotlib.pyplot as plt import grid_plot as gp #----------------------------------------------------------------------------- nzones = 8 # data that lives on the grid a = np.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.1, 0.5, 0.55]) gr = gp.FVGrid(nzones) plt.clf() gr.draw_grid() gr.label_center(nzones//2, r"$i$") gr.label_center(nzones//2-1, r"$i-1$") gr.label_center(nzones//2+1, r"$i+1$") gr.label_center(nzones//2-2, r"$i-2$") gr.label_center(nzones//2+2, r"$i+2$") pc = gp.PiecewiseConstant(gr, a) for n in range(nzones): pc.draw_cell_avg(n, color="r") gr.clean_axes() plt.ylim(-0.25, 1.2) plt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95) f = plt.gcf() f.set_size_inches(8.0,2.0) plt.savefig("piecewise-constant.pdf") plt.savefig("piecewise-constant.png") #------------- PLM ------------- plt.clf() gr.draw_grid() gr.label_center(nzones//2, r"$i$") gr.label_center(nzones//2-1, r"$i-1$") gr.label_center(nzones//2+1, r"$i+1$") gr.label_center(nzones//2-2, r"$i-2$") gr.label_center(nzones//2+2, r"$i+2$") # not limited and limited pl_n = gp.PiecewiseLinear(gr, a, nolimit=1) pl_y = gp.PiecewiseLinear(gr, a) for n in range(nzones): pc.draw_cell_avg(n, color="0.5") for n in range(2, nzones-2): pl_n.draw_slope(n, color="r", ls=":") pl_y.draw_slope(n, color="r") gr.clean_axes() plt.ylim(-0.25, 1.2) plt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95) f = plt.gcf() f.set_size_inches(8.0,2.0) plt.savefig("piecewise-linear.pdf") plt.savefig("piecewise-linear.png") #------------- PPM ------------- plt.clf() gr.draw_grid() gr.label_center(nzones//2, r"$i$") gr.label_center(nzones//2-1, r"$i-1$") gr.label_center(nzones//2+1, r"$i+1$") gr.label_center(nzones//2-2, r"$i-2$") gr.label_center(nzones//2+2, r"$i+2$") for n in range(nzones): pc.draw_cell_avg(n, color="0.5") # compute the parabolic coefficients -- limited and not limited pm_n = gp.PiecewiseParabolic(gr, a, nolimit=1) pm_y = gp.PiecewiseParabolic(gr, a) for n in range(2, nzones-2): pm_n.draw_parabola(n, color="r", ls=":") pm_y.draw_parabola(n, color="r") gr.clean_axes() plt.ylim(-0.25, 1.2) plt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95) f = plt.gcf() f.set_size_inches(8.0,2.0) plt.savefig("piecewise-parabolic.pdf") plt.savefig("piecewise-parabolic.png") ================================================ FILE: figures/Euler/rarefaction_cartoon.py ================================================ import numpy as np import matplotlib.pyplot as plt import matplotlib as mpl mpl.rcParams['mathtext.fontset'] = 'cm' mpl.rcParams['mathtext.rm'] = 'serif' # font sizes mpl.rcParams['font.size'] = 12 mpl.rcParams['legend.fontsize'] = 'large' mpl.rcParams['figure.titlesize'] = 'medium' class RiemannWaves(object): """draw a diagram with a left rarefaction and 2 other waves """ def __init__(self, rpos=-1): """rpos is position of rarefaction, -1 is left of the interface, +1 is right of the interface """ self.rpos = rpos # these control the spacing of the waves in the figure self.h = 1.0 # this controls the spacing of the structure in the rarefaction self.rare_theta = np.radians(2.5) if rpos < 0: self.head = np.radians(-45) self.center = np.radians(5) self.right = np.radians(40) elif rpos == 0: self.head = 0 - 2.5*self.rare_theta self.center = np.radians(30) self.right = np.radians(45) else: self.head = np.radians(5) self.center = np.radians(35) self.right = np.radians(50) self.tail = self.head + 5*self.rare_theta def draw(self, output="plot.png", label=None): """ draw the wave structure """ # each wave has a point on the origin. The maximum height we # want to draw to is self.h -- we'll do the center and right xc = self.h*np.tan(self.center) x3 = self.h*np.tan(self.right) plt.clf() # draw the waves plt.plot([0, xc], [0, self.h], color="C0") plt.plot([0, x3], [0, self.h], color="C0") # rarefaction for t in np.arange(self.head, self.tail+self.rare_theta, self.rare_theta): xr = self.h*np.tan(t) plt.plot([0, xr], [0, self.h], color="C0") xhead = self.h*np.tan(self.head) xtail = self.h*np.tan(self.tail) #+self.rare_theta) # draw the grid L = 1.1*max(abs(xhead), abs(x3)) plt.plot([-L, L], [0, 0], color="k", zorder=-100) plt.plot([0, 0], [0, 1.2*self.h], color="k", zorder=-100) plt.text(xhead, self.h, r"$\lambda_\mathrm{head}$", horizontalalignment="right", color="C0") plt.text(xtail, self.h, r"$\lambda_\mathrm{tail}$", horizontalalignment="left", color="C0") dx = abs(0.75*self.h*np.tan(0.5*(self.head + self.tail))) if self.rpos == -1: fac = 0.4 elif self.rpos == 0: fac = 1.0 else: fac = 1.0 xh = 0.75*self.h*np.tan(self.head) xt = 0.75*self.h*np.tan(self.tail) xc = 0.75*self.h*np.tan(self.center) xr = 0.75*self.h*np.tan(self.right) if dx == 0: dx = abs(xh) xL = xh - fac*dx xLstar = 0.5*(xt + xc) xRstar = 0.5*(xc + xr) xR = xr + fac*dx plt.text(xL, 0.65*self.h, r"$L$", horizontalalignment="right", color="C0") plt.text(xLstar, 0.75*self.h, r"$L_\star$", horizontalalignment="center", color="C0") plt.text(xRstar, 0.75*self.h, r"$R_\star$", horizontalalignment="center", color="C0") plt.text(xR, 0.65*self.h, r"$R$", horizontalalignment="left", color="C0") f = plt.gcf() # label? if label is not None: plt.text(0.1, 0.9, label, transform=f.transFigure) plt.axis("off") plt.subplots_adjust(left=0.02, right=0.95, bottom=0.05, top=0.95) f.set_size_inches(5.0, 3.5) plt.tight_layout() plt.savefig(output, bbox_inches="tight", pad_inches=0) if __name__ == "__main__": rw = RiemannWaves(rpos=-1) rw.draw("rarefaction_left.pdf", label="(a)") rw = RiemannWaves(rpos=0) rw.draw("rarefaction_center.pdf", label="(b)") rw = RiemannWaves(rpos=1) rw.draw("rarefaction_right.pdf", label="(c)") ================================================ FILE: figures/Euler/riemann-states-q.py ================================================ import numpy as np import matplotlib.pyplot as plt import grid_plot as gp def riemann(with_time=True): # grid info xmin = 0.0 xmax = 1.0 nzones = 2 ng = 0 gr = gp.FVGrid(nzones, xmin=xmin, xmax=xmax) plt.clf() #------------------------------------------------------------------------ # plot a domain without ghostcells gr.draw_grid(emphasize_end=True) # label a few gr.label_center(0, r"$i$", fontsize="medium") gr.label_center(1, r"$i+1$", fontsize="medium") gr.label_edge(1, r"$i+1/2$", fontsize="medium") if with_time: gr.mark_cell_left_state(1, r"$q_{i+1/2,R}^{n+1/2}$", fontsize="large", color="b") gr.mark_cell_right_state(0, r"$q_{i+1/2,L}^{n+1/2}$", fontsize="large", color="b") else: gr.mark_cell_left_state(1, r"$q_{i+1/2,R}$", fontsize="large", color="b") gr.mark_cell_right_state(0, r"$q_{i+1/2,L}$", fontsize="large", color="b") gr.label_cell_center(0, r"$q_i$") gr.label_cell_center(1, r"$q_{i+1}$") # flux plt.arrow(gr.xl[ng+nzones//2]-0.25*gr.dx, 1.05, 0.5*gr.dx, 0, shape='full', head_width=0.075, head_length=0.05, lw=1, width=0.03, edgecolor="none", facecolor="red", length_includes_head=True, zorder=100) if with_time: plt.text(gr.xl[ng+nzones//2], 1.15, r"$F(U(q_{i+1/2}^{n+1/2}))$", color="red", horizontalalignment="center") else: plt.text(gr.xl[ng+nzones//2], 1.15, r"$F(U(q_{i+1/2}))$", color="red", horizontalalignment="center") gr.clean_axes(padding=False) plt.ylim(-0.25, 1.25) plt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95) f = plt.gcf() f.set_size_inches(6.0,2.25) plt.tight_layout() if with_time: plt.savefig("riemann_comp_q.pdf") else: plt.savefig("riemann_comp_q_mol.pdf") if __name__== "__main__": riemann() riemann(with_time=False) ================================================ FILE: figures/Euler/riemann-states.py ================================================ import numpy as np import matplotlib.pyplot as plt import grid_plot as gp def riemann(with_time=True): # grid info xmin = 0.0 xmax = 1.0 nzones = 2 ng = 0 gr = gp.FVGrid(nzones, xmin=xmin, xmax=xmax) plt.clf() #------------------------------------------------------------------------ # plot a domain without ghostcells gr.draw_grid(emphasize_end=True) # label a few gr.label_center(0, r"$i$", fontsize="medium") gr.label_center(1, r"$i+1$", fontsize="medium") gr.label_edge(1, r"$i+\myhalf$", fontsize="medium") if with_time: gr.mark_cell_left_state(1, r"$U_{i+\myhalf,R}^{n+\myhalf}$", fontsize="large", color="b") gr.mark_cell_right_state(0, r"$U_{i+\myhalf,L}^{n+\myhalf}$", fontsize="large", color="b") else: gr.mark_cell_left_state(1, r"$U_{i+\myhalf,R}$", fontsize="large", color="b") gr.mark_cell_right_state(0, r"$U_{i+\myhalf,L}$", fontsize="large", color="b") gr.label_cell_center(0, r"$U_i$") gr.label_cell_center(1, r"$U_{i+1}$") # flux plt.arrow(gr.xl[ng+nzones//2]-0.25*gr.dx, 1.05, 0.5*gr.dx, 0, shape='full', head_width=0.075, head_length=0.05, lw=1, width=0.03, edgecolor="none", facecolor="red", length_includes_head=True, zorder=100) if with_time: plt.text(gr.xl[ng+nzones//2], 1.15, r"$F(U_{i+\myhalf}^{n+\myhalf})$", color="red", horizontalalignment="center") else: plt.text(gr.xl[ng+nzones//2], 1.15, r"$F(U_{i+\myhalf})$", color="red", horizontalalignment="center") gr.clean_axes(padding=False) plt.ylim(-0.25, 1.25) plt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95) f = plt.gcf() f.set_size_inches(6.0,2.25) plt.tight_layout() if with_time: plt.savefig("riemann_comp.pdf") else: plt.savefig("riemann_comp_mol.pdf") if __name__== "__main__": riemann() riemann(with_time=False) ================================================ FILE: figures/Euler/riemann-waves-jump.py ================================================ import matplotlib.pyplot as plt import grid_plot as gp def simplegrid(): # grid info xmin = 0.0 xmax = 1.0 nzones = 2 ng = 0 gr = gp.FVGrid(nzones, xmin=xmin, xmax=xmax) #------------------------------------------------------------------------ # plot a domain without ghostcells gr.draw_grid(draw_end=False, edge_ticks=False) gr.label_center(0, r"$i$") gr.label_center(1, r"$i+1$") gr.label_edge(1, r"$q_{i+\myhalf}$") # draw waves # u - c plt.plot([gr.xr[0], gr.xr[0]-0.75*gr.dx], [0,1.0], color="C0", ls="-") plt.text(gr.xr[0]-0.75*gr.dx, 1.0+0.05, "$\lambda^{(-)} =\, u - c$", horizontalalignment="center") # u plt.plot([gr.xr[0], gr.xr[0]-0.2*gr.dx], [0,1.0], color="C0", ls="-") plt.text(gr.xr[0]-0.2*gr.dx, 1.0+0.05, "$\lambda^{(\circ)} =\, u$", horizontalalignment="center") # u + c plt.plot([gr.xr[0], gr.xr[0]+0.4*gr.dx], [0,1.0], color="C0", ls="-") plt.text(gr.xr[0]+0.4*gr.dx, 1.0+0.05, "$\lambda^{(+)} =\, u + c$", horizontalalignment="center") plt.plot([gr.xl[0], gr.xr[0]], [0.3, 0.3], color="C1", linewidth=2) plt.text(gr.xc[0], 0.33, r"$\langle q \rangle_i$", color="C1") plt.plot([gr.xl[1], gr.xr[1]], [0.6, 0.6], color="C1", linewidth=2) plt.text(gr.xc[1], 0.63, r"$\langle q \rangle_{i+1}$", color="C1") gr.clean_axes(padding=False) plt.ylim(-0.2,1.2) plt.tight_layout() f = plt.gcf() f.set_size_inches(6, 3.5) plt.savefig("riemann-waves-jump.png", dpi=150) if __name__== "__main__": simplegrid() ================================================ FILE: figures/Euler/riemann-waves.py ================================================ import matplotlib.pyplot as plt import grid_plot as gp def simplegrid(): # grid info xmin = 0.0 xmax = 1.0 nzones = 2 ng = 0 gr = gp.FVGrid(nzones, xmin=xmin, xmax=xmax) #------------------------------------------------------------------------ # plot a domain without ghostcells gr.draw_grid(draw_end=False, edge_ticks=False) gr.label_center(0, r"$i$") gr.label_center(1, r"$i+1$") gr.label_edge(1, r"$q_{i+\myhalf}$") # draw waves # u - c plt.plot([gr.xr[0], gr.xr[0]-0.75*gr.dx], [0,1.0], color="C0", ls="-") plt.text(gr.xr[0]-0.75*gr.dx, 1.0+0.05, "$\lambda^{(-)} =\, u - c$", horizontalalignment="center") # u plt.plot([gr.xr[0], gr.xr[0]-0.2*gr.dx], [0,1.0], color="C0", ls="-") plt.text(gr.xr[0]-0.2*gr.dx, 1.0+0.05, "$\lambda^{(\circ)} =\, u$", horizontalalignment="center") # u + c plt.plot([gr.xr[0], gr.xr[0]+0.4*gr.dx], [0,1.0], color="C0", ls="-") plt.text(gr.xr[0]+0.4*gr.dx, 1.0+0.05, "$\lambda^{(+)} =\, u + c$", horizontalalignment="center") # label regions plt.text(gr.xr[0]-0.5*gr.dx, 0.3, r"$L$", horizontalalignment="center", color="C1") plt.text(gr.xr[0]-0.5*gr.dx, 0.22, r"$(\rho_L, u_L, p_L)$", horizontalalignment="center", color="C1", fontsize="small") plt.text(gr.xr[0]-0.33*gr.dx, 0.75, r"$L_*$", horizontalalignment="center", color="C1") plt.text(gr.xr[0]-0.33*gr.dx, 0.67, r"$(\rho_{\star L}, u_\star, p_\star)$", horizontalalignment="center", color="C1", fontsize="small") plt.text(gr.xr[0]+0.1*gr.dx, 0.75, r"$R_*$", horizontalalignment="center", color="C1") plt.text(gr.xr[0]+0.1*gr.dx, 0.67, r"$(\rho_{\star R}, u_\star, p_\star)$", horizontalalignment="center", color="C1", fontsize="small", zorder=1000) plt.text(gr.xr[0]+0.3*gr.dx, 0.3, r"$R$", horizontalalignment="center", color="C1") plt.text(gr.xr[0]+0.3*gr.dx, 0.22, r"$(\rho_R, u_R, p_R)$", horizontalalignment="center", color="C1", fontsize="small") gr.clean_axes(padding=False) plt.ylim(-0.2,1.2) plt.tight_layout() f = plt.gcf() f.set_size_inches(6,3.5) plt.savefig("riemann-waves.pdf") if __name__== "__main__": simplegrid() ================================================ FILE: figures/Euler/riemann_cartoon.py ================================================ import numpy as np import matplotlib.pyplot as plt import matplotlib as mpl mpl.rcParams['mathtext.fontset'] = 'cm' mpl.rcParams['mathtext.rm'] = 'serif' # font sizes mpl.rcParams['font.size'] = 12 mpl.rcParams['legend.fontsize'] = 'large' mpl.rcParams['figure.titlesize'] = 'medium' class RiemannWaves(object): """draw a diagram with the 3 waves for the Euler Riemann problem""" def __init__(self, left=-1, contact=-1, right=1): """for each of the waves, -1 is left of the interface, +1 is right of the interface""" self.left = left self.contact = contact self.right = right # these control the spacing of the waves in the figure self.theta = np.radians(25) self.h = 1.0 def draw(self, output="plot.png", label=None): """ draw the wave structure """ # by default, we'll make the waves theta degrees apart if self.right < 0: angle3 = -0.75*self.theta elif self.right > 0 and self.contact < 0: angle3 = 0.534*self.theta elif self.right > 0 and self.contact > 0 and self.left < 0: angle3 = 0.966*self.theta else: angle3 = 2.25*self.theta anglec = angle3 - 0.75*self.theta angle1 = anglec - 0.75*self.theta print(np.degrees(angle1), np.degrees(anglec), np.degrees(angle3)) # each wave has a point on the origin. The maximum height we want to draw to is self.h x1 = self.h*np.tan(angle1) xc = self.h*np.tan(anglec) x3 = self.h*np.tan(angle3) print(x1, xc, x3) plt.clf() # draw the waves plt.plot([0, x1], [0, self.h], color="C0") plt.plot([0, xc], [0, self.h], color="C0") plt.plot([0, x3], [0, self.h], color="C0") # label regions x1h = 0.75*self.h*np.tan(angle1) xch = 0.75*self.h*np.tan(anglec) x3h = 0.75*self.h*np.tan(angle3) dx = max(xch - x1h, x3h - xch) xL = x1h - 0.6*dx xLstar = 0.5*(x1h + xch) xRstar = 0.5*(xch + x3h) xR = x3h + 0.6*dx plt.text(xL, 0.65*self.h, r"$L$", horizontalalignment="right", color="C0") plt.text(xLstar, 0.75*self.h, r"$L_\star$", horizontalalignment="center", color="C0") plt.text(xRstar, 0.75*self.h, r"$R_\star$", horizontalalignment="center", color="C0") plt.text(xR, 0.65*self.h, r"$R$", horizontalalignment="left", color="C0") # draw the grid L = 1.1*max(abs(x1), abs(x3)) plt.plot([-L, L], [0, 0], color="k") plt.plot([0, 0], [0, 1.2*self.h], color="k") f = plt.gcf() # label? if label is not None: plt.text(0.1, 0.9, label, transform=f.transFigure) plt.axis("off") plt.subplots_adjust(left=0.02, right=0.95, bottom=0.05, top=0.95) f.set_size_inches(5.0, 3.5) plt.tight_layout() plt.savefig(output, bbox_inches="tight", pad_inches=0) if __name__ == "__main__": rw = RiemannWaves(left=-1, contact=-1, right=-1) rw.draw("riemann_waves_ifc_R.pdf", label="(a)") rw = RiemannWaves(left=-1, contact=-1, right=1) rw.draw("riemann_waves_ifc_Rstar.pdf", label="(b)") rw = RiemannWaves(left=-1, contact=1, right=1) rw.draw("riemann_waves_ifc_Lstar.pdf", label="(c)") rw = RiemannWaves(left=1, contact=1, right=1) rw.draw("riemann_waves_ifc_L.pdf", label="(d)") ================================================ FILE: figures/Euler/states.py ================================================ import math import numpy as np import matplotlib.pyplot as plt import grid_plot as gp def riemann(): # grid info xmin = 0.0 xmax = 1.0 nzones = 1 ng = 0 gr = gp.FVGrid(nzones, xmin=xmin, xmax=xmax) #------------------------------------------------------------------------ # plot a domain without ghostcells gr.draw_grid() gr.label_center(0, r"$i$") gr.label_cell_center(0, r"$q_i$") gr.mark_cell_left_state(0, r"$q_{i-1/2,R}^{n+1/2}$", color="r") gr.mark_cell_right_state(0, r"$q_{i+1/2,L}^{n+1/2}$", color="r") plt.arrow(gr.xc[0]-0.05*gr.dx, 0.5, -0.13*gr.dx, 0, shape='full', head_width=0.075, head_length=0.05, lw=1, width=0.01, edgecolor="none", facecolor="r", length_includes_head=True, zorder=100) plt.arrow(gr.xc[0]+0.05*gr.dx, 0.5, 0.13*gr.dx, 0, shape='full', head_width=0.075, head_length=0.05, lw=1, width=0.01, edgecolor="none", facecolor="r", length_includes_head=True, zorder=100) plt.xlim(gr.xl[0]-0.25*gr.dx,gr.xr[2*ng+nzones-1]+0.25*gr.dx) # plt.ylim(-0.25, 0.75) plt.axis("off") plt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95) f = plt.gcf() f.set_size_inches(4.0,2.5) plt.savefig("states.png") plt.savefig("states.pdf") if __name__== "__main__": riemann() ================================================ FILE: figures/Euler/update.sh ================================================ #!/bin/sh DEST=../../Euler # riemann-states.py makes riemann_comp.pdf python3 riemann-states.py cp -f riemann_comp.pdf ${DEST} # ppm.py makes piecewise-constant.pdf piecewise-linear.pdf piecewise-parabolic.pdf python3 ppm.py cp -f piecewise-constant.pdf piecewise-linear.pdf piecewise-parabolic.pdf ${DEST} # states.py makes states.pdf python3 states.py cp -f states.pdf ${DEST} ================================================ FILE: figures/advection/2dFD.py ================================================ import math import numpy as np import matplotlib.pyplot as plt import grid_plot as gp def simplegrid(): nzones = 5 gr = gp.FDGrid2d(nzones, nzones, ng=0) # plot a domain without ghostcells gr.draw_grid() #------------------------------------------------------------------------ # label gr.label_cell_center(nzones/2, nzones/2, r"$a_{i,j}$", fontsize="x-large", color="r") # grid labels gr.label_center_x(nzones/2-2, r"$i-2$") gr.label_center_x(nzones/2-1, r"$i-1$") gr.label_center_x(nzones/2, r"$i$") gr.label_center_x(nzones/2+1, r"$i+1$") gr.label_center_x(nzones/2+2, r"$i+2$") gr.label_center_y(nzones/2-2, r"$j-2$") gr.label_center_y(nzones/2-1, r"$j-1$") gr.label_center_y(nzones/2, r"$j$") gr.label_center_y(nzones/2+1, r"$j+1$") gr.label_center_y(nzones/2+2, r"$j+2$") # axes gr.clean_axes() plt.subplots_adjust(left=0.025,right=0.98,bottom=0.1,top=0.98) f = plt.gcf() f.set_size_inches(7.0,7.0) plt.savefig("2dFD.png") if __name__== "__main__": simplegrid() ================================================ FILE: figures/advection/2dFV.py ================================================ import math import numpy as np import matplotlib.pyplot as plt import grid_plot as gp def simplegrid(): nzones = 5 gr = gp.FVGrid2d(nzones, nzones, ng=0) # plot a domain without ghostcells gr.draw_grid() #------------------------------------------------------------------------ # label gr.label_cell_center(nzones/2, nzones/2, r"$a_{i,j}$", fontsize="x-large", color="r") gr.shade_cell(nzones/2, nzones/2) # grid labels gr.label_center_x(nzones/2-2, r"$i-2$") gr.label_center_x(nzones/2-1, r"$i-1$") gr.label_center_x(nzones/2, r"$i$") gr.label_center_x(nzones/2+1, r"$i+1$") gr.label_center_x(nzones/2+2, r"$i+2$") gr.label_center_y(nzones/2-2, r"$j-2$") gr.label_center_y(nzones/2-1, r"$j-1$") gr.label_center_y(nzones/2, r"$j$") gr.label_center_y(nzones/2+1, r"$j+1$") gr.label_center_y(nzones/2+2, r"$j+2$") # axes gr.clean_axes() plt.subplots_adjust(left=0.025,right=0.98,bottom=0.1,top=0.98) f = plt.gcf() f.set_size_inches(7.0,7.0) plt.savefig("2dFV.png") if __name__== "__main__": simplegrid() ================================================ FILE: figures/advection/2dgrid.py ================================================ import numpy as np import matplotlib.pyplot as plt import grid_plot as gp def simplegrid(): nzones = 3 gr = gp.FVGrid2d(nzones, nzones, ng=0) # plot a domain without ghostcells gr.draw_grid() #------------------------------------------------------------------------ # label gr.label_cell_center(nzones//2, nzones//2, r"$a_{i,j}$", fontsize="large") gr.label_cell_center(nzones//2+1, nzones//2, r"$a_{i+1,j}$", fontsize="large") gr.label_cell_center(nzones//2, nzones//2+1, r"$a_{i,j+1}$", fontsize="large") # i+1/2,j interface gr.mark_cell_left_state_x(nzones//2, nzones//2, r"$a^{n+\myhalf}_{i+\myhalf,j,L}$", color="b") gr.mark_cell_right_state_x(nzones//2+1, nzones//2, r"$a^{n+\myhalf}_{i+\myhalf,j,R}$", color="b") # i,j+1/2 interface gr.mark_cell_left_state_y(nzones//2, nzones//2, r"$a^{n+\myhalf}_{i,j+\myhalf,L}$", color="b") gr.mark_cell_right_state_y(nzones//2, nzones//2+1, r"$a^{n+\myhalf}_{i,j+\myhalf,R}$", color="b") # grid labels gr.label_center_x(nzones//2-1, r"$i-1$") gr.label_center_x(nzones//2, r"$i$") gr.label_center_x(nzones//2+1, r"$i+1$") gr.label_center_y(nzones//2-1, r"$j-1$") gr.label_center_y(nzones//2, r"$j$") gr.label_center_y(nzones//2+1, r"$j+1$") # axes gr.clean_axes() plt.subplots_adjust(left=0.02,right=0.98,bottom=0.02,top=0.98) f = plt.gcf() f.set_size_inches(7.0,7.0) plt.savefig("2dgrid.pdf") if __name__== "__main__": simplegrid() ================================================ FILE: figures/advection/2dgrid_hat.py ================================================ import numpy import matplotlib.pyplot as plt import grid_plot as gp def simplegrid(): nzones = 3 gr = gp.FVGrid2d(nzones, nzones, ng=0) # plot a domain without ghostcells gr.draw_grid() #------------------------------------------------------------------------ # label gr.label_cell_center(nzones//2, nzones//2, r"$a_{i,j}$", fontsize="large") gr.label_cell_center(nzones//2+1, nzones//2, r"$a_{i+1,j}$", fontsize="large") gr.label_cell_center(nzones//2, nzones//2+1, r"$a_{i,j+1}$", fontsize="large") gr.label_cell_center(nzones//2, nzones//2-1, r"$a_{i,j-1}$", fontsize="large") # i+1/2,j interface gr.mark_cell_left_state_x(nzones//2, nzones//2, r"$\hat{a}^{n+\myhalf}_{i+\myhalf,j,L}$", color="b") # i,j+1/2 interface gr.mark_cell_left_state_y(nzones//2, nzones//2, r"$\hat{a}^{n+\myhalf}_{i,j+\myhalf,L}$", color="b") gr.mark_cell_right_state_y(nzones//2, nzones//2+1, r"$\hat{a}^{n+\myhalf}_{i,j+\myhalf,R}$", color="b") # i,j-1/2 interface gr.mark_cell_left_state_y(nzones//2, nzones//2-1, r"$\hat{a}^{n+\myhalf}_{i,j-\myhalf,L}$", color="b") gr.mark_cell_right_state_y(nzones//2, nzones//2, r"$\hat{a}^{n+\myhalf}_{i,j-\myhalf,R}$", color="b") # grid labels gr.label_center_x(nzones//2-1, r"$i-1$") gr.label_center_x(nzones//2, r"$i$") gr.label_center_x(nzones//2+1, r"$i+1$") gr.label_center_y(nzones//2-1, r"$j-1$") gr.label_center_y(nzones//2, r"$j$") gr.label_center_y(nzones//2+1, r"$j+1$") # axes gr.clean_axes() plt.subplots_adjust(left=0.02,right=0.98,bottom=0.02,top=0.98) f = plt.gcf() f.set_size_inches(7.0,7.0) plt.savefig("2dgrid-hat.pdf") if __name__== "__main__": simplegrid() ================================================ FILE: figures/advection/2dgrid_transverse.py ================================================ import math import numpy import matplotlib.pyplot as plt import grid_plot as gp def simplegrid(): nzones = 3 gr = gp.FVGrid2d(nzones, nzones, ng=0) # plot a domain without ghostcells gr.draw_grid() #------------------------------------------------------------------------ # label gr.label_cell_center(nzones//2, nzones//2, r"$a_{i,j}$", fontsize="large") gr.label_cell_center(nzones//2+1, nzones//2, r"$a_{i+1,j}$", fontsize="large") gr.label_cell_center(nzones//2, nzones//2+1, r"$a_{i,j+1}$", fontsize="large") gr.label_cell_center(nzones//2, nzones//2-1, r"$a_{i,j-1}$", fontsize="large") # i+1/2,j interface gr.mark_cell_left_state_x(nzones//2, nzones//2, r"$\hat{a}^{n+\myhalf}_{i+\myhalf,j,L}$", color="b") # i,j+1/2 interface gr.mark_cell_state_y(nzones//2, nzones//2, r"${a}^T_{i,j+\myhalf}$") # i,j-1/2 interface gr.mark_cell_state_y(nzones//2, nzones//2-1, r"${a}^T_{i,j-\myhalf}$", off_sign=-1) # helpful line showing the transverse bits plt.plot([gr.xc[nzones//2], gr.xc[nzones//2]], [gr.yl[nzones//2]+0.025*gr.dy, gr.yr[nzones//2]-0.025*gr.dy], linestyle=":", color="0.5", zorder=1000) plt.plot([gr.xc[nzones//2], gr.xr[nzones//2]-0.26*gr.dx], [gr.yc[nzones//2], gr.yc[nzones//2]], linestyle=":", color="0.5", zorder=1000) plt.plot([gr.xr[nzones//2]-0.34*gr.dx, gr.xr[nzones//2]-0.26*gr.dx], [gr.yc[nzones//2]+0.04*gr.dy, gr.yc[nzones//2]], linestyle=":", color="0.5", zorder=1000) plt.plot([gr.xr[nzones//2]-0.34*gr.dx, gr.xr[nzones//2]-0.26*gr.dx], [gr.yc[nzones//2]-0.04*gr.dy, gr.yc[nzones//2]], linestyle=":", color="0.5", zorder=1000) # grid labels gr.label_center_x(nzones//2-1, r"$i-1$") gr.label_center_x(nzones//2, r"$i$") gr.label_center_x(nzones//2+1, r"$i+1$") gr.label_center_y(nzones//2-1, r"$j-1$") gr.label_center_y(nzones//2, r"$j$") gr.label_center_y(nzones//2+1, r"$j+1$") # axes gr.clean_axes() plt.subplots_adjust(left=0.02,right=0.98,bottom=0.02,top=0.98) f = plt.gcf() f.set_size_inches(7.0,7.0) plt.savefig("2dgrid-transverse.pdf") if __name__== "__main__": simplegrid() ================================================ FILE: figures/advection/characteristics.py ================================================ import numpy as np import matplotlib as mpl import matplotlib.pyplot as plt mpl.rcParams['mathtext.fontset'] = 'cm' mpl.rcParams['mathtext.rm'] = 'serif' def fun(x): state = np.zeros_like(xplot) state[x < 0.5] = 1.0 return state def funs(x): state = 0.5 + 0.25*np.sin(2.0*np.pi*x) return state npts_plot = 1000 xplot = np.linspace(0.0, 1.0, npts_plot) nchar = 20 xchar = np.linspace(0.0, 1.0, nchar) state = funs(xplot) print(type(state)) plt.subplot(211) plt.plot(xplot, state, lw=2) ax = plt.gca() ax.spines['right'].set_visible(False) ax.spines['top'].set_visible(False) ax.spines['right'].set_visible(False) ax.spines['top'].set_visible(False) ax.spines['right'].set_visible(False) ax.spines['top'].set_visible(False) ax.xaxis.set_ticks_position('bottom') ax.yaxis.set_ticks_position('left') plt.ylim(-0.1, 1.1) plt.xlabel(r"$x$", fontsize="large") plt.ylabel(r"$a$", fontsize="large") plt.subplot(212) uchar = 1.0 t = np.linspace(0.0, 1.0, 100) for n in range(nchar): xc = xchar[n] + uchar*t plt.plot(xc, t, color="0.5") ax = plt.gca() ax.spines['right'].set_visible(False) ax.spines['top'].set_visible(False) ax.spines['right'].set_visible(False) ax.spines['top'].set_visible(False) ax.spines['right'].set_visible(False) ax.spines['top'].set_visible(False) ax.xaxis.set_ticks_position('bottom') ax.yaxis.set_ticks_position('left') plt.xlabel(r"$x$", fontsize="large") plt.ylabel(r"$t$", fontsize="large") plt.xlim(0.0, 1.0) plt.ylim(0.0, 1.1) f = plt.gcf() f.set_size_inches(6.0, 8.0) plt.tight_layout() plt.savefig("advection-characteristics.png") ================================================ FILE: figures/advection/domains.py ================================================ import numpy as np import matplotlib.pylab as plt import matplotlib as mpl mpl.rcParams['mathtext.fontset'] = 'cm' mpl.rcParams['mathtext.rm'] = 'serif' mpl.rcParams['font.size'] = 12 mpl.rcParams['legend.fontsize'] = 'large' mpl.rcParams['figure.titlesize'] = 'medium' def domains(method="FTCS"): # grid info xl = 0.0 xc = 1.0 xr = 2.0 dx = xr - xc t0 = 0.0 t1 = 1.0 dt = t1 - t0 C = 0.6 xp = xc - C*dx plt.clf() # plot a square representing [x, x+dx] x [t, t+dt] # x-axis plt.arrow(0, 0, 2.5*dx, 0, shape="full", head_width=0.04, head_length=0.06, lw=1, width=0.005, facecolor="k", length_includes_head=True, zorder=100) plt.text(2.55*dx, 0, r"$x$", fontsize=16, verticalalignment="center") # x points labeled for p, l in [((xc, 0), r"$x_i$"), ((xl, 0), r"$x_{i-1}$"), ((xr, 0), r"$x_{i+1}$")]: plt.text(p[0], p[1]-0.07, l, fontsize=16, horizontalalignment="center", verticalalignment="top") plt.plot([p[0], p[0]], [p[1], p[1]-0.04], color="k") plt.text(xp, -0.07, r"$x_i - C \Delta x$", fontsize=16, horizontalalignment="center", verticalalignment="top") plt.plot([xp, xp], [0, -0.04], color="k") # points plt.scatter([xl, xc, xr, xc], [0, 0, 0, dt], color="C3", marker="o", s=40, zorder=1000, alpha=1.0) # data points eps = 0.02*dx for p, l in [((xl, 0), r"$a_{i-1}^n$"), ((xc, 0), r"$a_{i}^n$"), ((xr, 0), r"$a_{i+1}^n$"), ((xc, dt), r"$a_{i}^{n+1}$")]: plt.text(p[0]+eps, p[1]+eps, l, horizontalalignment="left", verticalalignment="bottom", fontsize=16, color="C3", zorder=1000) # time axis plt.arrow(0, 0, 0, 1.3*dt, shape="full", head_width=0.04, head_length=0.06, lw=1, width=0.005, facecolor="k", length_includes_head=True, zorder=100) plt.text(0, 1.35*dt, r"$t$", fontsize=16, horizontalalignment="center") plt.text(-0.25, dt, r"$t^{n+1}$", fontsize=16) plt.plot([-0.04, 0], [dt, dt], color="k") plt.text(-0.25, 0, r"$t^n$", fontsize=16) plt.plot([-0.04, 0], [0, 0], color="k") plt.plot([0, 2.0*dx], [dt, dt], ls=":", color="0.5") plt.plot([dx, dx], [0, dt], ls=":", color="0.5") plt.plot([2.0*dx, 2.0*dx], [0, dt], ls=":", color="0.5") # domain of dependence if method == "upwind": plt.fill([0, xc, xc, 0], [0, dt, 0, 0], color="C0", lw=2, alpha=0.5) elif method == "FTCS": plt.fill([0, xc, xr, 0], [0, dt, 0, 0], color="C0", lw=2, alpha=0.5) elif method == "downwind": plt.fill([xc, xc, xr, 0], [0, dt, 0, 0], color="C0", lw=2, alpha=0.5) # true domain of dependence plt.fill([xp, xc, xc, 0], [0, dt, 0, 0], color="C1", lw=2, alpha=0.5) # label plt.text(xr+0.1*dx, 0.5*dt, method, fontsize=16, horizontalalignment="left") plt.axis("off") plt.subplots_adjust(left=0.05, right=0.95, bottom=0.05, top=0.95) plt.xlim(-0.4,2.8*dx) plt.ylim(-0.1,1.4*dx) f = plt.gcf() f.set_size_inches(9.0,5.0) plt.tight_layout() plt.savefig(f"domains_{method}.pdf", bbox_inches="tight") if __name__== "__main__": domains(method="upwind") domains(method="FTCS") domains(method="downwind") ================================================ FILE: figures/advection/fd_ghost.py ================================================ import grid_plot as gp import numpy as np # plot a simple finite-difference grid #----------------------------------------------------------------------------- nzones = 9 # data that lives on the grid #a = np.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.15, 0.5, 0.55]) a = np.array([0.55, 0.3, 1.0, 0.9, 0.8, 0.25, 0.1, 0.5, 0.55]) gr = gp.FDGrid(nzones, ng=1) gr.draw_grid(draw_ghost=1) labels = ["-1", "0", "1", "", "i-1", "i", "i+1", "", "N-2", "N-1", "N"] for i in range(gr.ilo-gr.ng, gr.ng+gr.nx+1): if not labels[i] == "": gr.label_node(i, r"$%s$" % (labels[i]), fontsize="medium") # draw the data for i in range(gr.ilo, gr.ihi+1): gr.draw_data(i, a[i-gr.ng], color="r") gr.label_value(gr.ilo+4, a[gr.ilo+4-gr.ng], r"$a_i$", color="r") # label dx gr.label_dx(gr.ng+nzones//2) gr.clean_axes(pad_fac=0.1, show_ghost=True, ylim=(-0.5, 1.3)) fig = gr.fig fig.set_size_inches(10.0,3.0) fig.savefig("fd_ghost.pdf") ================================================ FILE: figures/advection/fv_ghost.py ================================================ import numpy as np import matplotlib.pyplot as plt import grid_plot as gp # plot a simple finite-difference grid #----------------------------------------------------------------------------- nzones = 7 ng = 2 # data that lives on the grid #a = np.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.15, 0.5, 0.55]) #a = np.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.1, 0.5, 0.55]) a = np.array([1.0, 0.9, 0.8, 0.25, 0.1, 0.5, 0.55]) gr = gp.FVGrid(nzones, ng) aa = gr.scratch_array() aa[gr.ilo:gr.ihi+1] = a cc = gp.PiecewiseConstant(gr, aa) plt.clf() gr.draw_grid(draw_ghost=1, emphasize_end=1) gr.label_center(ng+nzones//2, r"$i$") gr.label_center(ng+nzones//2-1, r"$i-1$") gr.label_center(ng+nzones//2+1, r"$i+1$") gr.label_center(gr.ilo, r"$\mathrm{lo}$") gr.label_center(gr.ilo-1, r"$\mathrm{lo-1}$") gr.label_center(gr.ilo-2, r"$\mathrm{lo-2}$") gr.label_center(gr.ihi, r"$\mathrm{hi}$") gr.label_center(gr.ihi+1, r"$\mathrm{hi+1}$") gr.label_center(gr.ihi+2, r"$\mathrm{hi+2}$") gr.label_edge(ng+nzones//2, r"$i-\sfrac{1}{2}$") gr.label_edge(ng+nzones//2+1, r"$i+\sfrac{1}{2}$") # draw the data for i in range(nzones): cc.draw_cell_avg(ng+i, color="r") cc.label_cell_avg(ng+nzones//2, r"$\langle a\rangle_i$", color="r") # label dx gr.label_dx(gr.ng+nzones//2) gr.clean_axes(show_ghost=True, pad_fac=0.02, ylim=(-0.5, 1.6)) f = plt.gcf() f.set_size_inches(10.0,2.5) plt.savefig("fv_ghost.pdf") ================================================ FILE: figures/advection/generalgrid.py ================================================ import math import numpy as np import matplotlib.pyplot as plt import grid_plot as gp nzones = 7 # data that lives on the grid ainit = np.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.15, 0.5]) gr = gp.FVGrid(nzones) a = gr.scratch_array() a[gr.ilo:gr.ihi+1] = ainit[:] pl_nolim = gp.PiecewiseLinear(gr, a, nolimit=1) pl = gp.PiecewiseLinear(gr, a) gr.draw_grid() gr.label_center(nzones//2, r"$i$") gr.label_center(nzones//2-1, r"$i-1$") gr.label_center(nzones//2+1, r"$i+1$") gr.label_center(nzones//2-2, r"$i-2$") gr.label_center(nzones//2+2, r"$i+2$") #labelEdge(gr, nzones//2, r"$i-1/2$") #labelEdge(gr, nzones//2+1, r"$i+1/2$") for n in range(nzones): pl.draw_cell_avg(n, ls=":", color="0.5") pl_nolim.draw_slope(n, color="0.5") pl.draw_slope(n, color="r") gr.clean_axes(ylim=(-0.25, 1.2)) f = plt.gcf() f.set_size_inches(8.0,2.0) plt.savefig("generalgrid.pdf") ================================================ FILE: figures/advection/rea-limitex.py ================================================ # reconstruct - evolve - average: demonstrate what happens when we don't # limit import numpy as np import matplotlib.pyplot as plt import grid_plot as gp #----------------------------------------------------------------------------- def evolve(pl, C, num, nolimit=1): # first frame -- the original cell-averages plt.clf() pl.gr.draw_grid() pl.gr.label_center(pl.gr.ng + pl.gr.nx//2, r"$i$", fontsize="medium") pl.gr.label_center(pl.gr.ng + pl.gr.nx//2-1, r"$i-1$", fontsize="medium") pl.gr.label_center(pl.gr.ng + pl.gr.nx//2+1, r"$i+1$", fontsize="medium") pl.gr.label_center(pl.gr.ng + pl.gr.nx//2-2, r"$i-2$", fontsize="medium") pl.gr.label_center(pl.gr.ng + pl.gr.nx//2+2, r"$i+2$", fontsize="medium") # draw cell averages for n in range(pl.gr.ilo, gr.ihi+1): pl.draw_cell_avg(n, color="r") pl.gr.clean_axes(ylim=(-0.25, 1.2)) f = plt.gcf() f.set_size_inches(8.0,2.0) if nolimit: plt.savefig("rea-nolimit-start_%3.3d.pdf" % (num)) else: plt.savefig("rea-start_%3.3d.pdf" % (num)) # second frame -- reconstruction # draw plt.clf() pl.gr.draw_grid() pl.gr.label_center(pl.gr.ng + pl.gr.nx//2, r"$i$", fontsize="medium") pl.gr.label_center(pl.gr.ng + pl.gr.nx//2-1, r"$i-1$", fontsize="medium") pl.gr.label_center(pl.gr.ng + pl.gr.nx//2+1, r"$i+1$", fontsize="medium") pl.gr.label_center(pl.gr.ng + pl.gr.nx//2-2, r"$i-2$", fontsize="medium") pl.gr.label_center(pl.gr.ng + pl.gr.nx//2+2, r"$i+2$", fontsize="medium") # draw cell averages and slopes for n in range(pl.gr.ilo, gr.ihi+1): pl.draw_cell_avg(n, color="0.5", ls=":") pl.draw_slope(n, color="r") plt.axis([pl.gr.xmin-0.5*pl.gr.dx,pl.gr.xmax+0.5*pl.gr.dx, -0.25, 1.2]) plt.axis("off") plt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95) f = plt.gcf() f.set_size_inches(8.0,2.0) if nolimit: plt.savefig("rea-nolimit-reconstruction_%3.3d.pdf" % (num)) else: plt.savefig("rea-reconstruction_%3.3d.pdf" % (num)) # third frame -- evolve # draw plt.clf() pl.gr.draw_grid() pl.gr.label_center(pl.gr.ng + pl.gr.nx//2, r"$i$", fontsize="medium") pl.gr.label_center(pl.gr.ng + pl.gr.nx//2-1, r"$i-1$", fontsize="medium") pl.gr.label_center(pl.gr.ng + pl.gr.nx//2+1, r"$i+1$", fontsize="medium") pl.gr.label_center(pl.gr.ng + pl.gr.nx//2-2, r"$i-2$", fontsize="medium") pl.gr.label_center(pl.gr.ng + pl.gr.nx//2+2, r"$i+2$", fontsize="medium") # draw cell averages and slopes for n in range(pl.gr.ilo, gr.ihi+1): #pl.draw_cell_avg(n, color="r") pl.draw_slope(n, color="0.5", ls=":") # evolve for n in range(pl.gr.ilo, pl.gr.ihi+1): pl.evolve_to_right(n, C, color="r") pl.gr.clean_axes(ylim=(-0.25, 1.2)) f = plt.gcf() f.set_size_inches(8.0,2.0) if nolimit: plt.savefig("rea-nolimit-evolve_%3.3d.pdf" % (num)) else: plt.savefig("rea-evolve_%3.3d.pdf" % (num)) #------------------------------------------------------------------------- # fourth frame -- re-average # left states (we don't need the right state when u > 0) al = pl.gr.scratch_array() for n in range(pl.gr.ilo, pl.gr.ihi+2): al[n] = pl.a[n-1] + 0.5*(1.0 - C)*pl.slope[n-1] # the Riemann problem just picks the left state. Do a conservative # update anew = pl.gr.scratch_array() anew[pl.gr.ilo:pl.gr.ihi+1] = pl.a[pl.gr.ilo:pl.gr.ihi+1] + \ C*(al[pl.gr.ilo:pl.gr.ihi+1] - al[pl.gr.ilo+1:pl.gr.ihi+2]) plt.clf() pl.gr.draw_grid() pl.gr.label_center(pl.gr.ng + pl.gr.nx//2, r"$i$", fontsize="medium") pl.gr.label_center(pl.gr.ng + pl.gr.nx//2-1, r"$i-1$", fontsize="medium") pl.gr.label_center(pl.gr.ng + pl.gr.nx//2+1, r"$i+1$", fontsize="medium") pl.gr.label_center(pl.gr.ng + pl.gr.nx//2-2, r"$i-2$", fontsize="medium") pl.gr.label_center(pl.gr.ng + pl.gr.nx//2+2, r"$i+2$", fontsize="medium") # show old evolved profiles and new averages for n in range(pl.gr.ilo, pl.gr.ihi+1): pl.evolve_to_right(n, C, color="0.5", ls=":") # draw new averages pl.a[:] = anew[:] pl.fill_zero_gradient() pl.calculate_slopes() for n in range(pl.gr.ilo, pl.gr.ihi+1): pl.draw_cell_avg(n, color="r") pl.gr.clean_axes(ylim=(-0.25, 1.2)) f = plt.gcf() f.set_size_inches(8.0,2.0) if nolimit: plt.savefig("rea-nolimit-final_%3.3d.pdf" % (num)) else: plt.savefig("rea-final_%3.3d.pdf" % (num)) return anew #----------------------------------------------------------------------------- ainit = np.array([1.0, 1.0, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1]) nzones = len(ainit) # CFL number C = 0.7 for nolimit in range(2): gr = gp.FVGrid(nzones, ng=4) a = gr.scratch_array() a[gr.ilo:gr.ihi+1] = ainit[:] pl = gp.PiecewiseLinear(gr, a, nolimit=nolimit) # loop for i in range(1,9): pl.fill_zero_gradient() print(i, pl.a[:]) evolve(pl, C, i, nolimit=nolimit) ================================================ FILE: figures/advection/rea.py ================================================ # reconstruct - evolve - average import numpy as np import matplotlib.pyplot as plt import grid_plot as gp #----------------------------------------------------------------------------- atemp = np.array([1.0, 1.0, 0.9, 0.8, 0.25, 0.1, 0.1, 0.1]) nzones = len(atemp) # CFL number C = 0.6 gr = gp.FVGrid(nzones, ng=4) a = gr.scratch_array() a[gr.ilo:gr.ihi+1] = atemp[:] pl = gp.PiecewiseLinear(gr, a) pl.fill_zero_gradient() #----------------------------------------------------------------------------- # first frame -- the original cell-averages plt.clf() gr.draw_grid() labels = ["$i-2$", "$i-1$", "$i$", "$i+1$", "$i+2$"] indices = [gr.ng+nzones//2-2, gr.ng+nzones//2-1, gr.ng+nzones//2, gr.ng+nzones//2+1, gr.ng+nzones//2+2] for i, l in zip(indices, labels): gr.label_center(i, l) # draw cell averages for n in range(gr.ilo, gr.ihi+1): pl.draw_cell_avg(n, color="r") gr.clean_axes(ylim=(-0.25, 1.2)) f = plt.gcf() f.set_size_inches(8.0,2.0) plt.savefig("rea-start.pdf") #----------------------------------------------------------------------------- # second frame -- reconstruction plt.clf() gr.draw_grid() for i, l in zip(indices, labels): gr.label_center(i, l) # draw cell averages and slopes for n in range(gr.ilo, gr.ihi+1): pl.draw_cell_avg(n, color="0.5", ls=":") pl.draw_slope(n, color="r") gr.clean_axes(ylim=(-0.25, 1.2)) f = plt.gcf() f.set_size_inches(8.0,2.0) plt.savefig("rea-reconstruction.pdf") #----------------------------------------------------------------------------- # third frame -- shade the portion that will come into the cell plt.clf() gr.draw_grid() for i, l in zip(indices, labels): gr.label_center(i, l) # draw cell averages and slopes for n in range(gr.ilo, gr.ihi+1): pl.draw_slope(n, color="r") # shade regions ii = gr.ng + nzones//2-1 pl.slope_trace_left(ii, C, color="0.75") pl.slope_trace_right(ii+1, 1.0-C, color="0.75") gr.clean_axes(ylim=(-0.25, 1.2)) f = plt.gcf() f.set_size_inches(8.0,2.0) plt.savefig("rea-trace.pdf") #----------------------------------------------------------------------------- # fourth frame -- evolve plt.clf() gr.draw_grid() for i, l in zip(indices, labels): gr.label_center(i, l) # draw cell averages and slopes for n in range(gr.ilo, gr.ihi+1): pl.draw_slope(n, color="0.75", ls=":") pl.evolve_to_right(n, C, color="r") gr.clean_axes(ylim=(-0.25, 1.2)) f = plt.gcf() f.set_size_inches(8.0,2.0) plt.savefig("rea-evolve.pdf") #----------------------------------------------------------------------------- # fifth frame -- re-average # left states (we don't need the right state when u > 0) al = np.zeros(2*gr.ng + gr.nx, dtype=np.float64) for n in range(gr.ilo, gr.ihi+2): al[n] = a[n-1] + 0.5*(1 - C)*pl.slope[n-1] # the Riemann problem just picks the right state. Do a conservative # update anew = gr.scratch_array() anew[gr.ilo:gr.ihi+1] = a[gr.ilo:gr.ihi+1] + \ C*(al[gr.ilo:gr.ihi+1] - al[gr.ilo+1:gr.ihi+2]) plt.clf() gr.draw_grid() for i, l in zip(indices, labels): gr.label_center(i, l) # show the evolved profiles from the old time for n in range(gr.ilo, gr.ihi+1): pl.evolve_to_right(n, C, color="0.5", ls=":") pl.a[:] = anew[:] pl.fill_zero_gradient() pl.calculate_slopes() for n in range(pl.gr.ilo, pl.gr.ihi+1): pl.draw_cell_avg(n, color="r") gr.clean_axes(ylim=(-0.25, 1.2)) f = plt.gcf() f.set_size_inches(8.0,2.0) plt.savefig("rea-final.pdf") ================================================ FILE: figures/advection/riemann.py ================================================ import numpy as np import matplotlib.pylab as plt import grid_plot as gp def riemann(with_time=True): # grid info xmin = 0.0 xmax = 1.0 nzones = 2 ng = 0 gr = gp.FVGrid(nzones, xmin=xmin, xmax=xmax) plt.clf() #------------------------------------------------------------------------ # plot a domain without ghostcells gr.draw_grid() gr.label_center(0, r"$i$", fontsize="medium") gr.label_center(1, r"$i+1$", fontsize="medium") gr.label_edge(1, r"$i+\myhalf$", fontsize="medium") plt.arrow(gr.xc[0]+0.05*gr.dx, 0.5, 0.12*gr.dx, 0, shape='full', head_width=0.05, head_length=0.025, lw=1, width=0.02, edgecolor="none", facecolor="r", length_includes_head=True, zorder=100) plt.arrow(gr.xc[1]-0.1*gr.dx, 0.5, -0.12*gr.dx, 0, shape='full', head_width=0.05, head_length=0.025, lw=1, width=0.02, edgecolor="none", facecolor="r", length_includes_head=True, zorder=100) if with_time: gr.mark_cell_left_state(1, r"$a_{i+\myhalf,R}^{n+\myhalf}$", fontsize="large", color="b") gr.mark_cell_right_state(0, r"$a_{i+\myhalf,L}^{n+\myhalf}$", fontsize="large", color="b") else: gr.mark_cell_left_state(1, r"$a_{i+\myhalf,R}$", fontsize="large", color="b") gr.mark_cell_right_state(0, r"$a_{i+\myhalf,L}$", fontsize="large", color="b") gr.label_cell_center(0, r"$a_i$") gr.label_cell_center(1, r"$a_{i+1}$") gr.clean_axes(pad_fac=0.125, ylim=(-0.25, 1.0)) f = plt.gcf() f.set_size_inches(7.0,2.0) plt.tight_layout() if with_time: plt.savefig("riemann-adv.pdf") else: plt.savefig("riemann-adv-mol.pdf") if __name__== "__main__": riemann() riemann(with_time=False) ================================================ FILE: figures/advection/riemann_bc.py ================================================ import numpy as np import matplotlib.pyplot as plt import grid_plot as gp def riemann(): # grid info xmin = 0.0 xmax = 1.0 nzones = 4 ng = 2 gr = gp.FVGrid(nzones, ng=ng, xmin=xmin, xmax=xmax) # interior and ghost cell initialization a = gr.scratch_array() a[gr.ilo:gr.ihi+1] = np.array([0.8, 0.4, 0.3, 0.5]) a[0:gr.ilo] = a[gr.ihi-1:gr.ihi+1] a[gr.ihi:2*gr.ng+gr.nx] = a[gr.ihi] pc = gp.PiecewiseConstant(gr, a) pl = gp.PiecewiseLinear(gr, a, nolimit=1) #------------------------------------------------------------------------ # plot a domain without ghostcells gr.draw_grid(draw_ghost=1, emphasize_end=1) gr.label_center(gr.ng-2, r"$\mathrm{lo-2}$", fontsize="medium") gr.label_center(gr.ng-1, r"$\mathrm{lo-1}$", fontsize="medium") gr.label_center(gr.ng, r"$\mathrm{lo}$", fontsize="medium") gr.label_center(gr.ng+1, r"$\mathrm{lo+1}$", fontsize="medium") gr.label_edge(gr.ng, r"$\mathrm{lo}-\myhalf$", fontsize="medium") # draw cell averages for n in range(0, gr.ng+gr.nx-1): pc.draw_cell_avg(n, color="0.5", ls=":") # draw slopes for n in range(gr.ilo-1, gr.ihi): pl.draw_slope(n, color="r") # compute the states to the left and right of lo-1/2 C = 0.5 # CFL print(a) print(pl.slope) al = a[gr.ilo-1] + 0.5*(1.0 - C)*pl.slope[gr.ilo-1] ar = a[gr.ilo] - 0.5*(1.0 + C)*pl.slope[gr.ilo] # L gr.mark_cell_right_state(ng-1, r"$a_{\mathrm{lo}-\myhalf,L}^{n+\myhalf}$", value=al, vertical="center", color="b") # R gr.mark_cell_left_state(ng, r"$a_{\mathrm{lo}-\myhalf,R}^{n+\myhalf}$", value=ar, vertical="top", color="b") plt.xlim(gr.xl[0]-0.025*gr.dx,gr.xr[ng+1]+0.15*gr.dx) plt.ylim(-0.25, 1.1) plt.axis("off") plt.subplots_adjust(left=0.025,right=0.95,bottom=0.05,top=0.95) f = plt.gcf() f.set_size_inches(8.0,2.0) plt.tight_layout() plt.savefig("riemann-bc.pdf") if __name__== "__main__": riemann() ================================================ FILE: figures/advection/riemann_mol.py ================================================ import numpy as np import matplotlib.pyplot as plt import grid_plot as gp def riemann(): # grid info xmin = 0.0 xmax = 1.0 nzones = 4 ng = 2 gr = gp.FVGrid(nzones, ng=ng, xmin=xmin, xmax=xmax) # interior and ghost cell initialization a = gr.scratch_array() a[gr.ilo:gr.ihi+1] = np.array([0.8, 0.7, 0.4, 0.5]) a[0:gr.ilo] = a[gr.ihi-1:gr.ihi+1] a[gr.ihi:2*gr.ng+gr.nx] = a[gr.ihi] pc = gp.PiecewiseConstant(gr, a) pl = gp.PiecewiseLinear(gr, a, nolimit=1) #------------------------------------------------------------------------ # plot a domain without ghostcells gr.draw_grid(draw_ghost=0, emphasize_end=0) zi = gr.ilo+1 gr.label_center(zi-1, r"${i-1}$", fontsize="medium") gr.label_center(zi, r"${i}$", fontsize="medium") gr.label_center(zi+1, r"${i+1}$", fontsize="medium") gr.label_edge(zi+1, r"${i}+\myhalf$", fontsize="medium") # draw cell averages for n in range(gr.ilo, gr.ihi+1): pc.draw_cell_avg(n, color="0.5", ls=":") # draw slopes for n in range(gr.ilo, gr.ihi+1): pl.draw_slope(n, color="r") # compute the states to the left and right of lo-1/2 al = a[zi] + 0.5*pl.slope[zi] ar = a[zi+1] - 0.5*pl.slope[zi+1] # L gr.mark_cell_right_state(zi, r"$q_{i+\myhalf,L}$", value=al, vertical="top", color="b", zorder=100) # R gr.mark_cell_left_state(zi+1, r"$q_{i+\myhalf,R}$", value=ar, vertical="top", color="b", zorder=100) plt.xlim(gr.xl[gr.ilo]-0.025*gr.dx, gr.xr[gr.ihi]+0.15*gr.dx) plt.ylim(-0.25, 1.1) plt.axis("off") plt.subplots_adjust(left=0.025,right=0.95,bottom=0.05,top=0.95) f = plt.gcf() f.set_size_inches(8.0,2.0) plt.tight_layout() plt.savefig("riemann-mol.pdf") if __name__== "__main__": riemann() ================================================ FILE: figures/advection/update.sh ================================================ #!/bin/sh DEST=../../advection # characteristics.py makes advection-characteristics.pdf python3 characteristics.py cp -f advection-characteristics.pdf ${DEST} # fd_ghost.py makes fd_ghost.pdf python3 fd_ghost.py cp -f fd_ghost.pdf ${DEST} # hydro_examples: fdadvect.py makes fdadvect-upwind.pdf # hydro_examples: fdadvect.py makes fdadvect-FTCS-C0_1.pdf, fdadvect-FTCS-C0_5.pdf # hydro_examples: fdadvect_implicit.py makes fdadvect-implicit.pdf # fv_ghost.py makes fv_ghost.pdf python3 fv_ghost.py cp -f fv_ghost.pdf ${DEST} # riemann.py makes riemann-adv.pdf python3 riemann.py cp -f riemann-adv.pdf ${DEST} # riemann_bc.py makes riemamm-bc.pdf python3 riemann_bc.py cp -f riemann-bc.pdf ${DEST} # hydro_examples: fv-advection.py makes fv-advect.pdf # rea-limitex.py makes rea*.pdf python3 rea-limitex.py cp -f rea*start_???.pdf ${DEST} # generalgrid.py makes generalgrid.pdf python3 generalgrid.py cp -f generalgrid.pdf ${DEST} # rea.py makes rea-*.pdf python3 rea.py cp -f rea-final.pdf rea-evolve.pdf rea-trace.pdf rea-reconstruction.pdf rea-start.pdf ${DEST} # hydro_examples: advection.py makes plm-converge.pdf, fv-gaussian-limiters.pdf fv-tophat-limiters.pdf # 2dgrid.py makes 2dgrid.pdf python3 2dgrid.py cp -f 2dgrid.pdf ${DEST} # 2dgrid_hat.py makes 2dgrid-hat.pdf python3 2dgrid_hat.py cp -f 2dgrid-hat.pdf ${DEST} # 2dgrid_transverse.py makes 2dgrid-transverse.pdf python3 2dgrid_transverse.py cp -f 2dgrid-transverse.pdf ${DEST} # pyro makes smooth_init.pdf smooth_final.pdf tophat_init.pdf tophat_final.pdf # tophat_rk4_cfl*.pdf ================================================ FILE: figures/burgers/characteristics.py ================================================ import numpy as np import matplotlib as mpl import matplotlib.pyplot as plt mpl.rcParams['mathtext.fontset'] = 'cm' mpl.rcParams['mathtext.rm'] = 'serif' def fun(x): state = np.zeros_like(x) state[x < 0.5] = 1.0 return state def funs(x): state = 0.5 + 0.25*np.sin(2.0*np.pi*x) return state def funr(x): state = np.ones_like(x) state[x < 0.5] = 0.2 return state def make_plot(icfun=None, label=""): npts_plot = 1000 xplot = np.linspace(0.0, 1.0, npts_plot) nchar = 20 xchar = np.linspace(0.0, 1.0, nchar) state = icfun(xplot) print(type(state)) plt.clf() plt.subplot(211) plt.plot(xplot, state, lw=2) ax = plt.gca() ax.spines['right'].set_visible(False) ax.spines['top'].set_visible(False) ax.spines['right'].set_visible(False) ax.spines['top'].set_visible(False) ax.spines['right'].set_visible(False) ax.spines['top'].set_visible(False) ax.xaxis.set_ticks_position('bottom') ax.yaxis.set_ticks_position('left') plt.ylim(-0.1, 1.1) plt.xlabel(r"$x$", fontsize="large") plt.ylabel(r"$u$", fontsize="large") plt.subplot(212) uchar = icfun(xchar) t = np.linspace(0.0, 1.0, 100) for n in range(nchar): xc = xchar[n] + uchar[n]*t plt.plot(xc, t, color="0.5") ax = plt.gca() ax.spines['right'].set_visible(False) ax.spines['top'].set_visible(False) ax.spines['right'].set_visible(False) ax.spines['top'].set_visible(False) ax.spines['right'].set_visible(False) ax.spines['top'].set_visible(False) ax.xaxis.set_ticks_position('bottom') ax.yaxis.set_ticks_position('left') plt.xlabel(r"$x$", fontsize="large") plt.ylabel(r"$t$", fontsize="large") plt.xlim(0.0, 1.0) plt.ylim(0.0, 1.1) f = plt.gcf() f.set_size_inches(6.0, 8.0) plt.tight_layout() plt.savefig(f"burgers-characteristics-{label}.png") if __name__ == "__main__": make_plot(funr, "rare") make_plot(funs, "shock") ================================================ FILE: figures/burgers/rh.py ================================================ import numpy as np import matplotlib.pylab as plt import matplotlib as mpl mpl.rcParams['mathtext.fontset'] = 'cm' mpl.rcParams['mathtext.rm'] = 'serif' mpl.rcParams['font.size'] = 12 mpl.rcParams['legend.fontsize'] = 'large' mpl.rcParams['figure.titlesize'] = 'medium' def rh(): # grid info xl = 0.0 xr = 1.0 dx = xr - xl xc = 0.5*(xl + xr) t0 = 0.0 t1 = 1.0 dt = t1 - t0 # plot a square representing [x, x+dx] x [t, t+dt] # x-axis plt.arrow(0, 0, 1.3*dx, 0, shape="full", head_width=0.04, head_length=0.06, lw=1, width=0.005, facecolor="k", length_includes_head=True, zorder=100) plt.text(1.35*dx, 0, r"$x$", fontsize=20, verticalalignment="center") plt.text(dx, -0.1, r"$x_r$", fontsize=20) plt.text(0, -0.1, r"$x_l$", fontsize=20) # time axis plt.arrow(0, 0, 0, 1.3*dt, shape="full", head_width=0.04, head_length=0.06, lw=1, width=0.005, facecolor="k", length_includes_head=True, zorder=100) plt.text(0, 1.35*dt, r"$t$", fontsize=20, horizontalalignment="center") plt.text(-0.17, dt, r"$t^{n+1}$", fontsize=20) plt.text(-0.17, 0, r"$t^n$", fontsize=20) # space-time volume plt.plot([dx,dx], [0,dt], ls="--", color="0.5", lw=2) plt.plot([0,dx], [dt,dt], ls="--", color="0.5", lw=2) # diagonal representing S plt.plot([0,dx], [0,dt], color="k", lw=5, solid_capstyle="butt") plt.annotate(r"shock: $S = \Delta x / \Delta t$", xy=(0.74*dx,0.76*dt), xytext=(0.2*dx, 1.2*dt), textcoords="data", fontsize=18, arrowprops=dict(arrowstyle="->", connectionstyle="arc3,rad=.2")) # states plt.text(0.66*dx, 0.33*dt, r"$u_r$", color="k", fontsize=20) plt.text(0.33*dx, 0.66*dt, r"$u_l$", color="k", fontsize=20) # fluxes plt.arrow(-0.1*dx, 0.5*dt, 0.2*dx, 0, shape="full", head_width=0.04, head_length=0.06, lw=1, width=0.005, edgecolor="r", facecolor="r", length_includes_head=True, zorder=100) plt.text(-0.12*dx, 0.5*dt, r"$f = f(u_l)$", horizontalalignment="right", verticalalignment="center", color="r", fontsize=18) plt.arrow(0.9*dx, 0.5*dt, 0.2*dx, 0, shape="full", head_width=0.04, head_length=0.06, lw=1, width=0.005, edgecolor="r", facecolor="r", length_includes_head=True, zorder=100) plt.text(1.12*dx, 0.5*dt, r"$f = f(u_r)$", horizontalalignment="left", verticalalignment="center", color="r", fontsize=18) plt.axis("off") plt.subplots_adjust(left=0.05, right=0.95, bottom=0.05, top=0.95) plt.xlim(-0.4,1.4*dx) plt.ylim(-0.1,1.4*dx) f = plt.gcf() f.set_size_inches(7.0,6.0) plt.savefig("rh.pdf") if __name__== "__main__": rh() ================================================ FILE: figures/burgers/update.sh ================================================ #!/bin/sh DEST=../../burgers # characteristics.py makes burgers-characteristics.pdf python3 characteristics.py cp -f burgers-characteristics.pdf ${DEST} # rh.py makes rh.pdf python3 rh.py cp -f rh.pdf ${DEST} # hydro_examples: burgers.py makes fv-burger-rarefaction.pdf fv-burger-sine.pdf ================================================ FILE: figures/finite-volume/ccfd.py ================================================ import math import numpy as np import matplotlib.pyplot as plt import grid_plot as gp # plot a simple finite-difference grid #----------------------------------------------------------------------------- nzones = 8 # data that lives on the grid #a = np.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.15, 0.5, 0.55]) a = np.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.1, 0.5, 0.55]) gr = gp.FVGrid(nzones) plt.clf() gr.draw_grid() gr.label_center(nzones//2, r"$i$", fontsize="medium") gr.label_center(nzones//2-1, r"$i-1$", fontsize="medium") gr.label_center(nzones//2+1, r"$i+1$", fontsize="medium") gr.label_center(nzones//2-2, r"$i-2$", fontsize="medium") gr.label_center(nzones//2+2, r"$i+2$", fontsize="medium") cc = gp.CellCentered(gr, a) # draw the data for i in range(nzones): cc.draw_data_point(i, color="r") cc.label_data_point(nzones//2, r"$f_i$", color="r") # label dx gr.label_center_dx(gr.ng+nzones//2) gr.clean_axes(ylim=(-0.5, 1.2)) f = plt.gcf() f.set_size_inches(10.0,3.0) plt.savefig("ccfd_grid.pdf") ================================================ FILE: figures/finite-volume/domain.py ================================================ import matplotlib.pyplot as plt def draw_box(ll, uu, nx, ny, gridColor="0.5", ng=0): # draw the frame plt.plot([ll[0], ll[0], uu[0], uu[0], ll[0]], [ll[1], uu[1], uu[1], ll[1], ll[1]], color="k", lw=2) # draw the x grid lines dx = (uu[0] - ll[0])/nx for n in range(1, nx): plt.plot([ll[0]+n*dx, ll[0]+n*dx], [ll[1], uu[1]], color=gridColor, ls=":", lw=1) # draw the y grid lines dy = (uu[1] - ll[1])/ny for n in range(1, ny): plt.plot([ll[0], uu[0]], [ll[1]+n*dy, ll[1]+n*dy], color=gridColor, ls=":", lw=1) # ghostcells? if ng > 0: xmin = ll[0]-ng*dx xmax = uu[0]+ng*dx ymin = ll[1]-ng*dy ymax = uu[1]+ng*dy plt.plot([xmin, xmin, xmax, xmax, xmin], [ymin, ymax, ymax, ymin, ymin], ls="--", color="r") plt.clf() draw_box([0., 0.], [1., 1.], 5, 5, ng=1) draw_box([1., 0.], [2., 1.], 5, 5) draw_box([2., 0.], [3., 1.], 5, 5) draw_box([0., 1.], [1., 2.], 5, 5) draw_box([1., 1.], [2., 2.], 5, 5) draw_box([2., 1.], [3., 2.], 5, 5) plt.xlim(-0.25,3.25) plt.ylim(-0.25,2.25) a = plt.gca() a.set_aspect("equal", "datalim") plt.axis("off") f = plt.gcf() f.set_size_inches(7.0,4.25) plt.savefig("domain.pdf", bbox_inches="tight") ================================================ FILE: figures/finite-volume/fd.py ================================================ import matplotlib.pyplot as plt import numpy as np import grid_plot as gp # plot a simple finite-difference grid #----------------------------------------------------------------------------- nzones = 8 # data that lives on the grid #a = np.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.15, 0.5, 0.55]) a = np.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.1, 0.5, 0.55]) gr = gp.FDGrid(nzones) plt.clf() gr.draw_grid() gr.label_node(nzones//2, r"$i$", fontsize="medium") gr.label_node(nzones//2-1, r"$i-1$", fontsize="medium") gr.label_node(nzones//2+1, r"$i+1$", fontsize="medium") gr.label_node(nzones//2-2, r"$i-2$", fontsize="medium") gr.label_node(nzones//2+2, r"$i+2$", fontsize="medium") # draw the data for i in range(nzones): gr.draw_data(i, a[i], color="r") gr.label_value(nzones//2, a[nzones//2], r"$f_i$", color="r") # label dx gr.label_dx(gr.ng+nzones//2) gr.clean_axes(ylim=(-0.5, 1.2)) f = plt.gcf() f.set_size_inches(10.0,3.0) plt.savefig("fd_grid.pdf") ================================================ FILE: figures/finite-volume/fv.py ================================================ import numpy as np import matplotlib.pyplot as plt import grid_plot as gp # plot a simple finite-difference grid #----------------------------------------------------------------------------- nzones = 8 # data that lives on the grid #a = np.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.15, 0.5, 0.55]) a = np.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.1, 0.5, 0.55]) gr = gp.FVGrid(nzones) pc = gp.PiecewiseConstant(gr, a) plt.clf() gr.draw_grid() gr.label_center(nzones//2, r"$i$", fontsize="medium") gr.label_center(nzones//2-1, r"$i-1$", fontsize="medium") gr.label_center(nzones//2+1, r"$i+1$", fontsize="medium") gr.label_center(nzones//2-2, r"$i-2$", fontsize="medium") gr.label_center(nzones//2+2, r"$i+2$", fontsize="medium") gr.label_edge(nzones//2, r"$i-\sfrac{1}{2}$", fontsize="small") gr.label_edge(nzones//2+1, r"$i+\sfrac{1}{2}$", fontsize="small") # draw the data for i in range(nzones): pc.draw_cell_avg(i, color="r", filled=True) pc.label_cell_avg(nzones//2, r"$\langle f\rangle_i$", color="r") # label dx gr.label_dx(gr.ng+nzones//2) gr.clean_axes(ylim=(-0.5, 1.2)) f = plt.gcf() f.set_size_inches(10.0,3.0) plt.savefig("fv_grid.pdf") ================================================ FILE: figures/finite-volume/simplegrid2.py ================================================ import math import matplotlib.pyplot as plt import numpy as np import grid_plot as gp def simplegrid(): nzones = 7 gr = gp.FVGrid(nzones, xmin=0, xmax=1) gr.draw_grid(edge_ticks=0) # label a few cell-centers gr.label_center(nzones/2, r"$i$") gr.label_center(nzones/2-1, r"$i-1$") gr.label_center(nzones/2+1, r"$i+1$") # label a few edges gr.label_edge(nzones/2, r"$i-1/2$") gr.label_edge(nzones/2+1, r"$i+1/2$") # sample data A = 0.4 a = A*np.ones(nzones, dtype=np.float64) pc = gp.PiecewiseConstant(gr, a) # draw an average quantity pc.draw_cell_avg(nzones/2, color="r") pc.label_cell_avg(nzones/2, r"$\,\langle a \rangle_i$", color="r") gr.clean_axes() plt.ylim(-0.25, 1.5) plt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95) f = plt.gcf() f.set_size_inches(10.0,2.5) plt.savefig("simplegrid2.png") plt.savefig("simplegrid2.pdf") if __name__== "__main__": simplegrid() ================================================ FILE: figures/finite-volume/simplegrid_gc.py ================================================ import numpy as np import matplotlib.pyplot as plt import grid_plot as gp def simplegrid(): # grid info nzones = 7 ng = 1 gr = gp.FVGrid(nzones, ng, xmin=0.0, xmax=1.0) gr.draw_grid(emphasize_end=1, edge_ticks=0, draw_ghost=1) # label a few gr.label_center(ng+nzones//2, r"$i$", fontsize="medium") gr.label_center(ng+nzones//2-1, r"$i-1$", fontsize="medium") gr.label_center(ng+nzones//2+1, r"$i+1$", fontsize="medium") gr.label_center(ng-1, r"$\mathrm{lo}-1$", fontsize="medium") gr.label_center(ng, r"$\mathrm{lo}$", fontsize="medium") gr.label_center(ng+nzones-1, r"$\mathrm{hi}$", fontsize="medium") gr.label_center(ng+nzones, r"$\mathrm{hi+1}$", fontsize="medium") # label dx gr.label_dx(ng+nzones//2-2) gr.clean_axes(show_ghost=True, ylim=(-0.5, 1.5), padding=True, pad_fac=0.05) #plt.xlim(gr.xl[0]-0.05*gr.dx,gr.xr[2*ng+nzones-1]+0.25*gr.dx) #plt.ylim(-0.5, 1.5) #plt.axis("off") #plt.subplots_adjust(left=0.025,right=0.95,bottom=0.05,top=0.95) f = plt.gcf() f.set_size_inches(10.0,2.5) plt.tight_layout() plt.savefig("simplegrid_gc.pdf", bbox_inches="tight") if __name__== "__main__": simplegrid() ================================================ FILE: figures/finite-volume/update.sh ================================================ #!/bin/sh # fd.py creates fd_grid.pdf python3 fd.py cp -f fd_grid.pdf ../../finite-volume/ # ccfd.py creates ccfd_grid.pdf python3 ccfd.py cp -f ccfd_grid.pdf ../../finite-volume/ # fv.py creates fv_grid.pdf python3 fv.py cp -f fv_grid.pdf ../../finite-volume/ # simplegrid_gc.py creates simplegrid_gc.pdf python3 simplegrid_gc.py cp -f simplegrid_gc.pdf ../../finite-volume/ # domain.py creates domain.pdf python3 domain.py cp -r domain.pdf ../../finite-volume/ ================================================ FILE: figures/gridFigs/fdrestrict.py ================================================ import numpy as np import matplotlib.pyplot as plt import grid_plot as gp # plot two stacked fv grids of different (2x) resolution to show prolongation #----------------------------------------------------------------------------- nf = 5 nc = 3 grf = gp.FDGrid(nf, voff=2.0) grc = gp.FDGrid(nc) plt.clf() grf.draw_grid() grc.draw_grid() grf.label_node(nf/2-2, r"$i-2$") grf.label_node(nf/2-1, r"$i-1$") grf.label_node(nf/2, r"$i$") grf.label_node(nf/2+1, r"$i+1$") grf.label_node(nf/2+2, r"$i+2$") grc.label_node(nc/2-1, r"$j-1$") grc.label_node(nc/2, r"$j$") grc.label_node(nc/2+1, r"$j+1$") grf.label_node_data(nf/2-2, r"$\phi_{i-2}^h$") grf.label_node_data(nf/2-1, r"$\phi_{i-1}^h$") grf.label_node_data(nf/2, r"$\phi_i^h$") grf.label_node_data(nf/2+1, r"$\phi_{i+1}^h$") grf.label_node_data(nf/2+2, r"$\phi_{i+2}^h$") grc.label_node_data(nc/2-1, r"$\phi_{j-1}^{2h}$") grc.label_node_data(nc/2, r"$\phi_{j}^{2h}$") grc.label_node_data(nc/2+1, r"$\phi_{j+1}^{2h}$") # connect the dots... plt.plot([grf.xc[nf/2-2], grf.xc[nf/2-2]], [-0.25, 3.25], ":", color="0.5") plt.plot([grf.xc[nf/2], grf.xc[nf/2]], [-0.25, 3.25], ":", color="0.5") plt.plot([grf.xc[nf/2+2], grf.xc[nf/2+2]], [-0.25, 3.25], ":", color="0.5") plt.axis([grf.xmin-0.5*grf.dx,grf.xmax+0.5*grf.dx, -0.5, 3.5]) plt.axis("off") plt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95) f = plt.gcf() f.set_size_inches(6.0,5.0) plt.savefig("fdrestrict.pdf") ================================================ FILE: figures/gridFigs/fvprolong.py ================================================ import matplotlib.pyplot as plt import grid_plot as gp # plot two stacked fv grids of different (2x) resolution to show prolongation #----------------------------------------------------------------------------- nf = 4 nc = nf/2 grf = gp.FVGrid(nf) grc = gp.FVGrid(nc, voff=2.0) plt.clf() grf.draw_grid() grc.draw_grid() grf.label_center(nf/2-2, r"$i-2$") grf.label_center(nf/2-1, r"$i-1$") grf.label_center(nf/2, r"$i$") grf.label_center(nf/2+1, r"$i+1$") grc.label_center(nc/2-1, r"$j-1$") grc.label_center(nc/2, r"$j$") grf.label_cell_center(nf/2-2, r"$\phi_{i-2}^h$") grf.label_cell_center(nf/2-1, r"$\phi_{i-1}^h$") grf.label_cell_center(nf/2, r"$\phi_i^h$") grf.label_cell_center(nf/2+1, r"$\phi_{i+1}^h$") grc.label_cell_center(nc/2-1, r"$\phi_{j-1}^{2h}$") grc.label_cell_center(nc/2, r"$\phi_{j}^{2h}$") # connect the dots... plt.plot([grf.xl[nf/2-2], grf.xl[nf/2-2]], [-0.25, 3.25], ":", color="0.5") plt.plot([grf.xl[nf/2], grf.xl[nf/2]], [-0.25, 3.25], ":", color="0.5") plt.plot([grf.xr[nf/2+1], grf.xr[nf/2+1]], [-0.25, 3.25], ":", color="0.5") plt.axis([grf.xmin-0.5*grf.dx,grf.xmax+0.5*grf.dx, -0.5, 3.5]) plt.axis("off") plt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95) f = plt.gcf() f.set_size_inches(6.0,5.0) plt.savefig("fvprolong.pdf") ================================================ FILE: figures/gridFigs/fvrestrict.py ================================================ import numpy as np import matplotlib.pyplot as plt import grid_plot as gp # plot two stacked fv grids of different (2x) resolution to show prolongation #----------------------------------------------------------------------------- nf = 4 nc = nf/2 grf = gp.FVGrid(nf, voff=2.0) grc = gp.FVGrid(nc) plt.clf() grf.draw_grid() grc.draw_grid() grf.label_center(nf/2-2, r"$i-2$") grf.label_center(nf/2-1, r"$i-1$") grf.label_center(nf/2, r"$i$") grf.label_center(nf/2+1, r"$i+1$") grc.label_center(nc/2-1, r"$j-1$") grc.label_center(nc/2, r"$j$") grf.label_cell_center(nf/2-2, r"$\phi_{i-2}^h$") grf.label_cell_center(nf/2-1, r"$\phi_{i-1}^h$") grf.label_cell_center(nf/2, r"$\phi_i^h$") grf.label_cell_center(nf/2+1, r"$\phi_{i+1}^h$") grc.label_cell_center(nc/2-1, r"$\phi_{j-1}^{2h}$") grc.label_cell_center(nc/2, r"$\phi_{j}^{2h}$") # connect the dots... plt.plot([grf.xl[nf/2-2], grf.xl[nf/2-2]], [-0.25, 3.25], ":", color="0.5") plt.plot([grf.xl[nf/2], grf.xl[nf/2]], [-0.25, 3.25], ":", color="0.5") plt.plot([grf.xr[nf/2+1], grf.xr[nf/2+1]], [-0.25, 3.25], ":", color="0.5") plt.axis([grf.xmin-0.5*grf.dx,grf.xmax+0.5*grf.dx, -0.5, 3.5]) plt.axis("off") plt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95) f = plt.gcf() f.set_size_inches(6.0,5.0) plt.savefig("fvrestrict.pdf") ================================================ FILE: figures/gridFigs/nested.py ================================================ import matplotlib.pyplot as plt import grid_plot as gp # plot two stacked fv grids of different (2x) resolution to show prolongation #----------------------------------------------------------------------------- gr = [] nf = 8 gr.append(gp.FVGrid(nf, ng=1, voff=0.0)) gr.append(gp.FVGrid(nf, ng=1, voff=0.0, xmin=0.25, xmax=0.75)) plt.clf() gr[0].draw_grid(emphasize_end=1, draw_ghost=0, edge_ticks=0, color="0.75") gr[1].draw_grid(emphasize_end=1, draw_ghost=0, edge_ticks=0) f = plt.gcf() f.set_size_inches(7.0,1.0) grf = gr[0] plt.xlim(grf.xmin-0.75*grf.dx,grf.xmax+0.25*grf.dx) plt.axis("off") plt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95) plt.savefig("nested1.pdf") #----------------------------------------------------------------------------- plt.clf() gr = [] gr.append(gp.FVGrid(nf, ng=1, voff=0.0)) gr.append(gp.FVGrid(nf, ng=1, voff=4.0)) gr[0].draw_grid(emphasize_end=1, draw_ghost=0, edge_ticks=0) gr[1].draw_grid(emphasize_end=1, draw_ghost=0, edge_ticks=0) # labels plt.text(gr[0].xmin-0.7*gr[0].dx, gr[0].voff+0.5, r"$\phi^{c,n}$", fontsize="medium") plt.text(gr[1].xmin-0.7*gr[1].dx, gr[1].voff+0.5, r"$\phi^{c,n+1}$", fontsize="medium") plt.arrow(gr[0].xmin-0.5*gr[0].dx, 0.5+3.0*gr[0].dx, 0.001, gr[1].voff-4.0*gr[0].dx, shape='full', head_width=0.025, head_length=0.25, lw=1, width=0.01, edgecolor="none", facecolor="r", length_includes_head=True, zorder=100) f = plt.gcf() f.set_size_inches(7.0,5.0) grf = gr[0] plt.xlim(grf.xmin-0.75*grf.dx,grf.xmax+0.25*grf.dx) plt.axis("off") plt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95) plt.savefig("nested2.pdf") #----------------------------------------------------------------------------- plt.clf() gr1 = [] gr1.append(gp.FVGrid(nf, ng=1, voff=0.0)) gr1.append(gp.FVGrid(nf, ng=1, voff=0.0, xmin=0.25, xmax=0.75)) gr2 = [] gr2.append(gp.FVGrid(nf, ng=1, voff=2.0)) gr2.append(gp.FVGrid(nf, ng=1, voff=2.0, xmin=0.25, xmax=0.75)) gr3 = [] gr3.append(gp.FVGrid(nf, ng=1, voff=4.0)) gr3.append(gp.FVGrid(nf, ng=1, voff=4.0, xmin=0.25, xmax=0.75)) gr1[0].draw_grid(emphasize_end=1, draw_ghost=0, edge_ticks=0, color="0.75") gr1[1].draw_grid(emphasize_end=1, draw_ghost=0, edge_ticks=0) gr2[0].draw_grid(emphasize_end=1, draw_ghost=0, edge_ticks=0, color="0.75") gr2[1].draw_grid(emphasize_end=1, draw_ghost=0, edge_ticks=0) gr3[0].draw_grid(emphasize_end=1, draw_ghost=0, edge_ticks=0, color="0.75") gr3[1].draw_grid(emphasize_end=1, draw_ghost=0, edge_ticks=0) plt.text(gr1[1].xmin-0.95*gr1[0].dx, gr1[1].voff+0.5, r"$\phi^{f,n}$", fontsize="11", horizontalalignment="left") plt.text(gr2[1].xmin-0.95*gr2[0].dx, gr2[1].voff+0.5, r"$\phi^{f,n+1/2}$", fontsize="11", horizontalalignment="left") plt.text(gr3[1].xmin-0.95*gr3[0].dx, gr3[1].voff+0.5, r"$\phi^{f,n+1}$", fontsize="11", horizontalalignment="left") plt.annotate(r"Dirichlet BCs $\phi^{c,n}$", xy=(gr1[1].xmin, gr1[1].voff), xycoords='data', xytext=(30,-25), textcoords="offset points", arrowprops=dict(arrowstyle="->", connectionstyle="angle,angleA=180,angleB=-90,rad=10", ec="0.5"), fontsize="small",color="0.5") plt.annotate(r"Dirichlet BCs $(\phi^{c,n} + \phi^{c,n+1})/2$", xy=(gr2[1].xmin, gr2[1].voff), xycoords='data', xytext=(30,-25), textcoords="offset points", arrowprops=dict(arrowstyle="->", connectionstyle="angle,angleA=180,angleB=-90,rad=10", ec="0.5"), fontsize="small",color="0.5") plt.annotate(r"Dirichlet BCs $\phi^{c,n+1}$", xy=(gr3[1].xmin, gr3[1].voff), xycoords='data', xytext=(30,-25), textcoords="offset points", arrowprops=dict(arrowstyle="->", connectionstyle="angle,angleA=180,angleB=-90,rad=10", ec="0.5"), fontsize="small",color="0.5") plt.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, shape='full', head_width=0.025, head_length=0.25, lw=1, width=0.01, edgecolor="none", facecolor="r", length_includes_head=True, zorder=100) plt.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, shape='full', head_width=0.025, head_length=0.25, lw=1, width=0.01, edgecolor="none", facecolor="r", length_includes_head=True, zorder=100) f = plt.gcf() f.set_size_inches(7.0,5.0) grf = gr[0] plt.xlim(grf.xmin-0.75*grf.dx,grf.xmax+0.25*grf.dx) plt.axis("off") plt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95) plt.savefig("nested3.pdf") ================================================ FILE: figures/incompressible/MAC-solve.py ================================================ import math import numpy import pylab class Grid2d: def __init__(self, nx, ny, xmin=0.0, xmax=1.0, ymin=0.0, ymax=1.0): self.xmin = xmin self.xmax = xmax self.ymin = ymin self.ymax = ymax self.nx = nx self.ny = ny self.dx = (self.xmax - self.xmin)/float(self.nx) self.dy = (self.ymax - self.ymin)/float(self.ny) self.xl = (numpy.arange(self.nx) )*self.dx self.xr = (numpy.arange(self.nx)+1)*self.dx self.xc = 0.5*(self.xl + self.xr) self.yl = (numpy.arange(self.ny) )*self.dy self.yr = (numpy.arange(self.ny)+1)*self.dy self.yc = 0.5*(self.yl + self.yr) def draw_grid(self): """ plot a domain without ghostcells """ # lines parallel to the x-axis n = 0 while n < self.ny: pylab.plot([self.xmin-0.25*self.dx,self.xmax+0.25*self.dx], [self.yl[n],self.yl[n]], color="k", lw=2) n += 1 pylab.plot([self.xmin-0.25*self.dx,self.xmax+0.25*self.dx], [self.yr[self.ny-1],self.yr[self.ny-1]], color="k", lw=2) # lines parallel to the y-axis n = 0 while n < self.nx: pylab.plot([self.xl[n],self.xl[n]], [self.ymin-0.25*self.dy,self.ymax+0.25*self.dy], color="k", lw=2) n += 1 pylab.plot([self.xr[self.nx-1],self.xr[self.nx-1]], [self.ymin-0.25*self.dy,self.ymax+0.25*self.dy], color="k", lw=2) # grid labels pylab.text(self.xc[self.nx/2], self.yl[0]-0.35*self.dy, r"$i$", horizontalalignment='center', fontsize="16") if self.nx/2-1 >= 0: pylab.text(self.xc[self.nx/2-1], self.yl[0]-0.35*self.dy, r"$i-1$", horizontalalignment='center', fontsize="16") if self.nx/2+1 < self.nx: pylab.text(self.xc[self.nx/2+1], self.yl[0]-0.35*self.dy, r"$i+1$", horizontalalignment='center', fontsize="16") pylab.text(self.xl[0]-0.35*self.dx, self.yc[self.ny/2], r"$j$", verticalalignment='center', fontsize="16") if self.ny/2-1 >= 0: pylab.text(self.xl[0]-0.35*self.dx, self.yc[self.ny/2-1], r"$j-1$", verticalalignment='center', fontsize="16") if self.ny/2+1 < self.ny: pylab.text(self.xl[0]-0.35*self.dx, self.yc[self.ny/2+1], r"$j+1$", verticalalignment='center', fontsize="16") def simplegrid(): g = Grid2d(3,3) pylab.subplot(121) g.draw_grid() #------------------------------------------------------------------------ # label pylab.text(g.xc[g.nx/2], g.yc[g.ny/2], r"$\phi_{i,j}$", fontsize="15", horizontalalignment='center', verticalalignment='center') # phi pylab.text(g.xc[g.nx/2+1], g.yc[g.ny/2]+0.05*g.dy, r"$\phi_{i+1,j}$", fontsize="15", horizontalalignment='center', verticalalignment='center') pylab.text(g.xc[g.nx/2-1], g.yc[g.ny/2]+0.05*g.dy, r"$\phi_{i-1,j}$", fontsize="15", horizontalalignment='center', verticalalignment='center') pylab.text(g.xc[g.nx/2], g.yc[g.ny/2+1]+0.05*g.dy, r"$\phi_{i,j+1}$", fontsize="15", horizontalalignment='center', verticalalignment='center') pylab.text(g.xc[g.nx/2], g.yc[g.ny/2-1]+0.05*g.dy, r"$\phi_{i,j-1}$", fontsize="15", horizontalalignment='center', verticalalignment='center') # MAC velocities # i+1/2,j interface pylab.scatter(g.xr[g.nx/2], g.yc[g.ny/2], marker="x", s=50) pylab.text(g.xr[g.nx/2]+0.05*g.dx, g.yc[g.ny/2], r"${u}^\mathrm{adv}_{i+1/2,j}$", fontsize="15", rotation=270, horizontalalignment='left', verticalalignment='center') # i-1/2,j interface pylab.scatter(g.xr[g.nx/2-1], g.yc[g.ny/2], marker="x", s=50) pylab.text(g.xr[g.nx/2-1]-0.05*g.dx, g.yc[g.ny/2], r"${u}^\mathrm{adv}_{i-1/2,j}$", fontsize="15", rotation=270, horizontalalignment='right', verticalalignment='center') # i,j+1/2 interface pylab.scatter(g.xc[g.nx/2], g.yr[g.ny/2], marker="x", s=50) pylab.text(g.xc[g.nx/2], g.yr[g.ny/2]+0.05*g.dy, r"${v}^\mathrm{adv}_{i,j+1/2}$", fontsize="15", horizontalalignment='center', verticalalignment='bottom') # i,j-1/2 interface pylab.scatter(g.xc[g.nx/2], g.yr[g.ny/2-1], marker="x", s=50) pylab.text(g.xc[g.nx/2], g.yr[g.ny/2-1]-0.05*g.dy, r"${v}^\mathrm{adv}_{i,j-1/2}$", fontsize="15", horizontalalignment='center', verticalalignment='top') # axes pylab.xlim(g.xl[0]-0.5*g.dx,g.xr[g.nx-1]+0.25*g.dx) pylab.ylim(g.yl[0]-0.5*g.dy,g.yr[g.ny-1]+0.25*g.dy) pylab.axis("off") pylab.subplots_adjust(left=0.02,right=0.98,bottom=0.02,top=0.98) ax = pylab.gca() ax.set_aspect("equal", "datalim") pylab.subplot(122) g.draw_grid() #------------------------------------------------------------------------ # label pylab.text(g.xc[g.nx/2], g.yc[g.ny/2]+0.1*g.dy, r"$(DG\phi)_{i,j}$", fontsize="15", horizontalalignment='center', verticalalignment='center') pylab.text(g.xc[g.nx/2], g.yc[g.ny/2]-0.1*g.dy, r"$(DU^\mathrm{adv})_{i,j}$", fontsize="15", horizontalalignment='center', verticalalignment='center') # Gphi velocities # i+1/2,j interface pylab.scatter(g.xr[g.nx/2], g.yc[g.ny/2], marker="x", s=50) pylab.text(g.xr[g.nx/2]+0.05*g.dx, g.yc[g.ny/2], r"${(G\phi)}_{i+1/2,j}$", fontsize="15", rotation=270, horizontalalignment='left', verticalalignment='center') # i-1/2,j interface pylab.scatter(g.xr[g.nx/2-1], g.yc[g.ny/2], marker="x", s=50) pylab.text(g.xr[g.nx/2-1]-0.05*g.dx, g.yc[g.ny/2], r"${(G\phi)}_{i-1/2,j}$", fontsize="15", rotation=270, horizontalalignment='right', verticalalignment='center') # i,j+1/2 interface pylab.scatter(g.xc[g.nx/2], g.yr[g.ny/2], marker="x", s=50) pylab.text(g.xc[g.nx/2], g.yr[g.ny/2]+0.05*g.dy, r"${(G\phi)}_{i,j+1/2}$", fontsize="15", horizontalalignment='center', verticalalignment='bottom') # i,j-1/2 interface pylab.scatter(g.xc[g.nx/2], g.yr[g.ny/2-1], marker="x", s=50) pylab.text(g.xc[g.nx/2], g.yr[g.ny/2-1]-0.05*g.dy, r"${(G\phi)}_{i,j-1/2}$", fontsize="15", horizontalalignment='center', verticalalignment='top') # axes pylab.xlim(g.xl[0]-0.5*g.dx,g.xr[g.nx-1]+0.25*g.dx) pylab.ylim(g.yl[0]-0.5*g.dy,g.yr[g.ny-1]+0.25*g.dy) pylab.axis("off") pylab.subplots_adjust(left=0.02,right=0.98,bottom=0.02,top=0.98) ax = pylab.gca() ax.set_aspect("equal", "datalim") f = pylab.gcf() f.set_size_inches(12.0,6.0) pylab.tight_layout() pylab.savefig("MAC_solve.png") pylab.savefig("MAC_solve.eps", bbox_inches="tight") if __name__== "__main__": simplegrid() ================================================ FILE: figures/incompressible/MAC.py ================================================ import math import numpy import pylab def simplegrid(): # grid info xmin = 0.0 xmax = 1.0 ymin = 0.0 ymax = 1.0 nzones = 3 ng = 0 dx = (xmax - xmin)/float(nzones) dy = (ymax - ymin)/float(nzones) xl = (numpy.arange(2*ng+nzones) - ng)*dx xr = (numpy.arange(2*ng+nzones)+1 - ng)*dx xc = 0.5*(xl + xr) yl = (numpy.arange(2*ng+nzones) - ng)*dy yr = (numpy.arange(2*ng+nzones)+1 - ng)*dy yc = 0.5*(yl + yr) #------------------------------------------------------------------------ # plot a domain without ghostcells # x lines n = 0 while (n < nzones): pylab.plot([xmin-0.25*dx,xmax+0.25*dx], [yl[n],yl[n]], color="k", lw=2) n += 1 pylab.plot([xmin-0.25*dx,xmax+0.25*dx], [yr[nzones-1],yr[nzones-1]], color="k", lw=2) # y lines n = 0 while (n < nzones): pylab.plot([xl[n],xl[n]], [ymin-0.25*dy,ymax+0.25*dy], color="k", lw=2) n += 1 pylab.plot([xr[nzones-1],xr[nzones-1]], [ymin-0.25*dy,ymax+0.25*dy], color="k", lw=2) #------------------------------------------------------------------------ # label pylab.text(xc[nzones/2], yc[nzones/2]+0.05*dy, r"$u_{i,j}$", fontsize="15", horizontalalignment='center', verticalalignment='center') pylab.text(xc[nzones/2], yc[nzones/2]-0.05*dy, r"$v_{i,j}$", fontsize="15", horizontalalignment='center', verticalalignment='center') pylab.text(xc[nzones/2+1], yc[nzones/2]+0.05*dy, r"$u_{i+1,j}$", fontsize="15", horizontalalignment='center', verticalalignment='center') pylab.text(xc[nzones/2+1], yc[nzones/2]-0.05*dy, r"$v_{i+1,j}$", fontsize="15", horizontalalignment='center', verticalalignment='center') pylab.text(xc[nzones/2-1], yc[nzones/2]+0.05*dy, r"$u_{i-1,j}$", fontsize="15", horizontalalignment='center', verticalalignment='center') pylab.text(xc[nzones/2-1], yc[nzones/2]-0.05*dy, r"$v_{i-1,j}$", fontsize="15", horizontalalignment='center', verticalalignment='center') pylab.text(xc[nzones/2], yc[nzones/2+1]+0.05*dy, r"$u_{i,j+1}$", fontsize="15", horizontalalignment='center', verticalalignment='center') pylab.text(xc[nzones/2], yc[nzones/2+1]-0.05*dy, r"$v_{i,j+1}$", fontsize="15", horizontalalignment='center', verticalalignment='center') pylab.text(xc[nzones/2], yc[nzones/2-1]+0.05*dy, r"$u_{i,j-1}$", fontsize="15", horizontalalignment='center', verticalalignment='center') pylab.text(xc[nzones/2], yc[nzones/2-1]-0.05*dy, r"$v_{i,j-1}$", fontsize="15", horizontalalignment='center', verticalalignment='center') # i+1/2,j interface pylab.scatter(xr[nzones/2], yc[nzones/2], marker="x", s=50) pylab.text(xr[nzones/2]+0.05*dx, yc[nzones/2], r"${u}^\mathrm{adv}_{i+1/2,j}$", fontsize="15", rotation=270, horizontalalignment='left', verticalalignment='center') # i-1/2,j interface pylab.scatter(xr[nzones/2-1], yc[nzones/2], marker="x", s=50) pylab.text(xr[nzones/2-1]-0.05*dx, yc[nzones/2], r"${u}^\mathrm{adv}_{i-1/2,j}$", fontsize="15", rotation=270, horizontalalignment='right', verticalalignment='center') # i,j+1/2 interface pylab.scatter(xc[nzones/2], yr[nzones/2], marker="x", s=50) pylab.text(xc[nzones/2], yr[nzones/2]+0.05*dy, r"${v}^\mathrm{adv}_{i,j+1/2}$", fontsize="15", horizontalalignment='center', verticalalignment='bottom') # i,j-1/2 interface pylab.scatter(xc[nzones/2], yr[nzones/2-1], marker="x", s=50) pylab.text(xc[nzones/2], yr[nzones/2-1]-0.05*dy, r"${v}^\mathrm{adv}_{i,j-1/2}$", fontsize="15", horizontalalignment='center', verticalalignment='top') # grid labels pylab.text(xc[nzones/2-1], yl[0]-0.35*dy, r"$i-1$", horizontalalignment='center', fontsize="16") pylab.text(xc[nzones/2], yl[0]-0.35*dy, r"$i$", horizontalalignment='center', fontsize="16") pylab.text(xc[nzones/2+1], yl[0]-0.35*dy, r"$i+1$", horizontalalignment='center', fontsize="16") pylab.text(xl[0]-0.35*dx, yc[nzones/2-1], r"$j-1$", verticalalignment='center', fontsize="16") pylab.text(xl[0]-0.35*dx, yc[nzones/2], r"$j$", verticalalignment='center', fontsize="16") pylab.text(xl[0]-0.35*dx, yc[nzones/2+1], r"$j+1$", verticalalignment='center', fontsize="16") # axes pylab.xlim(xl[0]-0.5*dx,xr[2*ng+nzones-1]+0.25*dx) pylab.ylim(yl[0]-0.5*dy,yr[2*ng+nzones-1]+0.25*dy) pylab.axis("off") pylab.subplots_adjust(left=0.02,right=0.98,bottom=0.02,top=0.98) f = pylab.gcf() f.set_size_inches(8.0,8.0) pylab.savefig("MAC.png") pylab.savefig("MAC.eps") if __name__== "__main__": simplegrid() ================================================ FILE: figures/intro/fft_simple_examples.py ================================================ from __future__ import print_function import matplotlib.pyplot as plt import matplotlib as mpl import numpy as np import math mpl.rcParams['mathtext.fontset'] = 'cm' mpl.rcParams['mathtext.rm'] = 'serif' # see: http://glowingpython.blogspot.com/2011/08/how-to-plot-frequency-spectrum-with.html # and # http://docs.scipy.org/doc/numpy/reference/routines.fft.html # Since our input data is real, the negative frequency components # don't include any new information, and are not interesting to us. # The rfft routines understand this, and rfft takes n real points and # returns n/2+1 complex output points. The corresponding inverse # knows this, and acts accordingly. # # these are the routines we want for real valued data # # note that the scipy version of rfft returns that data differently # # M. Zingale (2013-03-03) def single_freq_sine(npts, xmax, f_0): # a pure sine with no phase shift will result in pure imaginary # signal xx = np.linspace(0.0, xmax, npts, endpoint=False) return xx, np.sin(2.0*math.pi*f_0*xx) def single_freq_sine_plus_shift(npts, xmax, f_0): # a pure sine with no phase shift will result in pure imaginary # signal xx = np.linspace(0.0, xmax, npts, endpoint=False) return xx, np.sin(2.0*math.pi*f_0*xx + math.pi/4) def two_freq_sine(npts, xmax, f_0, f_1): # a pure sine with no phase shift will result in pure imaginary # signal xx = np.linspace(0.0, xmax, npts, endpoint=False) f = 0.5*(np.sin(2.0*math.pi*f_0*xx) + np.sin(2.0*math.pi*f_1*xx)) return xx, f def single_freq_cosine(npts, xmax, f_0): # a pure cosine with no phase shift will result in pure real # signal xx = np.linspace(0.0, xmax, npts, endpoint=False) f = np.cos(2.0*math.pi*f_0*xx) return xx, f def plot_FFT(xx, xmax, f, outfile): plt.clf() plt.rc("font", size=10) npts = len(xx) # Forward transform: f(x) -> F(k) fk = np.fft.rfft(f) # Normalization -- the '2' here comes from the fact that we are # neglecting the negative portion of the frequency space, since # the FFT of a real function contains redundant information, so # we are only dealing with 1/2 of the frequency space. # # technically, we should only scale the 0 bin by N, since k=0 is # not duplicated -- we won't worry about that for these plots norm = 2.0/npts fk = fk*norm fk_r = fk.real fk_i = fk.imag # the fftfreq returns the postive and negative (and 0) frequencies # the newer versions of numpy (>=1.8) have an rfftfreq() function # that really does what we want -- we'll use that here. k = np.fft.rfftfreq(npts) # to make these dimensional, we need to divide by dx. Note that # max(xx) is not the true length, since we didn't have a point # at the endpoint of the domain. kfreq = k*npts/(max(xx) + xx[1]) # Inverse transform: F(k) -> f(x) -- without the normalization fkinv = np.fft.irfft(fk/norm) plt.subplot(411) plt.plot(xx, f) plt.xlabel("x") plt.ylabel("$f(x)$") plt.xlim(0, xmax) plt.subplot(412) plt.plot(kfreq, fk_r, label=r"Re($\mathcal{F}$)") plt.plot(kfreq, fk_i, ls=":", label=r"Im($\mathcal{F}$)") plt.xlabel(r"$k$") plt.ylabel("$\mathcal{F}_k$") plt.legend(fontsize="small", frameon=False, ncol=2, loc="upper right") plt.subplot(413) plt.plot(kfreq, np.abs(fk)) plt.xlabel(r"$k$") plt.ylabel(r"$|\mathcal{F}_k|$") plt.subplot(414) plt.plot(xx, fkinv.real) plt.xlabel(r"$x$") plt.ylabel(r"$\mathcal{F}^{-1}(\mathcal{F}_k)$") plt.xlim(0, xmax) plt.tight_layout() plt.savefig(outfile) #----------------------------------------------------------------------------- npts = 256 #64 #256 f_0 = 0.2 xmax = 10.0/f_0 # FFT of sine xx, f = single_freq_sine(npts, xmax, f_0) plot_FFT(xx, xmax, f, "fft-sine.pdf") # FFT of cosine xx, f = single_freq_cosine(npts, xmax, f_0) plot_FFT(xx, xmax, f, "fft-cosine.pdf") # FFT of sine with pi/4 phase xx, f = single_freq_sine_plus_shift(npts, xmax, f_0) plot_FFT(xx, xmax, f, "fft-sine-phase.pdf") # FFT of two sines f_1 = 0.5 xx, f = two_freq_sine(npts, xmax, f_0, f_1) plot_FFT(xx, xmax, f, "fft-two-sines.pdf") ================================================ FILE: figures/intro/integrals.py ================================================ import numpy as np import matplotlib as mpl import matplotlib.pyplot as plt from matplotlib.patches import Polygon mpl.rcParams['mathtext.fontset'] = 'cm' mpl.rcParams['mathtext.rm'] = 'serif' def f(x): """ the function we are integrating """ #return 1.0 + x*0.25*np.sin(np.pi*x) return (1.0 + x*0.25*np.sin(5*np.pi*x)) * np.exp(-(x - 1.0)**2/0.5) plt.rcParams.update({'xtick.labelsize': 18, 'ytick.labelsize': 18, 'font.size': 18}) def plot_base(xp, fp, xfine, a, b, label_mid=False): fmax = fp.max() for xl in xp: plt.plot([xl,xl], [0.0, 1.2*fmax], ls="--", color="0.5", zorder=-1) plt.scatter(xp, fp, marker="o", color="r", zorder=100) plt.figtext(0.9, 0.05, '$x$', fontsize=20) plt.figtext(0.1, 0.9, '$y$', fontsize=20) ax = plt.gca() ax.spines['right'].set_visible(False) ax.spines['top'].set_visible(False) ax.xaxis.set_ticks_position('bottom') if label_mid: ax.set_xticks((a, (a+b)/2, b)) ax.set_xticklabels(('$a$', r'$\frac{(a+b)}{2}$', '$b$')) else: ax.set_xticks((a, b)) ax.set_xticklabels(('$a$', '$b$')) ax.set_yticks([]) plt.plot(xfine, f(xfine), "r", linewidth=2) plt.xlim(np.min(xfine), 1.05*np.max(xfine)) plt.ylim(ymin = 0) def rectangle(xp, fp, a, b): ax = plt.gca() integral = 0.0 for n in range(len(xp)-1): xl = xp[n] xr = xp[n+1] fl = fp[n] # shade region verts = [(xl, 0), (xl, fl), (xr, fl), (xr, 0)] ax.add_patch(Polygon(verts, facecolor="0.8", edgecolor="k")) # and bonus! actually compute the integral in this approximation integral += (xr - xl) * fl return integral def trapezoid(xp, fp, a, b): ax = plt.gca() integral = 0.0 for n in range(len(xp)-1): xl = xp[n] xr = xp[n+1] # shade region fl = f(xl) fr = f(xr) verts = [(xl, 0), (xl, fl), (xr, fr), (xr, 0)] ax.add_patch(Polygon(verts, facecolor="0.8", edgecolor="k")) integral += 0.5 * (xr - xl) * (fl + fr) return integral def simpsons(xp, fp, a, b): ax = plt.gca() integral = 0.0 for n in range(0, len(xp)-1, 2): # we need to handle the 1 bin case specially if len(xp) == 2: xl = xp[0] xr = xp[1] xm = 0.5 * (xl + xr) f0 = f(xl) f1 = f(xm) f2 = f(xr) else: f0 = fp[n] f1 = fp[n+1] f2 = fp[n+2] xl = xp[n] xr = xp[n+2] delta = 0.5*(xr - xl) A = (f0 - 2*f1 + f2)/(2*delta**2) B = -(f2 - 4*f1 + 3*f0)/(2*delta) C = f0 xsimp = np.linspace(xl, xr, 100) fsimp = A * (xsimp - xl)**2 + B * (xsimp - xl) + C simpvert = list(zip(xsimp, fsimp)) verts = [(xl, 0)] + simpvert + [(xr, 0)] ax.add_patch(Polygon(verts, facecolor="0.8", edgecolor="k")) integral += (xr - xl) / 6.0 * (f0 + 4 * f1 + f2) return integral def main(): N_fine = 200 # the number of bins to divide [a, b] N_bins = 16 xmin = 0.5 xmax = 1.5 dx_extra = 0.5 # add a bin on each end of the domain outside of the integral xmin_plot = xmin - dx_extra xmax_plot = xmax + dx_extra xfine = np.linspace(xmin_plot, xmax_plot, N_fine+2) xp = np.linspace(xmin, xmax, N_bins+1) # integral range a = xmin b = xmax # function points fp = f(xp) # rectangle method plt.clf() plot_base(xp, fp, xfine, a, b) I_r = rectangle(xp, fp, a, b) plt.savefig(f"rectangle_N{N_bins}.png", bbox_inches="tight") # trapezoid method plt.clf() plot_base(xp, fp, xfine, a, b) I_t = trapezoid(xp, fp, a, b) plt.savefig(f"trapezoid_N{N_bins}.png", bbox_inches="tight") # simpsons method plt.clf() xp_tmp = list(xp) fp_tmp = list(fp) label_mid = False # if N_bins is 1, we need an extra point for Simpsons if N_bins == 1: xp_tmp.append((a + b)/2) fp_tmp.append(f((a + b)/2)) label_mid = True plot_base(np.array(xp_tmp), np.array(fp_tmp), xfine, a, b, label_mid=label_mid) I_s = simpsons(xp, fp, a, b) plt.savefig(f"simpsons_N{N_bins}.png", bbox_inches="tight") print(f"integral approximations: {I_r}, {I_t}, {I_s}") if __name__ == "__main__": main() ================================================ FILE: figures/intro/rk4_plot.py ================================================ # make a plot of what the 4th-order Runge-Kutta is doing using the # example dy/dt = -y (this comes from Garcia). from __future__ import print_function import numpy as np import matplotlib.pyplot as plt import matplotlib as mpl mpl.rcParams['mathtext.fontset'] = 'cm' mpl.rcParams['mathtext.rm'] = 'serif' def rhs(y): """ return dy/dt """ return -y def exact(t): """ analytic solution """ return np.exp(-t) # frame 1 -- show the initial condition (y^n) y0 = 1.0 t0 = 0.0 dt = 0.75 tt = np.linspace(t0, t0+2.0*dt, 100) def start(): """ default starting point """ plt.plot(tt, exact(tt), label="analytic solution", color="k") # draw the timeline plt.plot(tt, 0*tt, color="k") # label the current point plt.scatter([t0], [y0], color="r") plt.text(t0, y0+0.03, r"$y^n$", horizontalalignment="left", color="r", fontsize=20) plt.plot([t0,t0], [0, y0], ls=":", color="0.5") plt.text(t0, -0.05, r"$t^n$", verticalalignment="top", horizontalalignment="center", fontsize=20) # illustrate where t^n+1 is plt.plot([t0+dt,t0+dt], [0, y0], ls=":", color="0.5") plt.text(t0+dt, -0.05, r"$t^{n+1}$", verticalalignment="top", horizontalalignment="center", fontsize=20) start() plt.axis("off") leg = plt.legend() ltext = leg.get_texts() plt.setp(ltext, fontsize='large') leg.draw_frame(0) plt.xlim(t0-0.05*dt, t0+1.5*dt) plt.tight_layout() plt.savefig("rk4_initial.pdf", bbox_inches='tight') # now draw the solution Euler would get slope = rhs(y0) tEuler = np.linspace(t0, t0+dt, 2) yEuler = y0 + slope*(tEuler-t0) plt.plot(tEuler, yEuler, label="Euler step") plt.scatter([tEuler[1]], [yEuler[1]], color="r") plt.text(tEuler[1]+0.015, yEuler[1], r"$y^{n+1}$", horizontalalignment="left", verticalalignment="center", color="r", fontsize=20) leg = plt.legend() ltext = leg.get_texts() plt.setp(ltext, fontsize='large') leg.draw_frame(0) plt.xlim(t0-0.25*dt, t0+1.5*dt) plt.tight_layout() plt.savefig("rk4_Euler.pdf", bbox_inches='tight') #---------------------------------------------------------------------------- # show k1 slope plt.clf() start() plt.axis("off") k1 = rhs(y0) tSlope = np.linspace(t0-0.2*dt, t0+0.2*dt, 2) ySlope = y0 + k1*(tSlope - t0) plt.plot(tSlope, ySlope, label=r"slope", color="g", ls="--", lw=2, zorder=10) plt.text(t0-0.1*dt, y0, r"$k_1$", color="g", fontsize="large") leg = plt.legend() ltext = leg.get_texts() plt.setp(ltext, fontsize='large') leg.draw_frame(0) plt.xlim(t0-0.25*dt, t0+1.5*dt) plt.tight_layout() plt.savefig("rk4_k1.pdf", bbox_inches='tight') #---------------------------------------------------------------------------- # follow k1 to define k2 ytmp = y0 + k1*0.5*dt k2 = rhs(ytmp) # draw the k1 half step plt.plot([t0, t0+0.5*dt], [y0, ytmp], color="b", label="half-dt k1 step") plt.scatter(t0+0.5*dt, ytmp, color="b") # draw slope there tSlope = np.linspace(t0+0.5*dt-0.2*dt, t0+0.5*dt+0.2*dt, 2) ySlope = ytmp + k2*(tSlope - (t0 + 0.5*dt)) plt.plot(tSlope, ySlope, color="g", ls="--", lw=2) plt.text(t0+0.5*dt-0.1*dt, ytmp, r"$k_2$", color="g", verticalalignment="top", fontsize="large") leg = plt.legend() ltext = leg.get_texts() plt.setp(ltext, fontsize='large') leg.draw_frame(0) plt.xlim(t0-0.25*dt, t0+1.5*dt) plt.tight_layout() plt.savefig("rk4_k2.pdf", bbox_inches='tight') #---------------------------------------------------------------------------- # follow k2 to define k3 ytmp = y0 + k2*0.5*dt k3 = rhs(ytmp) # draw k2 half step plt.plot([t0, t0+0.5*dt], [y0, ytmp], color="c", label="half-dt k2 step") plt.scatter(t0+0.5*dt, ytmp, color="c") # draw slope there ySlope = ytmp + k3*(tSlope - (t0 + 0.5*dt)) plt.plot(tSlope, ySlope, color="g", ls="--", lw=2) plt.text(t0+0.5*dt+0.05*dt, ytmp, r"$k_3$", color="g", verticalalignment="bottom", fontsize="large") leg = plt.legend() ltext = leg.get_texts() plt.setp(ltext, fontsize='large') leg.draw_frame(0) plt.xlim(t0-0.25*dt, t0+1.5*dt) plt.tight_layout() plt.savefig("rk4_k3.pdf", bbox_inches='tight') #---------------------------------------------------------------------------- # follow k3 to define k4 ytmp = y0 + k3*dt k4 = rhs(ytmp) # draw k3 full step plt.plot([t0, t0+dt], [y0, ytmp], color="0.5", label="full-dt k3 step") plt.scatter(t0+dt, ytmp, color="0.5") # draw slope there tSlope2 = np.linspace(t0+dt-0.2*dt, t0+dt+0.2*dt, 2) ySlope = ytmp + k4*(tSlope2 - (t0 + dt)) plt.plot(tSlope2, ySlope, color="g", ls="--", lw=2) plt.text(t0+dt-0.1*dt, ytmp, r"$k_4$", color="g", verticalalignment="top", fontsize="large") leg = plt.legend() ltext = leg.get_texts() plt.setp(ltext, fontsize='large') leg.draw_frame(0) plt.xlim(t0-0.25*dt, t0+1.5*dt) plt.tight_layout() plt.savefig("rk4_k4.pdf", bbox_inches='tight') #---------------------------------------------------------------------------- # final RK-4 step ynew = y0 + (dt/6.0)*(k1 + 2.0*k2 + 2.0*k3 + k4) # draw full RK-4 step plt.plot([t0, t0+dt], [y0, ynew], color="r", label="full 4th-order RK step") plt.scatter(t0+dt, ynew, color="r") plt.text(t0+1.05*dt, ynew+0.015, r"$y^{n+1}$", color = "r", horizontalalignment="left", fontsize=20) leg = plt.legend() ltext = leg.get_texts() plt.setp(ltext, fontsize='large') leg.draw_frame(0) plt.xlim(t0-0.25*dt, t0+1.5*dt) plt.tight_layout() plt.savefig("rk4_final.pdf", bbox_inches='tight') ================================================ FILE: figures/intro/update.sh ================================================ #!/bin/sh # integrals.py creates rectange.pdf trapezoid.pdf simpsons.pdf python3 integrals.py cp -f rectangle.pdf trapezoid.pdf simpsons.pdf ../../intro/ # rk4_plot.py creates rk4_k[1-4].pdf, rk4_final.pdf python3 rk4_plot.py cp -f rk4_k[1-4].pdf rk4_final.pdf ../../intro ================================================ FILE: figures/multigrid/2dgrid-mg.py ================================================ import math import numpy import pylab def simplegrid(): #------------------------------------------------------------------------- # prolongation #------------------------------------------------------------------------- # grid info xmin = 0.0 xmax = 1.0 ymin = 0.0 ymax = 1.0 nzones = 3 ng = 0 dx = (xmax - xmin)/float(nzones) dy = (ymax - ymin)/float(nzones) xl = (numpy.arange(2*ng+nzones) - ng)*dx xr = (numpy.arange(2*ng+nzones)+1 - ng)*dx xc = 0.5*(xl + xr) yl = (numpy.arange(2*ng+nzones) - ng)*dy yr = (numpy.arange(2*ng+nzones)+1 - ng)*dy yc = 0.5*(yl + yr) #------------------------------------------------------------------------ # plot a domain without ghostcells # x lines n = 0 while (n < nzones): pylab.plot([xmin-0.25*dx,xmax+0.25*dx], [yl[n],yl[n]], color="k", lw=2) n += 1 pylab.plot([xmin-0.25*dx,xmax+0.25*dx], [yr[nzones-1],yr[nzones-1]], color="k", lw=2) # y lines n = 0 while (n < nzones): pylab.plot([xl[n],xl[n]], [ymin-0.25*dy,ymax+0.25*dy], color="k", lw=2) n += 1 pylab.plot([xr[nzones-1],xr[nzones-1]], [ymin-0.25*dy,ymax+0.25*dy], color="k", lw=2) #------------------------------------------------------------------------ # label pylab.text(xc[nzones/2], yc[nzones/2], r"$\phi_{i,j}^c$", fontsize="18", horizontalalignment='center', verticalalignment='center', zorder=100, color="b") pylab.text(xc[nzones/2+1], yc[nzones/2], r"$\phi_{i+1,j}^c$", fontsize="18", horizontalalignment='center', verticalalignment='center', color="b") pylab.text(xc[nzones/2], yc[nzones/2+1], r"$\phi_{i,j+1}^c$", fontsize="18", horizontalalignment='center', verticalalignment='center', color="b") pylab.text(xc[nzones/2-1], yc[nzones/2], r"$\phi_{i-1,j}^c$", fontsize="18", horizontalalignment='center', verticalalignment='center', color="b") pylab.text(xc[nzones/2], yc[nzones/2-1], r"$\phi_{i,j-1}^c$", fontsize="18", horizontalalignment='center', verticalalignment='center', color="b") # shading ii = nzones/2; jj = nzones/2 pylab.fill([xl[ii], xl[ii], xr[ii], xr[ii], xl[ii]], [yl[jj], yr[jj], yr[jj], yl[jj], yl[jj]], color="0.85", zorder=-1) ii = nzones/2+1; jj = nzones/2 pylab.fill([xl[ii], xl[ii], xr[ii], xr[ii], xl[ii]], [yl[jj], yr[jj], yr[jj], yl[jj], yl[jj]], color="0.85", zorder=-1) ii = nzones/2-1; jj = nzones/2 pylab.fill([xl[ii], xl[ii], xr[ii], xr[ii], xl[ii]], [yl[jj], yr[jj], yr[jj], yl[jj], yl[jj]], color="0.85", zorder=-1) ii = nzones/2; jj = nzones/2+1 pylab.fill([xl[ii], xl[ii], xr[ii], xr[ii], xl[ii]], [yl[jj], yr[jj], yr[jj], yl[jj], yl[jj]], color="0.85", zorder=-1) ii = nzones/2; jj = nzones/2-1 pylab.fill([xl[ii], xl[ii], xr[ii], xr[ii], xl[ii]], [yl[jj], yr[jj], yr[jj], yl[jj], yl[jj]], color="0.85", zorder=-1) # fine cells ii = nzones/2; jj = nzones/2 pylab.plot([xc[ii], xc[ii]], [yl[ii], yr[ii]], linestyle="--", color="0.3") pylab.plot([xl[ii], xr[ii]], [yc[ii], yc[ii]], linestyle="--", color="0.3") pylab.text(xc[ii]-dx/4, yc[jj]-dy/4, r"$\phi_{--}^{f}$", fontsize="18", horizontalalignment='center', verticalalignment='center', zorder=100, color="r") pylab.text(xc[ii]-dx/4, yc[jj]+dy/4, r"$\phi_{-+}^{f}$", fontsize="18", horizontalalignment='center', verticalalignment='center', zorder=100, color="r") pylab.text(xc[ii]+dx/4, yc[jj]-dy/4, r"$\phi_{+-}^{f}$", fontsize="18", horizontalalignment='center', verticalalignment='center', zorder=100, color="r") pylab.text(xc[ii]+dx/4, yc[jj]+dy/4, r"$\phi_{++}^{f}$", fontsize="18", horizontalalignment='center', verticalalignment='center', zorder=100, color="r") # grid labels pylab.text(xc[nzones/2-1], yl[0]-0.35*dy, r"$i-1$", horizontalalignment='center', fontsize="16") pylab.text(xc[nzones/2], yl[0]-0.35*dy, r"$i$", horizontalalignment='center', fontsize="16") pylab.text(xc[nzones/2+1], yl[0]-0.35*dy, r"$i+1$", horizontalalignment='center', fontsize="16") pylab.text(xl[0]-0.35*dx, yc[nzones/2-1], r"$j-1$", verticalalignment='center', fontsize="16") pylab.text(xl[0]-0.35*dx, yc[nzones/2], r"$j$", verticalalignment='center', fontsize="16") pylab.text(xl[0]-0.35*dx, yc[nzones/2+1], r"$j+1$", verticalalignment='center', fontsize="16") # axes pylab.xlim(xl[0]-0.5*dx,xr[2*ng+nzones-1]+0.25*dx) pylab.ylim(yl[0]-0.5*dy,yr[2*ng+nzones-1]+0.25*dy) pylab.axis("off") pylab.subplots_adjust(left=0.02,right=0.98,bottom=0.02,top=0.98) f = pylab.gcf() f.set_size_inches(8.0,8.0) pylab.savefig("2dgrid-prolong.png") pylab.savefig("2dgrid-prolong.eps") if __name__== "__main__": simplegrid() ================================================ FILE: figures/multigrid/fv-fd_bnd.py ================================================ import matplotlib.pyplot as plt import numpy as np import grid_plot as gp # plot a simple finite-difference grid showing the boundary data # and a finite-volume grid with the same data, showing it's boundary #----------------------------------------------------------------------------- # finite-difference # data that lives on the grid #a = np.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.15, 0.5, 0.55]) a = np.array([0.3, 1.0, 0.9, 0.8, 0.25, 0.1, 0.2, 0.5, 0.55]) nzones = len(a) voff = 2.0 gr = gp.FDGrid(nzones, voff=voff) gr.draw_grid(emphasize_end=True) gr.label_node(nzones/2, r"$i$", fontsize="large") gr.label_node(nzones/2-1, r"$i-1$", fontsize="large") gr.label_node(nzones/2+1, r"$i+1$", fontsize="large") gr.label_node(nzones/2-2, r"$i-2$", fontsize="large") gr.label_node(nzones/2+2, r"$i+2$", fontsize="large") # draw the data for i in range(nzones): gr.draw_data(i, a[i], color="r") gr.label_value(nzones/2, a[nzones/2], r"$f_i$", color="r") # label dx plt.plot([gr.xc[gr.ng+nzones/2-1], gr.xc[gr.ng+nzones/2-1]], [-0.35+voff,-0.25+voff], color="k") plt.plot([gr.xc[gr.ng+nzones/2], gr.xc[gr.ng+nzones/2]], [-0.35+voff,-0.25+voff], color="k") plt.plot([gr.xc[gr.ng+nzones/2-1], gr.xc[gr.ng+nzones/2]], [-0.3+voff,-0.3+voff], color="k") plt.text(0.5*(gr.xc[gr.ng+nzones/2-1] + gr.xc[gr.ng+nzones/2]), -0.45+voff, r"$\Delta x$", horizontalalignment="center") # finite-volume av = 0.5*(a[0:nzones-1] + a[1:]) nzones = len(av) ng = 1 gr = gp.FVGrid(nzones, ng=1) gr.draw_grid(emphasize_end=True, draw_ghost=True) gr.label_center(ng+nzones/2, r"$i$", fontsize="large") gr.label_center(ng+nzones/2-1, r"$i-1$", fontsize="large") gr.label_center(ng+nzones/2+1, r"$i+1$", fontsize="large") gr.label_center(ng+nzones-1, r"$\mathrm{hi}$", fontsize="large") gr.label_center(ng+nzones, r"$\mathrm{hi+1}$", fontsize="large") gr.label_center(gr.ilo, r"$\mathrm{lo}$", fontsize="large") gr.label_center(gr.ilo-1, r"$\mathrm{lo-1}$", fontsize="large") a = gr.scratch_array() a[gr.ilo:gr.ihi+1] = av cc = gp.CellCentered(gr, a) # draw the data for i in range(gr.ilo, gr.ihi+1): cc.draw_data_point(i, color="r") cc.label_data_point(ng+nzones/2, r"$f_i$", color="r") # label dx plt.plot([gr.xl[gr.ng+nzones/2], gr.xl[gr.ng+nzones/2]], [-0.35,-0.25], color="k") plt.plot([gr.xr[gr.ng+nzones/2], gr.xr[gr.ng+nzones/2]], [-0.35,-0.25], color="k") plt.plot([gr.xl[gr.ng+nzones/2], gr.xr[gr.ng+nzones/2]], [-0.3,-0.3], color="k") plt.text(gr.xc[gr.ng+nzones/2], -0.45, r"$\Delta x$", horizontalalignment="center") # illustrate the boundaries plt.plot([gr.xl[gr.ilo], gr.xl[gr.ilo]], [-0.5, 2.0], ls=":", color="0.5") plt.plot([gr.xr[gr.ihi], gr.xr[gr.ihi]], [-0.5, 2.0], ls=":", color="0.5") plt.text(gr.xl[gr.ilo], -0.5, "left BC", horizontalalignment="center", verticalalignment="top") plt.text(gr.xr[gr.ihi], -0.5, "right BC", horizontalalignment="center", verticalalignment="top") plt.axis([gr.xmin-gr.dx,gr.xmax+gr.dx, -0.5, 3.2]) plt.axis("off") plt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95) f = plt.gcf() f.set_size_inches(10.0,6.0) plt.savefig("fv-fd_grid_bc.pdf") ================================================ FILE: figures/multigrid/fvrestrict.py ================================================ import numpy as np import matplotlib.pyplot as plt import grid_plot as gp # plot two stacked fv grids of different (2x) resolution to show prolongation #----------------------------------------------------------------------------- nf = 4 nc = nf//2 grf = gp.FVGrid(nf, voff=2.0) grc = gp.FVGrid(nc) plt.clf() grf.draw_grid() grc.draw_grid() grf.label_center(nf//2-2, r"$i-2$") grf.label_center(nf//2-1, r"$i-1$") grf.label_center(nf//2, r"$i$") grf.label_center(nf//2+1, r"$i+1$") grc.label_center(nc//2-1, r"$j-1$") grc.label_center(nc//2, r"$j$") grf.label_cell_center(nf//2-2, r"$\phi_{i-2}^f$") grf.label_cell_center(nf//2-1, r"$\phi_{i-1}^f$") grf.label_cell_center(nf//2, r"$\phi_i^f$") grf.label_cell_center(nf//2+1, r"$\phi_{i+1}^f$") grc.label_cell_center(nc//2-1, r"$\phi_{j-1}^{c}$") grc.label_cell_center(nc//2, r"$\phi_{j}^{c}$") # connect the dots... plt.plot([grf.xl[nf//2-2], grf.xl[nf//2-2]], [-0.25, 3.25], ":", color="0.5") plt.plot([grf.xl[nf//2], grf.xl[nf//2]], [-0.25, 3.25], ":", color="0.5") plt.plot([grf.xr[nf//2+1], grf.xr[nf//2+1]], [-0.25, 3.25], ":", color="0.5") plt.axis([grf.xmin-0.5*grf.dx,grf.xmax+0.5*grf.dx, -0.5, 3.5]) plt.axis("off") plt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95) f = plt.gcf() f.set_size_inches(6.0,5.0) plt.savefig("fvrestrict.pdf") ================================================ FILE: figures/multigrid/laplacian.py ================================================ import numpy as np import matplotlib.pyplot as plt import grid_plot as gp def laplace(): # grid info xmin = 0.0 xmax = 1.0 nzones = 3 ng = 0 gr = gp.FVGrid(nzones, xmin=xmin, xmax=xmax) #------------------------------------------------------------------------ # plot a domain without ghostcells gr.draw_grid(emphasize_end=True) # label a few gr.label_center(0, r"$i-1$", fontsize="medium") gr.label_center(1, r"$i$", fontsize="medium") gr.label_center(2, r"$i+1$", fontsize="medium") gr.label_edge(1, r"$i-\myhalf$", fontsize="medium") gr.label_edge(2, r"$i+\myhalf$", fontsize="medium") gr.label_cell_center(0, r"$\phi_{i-1}$") gr.label_cell_center(1, r"$\phi_{i}$", value=0.7) gr.label_cell_center(2, r"$\phi_{i+1}$") gr.mark_cell_edge(1, r"$\left .\frac{d\phi}{dx} \right |_{i-\myhalf}$", color="C0") gr.mark_cell_edge(2, r"$\left .\frac{d\phi}{dx} \right |_{i+\myhalf}$", color="C0") gr.label_cell_center(1, r"$\left .\frac{d^2\phi}{dx^2} \right |_{i}$", value=0.3, color="C1") gr.clean_axes(padding=False) plt.ylim(-0.25, 1.25) plt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95) f = plt.gcf() f.set_size_inches(6.0,2.25) plt.tight_layout() plt.savefig("laplacian.pdf") if __name__== "__main__": laplace() ================================================ FILE: figures/multigrid/mgtower.py ================================================ import matplotlib.pyplot as plt import grid_plot as gp # plot two stacked fv grids of different (2x) resolution to show prolongation #----------------------------------------------------------------------------- gr = [] nzones = [2, 4, 8, 16] for nf in nzones: gr.append(gp.FVGrid(nf, ng=1, voff=2.0*len(gr))) plt.clf() for g in gr: g.draw_grid(emphasize_end=1, draw_ghost=1, edge_ticks=0) f = plt.gcf() f.set_size_inches(7.0,5.0) grf = gr[0] plt.xlim(grf.xmin-1.1*grf.dx,grf.xmax+1.1*grf.dx) plt.axis("off") plt.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95) plt.savefig("mgtower.pdf") ================================================ FILE: figures/multigrid/red_black.py ================================================ import math import numpy import pylab import random # red black gauss seidel pattern class marker: def __init__(self, xc, yc, color): # a marker is indicated by its center (xc,yc). self.xc = xc self.yc = yc # keep track of the color self.color = color def RB(): # define the number of markers in x and y nx = 10 ny = 10 # define the length of a marker side L = 0.8 # create a list of marker objects, one at each grid location markers = [] color = 0 while (color <= 1): j = 0 while (j < ny): if (color == 0): ioff = j % 2 else: ioff = 1 - (j % 2) i = ioff while (i < nx): markers.append(marker(i, j, color)) i += 2 j += 1 color += 1 pylab.clf() # the margins are funny -- we pick them to ensure that the # plot size is an integer multiple of the number of markers in # each dimension pylab.subplots_adjust(left=0.1, right=0.9, bottom=0.1,top=0.9) # draw the current state n = 0 while (n < len(markers)): if (markers[n].color == 1): c = "r" else: c = "k" pylab.fill([markers[n].xc-L/2, markers[n].xc-L/2, markers[n].xc+L/2, markers[n].xc+L/2, markers[n].xc-L/2], [markers[n].yc-L/2, markers[n].yc+L/2, markers[n].yc+L/2, markers[n].yc-L/2, markers[n].yc-L/2], c) n += 1 ax = pylab.axis([-0.5,nx+0.5,-0.5,ny+0.5]) pylab.axis("off") f = pylab.gcf() f.set_size_inches(6.0,6.0) pylab.tight_layout() pylab.savefig("rb.png") pylab.savefig("rb.eps", bbox_inches="tight", pad_inches=0) if __name__== "__main__": RB() ================================================ FILE: figures/multigrid/smooth-separate.py ================================================ #!/usr/bin/env python """ an example of solving Poisson's equation via smoothing only. Here, we solve u_xx = sin(x) u = 0 on the boundary [0,1] The analytic solution is u(x) = -sin(x) + x sin(1) This version (separate) differs from smooth.py in that we implement the smoothing here directly, instead of using the MG solver. M. Zingale (2013-03-31) """ #from io import * import numpy as np import matplotlib.pyplot as plt import sys def true(x): # the analytic solution return -np.sin(x) + x*np.sin(1.0) def error(ilo, ihi, dx, r): # L2 norm of elements in r, multiplied by dx to normalize return np.sqrt(dx*np.sum((r[ilo:ihi+1]**2))) def f(x): # the righthand side return np.sin(x) def compute_residual(ilo, ihi, dx, phi, frhs): # compute r = f - L phi r = np.zeros(len(phi)) r[ilo:ihi+1] = frhs[ilo:ihi+1] - \ (phi[ilo+1:ihi+2] - 2.0*phi[ilo:ihi+1] + phi[ilo-1:ihi])/dx**2 return r def smooth_run(nx, method="GS"): xmin = 0.0 xmax = 1.0 ng = 1 # initialize the solution to zero. Put one ghost cell on either end phi = np.zeros(nx + 2*ng, dtype=np.float64) phinew = np.zeros_like(phi) ilo = ng ihi = ng + nx - 1 # coordinates of centers dx = (xmax - xmin)/nx x = (np.arange(nx+2*ng) - ng + 0.5)*dx + xmin # initialize the RHS using the function f frhs = f(x) # smooth n = np.arange(20000) + 1 e = [] r = [] print("source norm: ", error(ilo, ihi, dx, frhs)) print(np.sum(frhs[ilo:ihi+1])) for i in n: # fill the ghost cells phi[ilo-1] = -phi[ilo] phi[ihi+1] = -phi[ihi] if method == "Jacobi": phinew[ilo:ihi+1] = \ (-dx*dx*frhs[ilo:ihi+1] + phi[ilo+1:ihi+2] + phi[ilo-1:ihi])/2.0 phi[:] = phinew[:] elif method == "GS": # red-black Gauss-Seidel -- first do the odd, then even points phi[ilo:ihi+1:2] = \ 0.5*(-dx*dx*frhs[ilo:ihi+1:2] + phi[ilo+1:ihi+2:2] + phi[ilo-1:ihi:2]) # fill the ghost cells between red and black phi[ilo-1] = -phi[ilo] phi[ihi+1] = -phi[ihi] phi[ilo+1:ihi+1:2] = \ 0.5*(-dx*dx*frhs[ilo+1:ihi+1:2] + \ phi[ilo+2:ihi+2:2] + phi[ilo:ihi:2]) else: sys.exit("invalid method") # compute the true error (wrt the analytic solution) and residual e.append(error(ilo, ihi, dx, phi - true(x))) # compute the residual resid = compute_residual(ilo, ihi, dx, phi, frhs) r.append(error(ilo, ihi, dx, resid)) return n, np.array(r), np.array(e) # test the multigrid solver N = [16, 32, 64] c = ["r", "g", "b"] for nx in N: n, r, e = smooth_run(nx) color = c.pop() plt.plot(n, e, color=color, label = str(nx)) plt.plot(n, r, color=color, ls=":") ax = plt.gca() ax.set_xscale('log') ax.set_yscale('log') plt.xlabel("# of iterations") plt.ylabel("L2 norm of true error (solid) and residual (dotted)") plt.legend(frameon=False, fontsize="small") plt.savefig("smooth-error.png") ================================================ FILE: figures/weno/weno.py ================================================ import numpy from matplotlib import pyplot ng = 2 def stencil_5_pts(q): """ Do reconstruction to q_{i+1/2} using five point stencil. Parameters ---------- q : numpy array Scalar data to reconstruct Returns ------- qL : numpy array Reconstructed data - boundary points are zero """ qL = numpy.zeros_like(q) np = len(q) - 2 * ng for i in range(ng, np+ng): qL[i] = ( 2 * q[i-2] - 13 * q[i-1] + 47 * q[i] + 27 * q[i+1] - 3 * q[i+2] ) / 60 return qL def stencils_3_pts(q): """ Do reconstruction to q_{i+1/2} using three point stencils. Parameters ---------- q : numpy array Scalar data to reconstruct Returns ------- qL : numpy array Reconstructed data - boundary points are zero """ qL = numpy.zeros((3, len(q))) np = len(q) - 2 * ng for i in range(ng, np+ng): qL[2, i] = ( 2 * q[i-2] - 7 * q[i-1] + 11 * q[i ]) / 6 qL[1, i] = (-1 * q[i-1] + 5 * q[i ] + 2 * q[i+1]) / 6 qL[0, i] = ( 2 * q[i ] + 5 * q[i+1] - 1 * q[i+2]) / 6 return qL def weno3(q): """ Do WENO reconstruction, r=3 Parameters ---------- q : numpy array Scalar data to reconstruct Returns ------- qL : numpy array Reconstructed data - boundary points are zero """ order = 3 C = numpy.array([1, 6, 3]) / 10 a = numpy.array([[11, -7, 2], [2, 5, -1], [-1, 5, 2]]) / 6 sigma = numpy.array([[[40, 0, 0], [-124, 100, 0], [44, -76, 16] ], [[16, 0, 0], [-52, 52, 0], [20, -52, 16] ], [[16, 0, 0], [-76, 100, 0], [44, -124, 40] ] ]) / 12 qL = numpy.zeros_like(q) beta = numpy.zeros((3, len(q))) w = numpy.zeros_like(beta) np = len(q) - 2 * ng epsilon = 1e-16 for i in range(ng, np+ng): q_stencils = numpy.zeros(order) alpha = numpy.zeros(order) for k in range(order): for l in range(order): for m in range(l+1): beta[k, i] += sigma[k, l, m] * q[i+k-l] * q[i+k-m] alpha[k] = C[k] / (epsilon + beta[k, i]**2) for l in range(order): q_stencils[k] += a[k, l] * q[i+k-l] w[:, i] = alpha / numpy.sum(alpha) qL[i] = numpy.dot(w[:, i], q_stencils) return qL, beta, w # First, use a smooth sine function to check convergence of the # various reconstructions f_smooth = lambda x: numpy.sin(2 * numpy.pi * x) f_antiderivative = lambda x: -numpy.cos(2 * numpy.pi * x) / (2 * numpy.pi) x_plot = numpy.linspace(0, 1, 200) x_recon, dx = numpy.linspace(0, 1, 20, retstep=True) x_recon_interior = x_recon[ng:-ng] + dx / 2 s_plot = f_smooth(x_plot) s_recon = f_smooth(x_recon) recon_exact = f_smooth(x_recon_interior) s_5 = stencil_5_pts(s_recon) s_3 = stencils_3_pts(s_recon) # The fifth order case pyplot.figure() pyplot.plot(x_plot, s_plot, 'k--', lw=2) pyplot.plot(x_recon, s_recon, 'bo', ms=5) pyplot.plot(x_recon_interior, s_5[ng:-ng], 'r^', ms=5) pyplot.xlabel(r"$x$") pyplot.xlim(0, 1) # The third order cases fig, axes = pyplot.subplots(1, 3) for r in range(3): axes[r].plot(x_plot, s_plot, 'k--', lw=2) axes[r].plot(x_recon, s_recon, 'bo', ms=5) axes[r].plot(x_recon_interior, s_3[r, ng:-ng], 'r^', ms=5) axes[r].set_xlabel(r"$x$") axes[r].set_xlim(0, 1) fig.tight_layout() # Convergence check, using the lazy fix to get error norms. points = 20 * 2**numpy.arange(1, 7) error_1norm_5 = numpy.zeros((len(points),)) error_1norm_3 = numpy.zeros((3, len(points))) error_1norm_weno = numpy.zeros((len(points))) error_2norm_5 = numpy.zeros((len(points),)) error_2norm_3 = numpy.zeros((3, len(points))) error_2norm_weno = numpy.zeros((len(points),)) for i, pts in enumerate(points): x_recon, dx = numpy.linspace(0, 1, pts, retstep=True) x_recon_interior = x_recon[ng:-ng] + dx / 2 s_recon = (f_antiderivative(x_recon + dx/2) - f_antiderivative(x_recon - dx/2)) / dx recon_exact = f_smooth(x_recon_interior) s_5 = stencil_5_pts(s_recon) s_3 = stencils_3_pts(s_recon) s_weno, beta, w = weno3(s_recon) error_1norm_5[i] = numpy.linalg.norm(s_5[ng:-ng] - recon_exact, 1) / pts error_2norm_5[i] = numpy.linalg.norm(s_5[ng:-ng] - recon_exact, 2) / numpy.sqrt(pts) error_1norm_weno[i] = numpy.linalg.norm(s_weno[ng:-ng] - recon_exact, 1) / pts error_2norm_weno[i] = numpy.linalg.norm(s_weno[ng:-ng] - recon_exact, 2) / numpy.sqrt(pts) for r in range(3): error_1norm_3[r, i] = numpy.linalg.norm(s_3[r, ng:-ng] - recon_exact, 1) / pts error_2norm_3[r, i] = numpy.linalg.norm(s_3[r, ng:-ng] - recon_exact, 2) / numpy.sqrt(pts) # Plot everything x_plot = numpy.linspace(0, 1, 200) x_recon, dx = numpy.linspace(0, 1, 20, retstep=True) x_recon_interior = x_recon[ng:-ng] + dx / 2 s_plot = f_smooth(x_plot) s_recon = f_smooth(x_recon) recon_exact = f_smooth(x_recon_interior) s_5 = stencil_5_pts(s_recon) s_3 = stencils_3_pts(s_recon) s_weno, beta, w = weno3(s_recon) fig = pyplot.figure(figsize=(10,8)) ax1 = pyplot.subplot2grid((2,2),(0,0),colspan=2) ax2 = pyplot.subplot2grid((2,2),(1,0)) ax3 = pyplot.subplot2grid((2,2),(1,1)) ax1.plot(x_plot, s_plot, 'k--', lw=2) ax1.plot(x_recon, s_recon, 'bo', ms=7) ax1.plot(x_recon_interior, s_5[ng:-ng], 'r^', ms=7) ax1.set_xlabel(r"$x$") ax1.set_xlim(0, 1) ax1.set_ylabel("Smooth function") ax2.loglog(1/points, error_1norm_5, 'k^', ms=5, label='stencil 5') ax2.loglog(1/points, error_1norm_weno, 'r+', ms=5, label='WENO') for r in range(3): ax2.loglog(1/points, error_1norm_3[r,:], 'bo', ms=3, label="stencil 3 ({})".format(r)) ax2.loglog(1/points, error_1norm_3[0,0]*(points[0]/points)**3, label=r"$\propto \Delta x^3$") ax2.loglog(1/points, error_1norm_5[0]*(points[0]/points)**5, label=r"$\propto \Delta x^5$") ax2.legend() ax2.set_xlabel(r"$\Delta x$") ax2.set_ylabel(r"$|$Error$|_1$") ax3.loglog(1/points, error_2norm_5, 'k^', ms=5, label='stencil 5') ax3.loglog(1/points, error_2norm_weno, 'r+', ms=5, label='WENO') for r in range(3): ax3.loglog(1/points, error_2norm_3[r,:], 'bo', ms=3, label="stencil 3 ({})".format(r)) ax3.loglog(1/points, error_2norm_3[0,0]*(points[0]/points)**3, label=r"$\propto \Delta x^3$") ax3.loglog(1/points, error_2norm_5[0]*(points[0]/points)**5, label=r"$\propto \Delta x^5$") ax3.legend() ax3.set_xlabel(r"$\Delta x$") ax3.set_ylabel(r"$|$Error$|_2$") fig.tight_layout() pyplot.savefig("weno-convergence.pdf") # Just the convergence plots fig, axes = pyplot.subplots(1, 2) axes[0].loglog(1/points, error_1norm_5, 'k^', ms=5, label='stencil 5') axes[0].loglog(1/points, error_1norm_weno, 'r+', ms=5, label='WENO') for r in range(3): axes[0].loglog(1/points, error_1norm_3[r,:], 'bo', ms=3, label="stencil 3 ({})".format(r)) axes[0].loglog(1/points, error_1norm_3[0,0]*(points[0]/points)**3, label=r"$\propto \Delta x^3$") axes[0].loglog(1/points, error_1norm_5[0]*(points[0]/points)**5, label=r"$\propto \Delta x^5$") axes[0].legend() axes[0].set_xlabel(r"$\Delta x$") axes[0].set_ylabel(r"$|$Error$|_1$") axes[1].loglog(1/points, error_2norm_5, 'k^', ms=5, label='stencil 5') axes[1].loglog(1/points, error_2norm_weno, 'r+', ms=5, label='WENO') for r in range(3): axes[1].loglog(1/points, error_2norm_3[r,:], 'bo', ms=3, label="stencil 3 ({})".format(r)) axes[1].loglog(1/points, error_2norm_3[0,0]*(points[0]/points)**3, label=r"$\propto \Delta x^3$") axes[1].loglog(1/points, error_2norm_5[0]*(points[0]/points)**5, label=r"$\propto \Delta x^5$") axes[1].legend() axes[1].set_xlabel(r"$\Delta x$") axes[1].set_ylabel(r"$|$Error$|_2$") fig.tight_layout() # Now we work with a non-smooth function f_nonsmooth = lambda x: numpy.where(x<0.5, numpy.sin(2 * numpy.pi * x), numpy.cos(2 * numpy.pi * x)) x_plot = numpy.linspace(0, 1, 200) x_recon, dx = numpy.linspace(0, 1, 20, retstep=True) x_recon_interior = x_recon[ng:-ng] + dx / 2 s_plot = f_nonsmooth(x_plot) s_recon = f_nonsmooth(x_recon) recon_exact = f_nonsmooth(x_recon_interior) s_5 = stencil_5_pts(s_recon) s_3 = stencils_3_pts(s_recon) s_weno, beta, w = weno3(s_recon) # 5 point stencil pyplot.figure() pyplot.plot(x_plot, s_plot, 'k--', lw=2) pyplot.plot(x_recon, s_recon, 'bo', ms=5) pyplot.plot(x_recon_interior, s_5[ng:-ng], 'r^', ms=5) pyplot.xlabel(r"$x$") pyplot.xlim(0, 1) # 3 point stencils fig, axes = pyplot.subplots(1, 3) for r in range(3): axes[r].plot(x_plot, s_plot, 'k--', lw=2) axes[r].plot(x_recon, s_recon, 'bo', ms=5) axes[r].plot(x_recon_interior, s_3[r, ng:-ng], 'r^', ms=5) axes[r].set_xlabel(r"$x$") axes[r].set_xlim(0, 1) fig.tight_layout() # WENO reconstruction pyplot.figure() pyplot.plot(x_plot, s_plot, 'k--', lw=2) pyplot.plot(x_recon, s_recon, 'bo', ms=5) pyplot.plot(x_recon_interior, s_weno[ng:-ng], 'r^', ms=5) pyplot.xlabel(r"$x$") pyplot.xlim(0, 1) # The smoothness indicators and the weights colors=['r','b','g'] markers=['o','^','s'] optimal_C = numpy.array([1, 6, 3]) / 10 fig, axes = pyplot.subplots(2, 1, sharex=True, figsize=(6,10)) fig.subplots_adjust(hspace=0) for r, (color, marker) in enumerate(zip(colors, markers)): axes[0].scatter(x_recon_interior, beta[r, ng:-ng], marker=marker, color=color, label=r"Jiang-Shu $\beta_{}$".format(r)) axes[1].scatter(x_recon_interior, w[r, ng:-ng], marker=marker, color=color, label=r"WENO weight $\omega_{}$".format(r)) axes[1].plot(x_recon, optimal_C[r] * numpy.ones_like(x_recon), linestyle='--', color=color, lw=2, label=r"Optimal weight $C_{}$".format(r)) axes[1].set_xlabel(r"$x$") axes[0].set_ylabel(r"$\beta$") axes[0].set_xlim(0,1) axes[1].set_xlim(0,1) axes[0].legend() axes[1].legend() fig.tight_layout() # Plot everything fig, axes = pyplot.subplots(3, 2, sharex = True, figsize=(12,14)) fig.subplots_adjust(hspace=0) # 5 point stencil axes[0,0].plot(x_plot, s_plot, 'k--', lw=2) axes[0,0].plot(x_recon, s_recon, 'kx', ms=7) axes[0,0].plot(x_recon_interior, s_5[ng:-ng], 'r^', ms=7) axes[0,0].set_xlim(0, 1) axes[0,0].set_ylabel('5 point stencil') # 3 point stencils colors=['r','b','g'] markers=['o','^','s'] axes[1,0].plot(x_plot, s_plot, 'k--', lw=2) axes[1,0].plot(x_recon, s_recon, 'kx', ms=7) for r, (color, marker) in enumerate(zip(colors, markers)): axes[1,0].plot(x_recon_interior, s_3[r, ng:-ng], '{}{}'.format(color, marker), ms=7) axes[1,0].set_xlim(0, 1) axes[1,0].set_ylabel('3 point stencils') # WENO reconstruction axes[2,0].plot(x_plot, s_plot, 'k--', lw=2) axes[2,0].plot(x_recon, s_recon, 'kx', ms=7) axes[2,0].plot(x_recon_interior, s_weno[ng:-ng], 'r^', ms=7) axes[2,0].set_xlabel(r"$x$") axes[2,0].set_xlim(0, 1) axes[2,0].set_ylabel('WENO stencils') # Shift the right axes for r in range(3): axes[r,1].yaxis.tick_right() axes[r,1].yaxis.set_label_position("right") # The function again axes[0,1].plot(x_plot, s_plot, 'k-', lw=2) axes[0,1].set_ylabel('Non-smooth function') # The smoothness indicators and the weights colors=['r','b','g'] markers=['o','^','s'] optimal_C = numpy.array([1, 6, 3]) / 10 for r, (color, marker) in enumerate(zip(colors, markers)): axes[1,1].scatter(x_recon_interior, beta[r, ng:-ng], marker=marker, color=color, label=r"Jiang-Shu $\beta_{}$".format(r)) axes[2,1].scatter(x_recon_interior, w[r, ng:-ng], marker=marker, color=color, label=r"WENO weight $\omega_{}$".format(r)) axes[2,1].plot(x_recon, optimal_C[r] * numpy.ones_like(x_recon), linestyle='--', color=color, lw=2, label=r"Optimal weight $C_{}$".format(r)) axes[2,1].set_xlabel(r"$x$") axes[1,1].set_ylabel(r"$\beta$") axes[2,1].set_ylabel(r"$C$, $\omega$") axes[1,1].set_xlim(0,1) axes[2,1].set_xlim(0,1) axes[1,1].legend() axes[2,1].legend() fig.tight_layout() pyplot.savefig("weno-weights.pdf") ================================================ FILE: finite-volume/discretizations.eps ================================================ %!PS-Adobe-3.0 EPSF-3.0 %%BoundingBox: 0 0 792 612 %%Pages: 0 %%Creator: LibreOffice 3.6 %%Title: none %%CreationDate: none %%LanguageLevel: 2 %%EndComments %%BeginProlog %%BeginResource: procset SDRes-Prolog 1.0 0 /b4_inc_state save def /dict_count countdictstack def /op_count count 1 sub def userdict begin 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit[] 0 setdash newpath /languagelevel where {pop languagelevel 1 ne {false setstrokeadjust false setoverprint} if} if /bdef {bind def} bind def /c {setrgbcolor} bdef /l {neg lineto} bdef /rl {neg rlineto} bdef /lc {setlinecap} bdef /lj {setlinejoin} bdef /lw {setlinewidth} bdef /ml {setmiterlimit} bdef /ld {setdash} bdef /m {neg moveto} bdef /ct {6 2 roll neg 6 2 roll neg 6 2 roll neg curveto} bdef /r {rotate} bdef /t {neg translate} bdef /s {scale} bdef /sw {show} bdef /gs {gsave} bdef /gr {grestore} bdef /f {findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end /NFont exch definefont pop /NFont findfont} bdef /p {closepath} bdef /sf {scalefont setfont} bdef /ef {eofill}bdef /pc {closepath stroke}bdef /ps {stroke}bdef /pum {matrix currentmatrix}bdef /pom {setmatrix}bdef /bs {/aString exch def /nXOfs exch def /nWidth exch def currentpoint nXOfs 0 rmoveto pum nWidth aString stringwidth pop div 1 scale aString show pom moveto} bdef %%EndResource %%EndProlog %%BeginSetup %%EndSetup %%Page: 1 1 %%BeginPageSetup %%EndPageSetup pum 0.02834 0.02834 s 0 -21590 t /tm matrix currentmatrix def gs tm setmatrix -1077 -1102 t 1 1 s 1077 1102 m 29016 1102 l 29016 22691 l 1077 22691 l 1077 1102 l eoclip newpath gs 0 0 m 27939 0 l 27939 21589 l 0 21589 l 0 0 l eoclip newpath 0.812 0.906 0.960 c 20294 13473 m 21783 14113 22323 16055 21543 17957 ct 20762 19859 19028 20822 17539 20183 ct 16050 19543 15511 17602 16292 15700 ct 17074 13798 18806 12833 20294 13473 ct p ef pum 14022 3458 t 0.003 0.003 1.000 c 344 0 m 344 -56 l 316 -12 275 10 220 10 ct 185 10 152 0 123 -19 ct 93 -39 70 -66 54 -101 ct 37 -136 29 -176 29 -222 ct 29 -266 37 -306 51 -342 ct 66 -378 88 -406 118 -425 ct 147 -444 180 -454 217 -454 ct 243 -454 267 -448 288 -437 ct 309 -426 326 -411 339 -393 ct 339 -613 l 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count op_count sub {pop} repeat countdictstack dict_count sub {end} repeat b4_inc_state restore %%PageTrailer %%Trailer %%EOF ================================================ FILE: finite-volume/finite-volume.tex ================================================ \label{ch:fv} \section{Discretization} The physical systems we model are described by continuous mathematical functions, $f(x,t)$ and their derivatives in space and time. To represent this continuous system on a computer we must discretize it---convert the continuous function into a discrete number of points in space at one or more discrete instances in time. There are many different discretization methods used throughout the physical sciences, engineering, and applied mathematics fields, each with their own strengths and weaknesses. Broadly speaking, we can divide these methods into grid-based and gridless methods. %% \begin{figure}[h] %% \centering %% \includegraphics[width=\linewidth]{discretizations} %% \caption[Taxonomy of different discretizations] %% {\label{fig:disc} Taxonomy of different discretizations.} %% \end{figure} Gridless methods include those which represent the function as a superposition of continuous basis functions (e.g.\ sines and cosines). This is the fundamental idea behind {\em spectral methods}. A different class of methods are those that use discrete particles to represent the mass distribution and produce continuous functions by integrating over these particles with a suitable kernel---this is the basis of {\em smoothed particle hydrodynamics} (SPH)~\cite{SPH}. SPH is a very popular method in astrophysics. For grid-based methods, we talk about both the style of the grid (structured vs.\ unstructured) and the discretization method, e.g.\ the finite-difference, finite-volume, and finite-element methods. Structured grids are logically Cartesian. This means that you can reference the location of any cell in the computational domain via an integer index in each spatial dimension. From a programming standpoint, the grid structure can be represented exactly by a multi-dimensional array. Unstructured grids don't have this simple pattern. A popular type of unstructured grid is created using triangular cells (in 2-d) or tetrahedra (in 3-d). \MarginPar{add figure} The main advantage of these grids is that you can easily represent irregularly-shaped domains. The disadvantage is that the data structures required to describe the grid are more complicated than a simple array (and tend to have more inefficient memory access). \ifdefined\debugmode Finite element methods are widely used in engineering together with unstructured grids. A type of finite-element method called a {\em Galerkin method} can be formed by integrating the continuous function against a compact basis set represented by tetrahedra. These methods work well with the irregular geometries that arise in engineering. \fi Once a grid is established, the system of PDEs is converted into a system of discrete equations on the grid. Finite-difference and finite-volume methods can both be applied to {\em structured grids}. The main difference between these methods is that the finite-difference methods build from the differential form of PDEs while the finite-volume methods build from the integral form of the PDEs. The attractiveness of finite-volume methods is that conservation is a natural consequence of the discretization---this is why they are popular in astrophysics. In these notes, we will focus on finite-volume techniques on structured grids. \section{Grid basics} The grid is the fundamental object for representing continuous functions in a discretized fashion, making them amenable to computation. In astrophysics, we typically use structured grids---these are logically Cartesian, meaning that the position of a quantity on the grid can be specified by a single integer index in each dimension. This works for our types of problems because we don't have irregular geometries---we typically use boxes, disks, or spheres. We represent derivatives numerically by discretizing the domain into a finite number of zones (a numerical grid). This converts a continuous derivative into a difference of discrete data. Different approximations have different levels of accuracy. There are two main types of structured grids used in astrophysics: {\em finite-difference} and {\em finite-volume}. These differ in way the data is represented. On a finite-difference grid, the discrete data is associated with a specific point in space. On a finite-volume grid, the discrete data is represented by averages over a control volume. Nevertheless, these methods can often lead to very similar looking discrete equations. Consider the set of grids show in Figure~\ref{fig:grids}. On the top is a classic finite-difference grid. The discrete data, $f_i$, are stored as points regularly spaced in $x$. With this discretization, the spatial locations of the points are simply $x_i = i \Delta x$, where $i = 0, \ldots, N$\footnote{When you see $f_{i+1}$, you can think of this as meaning $f((i+1)\Delta x)$ or $f(x_i + \Delta x)$}. Note that for a finite-sized domain, we would put a grid point precisely on the physical boundary at each end. The middle grid is also finite-difference, but now we imagine first dividing the domain into $N$ cells or zones, and we store the discrete data, $f_i$, at the center of the zone. This is often called a {\em cell-centered finite-difference} grid. The physical coordinate of the zone centers (where the data lives) are: $x_i = (i + \myhalf)\Delta x$, where $i = 0, \ldots, N-1$. Note that now for a finite-sized domain, the left edge of the first cell will be on the boundary and the first data value will be associated at a point $\Delta x/2$ inside the boundary. A similar situation arises at the right physical boundary. Some finite-difference schemes stagger the variables, e.g., putting velocity on the boundaries and density at the center. Finally, the bottom grid is a finite-volume grid. The layout looks identical to the cell-centered finite difference grid, except now instead of the discrete data being associated at a single point in space, 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, we will typically talk about the average of $f$, $\favg{f}_i$, over the zone, and represent this by a horizontal. The total amount of $f$ in the zone is then simply $\Delta x \favg{f}_i$. We label the left and right edges of a zone with half-integer indices $i-\myhalf$ and $i+\myhalf$. The physical coordinate of the center of the zone is the same as in the cell-centered finite-difference case. % the scripts to generate these figures are in figures/finite-volume % as fd.py, ccfd.py, and fv.py \begin{figure}[t] \centering \includegraphics[width=\linewidth]{fd_grid} \\ \includegraphics[width=\linewidth]{ccfd_grid} \\ \includegraphics[width=\linewidth]{fv_grid} \caption[Types of structured grids]{\label{fig:grids} Different types of structured grids showing the same data. Top: a finite-difference grid---the discrete data are associated with a specific point in space. Middle: a cell-centered finite-difference grid---again the data is at a specific point, but now we imagine the domain divided into zone with the data living at the center of each zone. Bottom: a finite-volume grid---here the domain is divided into zones and we store the average value of the function within each zone.} \end{figure} In all cases, for a {\em regular} structured grid, we take $\Delta x$ to be constant. For the finite-difference grids, the discrete value at each point is obtained from the continuous function $f(x)$ as: \begin{equation} f_i = f(x_i) \end{equation} \section{Finite-volume grids} In the finite-volume discretization, $f_i$ represents the average of $f(x,t)$ over the interval $x_{i-\myhalf}$ to $x_{i+\myhalf}$, where the half-integer indices denote the zone edges (i.e.\ $x_{i-\myhalf} = x_i - \Delta x/2$): \begin{equation} \label{eq:fv_cc-a} \favg{f}_i = \frac{1}{\Delta x} \int_{x_{i-\myhalf}}^{x_{i+\myhalf}} f(x) dx \end{equation} The lower panel of Figure~\ref{fig:grids} shows a finite-volume grid, with the half-integer indices bounding zone $i$ marked. % % figure created with figures/finite-volume/simplegrid2.py %% \begin{figure}[t] %% \centering %% \includegraphics[width=\linewidth]{simplegrid2} %% \caption[A simple 1-d finite-volume grid]{\label{fig:simplefv} A simple 1-d finite-volume %% grid with the zone-centered and zone-edge indexes. Here we show %% only the data associated with zone $i$.} %% \end{figure} % Here we've drawn $\favg{f}_i$ as a horizontal line spanning the entire zone---this is to represent that it is an average within the volume defined by the zone edges. % To second-order accuracy, \begin{equation} \label{eq:fv_cc} \favg{f}_i = \frac{1}{\Delta x} \int_{x_{i-\myhalf}}^{x_{i+\myhalf}} f(x) dx \sim f(x_i) \end{equation} \begin{exercise}[Finite-volume vs.\ finite-difference centering] {Show that Eq.~\ref{eq:fv_cc} is true to $O(\Delta x^2)$ by expanding $f(x)$ as a Taylor series in the integral, e.g., as: \begin{align} f(x) = f(x_i) &+ f^\prime(x_i) (x - x_i) + \frac{1}{2} f^{\prime\prime}(x_i) (x - x_i)^2 \nonumber \\ % &+ \frac{1}{6} f^{\prime\prime\prime}(x_i) (x - x_i)^3 + \ldots \end{align} } \end{exercise} This means that we can treat the average of $f$ over a zone as simply $f(x)$ evaluated at the zone center if we only want second-order accuracy. Using the subscript notation, we can express the average of the zone to the right as $\favg{f}_{i+1}$. When we want to interpolate data on a finite-volume grid, we want to construct an interpolating polynomial that is conservative. A {\em conservative interpolant} reconstructs a continuous functional form, $f(x)$, from a collection of cell-averages subject to the requirement that when $f(x)$ is averaged over a cell, it returns the original cell-average. \begin{exercise}[Conservative interpolation] {\label{ex:consinterp} Consider three cell averages: $\favg{f}_{i-1}$, $\favg{f}_{i}$, $\favg{f}_{i+1}$. Fit a quadratic polynomial through these points, \begin{equation} f(x) = A (x - x_i)^2 + B (x - x_i) + C \end{equation} where the coefficients, $A$, $B$, and $C$ are found by applying the constraints: \begin{align} \favg{f}_{i-1} &= \frac{1}{\Delta x} \int_{x_{i-\mythreehalf}}^{x_{i-\myhalf}} f(x) dx \\ \favg{f}_{i} &= \frac{1}{\Delta x} \int_{x_{i-\myhalf}}^{x_{i+\myhalf}} f(x) dx \\ \favg{f}_{i+1} &= \frac{1}{\Delta x} \int_{x_{i+\myhalf}}^{x_{i+\mythreehalf}} f(x) dx \end{align} Show that the conservative interpolant is: \begin{align} f(x) = &\frac{\favg{f}_{i-1} - 2 \favg{f}_i + \favg{f}_{i+1}}{2\Delta x^2} (x-x_i)^2 + \nonumber \\ &\frac{\favg{f}_{i+1} - \favg{f}_{i-1}} {2\Delta x} (x-x_i) + \nonumber \\ &\frac{-\favg{f}_{i-1} + 26 \favg{f}_i -\favg{f}_{i+1}}{24} \end{align} } \end{exercise} The {\sf Jupyter} notebook \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/finite-volume/conservative-interpolation.ipynb}{conservative-interpolation.ipynb}} shows how to derive these interpolants using {\sf SymPy}, and gives higher-order interpolants. \subsection{Differences and order of accuracy} In practice, when computing derivatives in a finite-volume discretization, we can use the second-order centered difference from \S~\ref{ch:intro:diff} treating the finite-volume data as living at the cell-centers and still be second-order accurate. For higher accuracy, we can fit a {\em conservative interpolant} (as illustrated in exercise~\ref{ex:consinterp}) to a collection of points and then differentiate the interpolant itself. \MarginPar{show convergence plot} Notice that the righthand side of all derivative approximations involve some linear combinations of $f_i$'s. We call this the {\em stencil}. The {\em width} of the stencil is a measure of how many zones on either side of zone $i$ we reach when computing our approximation. For example, a second derivative can be discretized as: \begin{equation} \left . \frac{d^2 f}{dx^2} \right |_i = \frac{f_{i+1} - 2 f_i + f_{i-1}}{\Delta x^2} \end{equation} The stencil on the righthand side encompasses 3 zones. \subsection{Conservation} The finite-volume grid is useful when dealing with conservation laws. Consider the following system: \begin{equation} \frac{\partial \Uc}{\partial t} + \nabla \cdot \Fb(\Uc) = 0 \end{equation} where $\Uc$ is a vector of conserved quantities and $\Fb(\Uc)$ is the flux of each quantity. This type of system arises, for example, in fluid flow, where the system will represent conservation of mass, momentum, and energy. Consider one-dimension. Integrating this system over a zone, and normalizing by $\Delta x$, we have: \begin{equation} \frac{1}{\Delta x} \int_{x_{i-\myhalf}}^{x_{i+\myhalf}} \ddt{\Uc} dx = -\frac{1}{\Delta x} \int_{x_{i-\myhalf}}^{x_{i+\myhalf}} \frac{\partial \Fb}{\partial x} dx \end{equation} On the left, we can take the time derivative out of the integral, and we are left with the definition of a zone average, so this becomes simply $\partial \favg{\Uc}_i/\partial t$. On the right, we apply the divergence theorem, giving: \begin{equation} \frac{\partial \favg{\Uc}_i}{\partial t} = - \frac{1}{\Delta x} \left \{ \left . \Fb(\Uc) \right |_{x_{i+\myhalf}} - \left . \Fb(\Uc) \right |_{x_{i-\myhalf}} \right \} \end{equation} Independent of how we discretize in time, notice that we have the cell-average on the left and that the righthand side is simply a difference of fluxes through the interfaces of the zone. For the $i+1$ zone, the update would be: \begin{equation} \label{eq:consup} \frac{\partial \favg{\Uc}_{i+1}}{\partial t} = - \frac{1}{\Delta x} \left \{ \left . \Fb(\Uc) \right |_{x_{i+\mythreehalf}} - \left . \Fb(\Uc) \right |_{x_{i+\myhalf}} \right \} \end{equation} Notice that this shares the flux at the $x_{i+\myhalf}$ interface, but with the opposite sign. \MarginPar{illustrate this} When all of the updates are done, the flux through each boundary adds to one zone and subtracts from its neighbor, exactly conserving (to round-off error) the quantity $\Uc$. This cancellation of the sums is an example of a {\em telescoping property}, and is the main reason why finite-volume methods are attractive---conserved quantities are conserved to machine (roundoff) precision. Note that conservation is not the same as accuracy. We can construct the fluxes for our discretized equation by calling a random number generator and we'd still be conservative, but not at all accurate. \subsection{Boundary conditions with finite-volume grids} \label{sec:fv:bcs} Imagine that we wish to compute the derivative in every zone using: \begin{equation} \left . \frac{\partial f}{\partial x} \right |_i = \frac{f_i - f_{i-1}}{\Delta x} \enskip . \label{eq:onesideddiff} \end{equation} If we denote the index corresponding to the leftmost zone in our domain as `lo', then when we try to compute ${\partial f}/{\partial x} |_\mathrm{lo}$, we will ``fall-off'' the grid, i.e., we need a value of $f$ for zone $\mathrm{lo}-1$, which is outside the domain. This is where boundary conditions for our grid come into play. We implement boundary conditions by extending the computational grid beyond the physical domain (see Figure~\ref{fig:fv_gc}). These additional zones are called {\em ghost cells}. They exist solely to handle the boundary conditions and allow us to use the same update equation (e.g.\ Eq.~\ref{eq:onesideddiff}) for all zones in the domain. % figure created with simplegrid-gc.py \begin{figure}[t] \centering \includegraphics[width=\linewidth]{simplegrid_gc} \caption[A simple 1-d finite-volume grid with ghost cells] {\label{fig:fv_gc} A simple 1-d finite-volume grid with a single ghost cell at each end of the domain. The domain boundaries are indicated by the heavy vertical lines. Here there are $\mathrm{hi}-\mathrm{lo}+1$ zones used in the discretization of the domain, with the first zone in the domain labeled $\mathrm{lo}$ and the last in the domain labeled $\mathrm{hi}$.} \end{figure} The number of ghostcells needed for the grid depends on how wide the stencils used in the computation are. The wider the stencil, the more ghostcells are needed. Periodic boundary conditions would be implemented as: \begin{eqnarray} f_{\mathrm{hi}+1} &=& f_\mathrm{lo} \\ f_{\mathrm{lo}-1} &=& f_\mathrm{hi} \end{eqnarray} A simple outflow (zero-gradient) boundary condition would be implemented as: \begin{eqnarray} f_{\mathrm{hi}+1} &=& f_\mathrm{hi} \\ f_{\mathrm{lo}-1} &=& f_\mathrm{lo} \end{eqnarray} \section{Numerical implementation details} The computational grids used for finite-volume techniques maps nicely onto multidimensional arrays. For the cell-average data, a typical array would be dimensioned as \begin{lstlisting}[language=Python] ilo = ng ihi = nx + ng - 1 state = np.zeros((nx + 2*ng, nvar)) \end{lstlisting} in python (using NumPy), where {\tt nx} is the number of zones in the domain and {\tt ng} is the number of ghost cells. For some problems, there might be more than one state variable, so {\tt nvar}, is the number of state variables that live in each zone. We would then loop from {\tt ilo} to {\tt ihi} to access the state data in the grid interior. Fluxes are also stored in arrays, but since we cannot index an array with half-integer indices, the standard convention is that the index {\tt i} in an array refers to the flux on the {\em left} edge of that zone. I.e., \begin{equation*} \mathtt{flux[i]} \quad \longleftrightarrow \quad F_{i-1/2} \end{equation*} \section{Going further} \begin{itemize} \item {\em Domain decomposition}: when running on a parallel computer, the work is divided up across processor using domain decomposition. Here, we break the computational domain into smaller sub-domains, and put one (or more) sub-domains on each processor. Each sub-domain has its own perimeter of ghost cells. These are now filled by copying information from the neighboring sub-domains or using the physical boundary conditions for the full domain, depending on where the ghost cells lie. Figure~\ref{fig:domain} shows a simple decomposition of a domain into 6 sub-domains. % create with figures/finite-volume/domain.py \begin{figure} \centering \includegraphics[width=\linewidth]{domain} \caption[Domain decomposition example] {\label{fig:domain} Domain decomposition of the computational domain into 6 separate sub-domains. Each sub-domain here has $5\times 5$ zones. For one of the sub-domains, the perimeter of ghost cells is illustrated as the red boundary.} \end{figure} \item {\em AMR for structured grids}: adaptive mesh refinement uses a hierarchy of grids to focus resolution in regions of complex flow. For finite-volume codes, the standard reference for AMR is Berger \& Colella~\cite{berger-colella}. Each level is an even integer multiple finer in resolution, and the grid cells line up with one another (i.e.\ in two-dimensions, four fine cells will be completely enclosed by a single coarse cell, when using a jump in resolution of 2$\times$.) This provides a natural way to enforce conservation. At coarse-fine interfaces, corrections are done to ensure consistency. \item {\em Mapped grids}: mapped grids are (usually) logically Cartesian grids that use a function (map) to transform from a rectangular mesh (that corresponds to the array in memory) to a general curvilinear grid. These maintain the performance characteristics of structured grids, since any zone can be directly indexed with an integer for each spatial dimension. \MarginPar{reference the Calhoun paper} \item {\em Lagrangian grids}: a hybrid of particle and grid methods is provided by methods that move particles in a Lagrangian fashion and use a Voronoi tessellation of these particles to define the grid that finite-volume methods are applied to. See, for example, the Arepo code paper~\cite{arepo}. \end{itemize} ================================================ FILE: finite-volume/main.tex ================================================ \documentclass[11pt]{article} % margins \usepackage[margin=0.75in]{geometry} % figures \usepackage{graphicx} % font \usepackage{mathpazo} \usepackage{helvet} % coloring \usepackage{color} \definecolor{mygray}{gray}{0.5} % footer \usepackage{fancyhdr} \pagestyle{fancy} \fancyfoot[LO,LE]{\footnotesize\sffamily \color{mygray} M.\ Zingale---Notes on Grids} \fancyfoot[RO,RE]{\footnotesize \sffamily \color{mygray} (\today)} \fancyfoot[CO,CE]{\thepage} \fancyhead{} \renewcommand{\headrulewidth}{0.0pt} \renewcommand{\footrulewidth}{0.0pt} % captions \usepackage{caption} \renewcommand{\captionfont}{\footnotesize} \renewcommand{\captionlabelfont}{\footnotesize} \setlength{\captionmargin}{5em} \usepackage{sectsty} \allsectionsfont{\sffamily} \newcounter{exercise} \usepackage{changepage} \newenvironment{exercise}% environment name {% begin code \begin{adjustwidth}{1.0cm}{0.5cm} \par\vspace{\baselineskip}\noindent \refstepcounter{exercise}% \textbf{Exercise \theexercise\ }\begin{itshape}% %\par\vspace{\baselineskip}\noindent\ignorespaces }% {% end code \end{itshape}\end{adjustwidth}\vspace{\baselineskip}\ignorespacesafterend } \begin{document} \begin{center} {\LARGE \textsf{\textbf{ Notes on Grids}} } \end{center} \input finite-volume \bibliographystyle{plain} \bibliography{../refs} \end{document} ================================================ FILE: finite-volume/old-scripts/domain-1d.py ================================================ import pylab import numpy def drawBox(ll, uu, nx, ny, gridColor="0.5", ng=0): # draw the frame pylab.plot([ll[0], ll[0], uu[0], uu[0], ll[0]], [ll[1], uu[1], uu[1], ll[1], ll[1]], color="k", lw=2) # draw the x grid lines dx = (uu[0] - ll[0])/nx n = 1 while (n < nx): pylab.plot([ll[0]+n*dx, ll[0]+n*dx], [ll[1], uu[1]], color=gridColor, ls=":") n += 1 # draw the y grid lines dy = (uu[1] - ll[1])/ny n = 1 while (n < ny): pylab.plot([ll[0], uu[0]], [ll[1]+n*dy, ll[1]+n*dy], color=gridColor, ls=":") n += 1 # ghostcells? if (ng > 0): print "here" xmin = ll[0]-ng*dx xmax = uu[0]+ng*dx ymin = ll[1]-ng*dy ymax = uu[1]+ng*dy pylab.plot([xmin, xmin, xmax, xmax, xmin], [ymin, ymax, ymax, ymin, ymin], ls="--", color="r") pylab.clf() drawBox([0., 0.], [1., 6.], 5, 30) drawBox([1., 0.], [2., 6.], 5, 30) drawBox([2., 0.], [3., 6.], 5, 30) drawBox([3., 0.], [4., 6.], 5, 30, ng=1) drawBox([4., 0.], [5., 6.], 5, 30) drawBox([5., 0.], [6., 6.], 5, 30) pylab.xlim(-0.25,6.25) pylab.ylim(-0.25,6.25) a = pylab.gca() a.set_aspect("equal", "datalim") pylab.axis("off") f = pylab.gcf() f.set_size_inches(7.0,7.0) pylab.tight_layout() pylab.savefig("domain_1d.png") pylab.savefig("domain_1d.eps", bbox_inches="tight") ================================================ FILE: finite-volume/old-scripts/simplegrid.py ================================================ import math import numpy import pylab import grid_plot_util as gpu def simplegrid(): # grid info nzones = 7 ng = 1 gr = gpu.grid(nzones, ng, xmin=0.0, xmax=1.0) #------------------------------------------------------------------------ gpu.drawGrid(gr, emphasizeEnd=1, edgeTicks=0) # label a few gpu.labelCenter(gr, ng+nzones/2, r"$i$") gpu.labelCenter(gr, ng+nzones/2-1, r"$i-1$") gpu.labelCenter(gr, ng+nzones/2+1, r"$i+1$") pylab.xlim(gr.xl[0]-0.5*gr.dx,gr.xr[2*ng+nzones-1]+0.5*gr.dx) # label dx pylab.plot([gr.xl[ng+nzones/2-2], gr.xl[ng+nzones/2-2]], [-0.35,-0.25], color="k") pylab.plot([gr.xr[ng+nzones/2-2], gr.xr[ng+nzones/2-2]], [-0.35,-0.25], color="k") pylab.plot([gr.xl[ng+nzones/2-2], gr.xr[ng+nzones/2-2]], [-0.3,-0.3], color="k") pylab.text(gr.xc[ng+nzones/2-2], -0.5, r"$\Delta x$", horizontalalignment="center") pylab.subplots_adjust(left=0.05,right=0.95,bottom=0.05,top=0.95) pylab.ylim(-0.5, 1.5) pylab.axis("off") f = pylab.gcf() f.set_size_inches(10.0,2.5) pylab.savefig("simplegrid.png") pylab.savefig("simplegrid.eps") if __name__== "__main__": simplegrid() ================================================ FILE: gravity/gravity.tex ================================================ \section{Self-gravity and hydrodynamics} \section{Monopole approximation} -- radial sampling -- time-centering -- predictor -- RK formalism (solve each stage) -- what to use as center? \section{Multipole expansions} \section{Full Poisson gravity} -- isolated boundaries \subsection{Special case: triply periodic and FFTs} \section{Conservative formulations} ================================================ FILE: higher-order/higher-order-burgers.tex ================================================ \section{WENO methods, nonlinear equations, and flux-splitting} \label{sec:weno-burgers-fvs} %\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.} For a nonlinear scalar conservation law \begin{equation} \label{eq:scalar_conslaw} u_t + f(u)_x = 0 \end{equation} the WENO method introduced in subsection~\ref{sec:WENO} above does not work, as we have assumed the characteristic information travels in one direction only. We have also reconstructed the variable $q$ and from that constructed the ``flux'' which we feed into the differencing formula of equation~\eqref{sec:ho-fd} to update the solution: this will not work either. In the nonlinear case, we instead reconstruct the \emph{flux} directly. That is, given the state $u_i$ at location $x_i$, we compute the flux $f_i = f(u_i)$ and reconstruct the flux $f_{i+\myhalf}$ using the WENO reconstruction above. This has the significant problem that we have no characteristic information for the flux: does it propagate to the left or to the right? To get around this we introduce \emph{flux splitting}. We write \begin{equation} \label{eq:flux-splitting} f(u) = f^{(+)}(u) + f^{(-)}(u) \end{equation} where we choose the functions such that the information contained in $f^{(+)}$ propagates to the right and that in $f^{(-)}$ propagates to the left. There are many ways of doing this, but the simplest is the \emph{Lax-Friedrichs} flux splitting \begin{equation} \label{eq:lf-flux-split} f^{(\pm)}(u) = \tfrac{1}{2} \left( f(u) \pm \alpha u \right) \end{equation} where $\alpha \ge \max | \partial_u f |$. With $\alpha$ being larger than the maximum propagation speed, this means that \begin{equation} \label{eq:lf-flux-split-speeds} \partial_u f^{(+)} \ge 0, \quad \partial_u f^{(-)} \le 0, \end{equation} and the characteristic information contained in $f^{(\pm)}$ will propagate as required. The simplest flux-split algorithm is then \begin{enumerate} \item Compute the maximum characteristic speed $\alpha$ over the entire computational grid; \item Compute the split fluxes $f^{(\pm)}$ from equation~\eqref{eq:lf-flux-split}; \item Reconstruct $f^{(+)}$ to the right using, e.g., the WENO algorithm of section~\ref{sec:WENO}; \item Reconstruct $f^{(-)}$ to the left using the same method; \item Compute $f_{i+\myhalf} = f^{(+)}_{i+\myhalf} + f^{(-)}_{i+\myhalf}$; \item Update the state using equation~\eqref{eq:consup}. \end{enumerate} \begin{figure}[t] \centering % figure generated by hydro_examples/advection/weno.py \includegraphics[width=0.8\linewidth]{weno-vs-plm-burger} \caption[Comparing PLM and WENO methods for Burgers' equation] {\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.\\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/advection/weno_burgers.py}{weno\_burgers.py}}} \end{figure} % As a direct comparison between the piecewise linear method used at the start of this chapter, and the more complex WENO methods introduced here, we start with looking at a shock forming from smooth initial data. Figure~\ref{fig:weno-vs-plm-burger} shows the evolution of the sinusoidal initial data as used in figure~\ref{fig:burgers-shock}, but only showing the result at one time, shortly after characteristic crossing leads to a shock. We see that there is very little difference between the different numerical methods. The characteristic focusing means that all the methods have a similar number of points within the shock. The piecewise linear method appears to capture the edges of the shock slightly more accurate than the (higher order) WENO method with $r=3$. This is likely as the piecewise linear method is using the exact solution to the Riemann Problem, whilst the WENO method is using the more diffusive Lax-Friedrichs flux splitting. At the higher $r=5$ order there is a marginal advantage to the WENO method. \begin{figure}[t] \centering % figure generated by hydro_examples/advection/weno.py \includegraphics[width=0.8\linewidth]{weno-converge-burgers} \caption[WENO convergence rates for Burgers' equation] {\label{fig:weno-converge-burgers} WENO solutions from evolving Burgers' equation for an initial Gaussian profile, stopping before the solution is discontinuous. 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 seventh order convergence briefly, before the error from the time integrator dominates. \\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/advection/weno_burgers.py}{weno\_burgers.py}}} \end{figure} % The advantages of the WENO method arise when the profile is still continuous, as in the rarefaction problem, or at times before shocks have formed. To confirm that high-order convergence is still possible for a nonlinear problem, an initially Gaussian profile is evolved for a short time and compared to the exact solution (computed from characteristic tracing). The resulting error convergence is shown in figure~\ref{fig:weno-converge-burgers}. We see the expected fifth and seventh order convergence rates, at least for some of the resolutions. At too low a resolution the higher order terms in the error expansion affect the solution. At too high a resolution, for the higher-order method, the time integration error starts to dominate. ================================================ FILE: higher-order/higher-order-euler.tex ================================================ \section{WENO methods for the Euler equations} \label{sec:weno-euler} When dealing with nonlinear systems the flux split method introduced in section~\ref{sec:weno-burgers-fvs} generalizes directly. The simplest, and most diffusive flux splitting uses the maximum characteristic speed $\alpha$. For a system, such as the Euler equations, the simplest method is to compute $\alpha$ by maximizing over all characteristics. The directional fluxes $\Fb^{(\pm)}(u)$ can then be reconstructed using a high order WENO scheme component by component. However, this approach can lead to significant oscillations, even in moderate tests. Rather than doing the reconstruction of the components of the directional fluxes, a safer approach is to work with characteristic variables. The discussion in chapter~\ref{ch:compressible-theory} suggests that, by computing the left and right eigenvectors of the appropriate Jacobian matrix $\Ab = \partial \Fb/\partial \Uc$, we can project the directional fluxes into the directional \emph{characteristic} fluxes (using the left eigenvectors), reconstruct these characteristic fluxes using a high order method, and then compute the reconstructed directional fluxes required (using the right eigenvectors). As the characteristic fluxes should contain information about only a single wave at a time, this \emph{should} give similar results to the scalar case and reduce problems from close waves from different families. The Jacobian matrix needs to be computed separately for each point at which the flux needs computing. The best choice of state from which the Jacobian is computed is not clear: often the arithmetic average of neighbouring states is used, but more complex choices can give better results. A direct comparison of component-wise and characteristic-wise flux-vector split WENO methods, applied to the Sod test, is given in figure~\ref{fig:weno-euler-r3} for $r=3$ and in figure~\ref{fig:weno-euler-r5} for $r=5$. The comparison to the PPM method in figure~\ref{fig:Euler:sod:ppm} is particularly instructive. The PPM method is specifically designed for hydrodynamic problems, and is much better at cleanly capturing the discontinuities, especially the contact. The WENO approach will have a higher order of accuracy, which will be clearest on tests with more smooth variation. The oscillations introduced by the WENO methods are more pronounced as the order is increased, but are reduced by the use of characteristic-wise reconstruction. This is most clearly seen in the results for the internal energy in the $r=5$ case shown in figure~\ref{fig:weno-euler-r5}. \begin{figure}[t] \centering % figure generated by hydro_examples/compressible/weno-euler.py \includegraphics[width=0.8\linewidth]{weno-euler-r3} \caption[WENO $r=3$ for the Sod test] {\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. \\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/compressible/weno_euler.py}{weno\_euler.py}}} \end{figure} % \begin{figure}[t] \centering % figure generated by hydro_examples/compressible/weno-euler.py \includegraphics[width=0.8\linewidth]{weno-euler-r5} \caption[WENO $r=5$ for the Sod test] {\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. \\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/compressible/weno_euler.py}{weno\_euler.py}}} \end{figure} % Another illustration of the advantages of a numerical method, such as PPM, that is specifically designed for the Euler equations, is shown in the double rarefaction test in figure~\ref{fig:weno-euler-rarefaction-r3}. The advantages of the higher order methods are shown by how well the WENO schemes capture the edges of the rarefaction waves, even with so few points and the diffusive Lax-Friedrichs flux splitting. However, the artificial heating effect seen at the trivial contact at the center of the domain is about as bad as that from piecewise constant reconstruction, and nowhere near as good as PPM. Whilst increasing the reconstruction order has a small impact, a less diffusive flux splitting would be needed to approach PPM's performance. \begin{figure}[t] \centering % figure generated by hydro_examples/compressible/weno-euler.py \includegraphics[width=0.8\linewidth]{weno-euler-rarefaction-r3} \caption[WENO $r=3$ for the double rarefaction test] {\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.\\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/compressible/weno_euler.py}{weno\_euler.py}}} \end{figure} % \subsection{Extensions} The main advantage of the WENO method is that it retains its high order when extending to multiple dimensions using dimensional splitting. From the finite-difference form there are no transverse Riemann problems to solve. From using the Method of Lines there is no issue about ordering the dimensional sweeps: the updates in each direction are computed separately but applied together in the time integrator. This formally retains the high-order accuracy but can lose significant absolute accuracy -- compare the advection of a top-hat function with lower order methods using transverse Riemann problem solutions. The global Lax-Friedrichs flux splitting above can be excessively diffusive. A local Lax-Friedrichs flux splitting, where $\alpha$ is computed at each point by maximising over the characteristic speed within the stencil at that point, is less diffusive but slightly more expensive. Roe-style flux splittings are possible but not always stable. ================================================ FILE: higher-order/weno-coefficients.ipynb ================================================ { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# WENO coefficients" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The starting point for a WENO code is a standard Lagrange interpolating polynomial.\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", "The Lagrange interpolation polynomial $p_{r, k}(x)$ is given by (equation (2.11) of Shu's review with notation and order altered)\n", "$$\n", " p_{r, k}(x) = \\sum_{m=0}^r \\sum_{j=0}^{m-1} q_{i-r+k+1+j} \\frac{1}{\\Delta x^r} \\frac{\\sum_{l=0, l \\ne m}^r \\prod_{s=0, s \\ne l, m}^r \\left( x - x_{i-r+k+1+s+1/2} \\right)}{\\prod_{l=0, l \\ne m}^r \\left( m - l \\right)}.\n", "$$\n", "\n", "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", "$$\n", " p_{r, k, i+1/2} = \\sum_{l=0}^{r-1} \\left( \\sum_{j=r-l}^r \\frac{\\sum_{m=0, m \\ne j}^r \\prod_{s=0, s \\ne j, m}^r \\left( r - k - s \\right)}{\\prod_{m=0, m \\ne j}^r \\left( j - m \\right)} \\right) q_{i+k-l} .\n", "$$\n", "\n", "Thus we write\n", "$$\n", " q_{r, k, i+1/2} = \\sum_{l=0}^{r-1} a_{r, k, l} q_{i+k-l}\n", "$$\n", "where the constants $a_{r, k, l}$ are given by\n", "$$\n", " a_{r, k, l} = \\sum_{j=r-l}^{r} \\frac{\\sum_{m=0, m \\ne j}^r \\prod_{s=0, s \\ne j, m}^r \\left( r - k - s \\right)}{\\prod_{m=0, m \\ne j}^r \\left( j - m \\right)}.\n", "$$" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true }, "outputs": [], "source": [ "import sympy\n", "sympy.init_printing()" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def a_coefficient(order_r, shift_k, index_l):\n", " coefficient = sympy.Rational(0, 1)\n", " for j in range(order_r - index_l, order_r+1):\n", " numerator = 0\n", " denominator = 1\n", " for m in range(order_r+1):\n", " if m == j:\n", " pass\n", " else:\n", " product = 1\n", " for s in range(order_r+1):\n", " if s == j or s == m:\n", " pass\n", " else:\n", " product *= order_r - shift_k - s\n", " numerator += product\n", " denominator *= j - m\n", " coefficient += sympy.Rational(numerator, denominator)\n", " return coefficient" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0 0 11/6\n", "0 1 -7/6\n", "0 2 1/3\n", "1 0 1/3\n", "1 1 5/6\n", "1 2 -1/6\n", "2 0 -1/6\n", "2 1 5/6\n", "2 2 1/3\n" ] } ], "source": [ "order_r = 3\n", "for k in range(order_r):\n", " for l in range(order_r):\n", " print(k, l, a_coefficient(order_r, k, l))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "So an individual stencil, of order $r$, shifted by $k$, will have the form\n", "$$\n", " q_{r, k, i+1/2} = \\sum_{l=0}^{r-1} a_{r, k, l} q_{i+k-l}\n", "$$\n", "which we can show symbolically:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [], "source": [ "q = sympy.symbols(r'q:{}'.format(2*order_r-1))" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def stencil(order_r, shift_k, q, offset=0):\n", " stencil_sum = sympy.Rational(0, 1)\n", " for l in range(order_r):\n", " stencil_sum += a_coefficient(order_r, shift_k, l) * q[order_r+shift_k-l-1-offset]\n", " return stencil_sum" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0 q0/3 - 7*q1/6 + 11*q2/6\n", "1 -q1/6 + 5*q2/6 + q3/3\n", "2 q2/3 + 5*q3/6 - q4/6\n" ] } ], "source": [ "for k in range(order_r):\n", " print(k, stencil(order_r, k, q))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "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:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "q0/30 - 13*q1/60 + 47*q2/60 + 9*q3/20 - q4/20\n" ] } ], "source": [ "print(stencil(2*order_r-1, order_r-1, q, offset=order_r-1))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, we can form a convex combination of each of the stencils of order $r$, so that \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", "We can match the stencils and so compute the $C_{r, k}$ coefficients:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[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" ] } ], "source": [ "C = sympy.symbols(r'C:{}'.format(order_r))\n", "sum_stencil = 0\n", "for k in range(order_r):\n", " sum_stencil += C[k] * stencil(order_r, k, q)\n", "eqns=[]\n", "optimal_stencil = stencil(2*order_r-1, order_r-1, q, offset=order_r-1)\n", "for k in range(order_r):\n", " eqns.append(sympy.Eq(optimal_stencil.expand().coeff(q[k]),\n", " sum_stencil.expand().coeff(q[k])))\n", "print(eqns)" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0 1/10\n", "1 3/5\n", "2 3/10\n" ] } ], "source": [ "sol=sympy.solve(eqns, C)\n", "for k in range(order_r):\n", " print(k, sol[C[k]])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We convert that into a function:" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def C_coefficients(order_r):\n", " coefficients = []\n", " q = sympy.symbols(r'q:{}'.format(2*order_r-1))\n", " C = sympy.symbols(r'C:{}'.format(order_r))\n", " sum_stencil = 0\n", " for k in range(order_r):\n", " sum_stencil += C[k] * stencil(order_r, k, q)\n", " eqns=[]\n", " optimal_stencil = stencil(2*order_r-1, order_r-1, q, offset=order_r-1)\n", " for k in range(order_r):\n", " eqns.append(sympy.Eq(optimal_stencil.expand().coeff(q[k]),\n", " sum_stencil.expand().coeff(q[k])))\n", " solution = sympy.solve(eqns, C)\n", " for k in range(order_r):\n", " coefficients.append(solution[C[k]])\n", " return coefficients" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1/24310\n", "36/12155\n", "504/12155\n", "2352/12155\n", "882/2431\n", "3528/12155\n", "1176/12155\n", "144/12155\n", "9/24310\n" ] } ], "source": [ "check_r = 9\n", "C = C_coefficients(check_r)\n", "for k in range(check_r):\n", " print(C[k])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "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", "The Jiang-Shu smoothness indicators are given by the sum of the integrals of the square of the derivatives of the interpolating polynomial:\n", "$$\n", " \\beta_{r, k, i+1/2} = \\sum_{m=1}^{r-1} \\Delta x^{2 m -1} \\int_{-\\Delta x / 2}^{+\\Delta x / 2} \\text{d} (\\xi - x_i) \\, \\frac{\\text{d}^m}{\\text{d} \\xi^m} p_{r, k}(\\xi-x_i).\n", "$$\n", "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 $\\Delta x = 1$." ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def interpolating_p(order_r, shift_k, xi_m_x, q):\n", " shift_r = order_r - shift_k - 1\n", " p = sympy.Rational(0, 1)\n", " for m in range(0, order_r+1):\n", " for j in range(0, m):\n", " numerator = 0\n", " denominator = 1\n", " for l in range(order_r+1):\n", " if l == m:\n", " pass\n", " else:\n", " product = 1\n", " for s in range(order_r+1):\n", " if s == l or s == m:\n", " pass\n", " else:\n", " product *= xi_m_x + shift_r - s + sympy.Rational(1, 2)\n", " numerator += product\n", " denominator *= m - l\n", " p += q[order_r-shift_r+j-1] * numerator / denominator\n", " return p" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def beta_integral(order_r, shift_k, q):\n", " xi_m_x = sympy.symbols(r'\\xi')\n", " half = sympy.Rational(1, 2)\n", " sum_m = 0\n", " for m in range(1, order_r):\n", " sum_m += sympy.integrate(sympy.diff(interpolating_p(order_r, shift_k, xi_m_x, q).simplify(), xi_m_x, m)**2, \n", " (xi_m_x, -half, half))\n", " return sum_m" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "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", "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", "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" ] } ], "source": [ "for k in range(order_r):\n", " print(k, beta_integral(order_r, k, q).expand())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "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", "$$\n", " \\beta_{r, k, i+1/2} = \\sum_{l=0}^{r-1} \\sum_{m=0}^{l} \\sigma_{r, k, l, m} q_{i+k-l} q_{i+k-m}.\n", "$$" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def beta_sigma(order_r, shift_k, q, sigma):\n", " sum_beta = 0\n", " for l in range(order_r):\n", " for m in range(l+1):\n", " sum_beta += sigma[l, m] * q[order_r+shift_k-l-1] * q[order_r+shift_k-m-1]\n", " return sum_beta.simplify()" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "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", "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", "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" ] } ], "source": [ "sigma = sympy.MatrixSymbol('\\sigma', order_r, order_r)\n", "for k in range(order_r):\n", " print(k, beta_sigma(order_r, k, q, sigma))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can now read off the sigma coefficients:" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def sigma_coefficients(order_r, shift_k):\n", " coefficients = []\n", " q = sympy.symbols(r'q:{}'.format(2*order_r-1))\n", " integral_beta = beta_integral(order_r, shift_k, q).expand()\n", " for l in range(order_r):\n", " l_coeffs = []\n", " for m in range(l+1):\n", " if l == m:\n", " l_coeffs.append(integral_beta.coeff(q[order_r+shift_k-l-1]**2))\n", " else:\n", " l_coeffs.append(integral_beta.coeff(q[order_r+shift_k-l-1]).coeff(q[order_r+shift_k-m-1]))\n", " coefficients.append(l_coeffs)\n", " return coefficients" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0 [[10/3], [-31/3, 25/3], [11/3, -19/3, 4/3]]\n", "1 [[4/3], [-13/3, 13/3], [5/3, -13/3, 4/3]]\n", "2 [[4/3], [-19/3, 25/3], [11/3, -31/3, 10/3]]\n" ] } ], "source": [ "for k in range(order_r):\n", " print(k, sigma_coefficients(order_r, k))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can now do the higher order terms, which are all tabulated in Gerolymos et al:" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "a coefficients\n", "0 0 25/12\n", "0 1 -23/12\n", "0 2 13/12\n", "0 3 -1/4\n", "1 0 1/4\n", "1 1 13/12\n", "1 2 -5/12\n", "1 3 1/12\n", "2 0 -1/12\n", "2 1 7/12\n", "2 2 7/12\n", "2 3 -1/12\n", "3 0 1/12\n", "3 1 -5/12\n", "3 2 13/12\n", "3 3 1/4\n", "C coefficients\n", "0 1/35\n", "1 12/35\n", "2 18/35\n", "3 4/35\n", "sigma coefficients\n", "0 [[2107/240], [-1567/40, 11003/240], [3521/120, -8623/120, 7043/240], [-309/40, 2321/120, -647/40, 547/240]]\n", "1 [[547/240], [-1261/120, 3443/240], [961/120, -2983/120, 2843/240], [-247/120, 267/40, -821/120, 89/80]]\n", "2 [[89/80], [-821/120, 2843/240], [267/40, -2983/120, 3443/240], [-247/120, 961/120, -1261/120, 547/240]]\n", "3 [[547/240], [-647/40, 7043/240], [2321/120, -8623/120, 11003/240], [-309/40, 3521/120, -1567/40, 2107/240]]\n" ] } ], "source": [ "order_r=4\n", "print(\"a coefficients\")\n", "for k in range(order_r):\n", " for l in range(order_r):\n", " print(k, l, a_coefficient(order_r, k, l))\n", "print(\"C coefficients\")\n", "C = C_coefficients(order_r)\n", "for k in range(order_r):\n", " print(k, C[k])\n", "print(\"sigma coefficients\")\n", "for k in range(order_r):\n", " print(k, sigma_coefficients(order_r, k))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We'll now write functions to generate a string representation of the coefficients:" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": true }, "outputs": [], "source": [ "import numpy\n", "\n", "def print_a(order_r):\n", " a_numerators = []\n", " numers = []\n", " denoms = []\n", " lcms = []\n", " for k in range(order_r):\n", " k_numers = []\n", " k_denoms = []\n", " for l in range(order_r):\n", " coeff = a_coefficient(order_r, k, l)\n", " k_numers.append(sympy.numer(coeff))\n", " k_denoms.append(sympy.denom(coeff))\n", " numers.append(k_numers)\n", " denoms.append(k_denoms)\n", " lcms.append(sympy.lcm_list(k_denoms))\n", " lcm = sympy.lcm_list(lcms)\n", " for k in range(order_r):\n", " common_numers = []\n", " for l in range(order_r):\n", " common_numers.append(numers[k][l] * sympy.Rational(lcm, denoms[k][l]))\n", " a_numerators.append(common_numers)\n", " max_len = len(str(numpy.max(numpy.abs(numpy.array(a_numerators)))))\n", " string1 = \"{{:>{}}}, \".format(max_len+1)\n", " string2 = \"{{:>{}}}],\\n\".format(max_len+1)\n", " spacing_string = \" \" + len(str(order_r))*\" \"\n", " a_string = \"a_{} = numpy.array([\\n\".format(order_r)\n", " for k in range(order_r):\n", " a_string += spacing_string\n", " a_string += \" [\"\n", " for l in range(order_r-1):\n", " a_string += string1.format(int(a_numerators[k][l]))\n", " a_string += string2.format(int(a_numerators[k][-1]))\n", " a_string += spacing_string\n", " a_string += \" ]) / {}\".format(int(lcm))\n", " return a_string\n", "\n", "\n", "def print_C(order_r):\n", " C_coeffs = C_coefficients(order_r)\n", " denoms = [sympy.denom(i) for i in C_coeffs]\n", " lcm = sympy.lcm_list(denoms)\n", " C_numerators = [sympy.numer(i) * sympy.Rational(lcm, sympy.denom(i)) for i in C_coeffs]\n", " max_len = len(str(numpy.max(numpy.abs(numpy.array(C_numerators)))))\n", " string1 = \"{{:>{}}}, \".format(max_len+1)\n", " string2 = \"{{:>{}}}\".format(max_len+1)\n", " C_string = \"C_{} = numpy.array([\".format(order_r)\n", " for k in range(order_r-1):\n", " C_string += string1.format(int(C_numerators[k]))\n", " C_string += string2.format(int(C_numerators[-1]))\n", " C_string += \" ]) / {}\".format(int(lcm))\n", " return C_string\n", "\n", "\n", "def print_sigma(order_r):\n", " sigma_numerators = []\n", " numers = []\n", " denoms = []\n", " lcms = []\n", " for k in range(order_r):\n", " k_numers = []\n", " k_denoms = []\n", " coeffs = sigma_coefficients(order_r, k)\n", " for l in range(order_r):\n", " l_numers = []\n", " l_denoms = []\n", " for m in range(l+1):\n", " l_numers.append(sympy.numer(coeffs[l][m]))\n", " l_denoms.append(sympy.denom(coeffs[l][m]))\n", " for m in range(l+1, order_r):\n", " l_numers.append(0)\n", " l_denoms.append(1)\n", " lcms.append(sympy.lcm_list(l_denoms))\n", " k_numers.append(l_numers)\n", " k_denoms.append(l_denoms)\n", " numers.append(k_numers)\n", " denoms.append(k_denoms)\n", " lcm = sympy.lcm_list(lcms)\n", " for k in range(order_r):\n", " k_common_numers = []\n", " for l in range(order_r):\n", " l_common_numers = []\n", " for m in range(order_r):\n", " l_common_numers.append(numers[k][l][m] * sympy.Rational(lcm, denoms[k][l][m]))\n", " k_common_numers.append(l_common_numers)\n", " sigma_numerators.append(k_common_numers)\n", " max_len = len(str(numpy.max(numpy.abs(numpy.array(sigma_numerators)))))\n", " string1 = \"{{:>{}}}, \".format(max_len+1)\n", " string2 = \"{{:>{}}}],\".format(max_len+1)\n", " string3 = \"{{:>{}}}]\".format(max_len+1)\n", " string4 = \"{{:>{}}}] ]\\n\".format(max_len+1)\n", " spacing_string = \" \" + len(str(order_r))*\" \"\n", " sigma_string = \"sigma_{} = numpy.array([\".format(order_r)\n", " start_k_string = \"\\n\" + spacing_string + \" [\"\n", " start_l_string = \"\\n\" + spacing_string + \" [\"\n", " end_k_string = \"\\n\" + spacing_string + \" ],\"\n", " end_l_string = \"\\n\" + spacing_string + \" ],\"\n", " for k in range(order_r-1):\n", " sigma_string += start_k_string\n", " for l in range(order_r-1):\n", " sigma_string += start_l_string\n", " for m in range(order_r-1):\n", " sigma_string += string1.format(int(sigma_numerators[k][l][m]))\n", " sigma_string += string2.format(int(sigma_numerators[k][l][-1]))\n", " sigma_string += start_l_string\n", " for m in range(order_r-1):\n", " sigma_string += string1.format(int(sigma_numerators[k][-1][m]))\n", " sigma_string += string3.format(int(sigma_numerators[k][-1][-1]))\n", " sigma_string += end_k_string\n", " sigma_string += start_k_string\n", " for l in range(order_r-1):\n", " sigma_string += start_l_string\n", " for m in range(order_r-1):\n", " sigma_string += string1.format(int(sigma_numerators[-1][l][m]))\n", " sigma_string += string2.format(int(sigma_numerators[-1][l][-1]))\n", " sigma_string += start_l_string\n", " for m in range(order_r-1):\n", " sigma_string += string1.format(int(sigma_numerators[-1][-1][m]))\n", " sigma_string += string3.format(int(sigma_numerators[-1][-1][-1]))\n", " sigma_string += \"\\n\" + spacing_string + \" ]\" + \"\\n\"\n", " if lcm == 1:\n", " sigma_string += \" ])\"\n", " else:\n", " sigma_string += \" ]) / {}\".format(int(lcm))\n", " return sigma_string" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# You really don't want to do this - it's only there to\n", "# check that the output from the functions is executable\n", "# Python code\n", "\n", "#exec(print_sigma(4))\n", "#sigma_4" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "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`." ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "collapsed": true }, "outputs": [], "source": [ "# Output the first \"few\" sets of coefficients to a file\n", "# This takes around 5 minutes on my machine...\n", "\n", "order_lo = 2\n", "order_hi = 7\n", "\n", "with open('./weno_coefficients.py', 'tw') as f:\n", " f.write(\"import numpy\\n\")\n", " f.write(\"\\n\\n\")\n", " for order in range(order_lo, order_hi+1):\n", " f.write(\"# Coefficients of order r={}\\n\".format(order))\n", " f.write(\"# On smooth solutions this should converge with order r={}\\n\".format(2*order-1))\n", " f.write(print_C(order))\n", " f.write(\"\\n\")\n", " f.write(print_a(order))\n", " f.write(\"\\n\")\n", " f.write(print_sigma(order))\n", " f.write(\"\\n\\n\")\n", " for var in \"C\", \"a\", \"sigma\":\n", " f.write(var + \"_all = {\\n\")\n", " for order in range(order_lo, order_hi):\n", " f.write((len(var)+9)*\" \" + \"{0} : {1}_{0},\\n\".format(order, var))\n", " f.write((len(var)+9)*\" \" + \"{0} : {1}_{0}\\n\".format(order_hi, var))\n", " f.write((len(var)+7)*\" \" + \"}\\n\\n\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we can use the coefficients, and repeat the convergence plot from the figures file." ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "collapsed": false }, "outputs": [], "source": [ "import weno_coefficients" ] }, { "cell_type": "code", "execution_count": 72, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def weno(order, q):\n", " \"\"\"\n", " Do WENO reconstruction\n", " \n", " Parameters\n", " ----------\n", " \n", " order : int\n", " The stencil width\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", " C = weno_coefficients.C_all[order]\n", " a = weno_coefficients.a_all[order]\n", " sigma = weno_coefficients.sigma_all[order]\n", "\n", " qL = numpy.zeros_like(q)\n", " beta = numpy.zeros((order, len(q)))\n", " w = numpy.zeros_like(beta)\n", " np = len(q) - 2 * order\n", " epsilon = 1e-16\n", " for i in range(order, np+order):\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" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "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." ] }, { "cell_type": "code", "execution_count": 82, "metadata": { "collapsed": false }, "outputs": [], "source": [ "f_smooth = lambda x: numpy.sin(2 * numpy.pi * x)\n", "f_antiderivative = lambda x: -numpy.cos(2 * numpy.pi * x) / (2 * numpy.pi)\n", "\n", "# Convergence check, using the lazy fix to get error norms.\n", "orders = numpy.arange(min(weno_coefficients.C_all.keys()), max(weno_coefficients.C_all.keys())+1)\n", "points = 4 * 2**numpy.arange(1, 7)\n", "error_1norm = numpy.zeros((len(points), len(orders)))\n", "error_2norm = numpy.zeros((len(points), len(orders)))\n", "for i, pts in enumerate(points):\n", " dx = 1 / pts\n", " x_recon_interior = numpy.arange(dx/2, 1, dx)\n", " recon_exact = f_smooth(x_recon_interior + dx/2)\n", " for j, order in enumerate(orders):\n", " x_recon = numpy.arange(dx * (1/2 - order), 1 + dx * order, dx)\n", " s_recon = (f_antiderivative(x_recon + dx/2) -\n", " f_antiderivative(x_recon - dx/2)) / dx\n", " s_weno = weno(order, s_recon)\n", " error_1norm[i, j] = numpy.linalg.norm(s_weno[order:-order] - recon_exact, 1) / pts\n", " error_2norm[i, j] = numpy.linalg.norm(s_weno[order:-order] - recon_exact, 2) / numpy.sqrt(pts)" ] }, { "cell_type": "code", "execution_count": 79, "metadata": { "collapsed": false }, "outputs": [], "source": [ "%matplotlib inline\n", "from matplotlib import pyplot" ] }, { "cell_type": "code", "execution_count": 87, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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TVFpamhOj/mvTpysVH3/jeXy8PpZdwHblAmOh2V93G4uFEI71d2OwUkrZbDa1bNkyVb9+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\nD2K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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "dxs = 1 / points\n", "fig, axes = pyplot.subplots(nrows=1, ncols=2, figsize=(10, 8))\n", "markers = '+x^osd<'\n", "colours = 'kbrgcm'\n", "for i, order in enumerate(orders):\n", " axes[0].loglog(dxs, error_1norm[:, i], label=r\"$r={}$\".format(order),\n", " linestyle=None, linewidth=0,\n", " marker=markers[i], mec=colours[i], markersize=6, mew=2)\n", " axes[0].loglog(dxs, error_1norm[0, i] * (dxs/dxs[0])**(2*order-1),\n", " linewidth=2, color=colours[i])\n", " axes[1].loglog(dxs, error_2norm[:, i], \n", " linestyle=None, linewidth=0,\n", " marker=markers[i], mec=colours[i], markersize=6, mew=2)\n", " axes[1].loglog(dxs, error_1norm[0, i] * (dxs/dxs[0])**(2*order-1),\n", " linewidth=2, color=colours[i],\n", " label=r\"$\\propto \\Delta x^{{{}}}$\".format(2*order-1))\n", "axes[0].set_xlabel(r\"$\\Delta x$\")\n", "axes[1].set_xlabel(r\"$\\Delta x$\")\n", "axes[0].set_ylabel(r\"$|$Error$|_1$\")\n", "axes[1].set_ylabel(r\"$|$Error$|_2$\")\n", "axes[0].set_ylim(1e-16, 1)\n", "axes[1].set_ylim(1e-16, 1)\n", "axes[0].legend()\n", "axes[1].legend()\n", "fig.tight_layout()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "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." ] } ], "metadata": { "anaconda-cloud": {}, "kernelspec": { "display_name": "Python [default]", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.4.5" } }, "nbformat": 4, "nbformat_minor": 0 } ================================================ FILE: hydro-test-problems/hydro-test-problems.tex ================================================ \section{Testing} There are a large number of standard hydrodynamics test problems that should be run on any implementation of an algorithm. These problems uncover strengths and weaknesses in your choice of algorithm and also simple coding bugs. An ideal test problem has an analytic solution that can be compared to directly. Convergence of your results is also an important test. Recall that for a finite-volume discretization, to second-order, the value of a function evaluated at the cell-center is the same as the average of that function over the zone. Usually this means that you can simply initialize your test problem using cell-center coordinates. But for problems that are not well aligned with your grid, e.g., a spherical initial function mapped onto your Cartesian grid, it is beneficial to try to initialize the average value in the zone. A common way to do this is to sub-divide a zone into a number of sub-zones, initialize each of the sub-zones, and then average the sub-zones back to the original zone. \MarginPar{show figure} \section{Shock tube problems} These tests are one-dimensional, and as such, they can provide good tests of symmetry in your code. Run the shock tube in the $x$, $y$, and $z$ directions separately and then compare the profiles---they should be identical (atleast to machine precision). If not, there may be a simple indexing bug, or something like that in your code. Since these tests start out with a discontinuity, they are not the best tests to use for convergence testing. Wherever there is an initiali discontinuity, the limiters will kick in and drop your method to first-order accurate. \subsection{Slow moving shock} \section{Advection} \section{Sedov} \section{Gresho vortex} The Gresho vortex is a vortex in with a stabilizing pressure gradient designed such that the overall structure is unchanging in time. It has a nice feature in that it allows you to set the Mach number of the flow as one of the parameters. \section{Odd-even decompiling} \section{Hydrostatic equilibrium} ================================================ FILE: hydro1d/hydro1d.tex ================================================ \label{app:hydro1d} %% \begin{center} %% \includegraphics[width=0.125\linewidth]{pyro-sm} %% \end{center} \section{Introduction} \hydrooned\ is a one-dimensional compressible hydrodynamics code written in modern Fortran. In particular, it implements the piecewise parabolic method described in \S~\ref{sec:hydro:ppm}, in both Cartesian and spherical geometries. It assumes a gamma-law equation of state and supports gravity. \section{Getting \hydrooned} \hydrooned\ can be downloaded from its github repository, \url{https://github.com/zingale/hydro1d} as: \begin{verbatim} git clone https://github.com/zingale/hydro1d \end{verbatim} Some details on the code can be found on its webpage: \url{http://zingale.github.io/hydro1d/}\, . As with the other codes detailed here, if you wish to contribute, fix bugs, etc., you can create issues or pull requests through the code's github page. \section{\hydrooned 's structure} There a few major data structures in \hydrooned, defined in the {\tt grid.f90} file: \begin{itemize} \item {\tt grid\_t} : this holds the grid information, with fields for the low and high indicies of the valid domain, and coordinate information (including interface areas and volumes, to support spherical geometry). \item {\tt gridvar\_t} : data that lives on a grid. This holds both a {\tt data} array and the {\tt grid\_t} that it is defined on. \item {\tt gridedgevar\_t} : like {\tt gridvar\_t}, but for data defined at the interfaces between zones. \end{itemize} The integer keys defined in the {\tt variables\_module} allow you to index the difference state fields in the data arrays of the grid variables. \section{Running \hydrooned} Each problem is in it's own directory, and the code is built and run there. For example, to run the Sod shock tube problem, do: \begin{verbatim} cd hydro1d/sod make ./hydro1d inputs-sod-xp \end{verbatim} As the code is built, object files and modules will be output into the {\tt \_build} subdirectory. {\tt make realclean} will clean up the objects. Things are setup for {\tt gfortran} by default---you will need to edit the {\tt Ghydro.mak} with different options for different compilers. Some bits of Fortran 2003 and 2008 are used, so an up-to-date compiler is needed. A number of runtime options can be set---these are listed in {\tt params.f90} and {\tt probparams.f90} (the latter is problem-specific parameters). \section{Problem setups} The following problem setups are provided: \begin{itemize} \item {\tt advect} : a simple advection test where a density profile is advected through periodic boundaries, in a uniform pressure medium (to suppress dynamics) \item {\tt hse} : a simple 1-d atmosphere in hydrostatic equilibrium. This test works to see if the atmosphere stays in HSE. \item {\tt sedov} : a 1-d spherical Sedov explosion. \item {\tt sod} : a shock tube setup for testing against exact Riemann solvers. \end{itemize} ================================================ FILE: hydro_examples/hydro_examples.tex ================================================ \label{app:hydroex} \section{Introduction} The \hydroex\ codes are simple 1-d solvers written in python that illustrate many of the ideas in these nodes. They are used for making many of the figures found throughout (wherever the ``\hydroexdoit{}'' note is found). Most are written for python 3.x. \section{Getting \hydroex} The \hydroex\ codes are hosted on github:\\ \url{https://github.com/zingale/hydro_examples} There are a few ways to get them. The simplest way is to just clone from github on the commandline: \begin{verbatim} git clone https://github.com/zingale/hydro_examples \end{verbatim} This will create a local git repo on your machine with all of the code. You can then run the scripts, ``as is'' without needing to go to github anymore. Periodically, there may be updates to the scripts, which you can obtain by issuing ``{\tt git pull}'' in the {\tt hydro\_examples/} directory. Note that if you have made any local changes to the scripts, \git\ may complain. The process of merging changes is descibed online in various places, e.g. ``\href{https://help.github.com/articles/resolving-a-merge-conflict-from-the-command-line}{Resolving a merge conflict from the command line}'' from github. Alternately, you can use github's web interface to fork it. Logon to github (or create an account) and click on the ``Fork'' icon on the \hydroex\ page. You can then interact with your version of the repo as needed. This method will allow you to push changes back to github, and, if you think they should be included in the main \hydroex\ repo, issue a pull-request. \section{\hydroex\ codes} The codes in \hydroex\ are organized into directories named after the chapters in these notes. Each directory is self-contained. The following scripts are available: \begin{itemize} \item {\tt advection/} \begin{itemize} \item {\tt advection.py}: a 1-d second-order linear advection solver with a wide range of limiters. \item {\tt fdadvect\_implicit.py}: a 1-d first-order implicit finite-difference advection solver using periodic boundary conditions. \item {\tt fdadvect.py}: a 1-d first-order explicit finite-difference linear advection solver using upwinded differencing. \end{itemize} \item {\tt burgers/} \begin{itemize} \item {\tt burgers.py}: a 1-d second-order solver for the inviscid Burgers' equation, with initial conditions corresponding to a shock and a rarefaction. \end{itemize} \item {\tt compressible/} \begin{itemize} \item {\tt cfl.py}: a simple brute force script used to find the stability limit of a multidimensional advection discretization. \item {\tt euler.ipynb}: an {\sf Jupyter} notebook using {\sf SymPy} that derives the eigensystem for the primitive-variable form of the Euler equations. \item {\tt euler-generaleos.ipynb}: an {\sf Jupyter} notebook using {\sf SymPy} that derives the eigensystem for the primitive-variable form of various forms of Euler equations with a general EOS (with either $(\rho e)$ or $\gamma_e$ augmenting the system) and for gray radiation hydrodynamics. Additionally, this notebook computes the full form of the interface states. \item {\tt riemann.py}: the main Riemann class for gamma-law Euler equations. The {\tt RiemannProblem} class has methods to find the star state, plot the Hugoniot curves, and sample the solution. \item {\tt riemann-2shock.py}: draw the Hugoniot curves in the $u$-$p$ plane for a pair of states that comprise a Riemann problem, {\em under the 2-shock approximation}. \item {\tt riemann-phase.py}: draw the Hugoniot curves in the $u$-$p$ plane for a pair of states that comprise a Riemann problem. \item {\tt riemann-sod.py}: solve the Riemann problem for the Sod initial conditions and plot the exact solution. \end{itemize} \item {\tt diffusion/} \begin{itemize} \item {\tt diff\_converge.py}: make a convergence plot (error vs.\ resolution) of different explicit and implicit diffusion methods. \item {\tt diffusion\_explicit.py}: solve the constant-diffusivity diffusion equation explicitly. The method is first-order accurate in time, but second-order in space. A Gaussian profile is diffused---the analytic solution is also a Gaussian. \item {\tt diffusion\_fo\_implicit.py}: solve the constant-diffusivity diffusion equation implicitly. Backwards-difference time-discretization is used, resulting in a first-order-in-time method. Spatial discretization is second order. A Gaussian profile is diffused. \item {\tt diffusion\_implicit.py}: solve the constant-diffusivity diffusion equation implicitly. Crank-Nicolson time-discretization is used, resulting in a second-order method. A Gaussian profile is diffused. \end{itemize} \item {\tt elliptic/} \begin{itemize} \item {\tt poisson\_fft.py}: an FFT solver for a 2-d Poisson problem with periodic boundaries. \end{itemize} \item {\tt finite-volume/} \begin{itemize} \item {\tt conservative-interpolation.ipynb}: an {\sf Jupyter} notebook using {\sf SymPy} that derives conservative interpolants. \end{itemize} \item {\tt incompressible/} \begin{itemize} \item {\tt project.py}: a simple example of using a projection to recover a divergence-free velocity field. \end{itemize} \item {\tt multigrid/} \begin{itemize} \item {\tt mg\_converge.py}: a convergence test of the multigrid solver. A Poisson problem is solved at various resolutions and compared to the exact solution. This demonstrates second-order accuracy. \item {\tt mg\_test.py}: a simple driver for the multigrid solver. This sets up and solves a Poisson problem and plots the behavior of the solution as a function of V-cycle number. \item{\tt multigrid.py}: a multigrid class for cell-centered data. This implements pure V-cycles. A square domain with $2^N$ zones ($N$ a positive integer) is required. \item {\tt patch1d.py}: a class for 1-d cell-centered data that lives on a grid. This manages the data, handles boundary conditions, and provides routines for prolongation and restriction to other grids. \item {\tt smooth.py}: solve a Poisson equation with smoothing only (but we still use the interface through the multigrid object). \end{itemize} \item {\tt multiphysics/} \begin{itemize} \item {\tt burgersvisc.py}: solve the viscous Burgers equation. The advective terms are treated explicitly with a second-order accurate method. The diffusive term is solved using an implicit Crank-Nicolson discretization. The overall coupling is second-order accurate. \item {\tt diffusion-reaction.py}: solve a diffusion-reaction equation that propagates a diffusive reacting front (flame). A simple reaction term is modeled. The diffusion is solved using a second-order Crank-Nicolson discretization. The reactions are evolved using the VODE ODE solver (via SciPy). 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Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. %%Creator: Converted from TrueType to type 3 by PPR 25 dict begin /_d{bind def}bind def /_m{moveto}_d /_l{lineto}_d /_cl{closepath eofill}_d /_c{curveto}_d /_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d /_e{exec}_d /FontName /Cmsy10 def /PaintType 0 def /FontMatrix[.001 0 0 .001 0 0]def /FontBBox[-29 -960 1123 779]def /FontType 3 def /Encoding [ /minus ] def /FontInfo 10 dict dup begin /FamilyName (cmsy10) def /FullName (cmsy10) def /Notice (Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. ) def /Weight (Regular) def /Version (1.1/12-Nov-94) def /ItalicAngle 0.0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def end readonly def /CharStrings 1 dict dup begin /minus{777 0 83 230 694 270 _sc 102 230 _m 96 230 92 232 88 236 _c 84 240 83 245 83 250 _c 83 254 84 259 88 263 _c 92 267 96 270 102 270 _c 676 270 _l 681 270 685 267 689 263 _c 692 259 694 254 694 250 _c 694 245 692 240 689 236 _c 685 232 681 230 676 230 _c 102 230 _l _cl}_d end readonly def /BuildGlyph {exch begin CharStrings exch 2 copy known not{pop /.notdef}if true 3 1 roll get exec end}_d /BuildChar { 1 index /Encoding get exch get 1 index /BuildGlyph get exec }_d FontName currentdict end definefont pop %!PS-Adobe-3.0 Resource-Font %%Title: cmr10 %%Copyright: Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. %%Creator: Converted from TrueType to type 3 by PPR 25 dict begin /_d{bind def}bind def /_m{moveto}_d /_l{lineto}_d /_cl{closepath eofill}_d /_c{curveto}_d /_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d /_e{exec}_d /FontName /Cmr10 def /PaintType 0 def /FontMatrix[.001 0 0 .001 0 0]def /FontBBox[-44 -250 1009 750]def /FontType 3 def /Encoding [ /v /two /parenright /d /plus /one /a /parenleft ] def /FontInfo 10 dict dup begin /FamilyName (cmr10) def /FullName (cmr10) def /Notice (Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. ) def /Weight (Regular) def /Version (1.1/12-Nov-94) def /ItalicAngle 0.0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def end readonly def /CharStrings 8 dict dup begin /v{{527 0 19 -10 508 431 _sc 244 0 _m 98 367 _l 92 379 82 387 69 391 _c 56 394 39 396 19 396 _c 19 431 _l 229 431 _l 229 396 _l 191 396 173 388 173 372 _c 173 369 173 367 174 366 _c 286 84 _l 387 339 _l 389 344 390 349 390 355 _c 390 367 385 377 375 385 _c 365 392 354 396 342 396 _c 342 431 _l 508 431 _l 508 396 _l 487 396 469 391 454 382 _c }_e{438 372 426 358 418 340 _c 283 0 _l 280 -6 275 -10 267 -10 _c 260 -10 _l 252 -10 246 -6 244 0 _c _cl}_e}_d /two{{500 0 50 0 449 666 _sc 50 0 _m 50 27 _l 50 28 50 30 52 32 _c 207 204 _l 230 229 249 250 264 268 _c 278 285 293 305 307 327 _c 321 349 333 372 341 396 _c 349 419 354 444 354 470 _c 354 497 349 523 339 548 _c 329 573 314 593 294 608 _c 274 623 249 631 221 631 _c 192 631 166 622 143 605 _c 119 587 103 565 94 537 _c 96 537 100 538 105 538 _c }_e{119 538 132 533 143 523 _c 153 513 159 500 159 484 _c 159 468 153 455 143 445 _c 132 434 119 429 105 429 _c 89 429 76 434 66 445 _c 55 456 50 469 50 484 _c 50 508 54 532 64 554 _c 73 576 86 595 104 613 _c 122 630 142 643 164 652 _c 186 661 210 666 236 666 _c 274 666 309 658 342 642 _c 375 626 401 603 420 573 _c 439 543 449 509 449 470 _c }_e{449 441 442 414 430 388 _c 417 362 401 338 381 317 _c 361 295 336 271 305 244 _c 274 217 254 199 244 190 _c 131 81 _l 227 81 _l 274 81 313 81 345 82 _c 377 82 394 84 396 86 _c 404 94 412 125 420 178 _c 449 178 _l 421 0 _l 50 0 _l _cl}_e}_d /parenright{{388 0 56 -249 291 750 _sc 65 -249 _m 59 -249 56 -246 56 -240 _c 56 -237 56 -235 58 -234 _c 172 -122 229 38 229 250 _c 229 462 172 623 60 733 _c 57 734 56 737 56 741 _c 56 743 57 745 59 747 _c 61 749 63 750 65 750 _c 74 750 _l 76 750 77 749 79 748 _c 127 710 167 664 199 610 _c 231 556 254 498 269 437 _c 283 376 291 314 291 250 _c 291 202 287 156 279 110 _c }_e{271 64 258 20 241 -24 _c 223 -68 201 -108 175 -146 _c 148 -184 116 -217 79 -247 _c 77 -248 76 -249 74 -249 _c 65 -249 _l _cl}_e}_d /d{{555 0 33 -10 529 694 _sc 244 -10 _m 204 -10 168 0 136 21 _c 103 42 78 70 60 105 _c 42 139 33 176 33 216 _c 33 256 42 293 62 328 _c 82 362 109 390 143 411 _c 177 431 214 442 255 442 _c 279 442 302 436 324 426 _c 346 416 365 402 381 384 _c 381 592 _l 381 610 378 623 373 631 _c 367 639 360 643 350 645 _c 340 647 324 648 303 648 _c 303 683 _l 451 694 _l 451 91 _l }_e{451 73 453 60 459 52 _c 464 44 472 40 482 38 _c 492 36 507 35 529 35 _c 529 0 _l 378 -10 _l 378 52 _l 360 32 340 16 316 6 _c 292 -4 268 -10 244 -10 _c 142 87 _m 153 65 168 47 187 34 _c 206 21 227 15 250 15 _c 278 15 303 23 326 39 _c 349 55 366 75 378 101 _c 378 341 _l 370 355 360 368 348 380 _c 336 391 323 400 308 406 _c }_e{293 412 277 416 261 416 _c 226 416 198 406 177 386 _c 156 366 141 341 133 311 _c 124 280 120 248 120 215 _c 120 188 121 165 124 145 _c 126 125 132 105 142 87 _c _cl}_e}_d /plus{{777 0 56 -82 721 583 _sc 75 230 _m 69 230 65 232 61 236 _c 57 240 56 245 56 250 _c 56 254 57 259 61 263 _c 65 267 69 270 75 270 _c 369 270 _l 369 565 _l 369 570 371 574 375 578 _c 379 581 383 583 389 583 _c 393 583 398 581 402 578 _c 406 574 409 570 409 565 _c 409 270 _l 703 270 _l 707 270 711 267 715 263 _c 719 259 721 254 721 250 _c 721 245 719 240 715 236 _c }_e{711 232 707 230 703 230 _c 409 230 _l 409 -64 _l 409 -69 406 -73 402 -77 _c 398 -80 393 -82 389 -82 _c 383 -82 379 -80 375 -77 _c 371 -73 369 -69 369 -64 _c 369 230 _l 75 230 _l _cl}_e}_d /one{500 0 87 0 421 666 _sc 93 0 _m 93 35 _l 176 35 218 45 218 67 _c 218 592 _l 183 575 139 567 87 567 _c 87 602 _l 168 602 230 623 272 666 _c 286 666 _l 288 666 291 665 293 663 _c 295 661 296 659 296 657 _c 296 67 _l 296 45 337 35 421 35 _c 421 0 _l 93 0 _l _cl}_d /a{{500 0 40 -10 493 448 _sc 40 98 _m 40 138 55 170 87 195 _c 118 219 155 237 199 247 _c 243 257 284 263 324 263 _c 324 304 _l 324 323 319 342 311 360 _c 303 378 291 392 275 404 _c 259 416 241 422 222 422 _c 178 422 144 412 121 392 _c 133 392 144 387 152 377 _c 160 367 165 356 165 344 _c 165 330 160 319 151 310 _c 141 300 130 296 117 296 _c 103 296 92 300 83 310 _c }_e{73 319 69 330 69 344 _c 69 379 85 405 117 422 _c 149 439 184 448 222 448 _c 249 448 276 442 304 431 _c 331 419 353 403 371 381 _c 388 359 397 334 397 306 _c 397 81 _l 397 68 399 56 405 45 _c 411 34 419 29 431 29 _c 442 29 450 34 456 45 _c 461 56 464 68 464 81 _c 464 145 _l 493 145 _l 493 81 _l 493 66 489 52 481 38 _c 473 24 463 14 450 6 _c }_e{436 -1 422 -5 407 -5 _c 387 -5 370 2 356 17 _c 342 32 334 50 333 71 _c 320 45 302 25 278 11 _c 254 -3 228 -10 201 -10 _c 175 -10 150 -6 126 0 _c 101 7 80 19 64 35 _c 48 51 40 72 40 98 _c 121 98 _m 121 74 129 55 147 39 _c 164 23 184 15 208 15 _c 229 15 248 20 266 31 _c 284 41 298 56 308 75 _c 318 93 324 113 324 134 _c 324 238 _l }_e{294 238 263 233 231 223 _c 199 213 173 197 152 176 _c 131 155 121 129 121 98 _c _cl}_e}_d /parenleft{{388 0 97 -249 332 750 _sc 310 -247 _m 272 -217 240 -184 214 -146 _c 187 -108 165 -67 148 -25 _c 131 17 118 62 110 109 _c 101 155 97 202 97 250 _c 97 298 101 345 110 392 _c 118 438 131 483 149 527 _c 166 570 188 611 215 649 _c 242 687 274 720 310 748 _c 310 749 311 750 315 750 _c 324 750 _l 326 750 327 749 329 747 _c 331 745 332 743 332 741 _c 332 738 331 736 330 735 _c }_e{297 703 270 668 248 632 _c 226 596 209 557 196 516 _c 183 474 174 432 168 388 _c 162 344 159 298 159 250 _c 159 38 215 -122 329 -232 _c 331 -234 332 -236 332 -240 _c 332 -242 331 -244 329 -246 _c 327 -248 326 -249 324 -249 _c 315 -249 _l 311 -249 310 -248 310 -247 _c _cl}_e}_d end readonly def /BuildGlyph {exch begin CharStrings exch 2 copy known not{pop /.notdef}if true 3 1 roll get exec end}_d /BuildChar { 1 index /Encoding get exch get 1 index /BuildGlyph get exec }_d FontName currentdict end definefont pop end %%EndProlog mpldict begin -115.609 185.798 translate 843.217 420.405 0 0 clipbox gsave 0 0 m 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-3.076406 moveto /i glyphshow 35.980225 -3.076406 moveto /comma glyphshow 39.092627 -3.076406 moveto /j glyphshow /Cmr10 findfont 10.5 scalefont setfont 45.253174 -3.076406 moveto /plus glyphshow 54.074951 -3.076406 moveto /one glyphshow /Cmmi10 findfont 10.5 scalefont setfont 59.324951 -3.076406 moveto /slash glyphshow /Cmr10 findfont 10.5 scalefont setfont 64.574951 -3.076406 moveto /two glyphshow grestore gsave 623.903667 162.758783 translate 0.000000 rotate /Cmr10 findfont 15.0 scalefont setfont 0.000000 0.750000 moveto /parenleft glyphshow /Cmmi10 findfont 15.0 scalefont setfont 5.822754 0.750000 moveto /G glyphshow 17.614746 0.750000 moveto /phi glyphshow /Cmr10 findfont 15.0 scalefont setfont 26.542969 0.750000 moveto /parenright glyphshow /Cmmi10 findfont 10.5 scalefont setfont 32.365723 -3.076406 moveto /i glyphshow 35.980225 -3.076406 moveto /comma glyphshow 39.092627 -3.076406 moveto /j glyphshow /Cmsy10 findfont 10.5 scalefont setfont 43.409521 -3.076406 moveto /minus glyphshow /Cmr10 findfont 10.5 scalefont setfont 51.566504 -3.076406 moveto /one glyphshow /Cmmi10 findfont 10.5 scalefont setfont 56.816504 -3.076406 moveto /slash glyphshow /Cmr10 findfont 10.5 scalefont setfont 62.066504 -3.076406 moveto /two glyphshow grestore end showpage ================================================ FILE: incompressible/incompressible.tex ================================================ \label{ch:incompressible} \section{Incompressible flow} As a fluid parcel advects through a domain, it compresses and expands due to a variety of effects (stratification, local heat release, acoustic/shock waves). The Lagrangian derivative of the density captures the changes in the fluid, and is a measure of its compressibility. From the continuity equation, we see: \begin{equation} -\frac{1}{\rho}\frac{D\rho}{Dt} = \nabla \cdot \Ub \end{equation} Note that for $\nabla \cdot \Ub > 0$, we have $-(1/\rho) (D\rho/Dt) > 0$, which means that $\rho$ gets smaller---this is expansion of the Lagrangian fluid element. A fluid in which the density (and therefore volume) of a fluid element is not allowed to change is called {\em incompressible}. An incompressible fluid obeys the velocity constraint: \begin{equation} \nabla \cdot \Ub = 0 \end{equation} (since $D\rho / Dt = 0$). The incompressible fluid approximation is a reasonable approximation when the Mach number of the fluid is small ($\ll 1$). To complete the system, we add the momentum equation. If we take the density to be constant everywhere in the domain (not just in our fluid element), then we have: \begin{equation} \frac{\partial \Ub}{\partial t} + \Ub \cdot \nabla \Ub + \nabla p = 0 \end{equation} Note that $p$ here looks like a pressure, but it is not subject to any equation of state. This system is closed as written. The value of $p$ is determined such that the velocity constraint is satisfied. We can gain more insight into the applicability of the incompressible equations by doing an asymptotic expansion. Starting with the momentum equation from the Euler equations: \begin{equation} \frac{\partial (\rho {\bf U})}{\partial t} + {\nabla}\cdot (\rho {\bf U}{\bf U}) + {\nabla}p = 0 \end{equation} we can introduce dimensionless variables: \begin{equation} \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} \end{equation} and we find the dimensionless evolution equation: \begin{equation} \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 \end{equation} Now we introduce an expansion of pressure in terms of Mach number: \begin{equation} p = p_0 + p_1 M + p_2 M^2 \end{equation} Grouping terms by power of Mach number, we find: \begin{equation} \nabla p_0 = \nabla p_1 = 0 \end{equation} and \begin{equation} \frac{\partial (\rho {\bf U})}{\partial t} + {\nabla}\cdot (\rho {\bf U}{\bf U}) + {\nabla}p_2 = 0 \end{equation} Only $p_2$, which is sometimes called the {\em dynamical pressure} appears in the dynamics. All of the thermodynamic content is in $p_0$, consistent with our argument previously that no equation of state is needed. Recalling that pressure evolves according to \begin{equation} \frac{\partial p}{\partial t} + \gamma p {\nabla}\cdot {\bf U} + {\bf U}\cdot {\nabla}p = 0 \end{equation} The general expression for the velocity divergence is: \begin{equation} {\nabla}\cdot {U} = -\frac{1}{\gamma p} \frac{Dp}{Dt} \end{equation} from our pressure expansion, we see that \begin{equation} \nabla \cdot {\bf U} \sim \mathcal{O}(M^2) \end{equation} This justifies our argument that the incompressible equations apply at low Mach number. \section{Projection methods} The basic idea behind a projection method is that any vector field can be decomposed into a divergence free part and the gradient of a scalar (this is sometimes called a {\em Hodge decomposition}). Given a velocity field $\Ub^\star$, we can express it in terms of the divergence free part $\Ub^d$ and a scalar, $\phi$ as: \begin{equation} \Ub^\star = \Ub^d + \nabla \phi \end{equation} Taking the divergence of each side, and noting that $\nabla \cdot \Ub^d = 0$, we have \begin{equation} \nabla \cdot \Ub^\star = \nabla^2 \phi \end{equation} This is an elliptic equation. Given suitable boundary conditions, we can solve for $\phi$ (for instance, using multigrid) and recover the divergence free part of $\Ub^\star$ as: \begin{equation} \Ub^d = \Ub^\star - \nabla \phi \end{equation} We call this operation of extracting the divergence free part of a velocity field a {\em projection}. This can be expressed by defining an operator $P$, such that $P\Ub^\star = \Ub^d$, and $(I - P)\Ub^\star = \nabla \phi$. From the momentum equation, we see that $\partial \Ub/\partial t + \nabla p$ is in the form of a divergence free term $+$ the gradient of a scalar. This means that advancing the velocity field subject to the constraint involves solving: \begin{equation} \Ub_t = P(\Ub_t + \nabla p) = P(-\Ub\cdot \nabla \Ub) \label{eq:projevolve} \end{equation} See Bell, Colella, and Howell~\cite{BCH} for a nice discussion of this. The original projection method for incompressible flows goes back to Chorin~\cite{chorin:1968}. Instead of evolving Eq.~\ref{eq:projevolve} directly, we break the update into pieces. The basic idea is to evolve the velocity advection equation without regard to the constraint, yielding a {\em provisional velocity} field which is then subjected to a projection to enforce the divergence-free constraint. Bell, Colella, and Glaz (BCG)~\cite{BCG} introduced a projection method that uses standard Godunov methods for the advection terms (much like is done with compressible flow) and then solves an elliptic equation to enforce the constraint. This division of the operations in the algorithm (advect then project) is a type of {\em fractional step} method. There are different ways to discretize the operators that make up the projection. We denote the discretized divergence as $D$ and the discretized gradient operation as $G$. For an exact projection, the discretized Laplacian, $L$, would be the same as applying $G$ and $D$ in succession (i.e. $L = DG$). Depending on our data centerings, we may prefer to discretize the Laplacian independently to $D$ and $G$, such that $L \ne DG$. This is called an {\em approximate projection}. Note that for an approximate projection, $P$ is not idempotent: $P^2 \Ub \ne P\Ub$. Many variations on this basic idea exist, using alternate forms of the projection, different grid centerings of the $\phi$ variable, and additional physics. Figure~\ref{fig:incomp:project} shows an example of a projection. The initial velocity field is specified as: \begin{align} \label{eq:incomp:proju} u &= -\sin(\pi x)^2 \sin(2\pi y) \\ \label{eq:incomp:projv} v &= \phantom{+} \sin(\pi y)^2 \sin(2\pi x) \end{align} which is divergence free and doubly-periodic. This is then modified by adding the gradient of a scalar, $\phi$, of the form: \begin{equation} \label{eq:incomp:projphi} \phi = \frac{1}{10} \cos(2\pi y) \cos(2\pi x) \end{equation} yielding a new velocity, $\Ub_p = \Ub + \nabla \phi$. The result is the middle panel in the figure, and is not divergence free. A projection is then done, to recover $\phi$ by solving \begin{equation} \nabla^2 \phi = \nabla \cdot \Ub_p \end{equation} For this figure, we use the standard 5-point Laplacian and a divergence built via centered-differences: \begin{equation} \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} \end{equation} Together, these represent an approximation projection. \begin{figure}[t] \centering \includegraphics[width=0.9\linewidth]{project-u} \caption[Example of a projection]{\label{fig:incomp:project} An example of an approximate projection, on a $128^2$ grid. The top shows the original velocity field, Eqs.~\ref{eq:incomp:proju} and \ref{eq:incomp:projv}. This was poluted by adding the gradient of a scalar, Eq.~\ref{eq:incomp:projphi} (middle) and then a projection was done to recover the original velocity field (bottom). In this case, the L$_2$ norm of the error on the velocity (comparing before and after projection) is $0.000189$. \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/incompressible/project.py}{project.py}}} \end{figure} \begin{exercise}[An approximate projection] Implement an approximate projection, using pure smoothing for solving the elliptic equation. Start with the velocity field described by Eqs.~\ref{eq:incomp:proju} and \ref{eq:incomp:projv} and the scalar from Eq.~\ref{eq:incomp:projphi}. Compute a poluted velocity field, $\Ub_p = \Ub + \nabla \phi$ and then project it to recover the original velocity field. \end{exercise} \section{Cell-centered approximate projection solver} Here we describe an incompressible algorithm that uses cell-centered data throughout---$\Ub$ and $p$ are both cell-centered. The projection at the end is an approximate projection. The basic algorithm flow is \begin{itemize} \item Create the time-centered advective velocities through the faces of the zones. \item Project the advective velocities such that they obey the velocity constraint \item Construct the time-centered interface states of all quantities on the faces of the zones using the advective velocity. \item Update the velocity to the new time. This is defines the provisional velocity field---it does not yet satisfy the constraint. \item Enforce the velocity constraint by projecting the velocity. \end{itemize} The description below is pieced together from a variety of sources. BCH describes a cell-centered method, but with an exact projection (with a larger, decoupled stencil). Almgren, Bell, and Szymczak (ABS)~\cite{ABS} describes an approximate projection method, but with a node-centered final projection. We follow this paper closely up until the projection. Martin and Colella~\cite{MartinColella} (and Martin's PhD thesis) method uses a cell-centered projection, as is described here. They go into additional effort to describe this for a refined grid. All of these method are largely alike, aside from how the discretization of the final projection is handled. \subsection{Advective velocity} In predicting the interface states, we first seek to construct the velocities through the interfaces. A key concept throughout the advection step is that once we have the normal velocities on the interfaces, we can use these to upwind left and right states of any quantity to get their interface value. The advective velocities we construct here, ${u}^\mathrm{adv}$ and ${v}^\mathrm{adv}$, will later be used to upwind the various states we predict to the interfaces. We only need the velocity through the interfaces, as shown in the figure \ref{fig:MAC}. This staggered grid arrangement is sometimes called a MAC grid. \begin{figure}[h] \centering \includegraphics[width=0.8\linewidth]{MAC} \caption[MAC grid for velocity] {\label{fig:MAC} The staggered `MAC' grid for the advective velocities.} \end{figure} We follow ABS. Our velocity evolution system (writing out the individual components of $\Ub$: $u$ and $v$) is \begin{eqnarray} \frac{\partial u}{\partial t} &=& -u \frac{\partial u}{\partial x} -v \frac{\partial u}{\partial y} -\frac{\partial p}{\partial x} = 0 \\ \frac{\partial v}{\partial t} &=& -u \frac{\partial v}{\partial x} -v \frac{\partial v}{\partial y} -\frac{\partial p}{\partial y} = 0 \end{eqnarray} Our goal in this step is to predict time-centered interface values of the normal velocity ($u$ on x-edges and $v$ on y-edges). The prediction follows from Taylor expanding the state to the interface (through $\Delta x/2$ or $\Delta y/2$) and to the half-time (through $\Delta t/2$). As with the regular advection, we can have left and right states which we will resolve by solving a Riemann problem. The left interface state of $u$ at $i+\myhalf,j$ is found as: \begin{eqnarray} u_{i+\myhalf,j,L}^{n+\myhalf} &=& u_{i,j} + \frac{\Delta x}{2} \left . \frac{\partial u}{\partial x} \right |_{i,j} + \frac{\Delta t}{2} \left . \frac{\partial u}{\partial t} \right |_{i,j}\\ &=& u_{i,j} + \frac{\Delta x}{2} \left . \frac{\partial u}{\partial x} \right |_{i,j} + \frac{\Delta t}{2} \left . \left (-u \frac{\partial u}{\partial x} -v \frac{\partial u}{\partial y} -\frac{\partial p}{\partial x} \right ) \right |_{i,j}\\ &=& u_{i,j} + \frac{\Delta x}{2} \left ( 1 - \frac{\Delta t}{\Delta x} u_{i,j} \right ) \left . \frac{\partial u}{\partial x}\right |_{i,j} - \frac{\Delta t}{2} \left ( v \frac{\partial u}{\partial y}\right )_{i,j} - \frac{\Delta t}{2} \left . \frac{\partial p}{\partial x} \right |_{i,j}\\ \end{eqnarray} We express ${\partial u}/{\partial x} |_{i,j}$ as $\Dux_{i,j} / \Delta x$, where $\Dux_{i,j}$ is the limited slope of $u$ in the $x$-direction in zone $i,j$. Our interface state is then: \begin{equation} u_{i+\myhalf,j,L}^{n+\myhalf} = \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}} - \underbrace{\frac{\Delta t}{2} \left ( v \frac{\partial u}{\partial y} \right )_{i,j}}_{\mathrm{transverse~term}} - \frac{\Delta t}{2} \left . \frac{\partial p}{\partial x} \right |_{i,j} \label{eq:uMAC_l} \end{equation} Similarly, for $v$ through the $y$ faces, we find: \begin{equation} v_{i,j+\myhalf,L}^{n+\myhalf} = \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}} - \underbrace{\frac{\Delta t}{2} \left ( u \frac{\partial v}{\partial x} \right )_{i,j}}_{\mathrm{transverse~term}} - \frac{\Delta t}{2} \left . \frac{\partial p}{\partial y} \right |_{i,j} \label{eq:vMAC_l} \end{equation} As indicated above (and following ABS and the similar notation used by Colella~\cite{colella:1990}), we denote the quantities that will be used to evaluate the transverse states (consisting only of the normal predictor) with a `$\hat{~}$'. These will be used to evaluate the transverse terms labeled above. We predict $u$ and $v$ to both the $x$ and $y$ interfaces, using only the normal part of the predictor. This gives us the left and right `hat' states on each interface. \noindent $u$ on $x$-interfaces: \begin{eqnarray} \hat{u}_{i+\myhalf,j,L}^{n+\myhalf} &=& u_{i,j} + \frac{1}{2} \left ( 1 - \frac{\Delta t}{\Delta x} u_{i,j} \right ) \Dux_{i,j} \\ % \hat{u}_{i+\myhalf,j,R}^{n+\myhalf} &=& u_{i+1,j} - \frac{1}{2} \left ( 1 + \frac{\Delta t}{\Delta x} u_{i+1,j} \right ) \Dux_{i+1,j} \end{eqnarray} \noindent $v$ on $x$-interfaces: \begin{eqnarray} \hat{v}_{i+\myhalf,j,L}^{n+\myhalf} &=& v_{i,j} + \frac{1}{2} \left ( 1 - \frac{\Delta t}{\Delta x} u_{i,j} \right ) \Dvx_{i,j} \\ % \hat{v}_{i,j+\myhalf,R}^{n+\myhalf} &=& v_{i+1,j} - \frac{1}{2} \left ( 1 + \frac{\Delta t}{\Delta x} u_{i+1,j} \right ) \Dvx_{i+1,j} \end{eqnarray} \noindent $u$ on $y$-interfaces: \begin{eqnarray} \hat{u}_{i,j+\myhalf,L}^{n+\myhalf} &=& u_{i,j} + \frac{1}{2} \left ( 1 - \frac{\Delta t}{\Delta y} v_{i,j} \right ) \Duy_{i,j} \\ % \hat{u}_{i,j+\myhalf,R}^{n+\myhalf} &=& u_{i,j+1} - \frac{1}{2} \left ( 1 + \frac{\Delta t}{\Delta y} v_{i,j+1} \right ) \Duy_{i,j+1} \end{eqnarray} \noindent $v$ on $y$-interfaces: \begin{eqnarray} \hat{v}_{i,j+\myhalf,L}^{n+\myhalf} &=& v_{i,j} + \frac{1}{2} \left ( 1 - \frac{\Delta t}{\Delta y} v_{i,j} \right ) \Dvy_{i,j} \\ % \hat{v}_{i,j+\myhalf,R}^{n+\myhalf} &=& v_{i,j+1} - \frac{1}{2} \left ( 1 + \frac{\Delta t}{\Delta y} v_{i,j+1} \right ) \Dvy_{i,j+1} \end{eqnarray} Note that the `right' state is constructed using the data to the right of the interface. Also note that in constructing these transverse velocities, we do not include the $p$ term. Next we find the advective velocity through each interface. The incompressible velocity equation looks like the inviscid Burger's equation, and the Riemann solver follows that construction. BCG provide the implementation used here (and throughout the incompressible literature). Also see Toro~\cite{toro:1997}. We denote the resulting velocities with the `adv' superscript, as these are the normal velocities used to advect the hat states. The Riemann problem solution is: \begin{equation} \mathcal{R}(q_L,q_R) = \left \{ \begin{array}{cl} q_L & \mathrm{~if~} q_L > 0, \qquad q_L + q_R > 0 \\ 0 & \mathrm{~if~} q_L \le 0, \qquad q_R \ge 0 \\ q_R & \mathrm{~otherwise} \end{array} \right . \end{equation} We solve this for each of the normal velocities, giving: \begin{eqnarray} \hat{u}^\mathrm{adv}_{i+\myhalf,j} &=& \mathcal{R}(\hat{u}_{i+\myhalf,j,L}^{n+\myhalf}, \hat{u}_{i+\myhalf,j,R}^{n+\myhalf}) \\ % \hat{v}^\mathrm{adv}_{i,j+\myhalf} &=& \mathcal{R}(\hat{v}_{i,j+\myhalf,L}^{n+\myhalf}, \hat{v}_{i,j+\myhalf,R}^{n+\myhalf}) \end{eqnarray} These advective velocities (sometimes called the {\em transverse velocities}) are used to resolve the left and right states of all the hat quantities by simple upwinding. For a $u$ or $v$ state on the $x$-interface, we upwind based on $\hat{u}^\mathrm{adv}$; and for a $u$ or $v$ state on the $y$-interface, we upwind based on $\hat{v}^\mathrm{adv}$. If we write the upwinding as: \begin{equation} \mathcal{U}[s^\mathrm{adv}](q_L, q_R) = \left \{ \begin{array}{cl} q_L & \mathrm{~if~} s^\mathrm{adv} > 0 \\ \frac{1}{2}(q_L + q_R) & \mathrm{~if~} s^\mathrm{adv} = 0 \\ q_R & \mathrm{~if~} s^\mathrm{adv} < 0 \\ \end{array} \right . \end{equation} Then the interface states are: \begin{eqnarray} \hat{u}_{i+\myhalf,j} &=& \mathcal{U}[\hat{u}_{i+\myhalf,j}^\mathrm{adv}] (\hat{u}_{i+\myhalf,j,L}^{n+\myhalf}, \hat{u}_{i+\myhalf,j,R}^{n+\myhalf} ) \\ \hat{v}_{i+\myhalf,j} &=& \mathcal{U}[\hat{u}_{i+\myhalf,j}^\mathrm{adv}] (\hat{v}_{i+\myhalf,j,L}^{n+\myhalf}, \hat{v}_{i+\myhalf,j,R}^{n+\myhalf} ) \\ \hat{u}_{i,j+\myhalf} &=& \mathcal{U}[\hat{v}_{i,j+\myhalf}^\mathrm{adv}] (\hat{u}_{i,j+\myhalf,L}^{n+\myhalf}, \hat{u}_{i,j+\myhalf,R}^{n+\myhalf} ) \\ \hat{v}_{i,j+\myhalf} &=& \mathcal{U}[\hat{v}_{i,j+\myhalf}^\mathrm{adv}] (\hat{v}_{i,j+\myhalf,L}^{n+\myhalf}, \hat{v}_{i,j+\myhalf,R}^{n+\myhalf} ) \end{eqnarray} Now we can construct the full left and right predictions for the normal velocities on each interface (Eqs.~\ref{eq:uMAC_l} and \ref{eq:vMAC_l}). This involves simply adding the transverse term to the hat quantities and adding the pressure gradient. \begin{equation} u_{i+\myhalf,j,L}^{n+\myhalf} = \hat{u}_{i+\myhalf,j,L}^{n+\myhalf} - \frac{\Delta t}{2} \left [ \frac{1}{2} \left (\hat{v}^\mathrm{adv}_{i,j-\myhalf} + \hat{v}^\mathrm{adv}_{i,j+\myhalf} \right ) \right ] \left ( \frac{\hat{u}_{i,j+\myhalf}^{n+\myhalf} - \hat{u}_{i,j-\myhalf}^{n+\myhalf}}{\Delta y} \right ) - \frac{\Delta t}{2} (Gp)^{(x),n-\myhalf}_{i,j} \end{equation} and \begin{equation} v_{i,j+\myhalf,L}^{n+\myhalf} = \hat{v}_{i,j+\myhalf,L}^{n+\myhalf} - \frac{\Delta t}{2} \left [ \frac{1}{2} \left (\hat{u}^\mathrm{adv}_{i-\myhalf,j} + \hat{u}^\mathrm{adv}_{i+\myhalf,j} \right ) \right ] \left ( \frac{\hat{v}_{i+\myhalf,j}^{n+\myhalf} - \hat{v}_{i-\myhalf,j}^{n+\myhalf}}{\Delta x} \right ) - \frac{\Delta t}{2} (Gp)^{(y),n-\myhalf}_{i,j} \end{equation} Here $ (Gp)^{(x),n-\myhalf}_{i,j}$ and $ (Gp)^{(y),n-\myhalf}_{i,j}$ are difference-approximations to $\nabla p$ in the $x$ and $y$ directions respectively. Note that they are lagged---these come from the projection at the end of the previous timestep. See BCG for a discussion. A similar construction is done for the right states at the interface. Finally, we do a Riemann solve (again, using the Burger's form of the Riemann problem) followed by upwinding to get the normal advective velocities. This is basically the $\mathcal{R}(.,.)$ operation followed by $\mathcal{U}(.,.)$. Together, it is: \begin{equation} u^\mathrm{adv}_{i+\myhalf,j} = \left \{ \begin{array}{cl} u_{i+\myhalf,j,L}^{n+\myhalf} & \mathrm{~if~} u_{i+\myhalf,j,L}^{n+\myhalf} > 0, \qquad u_{i+\myhalf,j,L}^{n+\myhalf} + u_{i+\myhalf,j,R}^{n+\myhalf} > 0 \\ \frac{1}{2} \left ( u_{i+\myhalf,j,L}^{n+\myhalf} + u_{i+\myhalf,j,R}^{n+\myhalf} \right ) & \mathrm{~if~} u_{i+\myhalf,j,L}^{n+\myhalf} \le 0, \qquad u_{i+\myhalf,j,R}^{n+\myhalf} \ge 0 \\ u_{i+\myhalf,j,R}^{n+\myhalf} & \mathrm{~otherwise} \end{array} \right . \end{equation} and similar for $v^\mathrm{adv}_{i,j+\myhalf}$. These velocities are sometimes referred to as the MAC velocities. \subsection{MAC projection} We could simply use these time-centered advective velocities to construct the fluxes through the interfaces and update to the new time level. However BCH showed that such a method is unstable for CFL $> 0.5$. The fix is to enforce the velocity constraint on these advective velocities. This involves projecting the velocity field onto the space that is divergence free. This projection is usually called the MAC projection. Once the MAC-projected advective velocities are computed, we can reconstruct the interface states using this divergence-free velocity field. Figure~\ref{fig:incomp:MACsolve} shows the location of the various quantities that participate it the MAC projection. \begin{figure}[t] \centering \includegraphics[width=\linewidth]{MAC_solve} \caption[MAC grid data centerings] {\label{fig:incomp:MACsolve} The centerings of the various components that make up the MAC projection.} \end{figure} The divergence of the MAC velocities is cell-centered and constructed as: \begin{equation} (D \Ub)_{i,j} = \frac{u^\mathrm{adv}_{i+\myhalf,j} - u^\mathrm{adv}_{i-\myhalf,j}} {\Delta x} + \frac{v^\mathrm{adv}_{i,j+\myhalf} - v^\mathrm{adv}_{i,j-\myhalf}} {\Delta y} \end{equation} We define a cell-centered $\phi$. $G\phi$ will then be edge-centered on a MAC grid, and $L\phi = DG\phi$ is again cell-centered. Since $L = DG$, this makes the MAC projection an exact projection. We solve \begin{equation} L\phi = D\Ub \end{equation} using multigrid V-cycles and then update the MAC velocities as: \begin{eqnarray} u^\mathrm{adv}_{i+\myhalf,j} &=& u^\mathrm{adv}_{i+\myhalf,j} - \frac{\phi_{i+1,j} - \phi_{i,j}}{\Delta x} \\ % v^\mathrm{adv}_{i,j+\myhalf} &=& v^\mathrm{adv}_{i,j+\myhalf} - \frac{\phi_{i,j+1} - \phi_{i,j}}{\Delta y} \end{eqnarray} \subsection{Reconstruct interface states} Next we redo the construction of the interface states. This procedure is identical to that above---construct the interface states $\hat{u}_{L,R}$, $\hat{v}_{L,R}$ on all edges, upwind based on $\hat{u}^\mathrm{adv}$, $\hat{v}^\mathrm{adv}$, and use these to construct the full states (including transverse terms). Now however, we construct the interface states of $u$ and $v$ on both the $x$ and $y$-interfaces (not just the normal component at each interface). Finally, instead of solving a Riemann problem to resolve the left and right states, we simply upwind using the MAC-projected $u^\mathrm{adv}$ and $v^\mathrm{adv}$. This results in the interface state $u^{n+\myhalf}_{i+\myhalf,j}$, $v^{n+\myhalf}_{i+\myhalf,j}$, $u^{n+\myhalf}_{i,j+\myhalf}$, $v^{n+\myhalf}_{i,j+\myhalf}$. The only reason we need to do this step over, instead of using the interface states that we predicted previously is we want to ensure that they are consistent with the MAC-projected advective velocities (and therefore, consistent with the constraint). \subsection{Provisional update} Once we have the time-centered interface states that are consistent with the MAC-projected advective velocities, we can update the velocities to the new time by discretizing the advective terms ($\Ub\cdot \nabla \Ub$). We express the advective terms for $u$ as $A^{(u),n+\myhalf}_{i,j}$ and those for $v$ as $A^{(v),n+\myhalf}_{i,j}$. These have the form: \begin{eqnarray} A^{(u),n+\myhalf}_{i,j} = && \frac{1}{2} \left ( u^\mathrm{adv}_{i-\myhalf,j} + u^\mathrm{adv}_{i+\myhalf,j} \right ) \frac{u^{n+\myhalf}_{i+\myhalf,j} - u^{n+\myhalf}_{i-\myhalf,j}}{\Delta x} + \nonumber \\ && \frac{1}{2} \left ( v^\mathrm{adv}_{i,j-\myhalf} + v^\mathrm{adv}_{i,j+\myhalf} \right ) \frac{u^{n+\myhalf}_{i,j+\myhalf} - u^{n+\myhalf}_{i,j-\myhalf}}{\Delta y} \\ % A^{(v),n+\myhalf}_{i,j} = && \frac{1}{2} \left ( u^\mathrm{adv}_{i-\myhalf,j} + u^\mathrm{adv}_{i+\myhalf,j} \right ) \frac{v^{n+\myhalf}_{i+\myhalf,j} - v^{n+\myhalf}_{i-\myhalf,j}}{\Delta x} \nonumber +\\ && \frac{1}{2} \left ( v^\mathrm{adv}_{i,j-\myhalf} + v^\mathrm{adv}_{i,j+\myhalf} \right ) \frac{v^{n+\myhalf}_{i,j+\myhalf} - v^{n+\myhalf}_{i,j-\myhalf}}{\Delta y} \end{eqnarray} The normal update for $u$, $v$ would include the $Gp$ term and appear as: \begin{eqnarray} u^{\star}_{i,j} &=& u^n_{i,j} - \Delta t A^{(u),n+\myhalf}_{i,j} - \Delta t (G p)^{(x),n-\myhalf}_{i,j} \\ v^{\star}_{i,j} &=& v^n_{i,j} - \Delta t A^{(v),n+\myhalf}_{i,j} - \Delta t (G p)^{(y),n-\myhalf}_{i,j} \end{eqnarray} Note that at this point, we don't have an updated $p$, so we use a lagged value from the previous step's projection. Alternately, we can note that for an exact projection, $Gp$, is the gradient of a scalar and would be removed by the projection, so we can omit it in this update, giving an alternate provisional update: \begin{eqnarray} u^{\star\star}_{i,j} &=& u^n_{i,j} - \Delta t A^{(u),n+\myhalf}_{i,j} \\ v^{\star\star}_{i,j} &=& v^n_{i,j} - \Delta t A^{(v),n+\myhalf}_{i,j} \end{eqnarray} Following the notation in Martin, we distinguish between these with an `$\star$' vs.\ `$\star\star$' \footnote{Note that these are identical to ABC~\cite{ABC} approximation projections (1) and (2) (a quick look at ABC might suggest the opposite, but note that their definition of $\Ub^\star$ already includes a $-Gp^{n-\myhalf}$ term, so by explicitly adding it back in, you are dealing with the case where $\Ub^\star$ was updated without any $Gp^{n-\myhalf}$ term, like the `$\star\star$' case above.)}. \subsection{Approximate projection} This provisional velocity field does not yet obey the constraint. To enforce the constraint, we need to do a projection. Here is where we have the flexibility on whether to include the $G p^{n-\myhalf}$ term. If we were doing an exact projection, then adding the gradient of a scalar would not change the divergence-free velocity field, so there would be no need to add it. BCH did an exact projection on a cell-centered grid. There, the divergence operator is: \begin{equation} (D \Ub)_{i,j} = \frac{u_{i+1,j} - u_{i-1,j}}{2\Delta x} + \frac{v_{i,j+1} - v_{i,j-1}}{2\Delta y} \label{eq:ccdiv} \end{equation} This gives a cell-centered $D\Ub$. If we want $\phi$ cell-centered, then the gradient, $G\phi$ must also be cell centered so $L\phi = DG\phi$ is cell-centered. This means that we must have \begin{eqnarray} (G\phi)^{(x)}_{i,j} &=& \frac{\phi_{i+1,j} - \phi_{i-1,j}}{2\Delta x} \\ (G\phi)^{(y)}_{i,j} &=& \frac{\phi_{i,j+1} - \phi_{i,j-1}}{2\Delta y} \end{eqnarray} The resulting Laplacian would then be a 5-point stencil that skips over the immediate neighbors: \begin{equation} (L\phi)_{i,j} = \frac{\phi_{i+2,j} -2\phi_{i,j} + \phi_{i-2,j}}{4 \Delta x^2} + \frac{\phi_{i,j+2} -2\phi_{i,j} + \phi_{i,j-2}}{4 \Delta y^2} \end{equation} This decomposes the domain into 4 distinct grids that are only linked together at the boundaries. While an exact projection, this decoupling can be undesirable. Approximate projections relax the idea that $L = DG$. In an exact projection, when you apply the projection operator, $P$, in succession, the result is unchanged ($P^2 = P$). This is not the case for an approximate projection. As a result, exactly what form you project matters. For an approximate projection, we can use the standard 5-point stencil for the Laplacian, \begin{equation} (L\phi)_{i,j} = \frac{\phi_{i+1,j} -2\phi_{i,j} + \phi_{i-1,j}}{\Delta x^2} + \frac{\phi_{i,j+1} -2\phi_{i,j} + \phi_{i,j-1}}{\Delta y^2} \end{equation} together with the cell-centered divergence above (Eq.~\ref{eq:ccdiv}). Rider~\cite{rider} and Almgren, Bell, and Crutchfield (ABC)~\cite{ABC} explore various forms of what to project when doing the approximate projection. For instance, do we include the $Gp^{n-\myhalf}$ term in the provisional velocity or not? Chorin noted that if viscosity is being modeled, then it is necessary to include it here to get second-order accuracy. Also, one can project the update to the velocity, $(\Ub^\star - \Ub^n)/\Delta t$ instead of just the new velocity, since $\Ub^n$ should already be divergence free. Rider argues that projecting the update is not desirable with approximate projections, since any error in $\Ub^n$ being divergence-free is carried forward to the new $\Ub^{n+1}$. One issue is that a cell-centered approximate projection cannot remove all sources of divergence (see Rider and Martin's PhD thesis). When projecting the new velocity, we scale by $\Delta t$ to get a quantity that has dimensions of pressure. The procedure for the projection differs slightly depending on whether we project $\Ub^\star$ or $\Ub^{\star\star}$: \begin{itemize} \item case I: projecting $\Ub^{\star}/\Delta t$. From the expression above, this looks like: \begin{equation} \frac{\Ub^\star}{\Delta t} = \frac{\Ub^n}{\Delta t} - A^{(u),n+\myhalf} - (G p)^{(x),n-\myhalf} \end{equation} Ideally, $\Ub^n$ is already divergence free, and $Gp^{n-\myhalf}$ is the gradient of a scalar, which will be removed, so the projection should pick out the divergence free portion of $A^{(u)}$. We solve: \begin{equation} L\phi = D (\Ub^\star/\Delta t) \end{equation} using multigrid V-cycles. We then find the new, divergence free velocity field as: \begin{equation} \Ub^{n+1} = \Ub^\star - \Delta t G\phi \end{equation} Since we already included $Gp^{n-\myhalf}$ in what we projected, $G\phi$ will be the correction, \begin{equation} G\phi = Gp^{n+\myhalf} - Gp^{n-\myhalf} \end{equation} or \begin{equation} Gp^{n+\myhalf} = Gp^{n-\myhalf} + G\phi \end{equation} (see Martin 2.5 or ABC). We store this for the next timestep. \item case II: projecting $\Ub^{\star\star}/\Delta t$. From the expression above, this looks like: \begin{equation} \frac{\Ub^{\star\star}}{\Delta t} = \frac{\Ub^n}{\Delta t} - A^{(u),n+\myhalf} \end{equation} There is no explicit $Gp^{n-\myhalf}$ term. We solve: \begin{equation} L\phi = D (\Ub^{\star\star}/\Delta t) \end{equation} using multigrid V-cycles. We then find the new, divergence free velocity field as: \begin{equation} \Ub^{n+1} = \Ub^{\star\star} - \Delta t G\phi \end{equation} Since there was no $Gp^{n-\myhalf}$ in what we projected, $p^{n+\myhalf} = \phi$, and \begin{equation} Gp^{n+\myhalf} = G\phi \end{equation} We store this for the next timestep. \end{itemize} One pathology of this form of the projection is that $(D\Ub)_{i,j}$ does not actually make use of the velocity field in zone $(i,j)$. This decoupling from the local zone can result in a checkerboarding pattern in the projected velocity field. \section{Boundary conditions} For the advection portion of the algorithm, the boundary conditions on $u$ and $v$ are implemented in the usual way, using ghost cells. For the projection, For a periodic domain, the boundary conditions on $\phi$ are likewise periodic. At a solid wall or inflow boundary, we already predicted the velocity that we want at the wall (in the advection step), and we do not want this value to change in the projection step. Since the correction is: \begin{equation} \Ub^{n+1} = \Ub^\star - \nabla \phi \end{equation} we want $\nabla \phi \cdot n = 0$. At outflow boundaries, we do not want to introduce any shear as we go through the boundary. This means that we do not want any tangential acceleration. Setting $\phi = 0$ on the boundary enforces $\nabla \phi \cdot t = 0$, where $t$ is the unit vector tangential to the boundary. See ABS for a discussion of the boundary conditions. \section{Bootstrapping} At step 0, we do not have a value of $Gp^{-\myhalf}$. To get an initial value for $Gp$, we run through the entire evolution algorithm starting with the initial data. At the end of a step, we reset $u$ and $v$ to the initial values and store the $Gp$ at the end of this step as $Gp^{-\myhalf}$. It is also common to precede this initialization by first projecting the velocity field to ensure it is divergence free. This way, we do not have to rely on the initial conditions to always set a divergence free velocity field. \section{Test problems} \subsection{Convergence test} Minion~\cite{minion:1996} introduced a simple test problem with an analytic solution. The velocity field is initialized as: \begin{eqnarray} u(x,y) &=& 1 - 2 \cos(2 \pi x) \sin(2 \pi y) \\ v(x,y) &=& 1 + 2 \sin(2 \pi x) \cos(2 \pi y) \end{eqnarray} The exact solution at some time $t$ is: \begin{eqnarray} u(x,y,t) &=& 1 - 2 \cos(2 \pi (x - t)) \sin(2 \pi (y - t)) \\ v(x,y,t) &=& 1 + 2 \sin(2 \pi (x - t)) \cos(2 \pi (y - t)) \end{eqnarray} Minion also gives the pressure, but this is not needed for the solution. This is run on a doubly-periodic unit square domain. The main utility of this set of initial conditions is that we can use the analytic solution to measure the convergence behavior of the algorithm. \section{Extensions} \begin{itemize} \item {\em Variable density incompressible}: Bell \& Marcus \cite{BellMarcus} describe how to extend these methods to variable density flows. This means that the density in the domain may not be constant, but within a fluid element, the density does not change. This can arise, for instance, in modeling the Rayleigh-Taylor instability. The basic idea follows the method described above. Now the mass continuity equation is also evolved: \begin{equation} \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \Ub) = 0 \end{equation} The density is predicted to the interfaces follow the same procedure above and upwinded using the MAC velocities. % For the projection, the decomposition is written as: \begin{equation} \Ub = \Ub^d + \frac{1}{\rho}\nabla \phi \end{equation} and the elliptic equation is now a variable-coefficient equation: \begin{equation} \nabla \cdot \frac{1}{\rho} \nabla \phi = \nabla \cdot \Ub \end{equation} \item {\em Viscosity}: When viscosity is included in the system, our momentum equation becomes: \begin{equation} \Ub_t + \Ub\cdot \nabla \Ub + \nabla p = \epsilon \nabla^2 \Ub \end{equation} The solution process for this equation is largely the same. Following BCH, first the advective term is computed by predicting the velocities to the interfaces, doing the MAC projection, and then forming $A^{n+\myhalf}$. Now there is an explicit viscosity term (at time-level $n$) present in the prediction, as a source term. The provision velocity update is no longer a simple flux update, but instead requires solving two decoupled diffusion-like equations (one for $u$ and one for $v$). These are differenced using Crank-Nicolson centering: \begin{eqnarray} \frac{u^\star - u^n}{\Delta t} &=& - A^{(u),n+\myhalf} - \nabla p^{(x),n-\myhalf} + \frac{\epsilon }{2} \nabla^2 (u^n + u^\star) \\ \frac{v^\star - v^n}{\Delta t} &=& - A^{(v),n+\myhalf} - \nabla p^{(y),n-\myhalf} + \frac{\epsilon }{2} \nabla^2 (v^n + v^\star) \end{eqnarray} This involves two separate multigrid solves. Once $\Ub^\star$ is found, the final projection is done as usual. \item {\em Low Mach number combustion}: In low-Mach number combustion flows, the fluid is allowed to respond to local heat release by expanding. The velocity constraint is derived by differentiating the equation of state along particle paths, leading to the appearance of a source term: \begin{equation} \nabla \cdot \Ub = S \end{equation} Here, $S$, incorporates the compressibility effects due to the heat release and diffusion. This system is used when modeling burning fronts (flames). This type of flow can be thought of as linking two incompressible states (the fuel and the ash) by the expansion across the interface. The solution technique largely follows that of the incompressible flow. One caveat, because the constraint now has a local heat source, $S$, doing the cell-centered divergence described above leads to a decoupling of $D\Ub$ from the local source, since the stencil of $D\Ub$ does not include the zone upon which it is centered. This system is described in detail in \cite{pember-flame, DayBell:2000,SNpaper}. \item {\em Stratified flows}: When the background is stratified, a different velocity constraint can arise, capturing the expansion of the fluid element due to the pressure change with altitude. For example, with an ideal gas, the equation of state can be recast as: \begin{equation} \nabla \cdot (p_0^{1/\gamma}) = 0 \end{equation} where $p_0(z)$ is the pressure as a function of height, representing the hydrostatic background, and $\gamma$ is the ratio of specific heats. For a general equation of state, $p_0^(1/\gamma)$ is replaced with something more complicated (see \cite{ABRZ:I,ABRZ:II,ABNZ:III} for the development of a low Mach hydrodynamics method for stratified astrophysical flows). \item {\em Nodal projection}: instead of discretizing the final projection using a cell-centered $\phi$, ABS use a node-centered $\phi$. While this is still an approximate projection, this discretization couples in the zone we are centered on, and is said to be able to do a better job removing pathological divergent velocity fields that cell-centered projections stumble on. \end{itemize} ================================================ FILE: incompressible/main.tex ================================================ \documentclass[11pt]{article} % margins \usepackage[margin=0.75in]{geometry} % figures \usepackage{graphicx} % font \usepackage{mathpazo} \usepackage{helvet} % coloring \usepackage{color} \definecolor{mygray}{gray}{0.5} % footer \usepackage{fancyhdr} \pagestyle{fancy} \fancyfoot[LO,LE]{\footnotesize\sffamily \color{mygray} M.\ Zingale---Notes on incompressible solvers} \fancyfoot[RO,RE]{\footnotesize \sffamily \color{mygray} (\today)} \fancyfoot[CO,CE]{\thepage} \fancyhead{} \renewcommand{\headrulewidth}{0.0pt} \renewcommand{\footrulewidth}{0.0pt} % captions \usepackage{caption} \renewcommand{\captionfont}{\footnotesize} \renewcommand{\captionlabelfont}{\footnotesize} \setlength{\captionmargin}{5em} \usepackage{sectsty} \allsectionsfont{\sffamily} % shortcuts \newcommand{\Dux}{\overline{\Delta u}^{(x)}} \newcommand{\Duy}{\overline{\Delta u}^{(y)}} \newcommand{\Dvx}{\overline{\Delta v}^{(x)}} \newcommand{\Dvy}{\overline{\Delta v}^{(y)}} \setlength{\marginparwidth}{0.75in} \newcommand{\MarginPar}[1]{\marginpar{\vskip-\baselineskip\raggedright\tiny\sffamily\hrule\smallskip{\color{red}#1}\par\smallskip\hrule}} \begin{document} \begin{center} {\LARGE \textsf{\textbf{ Notes on incompressible solvers}} } \end{center} \begin{quote} \noindent {\em These summarize methods for solving the incompressible hydrodynamics equations using an cell-centered approximate projection method.} \end{quote} \input incompressible \bibliographystyle{plain} \bibliography{../refs} \end{document} ================================================ FILE: instabilities/instabilities.tex ================================================ \section{Dimensionless numbers} Re Ra M Pr Rossby Taylor Atwood Turbulence and instabilities are ubitiquous in astrophysical flows. A proper treatment of them requires resolving the important lengthscales Furthermore, it is three-dimensional \section{Turbulence} Astrophysical flows are characterized by large Reynolds numbers. Turbulent eddies cascade between scales, owing to the nonlinear advective term in the momentum equation: $\mathbf{U} \cdot \nabla \mathbf{U}$. Burgers' turbulence? integral scale and intensity difference between 2- and 3-d subgrid scale modeling / bottleneck ILES / LES / DNS \section{Convection} MLT Ledoux internally driven vs. plates \section{Rayleigh-Taylor} linear growth eate alpha incompressible vs. compressible \section{Kelvin-Helmholtz} Galilean invariance \section{Reacting flow} LD fire-polishing length ================================================ FILE: intro/intro.tex ================================================ \section{What is simulation?} Astronomy is an observational science. Unlike in terrestrial physics, we do not have the luxury of being able to build a model system and do physical experimentation on it to understand the core physics. We have to take what nature gives us. Simulation enables us to build a model of a system and allows us to do virtual experiments to understand how this system reacts to a range of conditions and assumptions. It's tempting to think that one can download a simulation code, set a few parameters, maybe edit some initial conditions, run, and then have a virtual realization of some astrophysical system that you are interested in. Just like that. In practice, it is not this simple. All simulation codes make approximations---these start even before one turns to the computer, simply by making a choice of what equations are to be solved. The main approximation that we will follow here, is the {\em fluid approximation} (see Figure~\ref{fig:intro:fluid_scale}). We don't want to focus on the motions of the individual atoms, nuclei, electrons, and photons in our system, so we work on a scale that is much larger than the mean free path of the system. This allows us to describe the bulk properties of a fluid element, which in turn is small compared to the system of interest. Within the fluid approximation, additional approximations are made, both in terms of the physics included and how we represent a continuous fluid in the finite-memory of a computer (the {\em discretization} process). Typically, we have a system of PDEs, and we need to convert the continuous functional form of our system into a discrete form that can be represented in the finite memory of a computer. This introduces yet more approximation. %V\&V Blindly trusting the numbers that come out of the code is a recipe for disaster. You don't stop being a physicist the moment you execute the code---your job as a computational scientist is to make sure that the code is producing reasonable results, by testing it against known problems and your physical intuition. Simulations should be used to gain insight and provide a physical understanding. Because the systems we solve are so nonlinear, small changes in the code or the programming environment (compilers, optimization, etc.) can produce large differences in the numbers coming out of the code. That's not a reason to panic. As such it is best not to obsess about precise numbers, but rather the trends our simulations reveal. To really understand the limits of your simulations, you should do parameter and convergence studies. There is no ``\"uber-code''. Every algorithm begins with approximations and has limitations. Comparisons between different codes are important and common in our field (see, for example, \cite{frenk:1999,dimonte:2004,devalborro:2006}), and build confidence in the results that we are on the right track. % \begin{figure}[t] \centering \includegraphics[width=0.9\linewidth]{fluid_scale} \caption[The fluid scale.]{\label{fig:intro:fluid_scale} The fluid scale sits in an intermediate range---much smaller than the system of interest (a star in this case), but much larger than the mean free path.} \end{figure} % To really understand your simulations, you need to know what the code your are using is doing under the hood. This means understanding the core methods used in our field. These notes are designed to provide a basic tour of some of the more popular methods, referring to the key papers for full derivations and details. The best way to learn is to code up these methods for yourself. A companion python code, {\sf pyro} is available to help, and most of the exercises (or corresponding figures) have links to simple codes that are part of the \hydroex\ repository\footnote{look for the \hydroexsymb\ symbol.}. Descriptions and links to these codes are found in the appendices. \section{Numerical basics} We assume a familiarity with basic numerical methods, which we summarize below. Any book on numerical methods can provide a deeper discussion of these methods. Some good choices are the texts by Garcia~\cite{garcia}, Newman~\cite{newman}, and Pang~\cite{pang}. \subsection{Sources of error} With any algorithm, there are two sources of error we are concerned with: {\em roundoff error} and {\em truncation error}. Roundoff arises from the error inherent in representing a floating point number with a finite number of bits in the computer memory. An excellent introduction to the details of how computers represent numbers is provided in \cite{goldberg:1991}. \begin{exercise}[Floating point] In your choice of programming language, create a floating point variable and initialize it to 0.1. Now, print it out in full precision (you may need to use format specifiers in your language to get all significant digits the computer tracks). You should see that it is not exactly 0.1 to the computer---this is the floating point error. The number 0.1 is not exactly representable in the binary format used for floating point. \end{exercise} \begin{exercise}[Machine epsilon] To see roundoff error in action, write a program to find the value of $\epsilon$ for which $1 + \epsilon = 1$. Start with $\epsilon = 1$ and iterate, halving $\epsilon$ each iteration until $1 + \epsilon = 1$. This last value of $\epsilon$ for which this was not true is the {\em machine epsilon}. You will get a different value for single- vs.\ double-precision floating point arithmetic. \end{exercise} Some reorganization of algorithms can help minimize roundoff, e.g.\ avoiding the subtraction of two very large numbers by factoring as: \begin{equation} x^3 - y^3 = (x - y)(x^2 + xy + y^2) \enskip , \end{equation} but roundoff error will always be present at some level. Truncation error is a feature of an algorithm---we typically approximate an operator or function by expanding about some small quantity. When we throw away higher-order terms, we are truncating our expression, and introducing an error in the representation. If the quantity we expand about truly is small, then the error is small. A simple example is to consider the Taylor series representation of $\sin(x)$: \begin{equation} \sin(x) = \sum_{n=1}^\infty (-1)^{n-1} \frac{x^{2n-1}}{(2n-1)!} \end{equation} For $|x| \ll 1$, we can approximate this as: \begin{equation} \sin(x) \approx x - \frac{x^3}{6} \end{equation} in this case, our truncation error has the leading term $\propto x^5$, and we say that our approximation is $\mathcal{O}(x^5)$, or $5^\mathrm{th}$-order accurate. \begin{exercise}[Convergence and order-of-accuracy] We will be concerned with the order-of-accuracy of our methods, and a good way to test whether our method is behaving properly is to perform a convergence test. Consider our $5^\mathrm{th}$-order accurate approximation to $\sin(x)$ above. Pick a range of $x$'s ($< 1$), and compute the error in our approximation as \begin{equation*} \epsilon \equiv \sin(x) - [ x - x^3/6 ] \end{equation*} and show that as you cut $x$ in half, $|\epsilon|$ reduces by $2^5$---demonstrating $5^\mathrm{th}$-order accuracy. \end{exercise} This demonstration of measuring the error as we vary the size of our small parameter is an example of a {\em convergence test}. \subsection{Differentiation and integration} For both differentiation and integration, there are two cases we might encounter: \begin{enumerate} \item We have function values, $f_0$, $f_1$, $\ldots$, at a discrete number of points, $x_0$, $x_1$, $\ldots$, and we want to compute the derivative at a point or integration over a range of points. \item We know a function analytically and we want to construct a derivative or integral of this function numerically. \end{enumerate} In these notes, we will mainly be concerned with the first case. \subsubsection{Differentiation of discretely-sampled function} \label{ch:intro:diff} Consider a collection of equally spaced points, labeled with an index $i$, with the physical spacing between them denoted $\Delta x$. \MarginPar{show a figure} We can express the first derivative of a quantity $a$ at $i$ as: \begin{equation} \left . \frac{\partial a}{\partial x} \right |_i \approx \frac{a_i - a_{i-1}}{\Delta x} \end{equation} or \begin{equation} \left . \frac{\partial a}{\partial x} \right |_i \approx \frac{a_{i+1} - a_i}{\Delta x} \end{equation} % (Indeed, as $\Delta x \rightarrow 0$, this is the definition of a derivative we learned in calculus.) Both of these are {\em one-sided differences}. By Taylor expanding the data about $x_i$, we see \begin{equation} a_{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 \end{equation} Solving for ${\partial a}/{\partial x} |_i$, we see \begin{align} \left . \frac{\partial a}{\partial x} \right |_i &= \frac{a_i - a_{i-1}}{\Delta x} - \frac{1}{2}\Delta x \left . \frac{\partial^2 a}{\partial x^2} \right |_i + \ldots \\ % &= \frac{a_i - a_{i-1}}{\Delta x} + \mathcal{O}(\Delta x) \end{align} where $\mathcal{O}(\Delta x)$ indicates that the leading term in the error for this approximation is $\sim \Delta x$\footnote{in some texts, you see this $\mathcal{O}(\Delta x^n)$ referred to as ``big O notation''}. We say that this is {\em first order accurate}. This means that we are neglecting terms that scale as $\Delta x$ or to higher powers. This is fine if $\Delta x$ is small. This error is our truncation error---just as discussed above, it arises because our numerical approximation throws away higher order terms. The approximation ${\partial a}/{\partial x} |_i = ({a_{i+1} - a_i})/{\Delta x}$ has the same order of accuracy. \begin{exercise}[Truncation error] {Show that a centered difference, \begin{equation*} \left .\frac{\partial a}{\partial x} \right |_i = \frac{a_{i+1} - a_{i-1}}{2 \Delta x} \end{equation*} is second order accurate, i.e.\ its truncation error is $\mathcal{O}(\Delta x^2)$.} \end{exercise} % this figure was created by hydro_examples/basics/derivatives/derivatives.py \begin{figure}[t] \centering \includegraphics[width=0.8\linewidth]{derivs} \caption[Difference approximations to the derivative of $\sin(x)$] {\label{fig:derivs} A comparison of one-sided and centered difference approximations to the derivative of $\sin(x)$. \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/basic_numerics/derivatives/derivatives.py}{derivatives.py}}} \end{figure} Figure~\ref{fig:derivs} shows the left- and right-sided first-order differences and the central difference as approximations to $\sin(x)$. Generally speaking, higher-order methods have lower numerical error associated with them, and also involve a wider range of data points. Second- and higher-order derivatives can be constructed in the same fashion. \begin{exercise}[Second derivative] {Using the Taylor expansions for $a_{i+1}$ and $a_{i-1}$, find a difference approximation to the second derivative at $i$.} \end{exercise} \subsubsection{Differentiation of an analytic function} An alternate scenario is when you know the analytic form of the function, $f(x)$, and are free to choose the points where you evaluate it. Here you can pick a $\delta x$ and evaluate the derivative as \begin{equation} \left . \frac{df}{dx} \right |_{x=x_0} = \frac{f(x_0+\delta x) - f(x_0)}{\delta x} \end{equation} An optimal value for $\delta x$ requires a balance of truncation error (which wants a small $\delta x$) and roundoff error (which becomes large when $\delta x$ is close to machine $\epsilon$). Figure~\ref{fig:deriv_error} shows the error for the numerical derivative of $f(x) = \sin(x)$ at the point $x_0 = 1$, as a function of $\delta x$. A nice discussion of this is given in \cite{yak}. A good rule-of-thumb is to pick $\delta x \approx \sqrt{\epsilon}$, where $\epsilon$ is machine epsilon, to balance roundoff and truncation error. Comparing the result with different choices of $\delta x$ allows for error estimation and an improvement of the result by combining the estimates using $\delta x$ and $\delta x/2$ (this is the basis for a method called {\em Richardson extrapolation}). \begin{figure} \centering \includegraphics[width=0.8\linewidth]{deriv_error} \caption[Error in numerical derivatives] {\label{fig:deriv_error} Error in the numerical approximation of the derivative of $f(x) = \sin(x)$ at $x_0 = 1$ as a function of the spacing $\delta x$. For small $\delta x$, roundoff error dominates the error in the approximation. For large $\delta x$, truncation error dominates. \\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/basic_numerics/derivatives/deriv_error.py}{deriv\_error.py}}} \end{figure} \subsubsection{Integration} In numerical analysis, any integration method that is composed as a weighted sum of the function evaluated at discrete points is called a {\em quadrature rule}. If we have a function sampled at a number of equally-spaced points, $x_0 \equiv a, x_1, \ldots, x_N \equiv b$\footnote{Note that this is $N$ intervals and $N+1$ points}, we can construct a discrete approximation to an integral as: \begin{equation} I \equiv \int_a^b f(x) dx \approx \Delta x \sum_{i = 0}^{N-1} f(x_i) \end{equation} where $\Delta x \equiv (b-a)/N$ is the width of the intervals (see the top-left panel in Figure~\ref{fig:integration}). This is a very crude method, but in the limit that $\Delta x \rightarrow 0$ (or $N \rightarrow \infty$), this will converge to the true integral. This method is called the {\em rectangle rule}. Note that here we expressing the integral over the $N$ intervals using a simple quadrature rule in each interval. Summing together the results of the integral over each interval to get the result in our domain is called {\em compound} integration. We can get a more accurate answer for $I$ by interpolating between the points. The simplest case is to connect the sampled function values, $f(x_0), f(x_1), \ldots, f(x_N)$ with a line, creating a trapezoid in each interval, and then simply add up the area of all of the trapezoids: \begin{equation} I \equiv \int_a^b f(x) dx \approx \Delta x \sum_{i = 0}^{N-1} \frac{f(x_i) + f(x_{i+1})}{2} \end{equation} This is called the {\em trapezoid rule} (see the top-right panel in Figure~\ref{fig:integration}). Note here we assume that the points are equally spaced. One can keep going, but practically speaking, a quadratic interpolation is as high as one usually encounters. Fitting a quadratic polynomial requires three points. \begin{exercise}[Simpson's rule for integration] \label{ex:simpsons} Consider a function, $f(x)$, sampled at three equally-spaced points, $\alpha, \beta, \gamma$, with corresponding function values $f_\alpha, f_\beta, f_\gamma$. Derive the expression for Simpson's rule by fitting a quadratic $\hat{f}(x) = A(x - \alpha)^2 + B(x - \alpha) + C$ to the three points (this gives you $A$, $B$, and $C$), and then analytically integrating $\hat{f}(x)$ in the interval $[\alpha,\gamma]$. You should find \begin{equation} I = \frac{\gamma-\alpha}{6} (f_\alpha + 4f_\beta + f_\gamma) \end{equation} Note that $(\gamma-\alpha)/6 = \Delta x/3$ \end{exercise} For a number of samples, $N$, in $[a,b]$, we will consider every two intervals together. The resulting expression is: \begin{align} I &\equiv \int_a^b f(x) dx \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 ] \\ &= 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) \end{align} This method is called {\em Simpson's rule}. Note that for 2 intervals / 3 sample points ($N=2$), we only have 1 term in the sum, $(N-2)/2 = 0$, and we get the result derived in Exercise~\ref{ex:simpsons}. Figure~\ref{fig:integration} shows these different approximations for the case of two intervals (three points). % these figures were created with figures/intro/integrals.py \begin{figure} \centering \includegraphics[width=0.49\linewidth]{rectangle} \includegraphics[width=0.49\linewidth]{trapezoid} \\ \includegraphics[width=0.49\linewidth]{simpsons} \begin{minipage}[b]{0.49\linewidth} \caption[Integration rules]{\label{fig:integration} The rectangle rule (top left), trapezoid rule (top right) and Simpson's rule (left) for integration.} \end{minipage} \end{figure} Analogous expressions exist for the case of unequally-spaced points. The compound trapezoid rule converges as second-order over the interval $[a,b]$, while Simpson's rule converges as fourth-order. As with differentiation, if you are free to pick the points where you evaluate $f(x)$, you can get a much higher-order accurate result. {\em Gaussian quadrature} is a very powerful technique that uses the zeros of different polynomials as the evaluation points for the function to give extremely accurate results. See the text by Garcia~\cite{garcia} for a nice introduction to these methods. \subsection{Root finding} Often we want to find the root of a function (or the vector that zeros a vector of functions). The most popular method for root finding is the {\em Newton-Raphson method}. We want to find $x$, such that $f(x) = 0$. Start with an initial guess $x_0$ that you believe is close to the root, then you can improve the guess to the root by an amount $\delta x$ by looking at the Taylor expansion about $x_0$: \begin{equation} f(x_0 + \delta x) \sim f(x_0) + f^\prime(x_0) \delta x + \ldots = 0 \end{equation} Keeping only to $\mathcal{O}(\delta x)$, we can solve for the correction, $\delta x$: \begin{equation} \label{eq:intro:newtonsmethod} \delta x = -\frac{f(x_0)}{f^\prime(x_0)} \end{equation} This can be used to correct out guess as $x_0 \leftarrow x_0 + \delta x$, and we can iterate on this procedure until $\delta x$ falls below some tolerance. Figure~\ref{fig:newtonsmethod} illustrates this iteration. \begin{figure} \centering \includegraphics[width=0.49\linewidth]{newton_00} \includegraphics[width=0.49\linewidth]{newton_01} \\[1em] \includegraphics[width=0.49\linewidth]{newton_02} \includegraphics[width=0.49\linewidth]{newton_03} \caption[Convergence of Newton's method for root finding]{\label{fig:newtonsmethod} The convergence of Newton's method for finding the root. In each pane, the red point is the current guess for the root. The solid gray line is the extrapolation of the slope at the guess to the $x$-axis, which defines the next approximation to the root. The vertical dotted line to the function shows the new slope that will be used for extrapolation in the next iteration. \\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/basic_numerics/roots/roots_plot.py}{roots\_plot.py}}} \end{figure} The main ``got-ya'' with this technique is that you need a good initial guess. In the Taylor expansion, we threw away the $\delta x^2$ term, but if our guess was far from the root, this (and higher-orders) term may not be small. Obviously, the derivative must be non-zero in the region around the root that you search as well. \begin{exercise}[Newton's method] {Code up Newton's method for finding the root of a function $f(x)$ and test it on several different test functions.} \end{exercise} The {\em secant method} for root finding is essentially Newton-Raphson, but instead of using an analytic derivative, $f^\prime$ is estimated as a numerical difference. Newton's method has pathologies---it is possible to get into a cycle where you don't converge but simply pass through the same set of root approximations. Many other root finding methods exist, including {\em bisection}, which iteratively halves an interval known to contain a root by looking for the change in sign of the function $f(x)$. {\em Brent's method} combines several different methods to produce a robust procedure for root-finding. Most numerical analysis texts will give a description of these. \subsection{Norms} \label{intro:sec:norm} Often we will need to measure the ``size'' of an error to a discrete approximation. For example, imagine that we know the exact function, $f(x)$ and we have have an approximation, $f_i$ defined at $N$ points, $i = 0, \ldots, N-1$. The error at each point $i$ is $\epsilon_i = |f_i - f(x_i)|$. But this is $N$ separate errors---we want a single number that represents the error of our approximation. This is the job of a vector norm. There are many different norms we can define. For a vector ${\bf q}$, we write the norm as $\|q\|$. Often, we'll put a subscript after the norm brackets to indicate which norm was used. Some popular norms are: \begin{itemize} \item {\em inf norm}: \begin{equation} \| q \|_\infty = \max_i |q_i| \end{equation} \item {\em L1 norm}: \begin{equation} \| q \|_1 = \frac{1}{N} \sum_{i=0}^{N-1} |q_i| \end{equation} \item {\em L2 norm}: \begin{equation} \| q \|_2 = \left [ \frac{1}{N} \sum_{i=0}^{N-1} |q_i|^2 \right ]^{1/2} \end{equation} \item {\em general p-norm} \begin{equation} \| q \|_p = \left [ \frac{1}{N} \sum_{i=0}^{N-1} |q_i|^p \right ]^{1/p} \end{equation} \end{itemize} Note that these norms are defined such that they are normalized---if you double the number of elements ($N$), the normalization gives you a number that can still be meaningfully compared to the smaller set. For this reason, we will use these norms when we look at the convergence with resolution of our numerical methods. We'll look into how the choice of norm influences you convergence criterion, but inspection shows that the inf-norm is local---a element in your vector is given the entire weight, whereas the other norms are more global. \subsection{ODEs} Consider a system of first-order ordinary differential equations, \begin{equation} \dot{\bf y} = {\bf f}(t, {\bf y}(t)) \end{equation} If $k$ represents an index into the vector ${\bf y}$, then the $k^\mathrm{th}$ ODE is \begin{equation} \dot{y}_k = f_k(t, {\bf y}(t)) \end{equation} We want to solve for the vector ${\bf y}$ as a function of time. Note higher-order ODEs can always be written as a system of first-order ODEs by introducing new variables\footnote{As an example, consider Newton's second law, $\ddot{x} = F/m$. We can write this as a system of two ODEs by introducing the velocity, $v$, giving us: $\dot{v} = F/m$, $\dot{x} = v$.}. Broadly speaking, methods for integrating ODEs can be broken down into explicit and implicit methods. Explicit methods difference the system to form an update that uses only the current (and perhaps previous) values of the dependent variables in evaluating $f$. For example, a first-order explicit update ({\em Euler's method}) appears as: \begin{equation} y^{n+1}_k = y^n_k + \Delta t f_k(t^n, {\bf y}^n) \end{equation} where $\Delta t$ is the stepsize. Implicit methods instead evaluate the righthand side, ${\bf f}$ using the new-time value of ${\bf y}$ we are solving for. Perhaps the most popular explicit method for ODEs is the 4th-order Runge-Kutta method (RK4). This is a multistage method where several extrapolations to the midpoint and endpoint of the interval are made to estimate slopes and then a weighted average of these slopes are used to advance the solution. The various slopes are illustrated in Figure~\ref{fig:rk} and the overall procedure looks like: \begin{equation} y_k^{n+1} = y_k^n + \frac{\Delta t}{6} (k_1 + 2 k_2 + 2 k_3 + k_4) \end{equation} where the slopes are: \begin{align} k_1 &= f(t^n, y_k^n) \\ k_2 &= f(t^n + \tfrac{\Delta t}{2}, y_k^n + \tfrac{\Delta t}{2} k_1) \\ k_3 &= f(t^n + \tfrac{\Delta t}{2}, y_k^n + \tfrac{\Delta t}{2} k_2) \\ k_4 &= f(t^n + \Delta t, y_k^n + \Delta t k_3) \end{align} Note the similarity to Simpson's method for integration. This is fourth-order accurate overall. \afterpage{\clearpage} % rk4_plot.py creates rk4_k[1-4].pdf, rk4_final.pdf \begin{figure}[p] \centering \includegraphics[width=0.48\linewidth]{rk4_k1} \includegraphics[width=0.48\linewidth]{rk4_k2} \\ \includegraphics[width=0.48\linewidth]{rk4_k3} \includegraphics[width=0.48\linewidth]{rk4_k4} \\ \includegraphics[width=0.6\linewidth]{rk4_final} % \caption[The $4^\mathrm{th}$-order Runge-Kutta method] {\label{fig:rk} A graphical illustration of the four steps in the $4^\mathrm{th}$-order Runge-Kutta method. This example is integrating $dy/dt = -y$.} \end{figure} \begin{exercise}[ODE accuracy] Consider the orbit of Earth around the Sun. If we work in the units of astronomical units, years, and solar masses, then Newton's gravitational constant and the solar mass together are simply $G M = 4\pi^2$ (this should look familiar as Kepler's third law). We can write the ODE system describing the motion of Earth as: \begin{align} \dot{\bf x} &= {\bf v} \\ \dot{\bf v} &= -\frac{G M {\bf r}}{r^3} \end{align} If we take the coordinate system such that the Sun is at the origin, then, ${\bf x} = (x, y)^\intercal$ is the position of the Earth and ${\bf r} = x {\bf \hat{x}} + y {\bf \hat{y}}$ is the radius vector pointing from the Sun to the Earth. Take as initial conditions the planet at perihelion: \begin{align*} x_0 &= 0 \\ y_0 &= a (1-e) \\ ({\bf v}\cdot {\bf \hat{x}})_0 &= -\sqrt{\frac{GM}{a} \frac{1+e}{1-e}} \\ ({\bf v}\cdot {\bf \hat{y}})_0 &= 0 \end{align*} where $a$ is the semi-major axis and $e$ is the eccentricity of the orbit (these expressions can be found in any introductory astronomy text that discusses Kepler's laws). Integrate this system for a single orbital period with the first-order Euler and the RK4 method and measure the convergence by integrating at a number of different $\Delta t$'s. Note: you'll need to define some measure of error, you can consider a number of different metrics, e.g., the change in radius after a single orbit. \end{exercise} \MarginPar{show figure} The choice of the stepsize, $\Delta t$, in the method greatly affects the accuracy. In practice you want to balance the desire for accuracy with the expense of taking lots of small steps. A powerful technique for doing this is to use error estimation and an adaptive stepsize with your ODE integrator. This monitors the size of the truncation error and adjusts the stepsize, as needed, to achieve a desired accuracy. A nice introduction to how this works for RK4 is given in \cite{garcia}. \subsubsection{Implicit methods} Implicit methods difference the system in a way that includes the new value of the dependent variables in the evaluation of $f_k(t,{\bf y}(t))$---the resulting implicit system is usually solved using, for example, Newton-Raphson iteration. A first-order implicit update (called {\em backward Euler}) is: \begin{equation} \label{eq:backwardeuler} {\bf y}^{n+1} = {\bf y}^n + \Delta t {\bf f}(t^{n+1}, {\bf y}^{n+1}) \end{equation} This is more complicated to solve than the explicit methods above, and generally will require some linear algebra. If we take $\Delta t$ to be small, then the change in the solution, $\Delta {\bf y}$ will be small as well, and we can Taylor-expand the system. To solve this, we pick a guess, ${\bf y}^{n+1}_0$, that we think is close, to the solution and we will solve for a correction, $\Delta {\bf y}_0$ such that \begin{equation} \label{eq:impl_correction} {\bf y}^{n+1} = {\bf y}^{n+1}_0 + \Delta {\bf y}_0 \end{equation} Using this approximation, we can expand the righthand side vector, \begin{equation} \label{eq:impl_rhs_expansion} {\bf f}(t^{n+1},{\bf y}^{n+1}) = {\bf f}(t^{n+1}, {\bf y}^{n+1}_0) + \left . \frac{\partial {\bf f}}{\partial {\bf y}} \right |_0 \Delta {\bf y}_0 + \ldots \end{equation} Here we recognize the Jacobian matrix, ${\bf J} \equiv \partial {\bf f}/\partial {\bf y}$, \begin{equation} \renewcommand\arraystretch{1.5} {\bf J} = \left ( \begin{array}{ccccc} \partial f_1/\partial y_1 & \partial f_1/\partial y_2 & \partial f_1/\partial y_3 & \ldots & \partial f_1/\partial y_n \\ % \partial f_2/\partial y_1 & \partial f_2/\partial y_2 & \partial f_2/\partial y_3 & \ldots & \partial f_2/\partial y_n \\ % \vdots & \vdots & \vdots & \ddots & \vdots \\ % \partial f_n/\partial y_1 & \partial f_n/\partial y_2 & \partial f_n/\partial y_3 & \ldots & \partial f_n/\partial y_n \\ \end{array} \right ) \end{equation} % Putting Eqs.~\ref{eq:impl_correction} and \ref{eq:impl_rhs_expansion} into Eq.~\ref{eq:backwardeuler}, we have: \begin{equation} {\bf y}^{n+1}_0 + \Delta {\bf y}_0 = {\bf y}^n + \Delta t \left [ {\bf f}(t^{n+1}, {\bf y}^{n+1}_0) + \left .{\bf J}\right |_0\, \Delta {\bf y}_0 \right ] \end{equation} Writing this as a system for the unknown correction, $\Delta {\bf y}_0$, we have \begin{equation} \left ({\bf I} - \Delta t \left . {\bf J} \right |_0 \right ) \Delta {\bf y}_0 = {\bf y}^n - {\bf y}_0^{n+1} + \Delta t {\bf f}(t^{n+1}, {\bf y}^{n+1}_0) \end{equation} This is a linear system (a matrix $\times$ vector $=$ vector) that can be solved using standard matrix techniques (numerical methods for linear algebra can be found in most numerical analysis texts). After solving, we can correct our initial guess: \begin{equation} {\bf y}^{n+1}_1 = {\bf y}^{n+1}_0 + \Delta {\bf y}_0 \end{equation} Written this way, we see that we can iterate. To kick things off, we need a suitable guess---an obvious choice is ${\bf y}^{n+1}_0 = {\bf y}^n$. Then we correct this guess by iterating, with the $k$-th iteration looking like: \begin{align} \left ({\bf I} - \Delta t \left . {\bf J} \right |_{k-1} \right ) \Delta {\bf y}_{k-1} &= {\bf y}^n - {\bf y}_{k-1}^{n+1} + \Delta t {\bf f}(t^{n+1}, {\bf y}^{n+1}_{k-1}) \\ {\bf y}^{n+1}_k &= {\bf y}^{n+1}_{k-1} + \Delta {\bf y}_{k-1} \end{align} We will iterate until we find $\| \Delta {\bf y}_k\| < \epsilon \| {\bf y}^n \|$. Here $\epsilon$ is a small tolerance, and we use ${\bf y}^n$ to produce a reference scale for meaningful comparison. Note that here we use a vector norm to give a single number for comparison. Note that the role of the Jacobian here is the same as the first derivative in the scalar Newton's method for root finding (Eq.~\ref{eq:intro:newtonsmethod})---it points from the current guess to the solution. Sometimes an approximation to the Jacobian, which is cheaper to evaluate, may work well enough for the method to converge. Explicit methods are easier to program and run faster (for a given $ \Delta t$), but implicit methods work better for {\em stiff} problems---those characterized by widely disparate timescales over which the solution changes~\cite{byrnehindmarsh}\footnote{Defining whether a problem is stiff can be tricky (see \cite{byrnehindmarsh} for some definitions). For a system of ODEs, a large range in the eigenvalues of the Jacobian usually means it is stiff.}. A good example of this issue in astrophysical problems is with nuclear reaction networks (see, e.g., \cite{timmes_networks}). As with the explicit case, higher-order methods exist that can provide better accuracy at reduced cost. \subsection{FFTs} \label{sec:intro:ffts} The discrete Fourier transform converts a discretely-sampled function from real space to frequency space, and identifies the amount of power associate with discrete wavenumbers. This is useful for both analysis, as well as for solving certain linear problems (see, e.g., \S~\ref{elliptic:sec:fft}). For a function, $f(x)$ sampled at $N$ equally-spaced points (such that $f_n = f(x_n)$), the discrete Fourier transform, $\mathcal{F}_k$ is written as: \begin{equation} \mathcal{F}_k = \sum_{n=0}^{N-1} f_n e^{-2\pi i n k /N} \qquad k \in [0, N-1] \end{equation} The exponential in the sum brings in a real (cosine terms, symmetric functions) and imaginary (sine terms, antisymmetric functions) part. \begin{align} \mathrm{Re}(\mathcal{F}_k) &= \sum_{n=0}^{N-1} f_n \cos\left(\frac{2\pi n k}{N}\right) \\ \mathrm{Im}(\mathcal{F}_k) &= \sum_{n=0}^{N-1} f_n \sin\left(\frac{2\pi n k}{N}\right) \end{align} Alternately, it is sometimes useful to combine the real and imaginary parts into an amplitude and phase. The inverse transform is: \begin{equation} f_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] \end{equation} The $1/N$ normalization is a consequence of Parseval's theorem---the total power in real space must equal the total power in frequency space. One way to see that this must be the case is to consider the discrete transform of $f(x) = 1$, which should be a delta function. The FFT of a discrete function has the same amount of information as the original discrete function. Note that if $f(x)$ is real-valued, then the transform $\mathcal{F}_k$ has 2 numbers (the real and imaginary parts) for each of our original $N$ real values. This would mean that we have $2N$ pieces of information in frequency space where we only had $N$ pieces of information in real space. Since we cannot create information in frequency space where there was no corresponding real-space information, half of the $\mathcal{F}_k$'s are redundant (and $\mathcal{F}_{-k} = \mathcal{F}^\star_k$). Directly computing $\mathcal{F}_k$ for each $k$ takes $\mathcal{O}(N^2)$ operations. The fast Fourier transform (FFT) is a reordering of the sums in the discrete Fourier transform to reuse partial sums and compute the same result in $\mathcal{O}(N\log N)$ work. Many numerical methods books can give a good introduction to how to design an FFT algorithm. \begin{exercise}[FFTs] {Learn how to use a FFT library or the built-in FFT method in your programming language of choice. There are various ways to define the normalization in the FFT, that can vary from one library to the next. To ensure that you are doing things correctly, compute the following transforms: \begin{itemize} \item $\sin(2\pi k_0 x)$ with $k_0 = 0.2$. The transform should have all of the power in the imaginary component only at the frequency $0.2$. \item $\cos(2\pi k_0 x)$ with $k_0 = 0.2$. The transform should have all of the power in the real component only at the frequency $0.2$. \item $\sin(2\pi k_0 x + \pi/4)$ with $k_0 = 0.2$. The transform should have equal power in the real and imaginary components, only at the frequency $0.2$. Since the power is 1, the amplitude of the real and imaginary parts will be $1/\sqrt{2}$. \end{itemize} An example of the transform of a single-frequency sine wave with a phase shift is shown in Figure~\ref{fig:intro:fft}. } \end{exercise} The FFT assumes that a function is periodic and that the points are evenly spaced. If the function is not periodic, then a signal will show up in the FFT at a wavenumber corresponding to the size of the domain. Related to this, you need to ensure that your points are evenly spaced even at the boundary, e.g., if you have a function on $[0, 1]$ represented by 5 points, you want the points to be $\{0, 0.2, 0.4, 0.6, 0.8\}$ and not $\{0, 0.25, 0.5, 0.75, 1\}$. In the latter case, $0$ and $1$ are the same function value (due to periodicity), but the FFT assumes that all points are evenly spaced (even across the boundary), so it will think that there is a spacing of $0.25$ between these end points. \begin{figure}[t] \centering \includegraphics[width=\linewidth]{fft-sine-phase} \caption[Fourier transform of $f(x) = \sin(2\pi k_0 x + \pi/4)$]{\label{fig:intro:fft} The Fourier transform of a sine wave with a phase of $\pi/4$, $f(x) = \sin(2\pi k_0 x + \pi/4)$ with $k_0 = 0.2$. The top shows the original function. The second panel shows the real and imaginary components---we see all of the power is at our input wavenumber, split equally between the real and imaginary parts. The third pane shows the power (the absolute value of the transform). Finally, the bottom panel shows the inverse transform of our transform, giving us back our original function. \\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/basic_numerics/FFT/fft_simple_examples.py}{fft\_simple\_examples.py}}} \end{figure} %% \section{A Sample of Public Codes} %% There are a large number of freely-available codes for modeling astrophysical %% flows. ================================================ FILE: low_mach/low_mach.tex ================================================ \newcommand{\teta}{\tilde{\eta}} Incompressible flow represents the zero-Mach number limit of fluid flow---no compressibility effects are modeled. We can extend the ideas of incompressibe flow to allow us to model some compressibility effects, giving rise to low Mach number methods. \section{Low Mach divergence constraints} \label{ch:lm:constraints} The key idea in solvers for low Mach number flows is that, as a fluid element advects, its pressure remains the same as the background state. For an atmosphere, this background state is a hydrostatic profile. For a smallscale combustion flow, this background state is a spatially constant pressure (and constant-in-time if the domain is open). We'll denote this background pressure as $p_0(r,t)$, where $r$ is a radial coordinate. Previously, we derived the pressure evolution with reactive sources (see Eq.~\ref{eq:reacting:p}) by following the evolution of pressure as a fluid element moves (e.g., the Lagrangian derivative). For the low Mach flows, we want to see the evolution $p_0$ in the fluid element, which is expressed as: \begin{equation} \ddt{p_0} + \Ub \cdot \nabla p_0 + \Gamma_1 p_0 \nabla \cdot \Ub = \Gamma_1 p_0 \sigma \Hnuc \end{equation} We can rewrite this as a constraint on the velocity field by solving for $\nabla \cdot \Ub$: \begin{equation} \nabla \cdot \Ub + \frac{1}{\Gamma_1 p_0}\frac{Dp_0}{Dt} = \sigma \Hnuc \equiv S \end{equation} with \begin{equation} \sigma = \frac{\partial p_0/\partial T |_\rho}{\rho c_p \partial p_0/\partial \rho |_T} \end{equation} This is the general constraint equation for low Mach flow. Note that the only approximation we made is $p \rightarrow p_0$. This form of the constraint, for a general equation of state, was originally derived in \cite{ABRZ:I}. \subsubsection{Combustion limit} A useful limit is smallscale combustion. In an open domain, we can take $p_0$ as constant, so $Dp_0/Dt = 0$, and we are left with \begin{equation} \nabla \cdot \Ub = S \end{equation} This looks like the constraint for incompressible flow, but with a source to the divergence. This source captures the compressible effects due to local heat release---as a fluid parcel moves, the only changes to its density will come through local heat release. Methods for this type of constraint are discussed in \cite{pember-flame,DayBell:2000,SNpaper}. \subsubsection{Atmospheric case} Another interesting case is that of an atmosphere. If we consider an ideal gas, then $\Gamma_1 = \gamma = \mathrm{constant}$. A second appromation we take is that $p_0 = p_0(r)$---i.e., no time dependence to the atmosphere. Finally, we'll consider the case with no local heat sources ($S = 0$). Then we have \begin{equation} \nabla \cdot \Ub + \frac{1}{\gamma p_0} \Ub \cdot \nabla p_0 = 0 \end{equation} which is equivalent to \begin{equation} \nabla \cdot \left ( p_0^{1/\gamma} \Ub \right ) = 0 \end{equation} This constraint captures the changes in compressibilty due to the background stratification of an atmosphere. This form was originally derived for atmospheric flows by \cite{durran:1989} and generalized to stellar flows in \cite{ABRZ:I}. If the structure of the atmosphere is isentropic, then we know that $d\log p_0 = \gamma d\log \rho_0$, where we use $\rho_0$ to represent the density corresponding to $p_0$, and we can write this constraint as: \begin{equation} \nabla \cdot (\rho_0 \Ub) = 0 \end{equation} This is the traditional anelastic constraint. Extensions involving external heating sources or reactions are discussed in \cite{ABRZ:II,ABNZ:III}, and dealing with the non-constant $\Gamma_1$ is discussed in \cite{ABRZ:I,KP:2012}. The time-dependence of the background atmosphere is explored in \cite{ABRZ:II} for plane-parallel atmospheres, following the ideas from \cite{almgren:2000}, and in \cite{multilevel} for spherical, self-gravitating bodies. \section{Multigrid for Variable-Density Flows} The solution methodology for these low Mach number systems follows that of the incompressible flow, but with two additions. First, we need to incorporate a density (mass continuity) evolution equation---this will follow the same techniques we already saw for advection, as we'll see later. Next, we need to be able to enforce more general forms of the divergence constraint, which as we'll see in a moment, require us to solve a variable-coefficient elliptic equation. Our multigrid technique will need to be suitably modified. \label{sec:lm:vcelliptic} We now need to solve an elliptic equation of the form: \begin{equation} \nabla \cdot (\eta \nabla \phi) = f \end{equation} If we denote the discrete divergence and gradient operators as $D$ and $G$, then our operator will be $L_\eta \equiv D \eta G$. If we wish to use a cell-centered discretization for $\phi$, then using a standard centered-difference for $D$ and $G$ will result in a stencil that reaches two zones on either side of the current zone. This can lead to an odd-even decoupling. \MarginPar{cite the appropriate Bell paper} \MarginPar{there is a Pao and Colella (or Pao's thesis?) that also discusses issues with cell-centered} Instead, we again use an approximate projection. We discretize the variable-coefficient Laplacian as: \begin{align} (L_\eta \phi)_{i,j} = & \frac{\eta_{i+\myhalf,j} (\phi_{i+1,j} - \phi_{i,j}) - \eta_{i-\myhalf,j} (\phi_{i,j} - \phi_{i-1,j})}{\Delta x^2} + \nonumber \\ & \frac{\eta_{i,j+\myhalf} (\phi_{i,j+1} - \phi_{i,j}) - \eta_{i,j-\myhalf} (\phi_{i,j} - \phi_{i,j-1})}{\Delta y^2} \label{lm:eq:lap} \end{align} We can define the interface values of $\eta$ as the averages of the cell-centered values, e.g., \begin{equation} \eta_{i,j+\myhalf} = \frac{1}{2}(\eta_{i,j} + \eta_{i,j+1}) \end{equation} Our elliptic equation is then \begin{equation} (L_\eta \phi)_{i,j} = f_{i,j} \end{equation} The relaxation method for this operator again relies on isolating $\phi_{i,j}$, yielding: \begin{equation} \phi_{i,j} = \frac{\teta_{i+\myhalf,j}\phi_{i+1,j} + \teta_{i-\myhalf,j}\phi_{i-1,j} + \teta_{i,j+\myhalf}\phi_{i,j+1} + \teta_{i,j-\myhalf}\phi_{i,j-1} - f_{i,j} } {\teta_{i+\myhalf,j} + \teta_{i-\myhalf,j} + \teta_{i,j+\myhalf} + \teta_{i,j-\myhalf}} \label{lm:eq:smooth} \end{equation} with the shorthand that $\teta_{i\pm\myhalf,j} = \eta_{i\pm\myhalf,j}/\Delta x^2$ and $\teta_{i,j\pm\myhalf} = \eta_{i,j\pm\myhalf}/\Delta y^2$. To put this into our multigrid framework, there are three changes we need to make: \begin{itemize} \item The smoothing function needs to implement the more general smoothing described by Eq.~\ref{lm:eq:smooth}. \item The residual function needs to compute $(L_\eta \phi)_{i,j}$ according to Eq.~\ref{lm:eq:lap}, and then $r_{i,j} = f_{i,j} - (L_\eta \phi)_{i,j}$. \item The coefficients, $\eta$ should be averaged to the edges on the fine grid and then restricted down the multigrid hierarchy as edge-based quantities. \end{itemize} \subsection{Test problem} \subsubsection{Periodic} \label{sec:lm:periodicbcs} To test the solver, we need to devise a problem with a known analytic solution. The easiest way to do this is to pick an $\eta(x)$ and $\phi$ and then do the divergence and gradients to find the required righthand side, $f$. We'll use periodic BCs, and for our equation $\nabla \cdot ( \eta \nabla \phi ) = f$, the following provide a well-posed test problem: \begin{align} \eta &= 2 + \cos(2\pi x) \cos(2\pi y) \label{eq:vc:lap} \\ f &= -16.0 \pi^2 \left [ \cos(2\pi x)\cos(2\pi y) + 1 \right ] \sin(2\pi x)\sin(2 \pi y) \nonumber \end{align} with the solution: \begin{equation} \phi^\mathrm{true} = \sin(2 \pi x) \sin(2\pi y) \nonumber \\ \end{equation} There is an important caveat when dealing with a purely-periodic problem. Since there is no boundary values to ``anchor'' the solution, it is free to float. Solving the elliptic problem will give use the correct $\nabla \phi$, but the average of $\phi$ over the domain is unconstrained. For our algorithms, it is $\nabla \phi$ that matters (that is the forcing term that enters into the momentum equation). When for checking convergence, we want to compare to the exact solution. We therefore normalize $\phi$ by subtracting off its average value before computing the norm of the error with respect to the exact solution: \begin{equation} \epsilon = \| \phi_{i,j} - \bar{\phi} - \phi^\mathrm{true}_{i,j} \| \enskip , \end{equation} where \begin{equation} \bar{\phi} = \frac{1}{N_x N_y} \sum_{i,j} \phi_{i,j} \end{equation} As discussed in \S~\ref{sec:multigrid:other}, this can arise if the discrete form the righthand side, $f_{i,j}$ does not sum exactly to zero. Figure~\ref{fig:mg_vc} shows the solution to this problem with a $512^2$ grid. and the convergence of the solver described here is shown in Figure~\ref{fig:mg_vc_converge}. \begin{figure}[t] \centering \includegraphics[width=\linewidth]{mg_vc_periodic_test} \caption[Solution and error of a variable-coefficient Poisson problem]{\label{fig:mg_vc} Solution and error to the variable-coefficient Poisson problem defined in Eq.~\ref{eq:vc:lap}. This test can be run with \pyro\ {\tt multigrid/test\_mg\_vc\_periodic.py}.} \end{figure} \begin{figure}[t] \centering \includegraphics[width=0.8\linewidth]{mg_vc_periodic_converge} \caption[Convergence of the variable-coefficient Poisson solver]{\label{fig:mg_vc_converge} Convergence of the variable-coefficient multigrid solver for the test problem defined in Eq.~\ref{eq:vc:lap}. This test can be run with \pyro\ {\tt multigrid/test\_mg\_vc\_periodic.py}.} \end{figure} \subsubsection{Dirichlet} We can run the same problem with Dirichlet boundary conditions on $\phi$, and we are free to pick different boundary conditions for $\eta$, since it represents a different physical quantity. Since we only have homogeneous Dirichlet or Neumann BCs implemented, we'll run with Neumann BCs on $\eta$. \section{Atmospheric flows} \label{sec:lm:atm} \subsection{Equation Set} For atmospheric flows, we define a one-dimensional base state that is in hydrostatic equilibrium. In general, this base state can be time-dependent, expanding in response to heating in the atmosphere (see e.g. \cite{almgren:2000,ABRZ:II}. Here we'll consider only a time-independent state. We'll follow the procedure defined in \cite{multilevel}: we define $\rho_0$ as the lateral average of $\rho$: \begin{equation} {\rho_0}_j = \frac{1}{N_x} \sum_i \rho_{i,j} \end{equation} and then we define the base state pressure, $p_0$, by integrating the equation of hydrostatic equilibrium, $dp_0/dy = \rho g$, as: \begin{equation} {p_0}_{j+1} = {p_0}_j + \frac{\Delta y}{2} ({\rho_0}_j + {\rho_0}_{j+1}) g \end{equation} with an initial condition of \begin{equation} {p_0}_\mathrm{jlo} = \frac{1}{N_x} \sum_{i} p^\mathrm{initial}_{i,\mathrm{jlo}} \end{equation} The compressible momentum equation (written in terms of velocity is): \begin{equation} \rho \frac{\partial \Ub}{\partial t} + \rho \Ub \cdot \nabla \Ub + \nabla p = \rho \gb \end{equation} Subtracting off the base state, and defining the perturbational pressure (sometimes called the dynamic pressure) as $p^\prime = p - p_0$, and perturbational density as $\rho' = \rho - \rho_0$, we have: \begin{equation} \rho \frac{\partial \Ub}{\partial t} + \rho \Ub \cdot \nabla \Ub + \nabla p' = \rho' \gb \end{equation} or \begin{equation} \frac{\partial \Ub}{\partial t} + \Ub \cdot \nabla \Ub + \frac{1}{\rho} \nabla p^\prime = \frac{\rho^\prime}{\rho} \gb \end{equation} Several authors \cite{KP:2012,VLBWZ:2013} explored the idea of energy conservation in a low Mach number system and found that an additional term (which can look like a buoyancy) is needed in the low Mach number formulation, yielding: \begin{equation} \frac{\partial \Ub}{\partial t} + \Ub \cdot \nabla \Ub + \frac{\beta_0}{\rho} \nabla \left (\frac{p^\prime}{\beta_0} \right ) = \frac{\rho^\prime}{\rho} \gb \end{equation} Completing the system are the continuity equation, \begin{equation} \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \Ub) = 0 \end{equation} and the constraint, \begin{equation} \nabla \cdot (\beta_0 \Ub) = 0 \end{equation} with $\beta_0 = p_0^{1/\gamma}$. \subsection{Solution Procedure} \label{sec:lm:density} The general solution procedure is for a single step is: \begin{enumerate} \renewcommand{\labelenumi}{\Roman{enumi}.} \item {\em Predict $\Ub$ to the interfaces} \item {\em Enforce the divergence constraint on the interface $\Ub$'s (the MAC projection) to get $\Ub^\mathrm{adv}$.} Decomposing the velocity field as \begin{equation} \Ub^\star = \Ub^d + \frac{\beta_0}{\rho} \nabla \phi \end{equation} as suggested from the form of the momentum equation, our Poisson equation can be defined by multiplying by $\beta_0$ and taking the divergence, and using $\nabla \cdot (\beta_0 \Ub^d) = 0$, giving \begin{equation} \nabla \cdot \frac{\beta_0^2}{\rho} \nabla \phi = \nabla \cdot (\beta_0 \Ub^\star) \end{equation} Note that when written like this, $\phi$ has units of $p^\prime/\beta_0$. \MarginPar{there is a $\Delta t$ here too} For the MAC projection, we have edge-centered velocities (in their respective coordinate direction). We define an edge-centered $\beta_0$ as \begin{equation} {\beta_0}_{j+\myhalf} = \frac{1}{2} ( {\beta_0}_j + {\beta_0}_{j+1} ) \end{equation} (note that, since $\beta_0$ is one-dimensional, we only average in the vertical direction). The divergence term is then: \begin{equation} \left [ \nabla \cdot (\beta_0 \Ub) \right ]_{i,j}^\mathrm{adv} = {\beta_0}_j \frac{u^\mathrm{adv}_{i+\myhalf,j} - u^\mathrm{adv}_{i-\myhalf,j}}{\Delta x} + \frac{{\beta_0}_{j+\myhalf} v^\mathrm{adv}_{i,j+\myhalf} - {\beta_0}_{j-\myhalf} v^\mathrm{adv}_{i,j-\myhalf} }{\Delta y} \end{equation} We define the coefficient, $\eta$, in the Poisson equation as: \begin{equation} \eta_{i,j} = \frac{{\beta_0}_j^2}{\rho_{i,j}} \end{equation} and bring it to edges simply through averaging. Multigrid is used to solve for $\phi_{i,j}$. We then solve \begin{equation} (L_\eta\phi)_{i,j} = \left [ \nabla \cdot (\beta_0 \Ub) \right ]_{i,j}^\mathrm{adv} \end{equation} Once we solve for $\phi$, we correct the velocity as: \begin{equation} \Ub^\mathrm{new} = \Ub^\star - \frac{\beta_0}{\rho} \nabla \phi \end{equation} Since the MAC velocities are edge-centered, our correction appears as: \begin{align} u^\mathrm{adv}_{i+\myhalf,j} &= u^\mathrm{adv}_{i+\myhalf,j} - \left (\frac{\beta_0}{\rho}\right )_{i+\myhalf,j} \frac{\phi_{i+1,j} - \phi_{i,j}}{\Delta x} \\ % v^\mathrm{adv}_{i,j+\myhalf} &= v^\mathrm{adv}_{i,j+\myhalf} - \left (\frac{\beta_0}{\rho}\right )_{i,j+\myhalf} \frac{\phi_{i,j+1} - \phi_{i,j}}{\Delta y} \end{align} \item {\em Predict $\rho$ to the interfaces} We need to solve the continuity equation. We use the same techniques that were used for advection. Our equation is: \begin{equation} \rho_t + (\rho u)_x + (\rho v)_y = 0 \end{equation} The $x$-interface left state would be: \begin{align} \rho_{i+\myhalf,j,L}^{n+\myhalf} &= \rho_{i,j}^n + \frac{\Delta x}{2} \frac{\partial \rho}{\partial x} + \frac{\Delta t}{2} \frac{\partial \rho}{\partial t} + \ldots \nonumber \\ % &= \rho_{i,j}^n + \frac{\Delta x}{2} \frac{\partial \rho}{\partial x} + \frac{\Delta t}{2} \left [ -(\rho u)_x -(\rho v)_y \right ]_{i,j} \nonumber\\ % &= \underbrace{\rho_{i,j}^n + \frac{\Delta x}{2} \left ( 1 - \frac{\Delta t}{\Delta x} u_{i,j} \right ) \frac{\partial \rho}{\partial x}}_{\hat{\rho}_{i+\myhalf,j,L}^{n+\myhalf}} - \frac{\Delta t}{2} \left [\rho u_x \right ]_{i,j} - \frac{\Delta t}{2} \left [ (\rho v)_y \right ]_{i,j} \end{align} A similar construction would yield the right state at that interface, and the $y$-interface states. Since we already have the MAC-projected advected velocities at this point, we can use them in the construction of the interface states and the upwinding. As before, we split this into a normal part (the $\hat{\rho}$ term) and ``transverse'' part (which here includes the non-advective part of the divergence). We first construct the normal parts as \begin{equation} \hat{\rho}_{i+\myhalf,j,L}^{n+\myhalf} = \rho_{i,j}^n + \frac{1}{2} \left ( 1 - \frac{\Delta t}{\Delta x} u^\mathrm{adv}_{i+\myhalf,j} \right ) \overline{\Delta \rho}^{(x)}_{i,j} \end{equation} and then solve the Riemann problem (upwinding) for these: \begin{equation} \hat{\rho}_{i+\myhalf,j}^{n+\myhalf} = \mathcal{U}[u^\mathrm{adv}_{i+\myhalf,j}](\hat{\rho}_{i+\myhalf,j,L}^{n+\myhalf}, \hat{\rho}_{i+\myhalf,j,R}^{n+\myhalf}) = \left \{ \begin{array}{cl} \hat{\rho}_{i+\myhalf,j,L}^{n+\myhalf} & u^\mathrm{adv}_{i+\myhalf,j} > 0 \\ \hat{\rho}_{i+\myhalf,j,R}^{n+\myhalf} & u^\mathrm{adv}_{i+\myhalf,j} < 0 \end{array} \right . \end{equation} The same procedure is done for the $y$-interfaces. The full states are then constructed using these ``hat'' states. For example, \begin{align} \rho_{i+\myhalf,j,L}^{n+\myhalf} = \hat{\rho}_{i+\myhalf,j,L}^{n+\myhalf} &- \frac{\Delta t}{2} \rho_{i,j} \frac{u^\mathrm{adv}_{i+\myhalf,j} - u^\mathrm{adv}_{i-\myhalf,j}}{\Delta x} \nonumber \\ &- \frac{\Delta t}{2} \frac{\hat{\rho}_{i,j+\myhalf}^{n+\myhalf} v^\mathrm{adv}_{i,j+\myhalf} - \hat{\rho}_{i,j-\myhalf}^{n+\myhalf} v^\mathrm{adv}_{i,j-\myhalf}}{\Delta y} \end{align} Once the new states on both the $x$- and $y$-interfaces are constructed, we again upwind to find the final $\rho$ state on each interface: \begin{equation} \rho_{i+\myhalf,j}^{n+\myhalf} = \mathcal{U}[u^\mathrm{adv}_{i+\myhalf,j}](\rho_{i+\myhalf,j,L}^{n+\myhalf}, \rho_{i+\myhalf,j,R}^{n+\myhalf}) = \left \{ \begin{array}{cl} \rho_{i+\myhalf,j,L}^{n+\myhalf} & u^\mathrm{adv}_{i+\myhalf,j} > 0 \\ \rho_{i+\myhalf,j,R}^{n+\myhalf} & u^\mathrm{adv}_{i+\myhalf,j} < 0 \end{array} \right . \end{equation} \item {\em Do the conservative update of $\rho$} Once the interface states are found, we can conservatively-difference the continuity equation: \begin{align} \rho^{n+1}_{i,j} = \rho^n_{i,j} &- \frac{\Delta t}{\Delta x} \left (\rho_{i+\myhalf,j}^{n+\myhalf} u^\mathrm{adv}_{i+\myhalf,j} - \rho_{i-\myhalf,j}^{n+\myhalf} u^\mathrm{adv}_{i-\myhalf,j} \right ) \nonumber \\ &- \frac{\Delta t}{\Delta y} \left (\rho_{i,j+\myhalf}^{n+\myhalf} v^\mathrm{adv}_{i,j+\myhalf} - \rho_{i,j-\myhalf}^{n+\myhalf} v^\mathrm{adv}_{i,j-\myhalf} \right ) \end{align} \item {\em Update $\Ub^n$ to $\Ub^{n+1,\star}$} \item {\em Enforce the divergence constraint on $\Ub^{n+1}$} For the final projection, we have cell-centered velocties. We define the divergence term as: \begin{equation} \left [ \nabla \cdot (\beta_0 \Ub) \right ]_{i,j}^\star = {\beta_0}_j \frac{u^\star_{i+1,j} - u^\star_{i-1,j}}{2\Delta x} + \frac{{\beta_0}_{j+1} v^\star_{i,j+1} - {\beta_0}_{j-1} v^\star_{i,j-1} }{2\Delta y} \end{equation} \end{enumerate} \subsection{Timestep constraint} In addition to the advective timestep constraint, an additional constraint is needed if the velocity field is initialially zero. In \cite{ABNZ:III}, a constraint based on the buoyancy forcing is used. First, we compute the maxiumum buoyancy acceleration, \begin{equation} a_\mathrm{max} = \max \left \{ \left |\frac{\rho'\gb}{\rho} \right | \right \} \end{equation} and then \begin{equation} \Delta t_\mathrm{force} = \left ( \frac{2 \Delta x}{a_\mathrm{max}} \right )^{\myhalf} \end{equation} This is based simply on $\Delta x = (1/2) a t^2$---the time it takes for the buoyant force to accelerate a fluid element across a zone width. \subsection{Bootstrapping} First we need to make sure that the initial velocity field satisfies our constraint Next we need to find $\nabla {p^\prime}^{n-\myhalf}$ for the first step. \section{Combustion} Taking $p = p_0 + p^\prime$, with $p_0 = \mathrm{constant}$, the system becomes: \begin{align} \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \Ub) &= 0 \\ \frac{\partial \rho \Ub}{\partial t} + \nabla \cdot (\rho \Ub \Ub) + \nabla p^\prime &= 0 \\ \nabla \cdot \Ub &= S \end{align} \subsection{Species} \subsection{Constraint} Our constraint equation is $\nabla \cdot \Ub = S$. Decomposing the velocity field as \begin{equation} \Ub^\star = \Ub^d + \frac{1}{\rho} \nabla \phi \end{equation} our Poisson equation can be defined by taking the divergence, and using $\nabla \cdot \Ub^d = S$, giving \begin{equation} \nabla \cdot \frac{1}{\rho} \nabla \phi = \nabla \cdot \Ub^\star - S \end{equation} \subsection{Solution Procedure} The general solution procedure is for a single step is: \begin{itemize} \item React for $\Delta t/2$ \item Do the hydrodynamcis for $\Delta t$ \begin{itemize} \item Predict $\Ub$ to the interfaces \item Enforce the divergence constraint on the interface $\Ub$'s (the MAC projection) to get $\Ub^\mathrm{adv}$. \item Predict $\rho$ to the interfaces \item Do the conservative update of $\rho$ \item Update $\Ub^n$ to $\Ub^{n+1}$ \item Enforce the divergence constraint on $\Ub^{n+1}$ \end{itemize} \item React for $\Delta t/2$ \end{itemize} ================================================ FILE: low_mach/mg_vc_periodic_converge.eps ================================================ %!PS-Adobe-3.0 EPSF-3.0 %%Title: mg_vc_periodic_converge.eps %%Creator: matplotlib version 1.3.1, http://matplotlib.org/ %%CreationDate: Fri Mar 20 21:03:27 2015 %%Orientation: portrait %%BoundingBox: 54 180 558 612 %%EndComments %%BeginProlog /mpldict 8 dict def mpldict begin /m { moveto } bind def /l { lineto } bind def /r { rlineto } bind def /c { curveto } bind def /cl { closepath } bind def /box { m 1 index 0 r 0 exch r neg 0 r cl } bind def /clipbox { box clip newpath } bind def %!PS-Adobe-3.0 Resource-Font %%Title: Bitstream Vera Sans %%Copyright: Copyright (c) 2003 by Bitstream, Inc. 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106.292 l 286.572 107.207 l 274.851 107.207 l 274.851 106.292 l } bind def /p0_54 { newpath translate 274.851 107.207 m 286.572 107.207 l 286.572 108.123 l 274.851 108.123 l 274.851 107.207 l } bind def /p0_55 { newpath translate 274.851 108.123 m 286.572 108.123 l 286.572 109.039 l 274.851 109.039 l 274.851 108.123 l } bind def /p0_56 { newpath translate 274.851 109.039 m 286.572 109.039 l 286.572 109.954 l 274.851 109.954 l 274.851 109.039 l } bind def /p0_57 { newpath translate 274.851 109.954 m 286.572 109.954 l 286.572 110.87 l 274.851 110.87 l 274.851 109.954 l } bind def /p0_58 { newpath translate 274.851 110.87 m 286.572 110.87 l 286.572 111.786 l 274.851 111.786 l 274.851 110.87 l } bind def /p0_59 { newpath translate 274.851 111.786 m 286.572 111.786 l 286.572 112.702 l 274.851 112.702 l 274.851 111.786 l } bind def /p0_5a { newpath translate 274.851 112.702 m 286.572 112.702 l 286.572 113.617 l 274.851 113.617 l 274.851 112.702 l } bind def /p0_5b { newpath translate 274.851 113.617 m 286.572 113.617 l 286.572 114.533 l 274.851 114.533 l 274.851 113.617 l } bind def /p0_5c { newpath translate 274.851 114.533 m 286.572 114.533 l 286.572 115.449 l 274.851 115.449 l 274.851 114.533 l } bind def /p0_5d { newpath translate 274.851 115.449 m 286.572 115.449 l 286.572 116.364 l 274.851 116.364 l 274.851 115.449 l } bind def /p0_5e { newpath translate 274.851 116.364 m 286.572 116.364 l 286.572 117.28 l 274.851 117.28 l 274.851 116.364 l } bind def /p0_5f { newpath translate 274.851 117.28 m 286.572 117.28 l 286.572 118.196 l 274.851 118.196 l 274.851 117.28 l } bind def /p0_60 { newpath translate 274.851 118.196 m 286.572 118.196 l 286.572 119.111 l 274.851 119.111 l 274.851 118.196 l } bind def /p0_61 { newpath translate 274.851 119.111 m 286.572 119.111 l 286.572 120.027 l 274.851 120.027 l 274.851 119.111 l } bind def /p0_62 { newpath translate 274.851 120.027 m 286.572 120.027 l 286.572 120.943 l 274.851 120.943 l 274.851 120.027 l } bind def /p0_63 { newpath translate 274.851 120.943 m 286.572 120.943 l 286.572 121.858 l 274.851 121.858 l 274.851 120.943 l } bind def /p0_64 { newpath translate 274.851 121.858 m 286.572 121.858 l 286.572 122.774 l 274.851 122.774 l 274.851 121.858 l } bind def /p0_65 { newpath translate 274.851 122.774 m 286.572 122.774 l 286.572 123.69 l 274.851 123.69 l 274.851 122.774 l } bind def /p0_66 { newpath translate 274.851 123.69 m 286.572 123.69 l 286.572 124.606 l 274.851 124.606 l 274.851 123.69 l } bind def /p0_67 { newpath translate 274.851 124.606 m 286.572 124.606 l 286.572 125.521 l 274.851 125.521 l 274.851 124.606 l } bind def /p0_68 { newpath translate 274.851 125.521 m 286.572 125.521 l 286.572 126.437 l 274.851 126.437 l 274.851 125.521 l } bind def /p0_69 { newpath translate 274.851 126.437 m 286.572 126.437 l 286.572 127.353 l 274.851 127.353 l 274.851 126.437 l } bind def /p0_6a { newpath translate 274.851 127.353 m 286.572 127.353 l 286.572 128.268 l 274.851 128.268 l 274.851 127.353 l } bind 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134.678 l } bind def /p0_73 { newpath translate 274.851 135.594 m 286.572 135.594 l 286.572 136.509 l 274.851 136.509 l 274.851 135.594 l } bind def /p0_74 { newpath translate 274.851 136.509 m 286.572 136.509 l 286.572 137.425 l 274.851 137.425 l 274.851 136.509 l } bind def /p0_75 { newpath translate 274.851 137.425 m 286.572 137.425 l 286.572 138.341 l 274.851 138.341 l 274.851 137.425 l } bind def /p0_76 { newpath translate 274.851 138.341 m 286.572 138.341 l 286.572 139.257 l 274.851 139.257 l 274.851 138.341 l } bind def /p0_77 { newpath translate 274.851 139.257 m 286.572 139.257 l 286.572 140.172 l 274.851 140.172 l 274.851 139.257 l } bind def /p0_78 { newpath translate 274.851 140.172 m 286.572 140.172 l 286.572 141.088 l 274.851 141.088 l 274.851 140.172 l } bind def /p0_79 { newpath translate 274.851 141.088 m 286.572 141.088 l 286.572 142.004 l 274.851 142.004 l 274.851 141.088 l } bind def /p0_7a { newpath translate 274.851 142.004 m 286.572 142.004 l 286.572 142.919 l 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149.329 l 286.572 150.245 l 274.851 150.245 l 274.851 149.329 l } bind def /p0_83 { newpath translate 274.851 150.245 m 286.572 150.245 l 286.572 151.16 l 274.851 151.16 l 274.851 150.245 l } bind def /p0_84 { newpath translate 274.851 151.16 m 286.572 151.16 l 286.572 152.076 l 274.851 152.076 l 274.851 151.16 l } bind def /p0_85 { newpath translate 274.851 152.076 m 286.572 152.076 l 286.572 152.992 l 274.851 152.992 l 274.851 152.076 l } bind def /p0_86 { newpath translate 274.851 152.992 m 286.572 152.992 l 286.572 153.908 l 274.851 153.908 l 274.851 152.992 l } bind def /p0_87 { newpath translate 274.851 153.908 m 286.572 153.908 l 286.572 154.823 l 274.851 154.823 l 274.851 153.908 l } bind def /p0_88 { newpath translate 274.851 154.823 m 286.572 154.823 l 286.572 155.739 l 274.851 155.739 l 274.851 154.823 l } bind def /p0_89 { newpath translate 274.851 155.739 m 286.572 155.739 l 286.572 156.655 l 274.851 156.655 l 274.851 155.739 l } bind def /p0_8a { newpath translate 274.851 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609.242 105.248 l 609.242 104.332 l } bind def /p1_52 { newpath translate 609.242 105.248 m 620.972 105.248 l 620.972 106.164 l 609.242 106.164 l 609.242 105.248 l } bind def /p1_53 { newpath translate 609.242 106.164 m 620.972 106.164 l 620.972 107.081 l 609.242 107.081 l 609.242 106.164 l } bind def /p1_54 { newpath translate 609.242 107.081 m 620.972 107.081 l 620.972 107.997 l 609.242 107.997 l 609.242 107.081 l } bind def /p1_55 { newpath translate 609.242 107.997 m 620.972 107.997 l 620.972 108.914 l 609.242 108.914 l 609.242 107.997 l } bind def /p1_56 { newpath translate 609.242 108.914 m 620.972 108.914 l 620.972 109.83 l 609.242 109.83 l 609.242 108.914 l } bind def /p1_57 { newpath translate 609.242 109.83 m 620.972 109.83 l 620.972 110.747 l 609.242 110.747 l 609.242 109.83 l } bind def /p1_58 { newpath translate 609.242 110.747 m 620.972 110.747 l 620.972 111.663 l 609.242 111.663 l 609.242 110.747 l } bind def /p1_59 { newpath translate 609.242 111.663 m 620.972 111.663 l 620.972 112.579 l 609.242 112.579 l 609.242 111.663 l } bind def /p1_5a { newpath translate 609.242 112.579 m 620.972 112.579 l 620.972 113.496 l 609.242 113.496 l 609.242 112.579 l } bind def /p1_5b { newpath translate 609.242 113.496 m 620.972 113.496 l 620.972 114.412 l 609.242 114.412 l 609.242 113.496 l } bind def /p1_5c { newpath translate 609.242 114.412 m 620.972 114.412 l 620.972 115.329 l 609.242 115.329 l 609.242 114.412 l } bind def /p1_5d { newpath translate 609.242 115.329 m 620.972 115.329 l 620.972 116.245 l 609.242 116.245 l 609.242 115.329 l } bind def /p1_5e { newpath translate 609.242 116.245 m 620.972 116.245 l 620.972 117.162 l 609.242 117.162 l 609.242 116.245 l } bind def /p1_5f { newpath translate 609.242 117.162 m 620.972 117.162 l 620.972 118.078 l 609.242 118.078 l 609.242 117.162 l } bind def /p1_60 { newpath translate 609.242 118.078 m 620.972 118.078 l 620.972 118.994 l 609.242 118.994 l 609.242 118.078 l } bind def /p1_61 { newpath translate 609.242 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translate 609.242 126.326 m 620.972 126.326 l 620.972 127.242 l 609.242 127.242 l 609.242 126.326 l } bind def /p1_6a { newpath translate 609.242 127.242 m 620.972 127.242 l 620.972 128.159 l 609.242 128.159 l 609.242 127.242 l } bind def /p1_6b { newpath translate 609.242 128.159 m 620.972 128.159 l 620.972 129.075 l 609.242 129.075 l 609.242 128.159 l } bind def /p1_6c { newpath translate 609.242 129.075 m 620.972 129.075 l 620.972 129.991 l 609.242 129.991 l 609.242 129.075 l } bind def /p1_6d { newpath translate 609.242 129.991 m 620.972 129.991 l 620.972 130.908 l 609.242 130.908 l 609.242 129.991 l } bind def /p1_6e { newpath translate 609.242 130.908 m 620.972 130.908 l 620.972 131.824 l 609.242 131.824 l 609.242 130.908 l } bind def /p1_6f { newpath translate 609.242 131.824 m 620.972 131.824 l 620.972 132.741 l 609.242 132.741 l 609.242 131.824 l } bind def /p1_70 { newpath translate 609.242 132.741 m 620.972 132.741 l 620.972 133.657 l 609.242 133.657 l 609.242 132.741 l } 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609.242 140.072 l } bind def /p1_79 { newpath translate 609.242 140.989 m 620.972 140.989 l 620.972 141.905 l 609.242 141.905 l 609.242 140.989 l } bind def /p1_7a { newpath translate 609.242 141.905 m 620.972 141.905 l 620.972 142.821 l 609.242 142.821 l 609.242 141.905 l } bind def /p1_7b { newpath translate 609.242 142.821 m 620.972 142.821 l 620.972 143.738 l 609.242 143.738 l 609.242 142.821 l } bind def /p1_7c { newpath translate 609.242 143.738 m 620.972 143.738 l 620.972 144.654 l 609.242 144.654 l 609.242 143.738 l } bind def /p1_7d { newpath translate 609.242 144.654 m 620.972 144.654 l 620.972 145.571 l 609.242 145.571 l 609.242 144.654 l } bind def /p1_7e { newpath translate 609.242 145.571 m 620.972 145.571 l 620.972 146.487 l 609.242 146.487 l 609.242 145.571 l } bind def /p1_7f { newpath translate 609.242 146.487 m 620.972 146.487 l 620.972 147.404 l 609.242 147.404 l 609.242 146.487 l } bind def /p1_80 { newpath translate 609.242 147.404 m 620.972 147.404 l 620.972 148.32 l 609.242 148.32 l 609.242 147.404 l } bind def /p1_81 { newpath translate 609.242 148.32 m 620.972 148.32 l 620.972 149.236 l 609.242 149.236 l 609.242 148.32 l } bind def /p1_82 { newpath translate 609.242 149.236 m 620.972 149.236 l 620.972 150.153 l 609.242 150.153 l 609.242 149.236 l } bind def /p1_83 { newpath translate 609.242 150.153 m 620.972 150.153 l 620.972 151.069 l 609.242 151.069 l 609.242 150.153 l } bind def /p1_84 { newpath translate 609.242 151.069 m 620.972 151.069 l 620.972 151.986 l 609.242 151.986 l 609.242 151.069 l } bind def /p1_85 { newpath translate 609.242 151.986 m 620.972 151.986 l 620.972 152.902 l 609.242 152.902 l 609.242 151.986 l } bind def /p1_86 { newpath translate 609.242 152.902 m 620.972 152.902 l 620.972 153.819 l 609.242 153.819 l 609.242 152.902 l } bind def /p1_87 { newpath translate 609.242 153.819 m 620.972 153.819 l 620.972 154.735 l 609.242 154.735 l 609.242 153.819 l } bind def /p1_88 { newpath translate 609.242 154.735 m 620.972 154.735 l 620.972 155.651 l 609.242 155.651 l 609.242 154.735 l } bind def /p1_89 { newpath translate 609.242 155.651 m 620.972 155.651 l 620.972 156.568 l 609.242 156.568 l 609.242 155.651 l } bind def /p1_8a { newpath translate 609.242 156.568 m 620.972 156.568 l 620.972 157.484 l 609.242 157.484 l 609.242 156.568 l } bind def /p1_8b { newpath translate 609.242 157.484 m 620.972 157.484 l 620.972 158.401 l 609.242 158.401 l 609.242 157.484 l } bind def /p1_8c { newpath translate 609.242 158.401 m 620.972 158.401 l 620.972 159.317 l 609.242 159.317 l 609.242 158.401 l } bind def /p1_8d { newpath translate 609.242 159.317 m 620.972 159.317 l 620.972 160.233 l 609.242 160.233 l 609.242 159.317 l } bind def /p1_8e { newpath translate 609.242 160.233 m 620.972 160.233 l 620.972 161.15 l 609.242 161.15 l 609.242 160.233 l } bind def /p1_8f { newpath translate 609.242 161.15 m 620.972 161.15 l 620.972 162.066 l 609.242 162.066 l 609.242 161.15 l } bind def /p1_90 { newpath translate 609.242 162.066 m 620.972 162.066 l 620.972 162.983 l 609.242 162.983 l 609.242 162.066 l } bind def /p1_91 { newpath translate 609.242 162.983 m 620.972 162.983 l 620.972 163.899 l 609.242 163.899 l 609.242 162.983 l } bind def /p1_92 { newpath translate 609.242 163.899 m 620.972 163.899 l 620.972 164.816 l 609.242 164.816 l 609.242 163.899 l } bind def /p1_93 { newpath translate 609.242 164.816 m 620.972 164.816 l 620.972 165.732 l 609.242 165.732 l 609.242 164.816 l } bind def /p1_94 { newpath translate 609.242 165.732 m 620.972 165.732 l 620.972 166.648 l 609.242 166.648 l 609.242 165.732 l } bind def /p1_95 { newpath translate 609.242 166.648 m 620.972 166.648 l 620.972 167.565 l 609.242 167.565 l 609.242 166.648 l } bind def /p1_96 { newpath translate 609.242 167.565 m 620.972 167.565 l 620.972 168.481 l 609.242 168.481 l 609.242 167.565 l } bind def /p1_97 { newpath translate 609.242 168.481 m 620.972 168.481 l 620.972 169.398 l 609.242 169.398 l 609.242 168.481 l } bind def /p1_98 { newpath translate 609.242 169.398 m 620.972 169.398 l 620.972 170.314 l 609.242 170.314 l 609.242 169.398 l } bind def /p1_99 { newpath translate 609.242 170.314 m 620.972 170.314 l 620.972 171.231 l 609.242 171.231 l 609.242 170.314 l } bind def /p1_9a { newpath translate 609.242 171.231 m 620.972 171.231 l 620.972 172.147 l 609.242 172.147 l 609.242 171.231 l } bind def /p1_9b { newpath translate 609.242 172.147 m 620.972 172.147 l 620.972 173.063 l 609.242 173.063 l 609.242 172.147 l } bind def /p1_9c { newpath translate 609.242 173.063 m 620.972 173.063 l 620.972 173.98 l 609.242 173.98 l 609.242 173.063 l } bind def /p1_9d { newpath translate 609.242 173.98 m 620.972 173.98 l 620.972 174.896 l 609.242 174.896 l 609.242 173.98 l } bind def /p1_9e { newpath translate 609.242 174.896 m 620.972 174.896 l 620.972 175.813 l 609.242 175.813 l 609.242 174.896 l } bind def /p1_9f { newpath translate 609.242 175.813 m 620.972 175.813 l 620.972 176.729 l 609.242 176.729 l 609.242 175.813 l } 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l 609.242 183.144 l } bind def /p1_a8 { newpath translate 609.242 184.06 m 620.972 184.06 l 620.972 184.977 l 609.242 184.977 l 609.242 184.06 l } bind def /p1_a9 { newpath translate 609.242 184.977 m 620.972 184.977 l 620.972 185.893 l 609.242 185.893 l 609.242 184.977 l } bind def /p1_aa { newpath translate 609.242 185.893 m 620.972 185.893 l 620.972 186.81 l 609.242 186.81 l 609.242 185.893 l } bind def /p1_ab { newpath translate 609.242 186.81 m 620.972 186.81 l 620.972 187.726 l 609.242 187.726 l 609.242 186.81 l } bind def /p1_ac { newpath translate 609.242 187.726 m 620.972 187.726 l 620.972 188.643 l 609.242 188.643 l 609.242 187.726 l } bind def /p1_ad { newpath translate 609.242 188.643 m 620.972 188.643 l 620.972 189.559 l 609.242 189.559 l 609.242 188.643 l } bind def /p1_ae { newpath translate 609.242 189.559 m 620.972 189.559 l 620.972 190.475 l 609.242 190.475 l 609.242 189.559 l } bind def /p1_af { newpath translate 609.242 190.475 m 620.972 190.475 l 620.972 191.392 l 609.242 191.392 l 609.242 190.475 l } bind def /p1_b0 { newpath translate 609.242 191.392 m 620.972 191.392 l 620.972 192.308 l 609.242 192.308 l 609.242 191.392 l } bind def /p1_b1 { newpath translate 609.242 192.308 m 620.972 192.308 l 620.972 193.225 l 609.242 193.225 l 609.242 192.308 l } bind def /p1_b2 { newpath translate 609.242 193.225 m 620.972 193.225 l 620.972 194.141 l 609.242 194.141 l 609.242 193.225 l } bind def /p1_b3 { newpath translate 609.242 194.141 m 620.972 194.141 l 620.972 195.058 l 609.242 195.058 l 609.242 194.141 l } bind def /p1_b4 { newpath translate 609.242 195.058 m 620.972 195.058 l 620.972 195.974 l 609.242 195.974 l 609.242 195.058 l } bind def /p1_b5 { newpath translate 609.242 195.974 m 620.972 195.974 l 620.972 196.89 l 609.242 196.89 l 609.242 195.974 l } bind def /p1_b6 { newpath translate 609.242 196.89 m 620.972 196.89 l 620.972 197.807 l 609.242 197.807 l 609.242 196.89 l } bind def /p1_b7 { newpath translate 609.242 197.807 m 620.972 197.807 l 620.972 198.723 l 609.242 198.723 l 609.242 197.807 l } bind def /p1_b8 { newpath translate 609.242 198.723 m 620.972 198.723 l 620.972 199.64 l 609.242 199.64 l 609.242 198.723 l } bind def /p1_b9 { newpath translate 609.242 199.64 m 620.972 199.64 l 620.972 200.556 l 609.242 200.556 l 609.242 199.64 l } bind def /p1_ba { newpath translate 609.242 200.556 m 620.972 200.556 l 620.972 201.473 l 609.242 201.473 l 609.242 200.556 l } bind def /p1_bb { newpath translate 609.242 201.473 m 620.972 201.473 l 620.972 202.389 l 609.242 202.389 l 609.242 201.473 l } bind def /p1_bc { newpath translate 609.242 202.389 m 620.972 202.389 l 620.972 203.305 l 609.242 203.305 l 609.242 202.389 l } bind def /p1_bd { newpath translate 609.242 203.305 m 620.972 203.305 l 620.972 204.222 l 609.242 204.222 l 609.242 203.305 l } bind def /p1_be { newpath translate 609.242 204.222 m 620.972 204.222 l 620.972 205.138 l 609.242 205.138 l 609.242 204.222 l } bind def /p1_bf { newpath translate 609.242 205.138 m 620.972 205.138 l 620.972 206.055 l 609.242 206.055 l 609.242 205.138 l } bind def /p1_c0 { newpath translate 609.242 206.055 m 620.972 206.055 l 620.972 206.971 l 609.242 206.971 l 609.242 206.055 l } bind def /p1_c1 { newpath translate 609.242 206.971 m 620.972 206.971 l 620.972 207.888 l 609.242 207.888 l 609.242 206.971 l } bind def /p1_c2 { newpath translate 609.242 207.888 m 620.972 207.888 l 620.972 208.804 l 609.242 208.804 l 609.242 207.888 l } bind def /p1_c3 { newpath translate 609.242 208.804 m 620.972 208.804 l 620.972 209.72 l 609.242 209.72 l 609.242 208.804 l } bind def /p1_c4 { newpath translate 609.242 209.72 m 620.972 209.72 l 620.972 210.637 l 609.242 210.637 l 609.242 209.72 l } bind def /p1_c5 { newpath translate 609.242 210.637 m 620.972 210.637 l 620.972 211.553 l 609.242 211.553 l 609.242 210.637 l } bind def /p1_c6 { newpath translate 609.242 211.553 m 620.972 211.553 l 620.972 212.47 l 609.242 212.47 l 609.242 211.553 l } bind def /p1_c7 { newpath translate 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Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. %%Creator: Converted from TrueType to type 3 by PPR 25 dict begin /_d{bind def}bind def /_m{moveto}_d /_l{lineto}_d /_cl{closepath eofill}_d /_c{curveto}_d /_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d /_e{exec}_d /FontName /Cmr10 def /PaintType 0 def /FontMatrix[.001 0 0 .001 0 0]def /FontBBox[-44 -250 1009 750]def /FontType 3 def /Encoding [ /plus /one ] def /FontInfo 10 dict dup begin /FamilyName (cmr10) def /FullName (cmr10) def /Notice (Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. ) def /Weight (Regular) def /Version (1.1/12-Nov-94) def /ItalicAngle 0.0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def end readonly def /CharStrings 2 dict dup begin /plus{{777 0 56 -82 721 583 _sc 75 230 _m 69 230 65 232 61 236 _c 57 240 56 245 56 250 _c 56 254 57 259 61 263 _c 65 267 69 270 75 270 _c 369 270 _l 369 565 _l 369 570 371 574 375 578 _c 379 581 383 583 389 583 _c 393 583 398 581 402 578 _c 406 574 409 570 409 565 _c 409 270 _l 703 270 _l 707 270 711 267 715 263 _c 719 259 721 254 721 250 _c 721 245 719 240 715 236 _c }_e{711 232 707 230 703 230 _c 409 230 _l 409 -64 _l 409 -69 406 -73 402 -77 _c 398 -80 393 -82 389 -82 _c 383 -82 379 -80 375 -77 _c 371 -73 369 -69 369 -64 _c 369 230 _l 75 230 _l _cl}_e}_d /one{500 0 87 0 421 666 _sc 93 0 _m 93 35 _l 176 35 218 45 218 67 _c 218 592 _l 183 575 139 567 87 567 _c 87 602 _l 168 602 230 623 272 666 _c 286 666 _l 288 666 291 665 293 663 _c 295 661 296 659 296 657 _c 296 67 _l 296 45 337 35 421 35 _c 421 0 _l 93 0 _l _cl}_d end readonly def /BuildGlyph {exch begin CharStrings exch 2 copy known not{pop /.notdef}if true 3 1 roll get exec end}_d /BuildChar { 1 index /Encoding get exch get 1 index /BuildGlyph get exec }_d FontName currentdict end definefont pop %!PS-Adobe-3.0 Resource-Font %%Title: cmsy10 %%Copyright: Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. %%Creator: Converted from TrueType to type 3 by PPR 25 dict begin /_d{bind def}bind def /_m{moveto}_d /_l{lineto}_d /_cl{closepath eofill}_d /_c{curveto}_d /_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d /_e{exec}_d /FontName /Cmsy10 def /PaintType 0 def /FontMatrix[.001 0 0 .001 0 0]def /FontBBox[-29 -960 1123 779]def /FontType 3 def /Encoding [ /minus ] def /FontInfo 10 dict dup begin /FamilyName (cmsy10) def /FullName (cmsy10) def /Notice (Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. ) def /Weight (Regular) def /Version (1.1/12-Nov-94) def /ItalicAngle 0.0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def end readonly def /CharStrings 1 dict dup begin /minus{777 0 83 230 694 270 _sc 102 230 _m 96 230 92 232 88 236 _c 84 240 83 245 83 250 _c 83 254 84 259 88 263 _c 92 267 96 270 102 270 _c 676 270 _l 681 270 685 267 689 263 _c 692 259 694 254 694 250 _c 694 245 692 240 689 236 _c 685 232 681 230 676 230 _c 102 230 _l _cl}_d end readonly def /BuildGlyph {exch begin CharStrings exch 2 copy known not{pop /.notdef}if true 3 1 roll get exec end}_d /BuildChar { 1 index /Encoding get exch get 1 index /BuildGlyph get exec }_d FontName currentdict end definefont pop %!PS-Adobe-3.0 Resource-Font %%Title: cmmi10 %%Copyright: Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. %%Creator: Converted from TrueType to type 3 by PPR 25 dict begin /_d{bind def}bind def /_m{moveto}_d /_l{lineto}_d /_cl{closepath eofill}_d /_c{curveto}_d /_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d /_e{exec}_d /FontName /Cmmi10 def /PaintType 0 def /FontMatrix[.001 0 0 .001 0 0]def /FontBBox[-34 -250 1048 750]def /FontType 3 def /Encoding [ /phi /comma /i /f /j /c ] def /FontInfo 10 dict dup begin /FamilyName (cmmi10) def /FullName (cmmi10) def /Notice (Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. ) def /Weight (Regular) def /Version (1.1/12-Nov-94) def /ItalicAngle 0.0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def end readonly def /CharStrings 6 dict dup begin /phi{{595 0 47 -204 575 694 _sc 194 -194 _m 239 -13 _l 205 -13 173 -5 144 10 _c 114 25 91 47 73 75 _c 55 103 47 134 47 168 _c 47 204 56 238 74 271 _c 92 304 116 334 147 361 _c 178 387 211 408 247 423 _c 283 437 318 445 354 445 _c 414 688 _l 415 692 418 694 422 694 _c 434 694 _l 436 694 438 692 440 690 _c 442 688 443 686 443 684 _c 383 445 _l 407 445 431 440 455 431 _c }_e{479 421 500 408 518 392 _c 536 376 550 356 560 334 _c 570 312 575 288 575 263 _c 575 227 566 192 548 159 _c 530 126 506 96 476 70 _c 446 44 412 23 375 9 _c 337 -5 302 -13 268 -13 _c 222 -198 _l 220 -202 218 -204 214 -204 _c 202 -204 _l 199 -204 197 -202 196 -200 _c 194 -198 194 -196 194 -194 _c 245 12 _m 347 419 _l 330 419 312 415 294 408 _c 275 401 257 392 241 382 _c }_e{225 372 210 360 197 347 _c 180 331 165 311 153 289 _c 141 267 131 243 125 219 _c 119 195 116 171 116 147 _c 116 121 121 98 132 78 _c 142 58 157 42 177 30 _c 197 18 219 12 245 12 _c 275 12 _m 300 12 327 19 355 33 _c 383 47 406 64 425 84 _c 449 108 469 139 484 175 _c 498 211 506 247 506 284 _c 506 308 500 331 490 352 _c 479 372 464 389 444 401 _c }_e{424 413 402 419 377 419 _c 275 12 _l _cl}_e}_d /comma{277 0 84 -193 205 110 _sc 99 -179 _m 99 -176 100 -173 103 -171 _c 127 -147 145 -120 159 -90 _c 172 -60 179 -28 179 4 _c 179 16 _l 168 5 155 0 139 0 _c 123 0 110 5 100 16 _c 89 26 84 39 84 55 _c 84 71 89 84 100 94 _c 110 104 123 110 139 110 _c 163 110 180 99 190 77 _c 200 55 205 30 205 4 _c 205 -32 197 -67 183 -101 _c 168 -134 147 -164 120 -191 _c 117 -192 115 -193 114 -193 _c 110 -193 107 -191 104 -188 _c 100 -185 99 -182 99 -179 _c _cl}_d /i{{344 0 27 -10 296 661 _sc 78 72 _m 78 84 80 97 86 111 _c 167 326 _l 175 350 180 370 180 386 _c 180 406 172 416 158 416 _c 132 416 110 402 93 375 _c 75 347 63 317 55 284 _c 53 280 51 278 47 278 _c 35 278 _l 29 278 27 281 27 287 _c 27 290 _l 37 330 54 365 76 396 _c 98 426 126 442 160 442 _c 184 442 204 434 220 418 _c 236 402 245 383 245 359 _c }_e{245 346 242 333 237 320 _c 156 105 _l 146 83 142 63 142 45 _c 142 25 149 15 165 15 _c 191 15 212 29 230 57 _c 247 85 259 115 267 147 _c 268 151 271 153 275 153 _c 287 153 _l 289 153 291 152 293 150 _c 295 148 296 146 296 144 _c 296 143 295 142 295 141 _c 285 103 269 68 247 37 _c 224 5 196 -10 163 -10 _c 139 -10 119 -2 103 13 _c 86 29 78 48 78 72 _c }_e{191 606 _m 191 620 196 632 208 644 _c 220 655 233 661 247 661 _c 258 661 267 657 275 650 _c 282 643 286 634 286 624 _c 286 609 280 596 268 585 _c 256 573 243 568 229 568 _c 218 568 209 571 202 579 _c 194 586 191 595 191 606 _c _cl}_e}_d /f{{489 0 53 -204 552 705 _sc 97 -164 _m 109 -173 124 -178 142 -178 _c 166 -178 185 -151 200 -99 _c 206 -73 223 9 251 151 _c 297 396 _l 211 396 _l 204 396 201 400 201 409 _c 203 423 208 431 214 431 _c 303 431 _l 315 496 _l 319 516 322 533 326 548 _c 329 562 332 576 336 588 _c 340 600 345 614 353 630 _c 364 652 379 670 399 684 _c 418 698 439 705 462 705 _c 476 705 490 702 504 697 _c }_e{518 691 529 683 538 673 _c 547 662 552 649 552 635 _c 552 617 546 602 535 590 _c 523 578 510 572 494 572 _c 483 572 474 575 466 582 _c 458 588 455 597 455 608 _c 455 622 460 635 470 646 _c 480 657 492 663 507 665 _c 494 674 479 679 461 679 _c 451 679 441 674 432 665 _c 423 655 417 645 415 635 _c 411 617 401 571 387 497 _c 375 431 _l 478 431 _l 484 431 488 426 488 418 _c }_e{487 416 486 413 485 409 _c 484 405 483 401 481 399 _c 479 397 477 396 475 396 _c 368 396 _l 322 152 _l 316 116 309 80 301 44 _c 293 8 283 -29 269 -68 _c 255 -106 238 -139 216 -165 _c 194 -191 169 -204 140 -204 _c 117 -204 97 -197 79 -185 _c 61 -172 53 -155 53 -134 _c 53 -116 58 -101 69 -89 _c 80 -77 94 -71 111 -71 _c 121 -71 130 -74 138 -81 _c 146 -87 150 -96 150 -107 _c }_e{150 -121 144 -134 134 -146 _c 123 -158 111 -164 97 -164 _c _cl}_e}_d /j{{411 0 -12 -204 398 661 _sc -12 -141 _m -12 -125 -6 -111 4 -99 _c 15 -87 29 -81 45 -81 _c 55 -81 64 -84 72 -91 _c 79 -97 83 -106 83 -117 _c 83 -129 79 -141 71 -151 _c 63 -161 53 -168 41 -172 _c 53 -176 64 -178 76 -178 _c 104 -178 129 -165 150 -139 _c 170 -113 185 -83 194 -51 _c 288 324 _l 292 343 295 359 295 373 _c 295 401 285 416 265 416 _c 235 416 210 402 188 376 _c }_e{166 349 148 318 135 284 _c 133 280 131 278 127 278 _c 115 278 _l 109 278 107 281 107 287 _c 107 290 _l 121 330 142 365 170 396 _c 198 426 230 442 267 442 _c 295 442 317 433 335 416 _c 353 398 362 376 362 350 _c 362 340 361 331 359 322 _c 264 -57 _l 257 -84 244 -109 225 -131 _c 205 -153 182 -171 156 -184 _c 129 -197 102 -204 74 -204 _c 52 -204 32 -198 14 -188 _c }_e{-3 -177 -12 -161 -12 -141 _c 303 606 _m 303 620 308 633 320 644 _c 331 655 344 661 359 661 _c 370 661 379 657 387 650 _c 394 643 398 634 398 624 _c 398 609 392 596 380 585 _c 368 573 355 568 341 568 _c 330 568 321 571 314 579 _c 306 586 303 595 303 606 _c _cl}_e}_d /c{{432 0 40 -10 430 442 _sc 117 119 _m 117 90 124 65 139 45 _c 154 25 175 15 203 15 _c 243 15 281 24 318 42 _c 354 60 384 86 407 118 _c 409 120 411 121 415 121 _c 418 121 421 119 425 115 _c 428 111 430 108 430 105 _c 430 102 429 100 428 99 _c 404 65 371 38 330 19 _c 288 0 245 -10 201 -10 _c 169 -10 140 -2 116 12 _c 92 26 73 46 60 72 _c }_e{46 97 40 126 40 158 _c 40 203 52 248 77 292 _c 102 336 135 372 177 400 _c 219 428 263 442 309 442 _c 339 442 366 434 390 420 _c 414 405 426 384 426 356 _c 426 338 420 322 410 309 _c 399 296 385 290 367 290 _c 356 290 347 293 340 300 _c 332 306 329 315 329 326 _c 329 342 334 355 346 366 _c 357 377 370 383 386 383 _c 388 383 _l 380 394 368 402 354 408 _c }_e{339 413 324 416 308 416 _c 269 416 235 399 206 366 _c 176 332 154 292 139 246 _c 124 200 117 157 117 119 _c _cl}_e}_d end readonly def /BuildGlyph {exch begin CharStrings exch 2 copy known not{pop /.notdef}if true 3 1 roll get exec end}_d /BuildChar { 1 index /Encoding 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12.6 scalefont setfont 12.961055 8.118750 moveto /f glyphshow /Cmsy10 findfont 12.6 scalefont setfont 10.713867 -4.365625 moveto /minus glyphshow 20.502246 -4.365625 moveto /minus glyphshow grestore gsave 252.568000 337.796000 translate 0.000000 rotate /Cmmi10 findfont 18.0 scalefont setfont 0.000000 0.226062 moveto /phi glyphshow /Cmmi10 findfont 12.6 scalefont setfont 12.961055 8.118750 moveto /f glyphshow /Cmsy10 findfont 12.6 scalefont setfont 10.713867 -4.365625 moveto /minus glyphshow /Cmr10 findfont 12.6 scalefont setfont 22.714629 -4.365625 moveto /plus glyphshow grestore gsave 326.296000 264.068000 translate 0.000000 rotate /Cmmi10 findfont 18.0 scalefont setfont 0.000000 0.226062 moveto /phi glyphshow /Cmmi10 findfont 12.6 scalefont setfont 12.961055 8.118750 moveto /f glyphshow /Cmr10 findfont 12.6 scalefont setfont 12.926250 -4.365625 moveto /plus glyphshow /Cmsy10 findfont 12.6 scalefont setfont 23.516758 -4.365625 moveto /minus glyphshow grestore gsave 324.796000 337.796000 translate 0.000000 rotate /Cmmi10 findfont 18.0 scalefont setfont 0.000000 0.226062 moveto /phi glyphshow /Cmmi10 findfont 12.6 scalefont setfont 12.961055 8.118750 moveto /f glyphshow /Cmr10 findfont 12.6 scalefont setfont 12.926250 -4.365625 moveto /plus glyphshow 25.729141 -4.365625 moveto /plus glyphshow grestore end showpage ================================================ FILE: multigrid/GNUmakefile ================================================ EPStoPDF = epstopdf ALL: multigrid.pdf multigrid.pdf: main.tex multigrid.tex pdflatex -jobname=multigrid main.tex < /dev/null bibtex multigrid.aux pdflatex -jobname=multigrid main.tex < /dev/null pdflatex -jobname=multigrid main.tex < /dev/null clean: $(RM) *.aux *.log *.dvi *.bbl *.blg $(RM) *~ .PHONY: clean ================================================ FILE: multigrid/main.tex ================================================ \documentclass[11pt]{article} % margins \usepackage[margin=0.75in]{geometry} % figures \usepackage{graphicx} % font \usepackage{mathpazo} \usepackage{helvet} % coloring \usepackage{color} \definecolor{mygray}{gray}{0.5} % footer \usepackage{fancyhdr} \pagestyle{fancy} \fancyfoot[LO,LE]{\footnotesize \sffamily \color{mygray} M.\ Zingale---Notes on multigrid} \fancyfoot[RO,RE]{\footnotesize \sffamily \color{mygray} (\today)} \fancyfoot[CO,CE]{\thepage} \fancyhead{} \renewcommand{\headrulewidth}{0.0pt} \renewcommand{\footrulewidth}{0.0pt} % captions \usepackage{caption} \renewcommand{\captionfont}{\footnotesize} \renewcommand{\captionlabelfont}{\footnotesize} \setlength{\captionmargin}{5em} \usepackage{sectsty} \allsectionsfont{\sffamily} \begin{document} \begin{center} {\LARGE \textsf{\textbf{ Notes on multigrid}} } \end{center} \input multigrid \bibliographystyle{plain} \bibliography{../refs} \end{document} ================================================ FILE: multigrid/mg-converge.eps ================================================ %!PS-Adobe-3.0 EPSF-3.0 %%Title: mg-converge.eps %%Creator: matplotlib version 1.3.1, http://matplotlib.org/ %%CreationDate: Mon Jun 30 21:07:18 2014 %%Orientation: portrait %%BoundingBox: 18 180 594 612 %%EndComments %%BeginProlog /mpldict 11 dict def mpldict begin /m { moveto } bind def /l { lineto } bind def /r { rlineto } bind def /c { curveto } bind def /cl { closepath } bind def /box { m 1 index 0 r 0 exch r neg 0 r cl } bind def /clipbox { box clip newpath } bind def %!PS-Adobe-3.0 Resource-Font %%Title: cmmi10 %%Copyright: Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. %%Creator: Converted from TrueType to type 3 by PPR 25 dict begin /_d{bind def}bind def /_m{moveto}_d /_l{lineto}_d /_cl{closepath eofill}_d /_c{curveto}_d /_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d /_e{exec}_d /FontName /Cmmi10 def /PaintType 0 def /FontMatrix[.001 0 0 .001 0 0]def /FontBBox[-34 -250 1048 750]def /FontType 3 def /Encoding [ /x ] def /FontInfo 10 dict dup begin /FamilyName (cmmi10) def /FullName (cmmi10) def /Notice (Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. ) def /Weight (Regular) def /Version (1.1/12-Nov-94) def /ItalicAngle 0.0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def end readonly def /CharStrings 1 dict dup begin /x{{571 0 31 -10 526 442 _sc 78 29 _m 90 19 106 15 128 15 _c 148 15 167 25 183 45 _c 199 65 210 87 216 111 _c 261 288 _l 268 320 272 342 272 354 _c 272 370 267 385 258 397 _c 248 409 235 416 219 416 _c 198 416 178 409 160 396 _c 141 382 125 365 113 345 _c 100 325 91 304 86 284 _c 84 280 82 278 78 278 _c 66 278 _l 60 278 58 281 58 287 _c 58 290 _l }_e{64 314 75 338 91 362 _c 107 386 126 405 149 420 _c 171 434 195 442 221 442 _c 245 442 267 435 287 422 _c 307 409 321 391 329 369 _c 340 389 354 407 372 421 _c 390 435 409 442 431 442 _c 445 442 460 439 475 434 _c 490 429 502 421 512 411 _c 521 401 526 388 526 372 _c 526 355 520 340 509 328 _c 498 316 484 310 468 310 _c 457 310 448 313 441 320 _c }_e{433 327 430 336 430 346 _c 430 360 434 372 444 383 _c 454 393 465 400 479 402 _c 467 411 450 416 429 416 _c 407 416 389 406 373 386 _c 357 366 346 344 340 320 _c 296 143 _l 288 116 285 94 285 77 _c 285 60 289 45 299 33 _c 309 21 322 15 338 15 _c 369 15 397 28 421 56 _c 445 84 462 114 470 147 _c 471 151 474 153 478 153 _c 490 153 _l 492 153 494 152 496 150 _c }_e{498 148 499 146 499 144 _c 499 143 498 142 498 141 _c 488 101 468 66 438 36 _c 408 5 374 -10 336 -10 _c 311 -10 289 -3 269 9 _c 249 21 236 39 228 62 _c 217 42 202 25 184 11 _c 166 -3 146 -10 126 -10 _c 111 -10 96 -7 81 -3 _c 66 1 54 9 45 19 _c 35 29 31 43 31 59 _c 31 75 36 89 47 102 _c 57 114 71 121 88 121 _c }_e{98 121 107 117 115 111 _c 123 104 127 95 127 85 _c 127 71 122 58 113 48 _c 103 38 92 31 78 29 _c _cl}_e}_d end readonly def /BuildGlyph {exch begin CharStrings exch 2 copy known not{pop /.notdef}if true 3 1 roll get exec end}_d /BuildChar { 1 index /Encoding get exch get 1 index /BuildGlyph get exec }_d FontName currentdict end definefont pop %!PS-Adobe-3.0 Resource-Font %%Title: cmsy10 %%Copyright: Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. %%Creator: Converted from TrueType to type 3 by PPR 25 dict begin /_d{bind def}bind def /_m{moveto}_d /_l{lineto}_d /_cl{closepath eofill}_d /_c{curveto}_d /_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d /_e{exec}_d /FontName /Cmsy10 def /PaintType 0 def /FontMatrix[.001 0 0 .001 0 0]def /FontBBox[-29 -960 1123 779]def /FontType 3 def /Encoding [ /O ] def /FontInfo 10 dict dup begin /FamilyName (cmsy10) def /FullName (cmsy10) def /Notice (Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. ) def /Weight (Regular) def /Version (1.1/12-Nov-94) def /ItalicAngle 0.0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def end readonly def /CharStrings 1 dict dup begin /O{{796 0 57 -21 777 705 _sc 291 -21 _m 217 -21 160 4 119 56 _c 77 108 57 171 57 247 _c 57 345 89 438 155 526 _c 170 546 192 570 222 598 _c 251 626 281 650 311 671 _c 341 691 366 703 388 705 _c 393 705 _l 397 703 400 700 400 697 _c 400 694 398 691 396 687 _c 392 682 375 669 346 648 _c 316 627 300 615 297 613 _c 266 586 240 556 218 524 _c 194 489 176 452 164 412 _c }_e{152 372 146 331 146 289 _c 146 244 154 202 170 164 _c 186 125 211 94 244 71 _c 277 48 317 37 363 37 _c 411 37 455 49 496 73 _c 536 97 571 129 599 168 _c 627 207 648 251 664 299 _c 680 347 688 393 688 439 _c 688 475 682 509 671 540 _c 660 571 643 596 620 616 _c 596 636 567 646 531 646 _c 516 646 502 643 489 637 _c 467 625 446 604 426 574 _c }_e{406 543 390 512 377 480 _c 370 468 359 457 344 449 _c 329 440 315 435 301 434 _c 296 434 _l 294 434 293 435 291 437 _c 289 438 289 440 289 442 _c 289 443 289 445 290 447 _c 320 522 361 582 413 626 _c 441 650 470 669 501 683 _c 532 697 564 705 597 705 _c 603 705 _l 643 705 676 694 702 674 _c 728 653 747 626 759 592 _c 771 558 777 521 777 482 _c 777 438 771 396 759 354 _c }_e{747 312 731 273 710 237 _c 688 201 662 167 632 136 _c 600 105 566 78 531 55 _c 495 31 457 13 417 0 _c 376 -14 336 -21 296 -21 _c 291 -21 _l _cl}_e}_d end readonly def /BuildGlyph {exch begin CharStrings exch 2 copy known not{pop /.notdef}if true 3 1 roll get exec end}_d /BuildChar { 1 index /Encoding get exch get 1 index /BuildGlyph get exec }_d FontName currentdict end definefont pop %!PS-Adobe-3.0 Resource-Font %%Title: Bitstream Vera Sans %%Copyright: Copyright (c) 2003 by Bitstream, Inc. All Rights Reserved. %%Creator: Converted from TrueType to type 3 by PPR 25 dict begin /_d{bind def}bind def /_m{moveto}_d /_l{lineto}_d /_cl{closepath eofill}_d /_c{curveto}_d /_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d /_e{exec}_d /FontName /BitstreamVeraSans-Roman def /PaintType 0 def /FontMatrix[.001 0 0 .001 0 0]def /FontBBox[-183 -236 1287 928]def /FontType 3 def /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 /FontInfo 10 dict dup begin /FamilyName (Bitstream Vera Sans) def /FullName (Bitstream Vera Sans) def /Notice (Copyright (c) 2003 by Bitstream, Inc. All Rights Reserved. Bitstream Vera is a trademark of Bitstream, Inc.) def /Weight (Roman) def /Version (Release 1.10) def /ItalicAngle 0.0 def /isFixedPitch false def /UnderlinePosition -213 def /UnderlineThickness 143 def end readonly def /CharStrings 31 dict dup begin /space{318 0 0 0 0 0 _sc }_d /hyphen{361 0 49 234 312 314 _sc 49 314 _m 312 314 _l 312 234 _l 49 234 _l 49 314 _l _cl}_d /zero{636 0 66 -13 570 742 _sc 318 664 _m 267 664 229 639 203 589 _c 177 539 165 464 165 364 _c 165 264 177 189 203 139 _c 229 89 267 64 318 64 _c 369 64 407 89 433 139 _c 458 189 471 264 471 364 _c 471 464 458 539 433 589 _c 407 639 369 664 318 664 _c 318 742 _m 399 742 461 709 505 645 _c 548 580 570 486 570 364 _c 570 241 548 147 505 83 _c 461 19 399 -13 318 -13 _c 236 -13 173 19 130 83 _c 87 147 66 241 66 364 _c 66 486 87 580 130 645 _c 173 709 236 742 318 742 _c _cl}_d /one{636 0 110 0 544 729 _sc 124 83 _m 285 83 _l 285 639 _l 110 604 _l 110 694 _l 284 729 _l 383 729 _l 383 83 _l 544 83 _l 544 0 _l 124 0 _l 124 83 _l _cl}_d /two{{636 0 73 0 536 742 _sc 192 83 _m 536 83 _l 536 0 _l 73 0 _l 73 83 _l 110 121 161 173 226 239 _c 290 304 331 346 348 365 _c 380 400 402 430 414 455 _c 426 479 433 504 433 528 _c 433 566 419 598 392 622 _c 365 646 330 659 286 659 _c 255 659 222 653 188 643 _c 154 632 117 616 78 594 _c 78 694 _l 118 710 155 722 189 730 _c 223 738 255 742 284 742 _c }_e{359 742 419 723 464 685 _c 509 647 532 597 532 534 _c 532 504 526 475 515 449 _c 504 422 484 390 454 354 _c 446 344 420 317 376 272 _c 332 227 271 164 192 83 _c _cl}_e}_d /three{{636 0 76 -13 556 742 _sc 406 393 _m 453 383 490 362 516 330 _c 542 298 556 258 556 212 _c 556 140 531 84 482 45 _c 432 6 362 -13 271 -13 _c 240 -13 208 -10 176 -4 _c 144 1 110 10 76 22 _c 76 117 _l 103 101 133 89 166 81 _c 198 73 232 69 268 69 _c 330 69 377 81 409 105 _c 441 129 458 165 458 212 _c 458 254 443 288 413 312 _c 383 336 341 349 287 349 _c }_e{202 349 _l 202 430 _l 291 430 _l 339 430 376 439 402 459 _c 428 478 441 506 441 543 _c 441 580 427 609 401 629 _c 374 649 336 659 287 659 _c 260 659 231 656 200 650 _c 169 644 135 635 98 623 _c 98 711 _l 135 721 170 729 203 734 _c 235 739 266 742 296 742 _c 370 742 429 725 473 691 _c 517 657 539 611 539 553 _c 539 513 527 479 504 451 _c 481 423 448 403 406 393 _c _cl}_e}_d /four{636 0 49 0 580 729 _sc 378 643 _m 129 254 _l 378 254 _l 378 643 _l 352 729 _m 476 729 _l 476 254 _l 580 254 _l 580 172 _l 476 172 _l 476 0 _l 378 0 _l 378 172 _l 49 172 _l 49 267 _l 352 729 _l _cl}_d /five{{636 0 77 -13 549 729 _sc 108 729 _m 495 729 _l 495 646 _l 198 646 _l 198 467 _l 212 472 227 476 241 478 _c 255 480 270 482 284 482 _c 365 482 429 459 477 415 _c 525 370 549 310 549 234 _c 549 155 524 94 475 51 _c 426 8 357 -13 269 -13 _c 238 -13 207 -10 175 -6 _c 143 -1 111 6 77 17 _c 77 116 _l 106 100 136 88 168 80 _c 199 72 232 69 267 69 _c }_e{323 69 368 83 401 113 _c 433 143 450 183 450 234 _c 450 284 433 324 401 354 _c 368 384 323 399 267 399 _c 241 399 214 396 188 390 _c 162 384 135 375 108 363 _c 108 729 _l _cl}_e}_d /six{{636 0 70 -13 573 742 _sc 330 404 _m 286 404 251 388 225 358 _c 199 328 186 286 186 234 _c 186 181 199 139 225 109 _c 251 79 286 64 330 64 _c 374 64 409 79 435 109 _c 461 139 474 181 474 234 _c 474 286 461 328 435 358 _c 409 388 374 404 330 404 _c 526 713 _m 526 623 _l 501 635 476 644 451 650 _c 425 656 400 659 376 659 _c 310 659 260 637 226 593 _c }_e{192 549 172 482 168 394 _c 187 422 211 444 240 459 _c 269 474 301 482 336 482 _c 409 482 467 459 509 415 _c 551 371 573 310 573 234 _c 573 159 550 99 506 54 _c 462 9 403 -13 330 -13 _c 246 -13 181 19 137 83 _c 92 147 70 241 70 364 _c 70 479 97 571 152 639 _c 206 707 280 742 372 742 _c 396 742 421 739 447 735 _c 472 730 498 723 526 713 _c _cl}_e}_d /seven{636 0 82 0 551 729 _sc 82 729 _m 551 729 _l 551 687 _l 286 0 _l 183 0 _l 432 646 _l 82 646 _l 82 729 _l _cl}_d /eight{{636 0 68 -13 568 742 _sc 318 346 _m 271 346 234 333 207 308 _c 180 283 167 249 167 205 _c 167 161 180 126 207 101 _c 234 76 271 64 318 64 _c 364 64 401 76 428 102 _c 455 127 469 161 469 205 _c 469 249 455 283 429 308 _c 402 333 365 346 318 346 _c 219 388 _m 177 398 144 418 120 447 _c 96 476 85 511 85 553 _c 85 611 105 657 147 691 _c 188 725 245 742 318 742 _c }_e{390 742 447 725 489 691 _c 530 657 551 611 551 553 _c 551 511 539 476 515 447 _c 491 418 459 398 417 388 _c 464 377 501 355 528 323 _c 554 291 568 251 568 205 _c 568 134 546 80 503 43 _c 459 5 398 -13 318 -13 _c 237 -13 175 5 132 43 _c 89 80 68 134 68 205 _c 68 251 81 291 108 323 _c 134 355 171 377 219 388 _c 183 544 _m 183 506 194 476 218 455 _c }_e{242 434 275 424 318 424 _c 360 424 393 434 417 455 _c 441 476 453 506 453 544 _c 453 582 441 611 417 632 _c 393 653 360 664 318 664 _c 275 664 242 653 218 632 _c 194 611 183 582 183 544 _c _cl}_e}_d /L{557 0 98 0 552 729 _sc 98 729 _m 197 729 _l 197 83 _l 552 83 _l 552 0 _l 98 0 _l 98 729 _l _cl}_d /M{863 0 98 0 765 729 _sc 98 729 _m 245 729 _l 431 233 _l 618 729 _l 765 729 _l 765 0 _l 669 0 _l 669 640 _l 481 140 _l 382 140 _l 194 640 _l 194 0 _l 98 0 _l 98 729 _l _cl}_d /N{748 0 98 0 650 729 _sc 98 729 _m 231 729 _l 554 119 _l 554 729 _l 650 729 _l 650 0 _l 517 0 _l 194 610 _l 194 0 _l 98 0 _l 98 729 _l _cl}_d /a{{613 0 60 -13 522 560 _sc 343 275 _m 270 275 220 266 192 250 _c 164 233 150 205 150 165 _c 150 133 160 107 181 89 _c 202 70 231 61 267 61 _c 317 61 357 78 387 114 _c 417 149 432 196 432 255 _c 432 275 _l 343 275 _l 522 312 _m 522 0 _l 432 0 _l 432 83 _l 411 49 385 25 355 10 _c 325 -5 287 -13 243 -13 _c 187 -13 142 2 109 33 _c 76 64 60 106 60 159 _c }_e{60 220 80 266 122 298 _c 163 329 224 345 306 345 _c 432 345 _l 432 354 _l 432 395 418 427 391 450 _c 364 472 326 484 277 484 _c 245 484 215 480 185 472 _c 155 464 127 453 100 439 _c 100 522 _l 132 534 164 544 195 550 _c 226 556 256 560 286 560 _c 365 560 424 539 463 498 _c 502 457 522 395 522 312 _c _cl}_e}_d /b{{635 0 91 -13 580 760 _sc 487 273 _m 487 339 473 390 446 428 _c 418 466 381 485 334 485 _c 286 485 249 466 222 428 _c 194 390 181 339 181 273 _c 181 207 194 155 222 117 _c 249 79 286 61 334 61 _c 381 61 418 79 446 117 _c 473 155 487 207 487 273 _c 181 464 _m 199 496 223 520 252 536 _c 281 552 316 560 356 560 _c 422 560 476 533 518 481 _c 559 428 580 359 580 273 _c }_e{580 187 559 117 518 65 _c 476 13 422 -13 356 -13 _c 316 -13 281 -5 252 10 _c 223 25 199 49 181 82 _c 181 0 _l 91 0 _l 91 760 _l 181 760 _l 181 464 _l _cl}_e}_d /c{{550 0 55 -13 488 560 _sc 488 526 _m 488 442 _l 462 456 437 466 411 473 _c 385 480 360 484 334 484 _c 276 484 230 465 198 428 _c 166 391 150 339 150 273 _c 150 206 166 154 198 117 _c 230 80 276 62 334 62 _c 360 62 385 65 411 72 _c 437 79 462 90 488 104 _c 488 21 _l 462 9 436 0 410 -5 _c 383 -10 354 -13 324 -13 _c 242 -13 176 12 128 64 _c }_e{79 115 55 185 55 273 _c 55 362 79 432 128 483 _c 177 534 244 560 330 560 _c 358 560 385 557 411 551 _c 437 545 463 537 488 526 _c _cl}_e}_d /d{{635 0 55 -13 544 760 _sc 454 464 _m 454 760 _l 544 760 _l 544 0 _l 454 0 _l 454 82 _l 435 49 411 25 382 10 _c 353 -5 319 -13 279 -13 _c 213 -13 159 13 117 65 _c 75 117 55 187 55 273 _c 55 359 75 428 117 481 _c 159 533 213 560 279 560 _c 319 560 353 552 382 536 _c 411 520 435 496 454 464 _c 148 273 _m 148 207 161 155 188 117 _c 215 79 253 61 301 61 _c }_e{348 61 385 79 413 117 _c 440 155 454 207 454 273 _c 454 339 440 390 413 428 _c 385 466 348 485 301 485 _c 253 485 215 466 188 428 _c 161 390 148 339 148 273 _c _cl}_e}_d /e{{615 0 55 -13 562 560 _sc 562 296 _m 562 252 _l 149 252 _l 153 190 171 142 205 110 _c 238 78 284 62 344 62 _c 378 62 412 66 444 74 _c 476 82 509 95 541 113 _c 541 28 _l 509 14 476 3 442 -3 _c 408 -9 373 -13 339 -13 _c 251 -13 182 12 131 62 _c 80 112 55 181 55 268 _c 55 357 79 428 127 481 _c 175 533 241 560 323 560 _c 397 560 455 536 498 489 _c }_e{540 441 562 377 562 296 _c 472 322 _m 471 371 457 410 431 440 _c 404 469 368 484 324 484 _c 274 484 234 469 204 441 _c 174 413 156 373 152 322 _c 472 322 _l _cl}_e}_d /f{352 0 23 0 371 760 _sc 371 760 _m 371 685 _l 285 685 _l 253 685 230 678 218 665 _c 205 652 199 629 199 595 _c 199 547 _l 347 547 _l 347 477 _l 199 477 _l 199 0 _l 109 0 _l 109 477 _l 23 477 _l 23 547 _l 109 547 _l 109 585 _l 109 645 123 690 151 718 _c 179 746 224 760 286 760 _c 371 760 _l _cl}_d /g{{635 0 55 -207 544 560 _sc 454 280 _m 454 344 440 395 414 431 _c 387 467 349 485 301 485 _c 253 485 215 467 188 431 _c 161 395 148 344 148 280 _c 148 215 161 165 188 129 _c 215 93 253 75 301 75 _c 349 75 387 93 414 129 _c 440 165 454 215 454 280 _c 544 68 _m 544 -24 523 -93 482 -139 _c 440 -184 377 -207 292 -207 _c 260 -207 231 -204 203 -200 _c 175 -195 147 -188 121 -178 _c }_e{121 -91 _l 147 -105 173 -115 199 -122 _c 225 -129 251 -133 278 -133 _c 336 -133 380 -117 410 -87 _c 439 -56 454 -10 454 52 _c 454 96 _l 435 64 411 40 382 24 _c 353 8 319 0 279 0 _c 211 0 157 25 116 76 _c 75 127 55 195 55 280 _c 55 364 75 432 116 483 _c 157 534 211 560 279 560 _c 319 560 353 552 382 536 _c 411 520 435 496 454 464 _c 454 547 _l 544 547 _l }_e{544 68 _l _cl}_e}_d /i{278 0 94 0 184 760 _sc 94 547 _m 184 547 _l 184 0 _l 94 0 _l 94 547 _l 94 760 _m 184 760 _l 184 646 _l 94 646 _l 94 760 _l _cl}_d /l{278 0 94 0 184 760 _sc 94 760 _m 184 760 _l 184 0 _l 94 0 _l 94 760 _l _cl}_d /m{{974 0 91 0 889 560 _sc 520 442 _m 542 482 569 511 600 531 _c 631 550 668 560 711 560 _c 767 560 811 540 842 500 _c 873 460 889 403 889 330 _c 889 0 _l 799 0 _l 799 327 _l 799 379 789 418 771 444 _c 752 469 724 482 686 482 _c 639 482 602 466 575 435 _c 548 404 535 362 535 309 _c 535 0 _l 445 0 _l 445 327 _l 445 379 435 418 417 444 _c 398 469 369 482 331 482 _c }_e{285 482 248 466 221 435 _c 194 404 181 362 181 309 _c 181 0 _l 91 0 _l 91 547 _l 181 547 _l 181 462 _l 201 495 226 520 255 536 _c 283 552 317 560 357 560 _c 397 560 430 550 458 530 _c 486 510 506 480 520 442 _c _cl}_e}_d /n{634 0 91 0 549 560 _sc 549 330 _m 549 0 _l 459 0 _l 459 327 _l 459 379 448 417 428 443 _c 408 469 378 482 338 482 _c 289 482 251 466 223 435 _c 195 404 181 362 181 309 _c 181 0 _l 91 0 _l 91 547 _l 181 547 _l 181 462 _l 202 494 227 519 257 535 _c 286 551 320 560 358 560 _c 420 560 468 540 500 501 _c 532 462 549 405 549 330 _c _cl}_d /o{612 0 55 -13 557 560 _sc 306 484 _m 258 484 220 465 192 427 _c 164 389 150 338 150 273 _c 150 207 163 156 191 118 _c 219 80 257 62 306 62 _c 354 62 392 80 420 118 _c 448 156 462 207 462 273 _c 462 337 448 389 420 427 _c 392 465 354 484 306 484 _c 306 560 _m 384 560 445 534 490 484 _c 534 433 557 363 557 273 _c 557 183 534 113 490 63 _c 445 12 384 -13 306 -13 _c 227 -13 165 12 121 63 _c 77 113 55 183 55 273 _c 55 363 77 433 121 484 _c 165 534 227 560 306 560 _c _cl}_d /r{411 0 91 0 411 560 _sc 411 463 _m 401 469 390 473 378 476 _c 366 478 353 480 339 480 _c 288 480 249 463 222 430 _c 194 397 181 350 181 288 _c 181 0 _l 91 0 _l 91 547 _l 181 547 _l 181 462 _l 199 495 224 520 254 536 _c 284 552 321 560 365 560 _c 371 560 378 559 386 559 _c 393 558 401 557 411 555 _c 411 463 _l _cl}_d /s{{521 0 54 -13 472 560 _sc 443 531 _m 443 446 _l 417 458 391 468 364 475 _c 336 481 308 485 279 485 _c 234 485 200 478 178 464 _c 156 450 145 430 145 403 _c 145 382 153 366 169 354 _c 185 342 217 330 265 320 _c 296 313 _l 360 299 405 279 432 255 _c 458 230 472 195 472 151 _c 472 100 452 60 412 31 _c 372 1 316 -13 246 -13 _c 216 -13 186 -10 154 -5 _c }_e{122 0 89 8 54 20 _c 54 113 _l 87 95 120 82 152 74 _c 184 65 216 61 248 61 _c 290 61 323 68 346 82 _c 368 96 380 117 380 144 _c 380 168 371 187 355 200 _c 339 213 303 226 247 238 _c 216 245 _l 160 257 119 275 95 299 _c 70 323 58 356 58 399 _c 58 450 76 490 112 518 _c 148 546 200 560 268 560 _c 301 560 332 557 362 552 _c 391 547 418 540 443 531 _c }_e{_cl}_e}_d /t{392 0 27 0 368 702 _sc 183 702 _m 183 547 _l 368 547 _l 368 477 _l 183 477 _l 183 180 _l 183 135 189 106 201 94 _c 213 81 238 75 276 75 _c 368 75 _l 368 0 _l 276 0 _l 206 0 158 13 132 39 _c 106 65 93 112 93 180 _c 93 477 _l 27 477 _l 27 547 _l 93 547 _l 93 702 _l 183 702 _l _cl}_d /u{634 0 85 -13 543 547 _sc 85 216 _m 85 547 _l 175 547 _l 175 219 _l 175 167 185 129 205 103 _c 225 77 255 64 296 64 _c 344 64 383 79 411 110 _c 439 141 453 183 453 237 _c 453 547 _l 543 547 _l 543 0 _l 453 0 _l 453 84 _l 431 50 405 26 377 10 _c 348 -5 315 -13 277 -13 _c 214 -13 166 6 134 45 _c 101 83 85 140 85 216 _c _cl}_d /v{592 0 30 0 562 547 _sc 30 547 _m 125 547 _l 296 88 _l 467 547 _l 562 547 _l 357 0 _l 235 0 _l 30 547 _l _cl}_d end readonly def /BuildGlyph {exch begin CharStrings exch 2 copy known not{pop /.notdef}if true 3 1 roll get exec end}_d /BuildChar { 1 index /Encoding get exch get 1 index /BuildGlyph get exec }_d FontName currentdict end definefont pop %!PS-Adobe-3.0 Resource-Font %%Title: cmr10 %%Copyright: Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. %%Creator: Converted from TrueType to type 3 by PPR 25 dict begin /_d{bind def}bind def /_m{moveto}_d /_l{lineto}_d /_cl{closepath eofill}_d /_c{curveto}_d /_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d /_e{exec}_d /FontName /Cmr10 def /PaintType 0 def /FontMatrix[.001 0 0 .001 0 0]def /FontBBox[-44 -250 1009 750]def /FontType 3 def /Encoding [ /Delta /two /parenright /parenleft ] def /FontInfo 10 dict dup begin /FamilyName (cmr10) def /FullName (cmr10) def /Notice (Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. ) def /Weight (Regular) def /Version (1.1/12-Nov-94) def /ItalicAngle 0.0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def end readonly def /CharStrings 4 dict dup begin /Delta{833 0 46 0 786 716 _sc 54 0 _m 48 0 46 2 46 8 _c 46 8 46 9 46 9 _c 46 9 47 10 47 11 _c 391 705 _l 394 712 400 716 409 716 _c 423 716 _l 432 716 438 712 441 705 _c 785 11 _l 785 10 786 9 786 9 _c 786 9 786 8 786 8 _c 786 2 783 0 778 0 _c 54 0 _l 120 80 _m 647 80 _l 384 611 _l 120 80 _l _cl}_d /two{{500 0 50 0 449 666 _sc 50 0 _m 50 27 _l 50 28 50 30 52 32 _c 207 204 _l 230 229 249 250 264 268 _c 278 285 293 305 307 327 _c 321 349 333 372 341 396 _c 349 419 354 444 354 470 _c 354 497 349 523 339 548 _c 329 573 314 593 294 608 _c 274 623 249 631 221 631 _c 192 631 166 622 143 605 _c 119 587 103 565 94 537 _c 96 537 100 538 105 538 _c }_e{119 538 132 533 143 523 _c 153 513 159 500 159 484 _c 159 468 153 455 143 445 _c 132 434 119 429 105 429 _c 89 429 76 434 66 445 _c 55 456 50 469 50 484 _c 50 508 54 532 64 554 _c 73 576 86 595 104 613 _c 122 630 142 643 164 652 _c 186 661 210 666 236 666 _c 274 666 309 658 342 642 _c 375 626 401 603 420 573 _c 439 543 449 509 449 470 _c }_e{449 441 442 414 430 388 _c 417 362 401 338 381 317 _c 361 295 336 271 305 244 _c 274 217 254 199 244 190 _c 131 81 _l 227 81 _l 274 81 313 81 345 82 _c 377 82 394 84 396 86 _c 404 94 412 125 420 178 _c 449 178 _l 421 0 _l 50 0 _l _cl}_e}_d /parenright{{388 0 56 -249 291 750 _sc 65 -249 _m 59 -249 56 -246 56 -240 _c 56 -237 56 -235 58 -234 _c 172 -122 229 38 229 250 _c 229 462 172 623 60 733 _c 57 734 56 737 56 741 _c 56 743 57 745 59 747 _c 61 749 63 750 65 750 _c 74 750 _l 76 750 77 749 79 748 _c 127 710 167 664 199 610 _c 231 556 254 498 269 437 _c 283 376 291 314 291 250 _c 291 202 287 156 279 110 _c }_e{271 64 258 20 241 -24 _c 223 -68 201 -108 175 -146 _c 148 -184 116 -217 79 -247 _c 77 -248 76 -249 74 -249 _c 65 -249 _l _cl}_e}_d 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grestore gsave 486.208094 377.280000 translate 0.000000 rotate /Cmsy10 findfont 14.4 scalefont setfont 0.000000 0.173063 moveto /O glyphshow /Cmr10 findfont 14.4 scalefont setfont 11.453476 0.173063 moveto /parenleft glyphshow 17.039673 0.173063 moveto /Delta glyphshow /Cmmi10 findfont 14.4 scalefont setfont 29.027176 0.173063 moveto /x glyphshow /Cmr10 findfont 10.08 scalefont setfont 37.241348 6.295312 moveto /two glyphshow /Cmr10 findfont 14.4 scalefont setfont 46.885561 0.173063 moveto /parenright glyphshow grestore end showpage ================================================ FILE: multigrid/mg_error_vs_cycle.eps ================================================ %!PS-Adobe-3.0 EPSF-3.0 %%Title: mg_error_vs_cycle.eps %%Creator: matplotlib version 1.2.0, http://matplotlib.org/ %%CreationDate: Mon Apr 8 13:48:56 2013 %%Orientation: portrait %%BoundingBox: 18 180 594 612 %%EndComments %%BeginProlog /mpldict 10 dict def mpldict begin /m { moveto } bind def /l { lineto } bind def /r { rlineto } bind def /c { curveto } bind def /cl { closepath } bind def /box { m 1 index 0 r 0 exch r neg 0 r cl } bind def /clipbox { box clip newpath } bind def %!PS-Adobe-3.0 Resource-Font %%Title: cmsy10 %%Copyright: Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. %%Creator: Converted from TrueType to type 3 by PPR 25 dict begin /_d{bind def}bind def /_m{moveto}_d /_l{lineto}_d /_cl{closepath eofill}_d /_c{curveto}_d /_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d /_e{exec}_d /FontName /Cmsy10 def /PaintType 0 def /FontMatrix[.001 0 0 .001 0 0]def /FontBBox[-29 -960 1123 779]def /FontType 3 def /Encoding [ /bar ] def /FontInfo 10 dict dup begin /FamilyName (cmsy10) def /FullName (cmsy10) def /Notice (Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. ) def /Weight (Regular) def /Version (1.1/12-Nov-94) def /ItalicAngle 0.0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def end readonly def /CharStrings 1 dict dup begin /bar{277 0 119 -249 159 750 _sc 119 -231 _m 119 732 _l 119 737 121 741 125 745 _c 129 748 133 750 139 750 _c 143 750 148 748 152 745 _c 156 741 159 737 159 732 _c 159 -231 _l 159 -236 156 -240 152 -244 _c 148 -247 143 -249 139 -249 _c 133 -249 129 -247 125 -244 _c 121 -240 119 -236 119 -231 _c _cl}_d end readonly def /BuildGlyph {exch begin CharStrings exch 2 copy known not{pop /.notdef}if true 3 1 roll get exec end}_d /BuildChar { 1 index /Encoding get exch get 1 index /BuildGlyph get exec }_d FontName currentdict end definefont pop %!PS-Adobe-3.0 Resource-Font %%Title: DejaVu Sans %%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 %%Creator: Converted from TrueType to type 3 by PPR 25 dict begin /_d{bind def}bind def /_m{moveto}_d /_l{lineto}_d /_cl{closepath eofill}_d /_c{curveto}_d /_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d /_e{exec}_d /FontName /DejaVuSans def /PaintType 0 def /FontMatrix[.001 0 0 .001 0 0]def /FontBBox[-1021 -415 1681 1167]def /FontType 3 def /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 /FontInfo 10 dict dup begin /FamilyName (DejaVu Sans) def /FullName (DejaVu Sans) def /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 /Weight (Book) def /Version (Version 2.33) def /ItalicAngle 0.0 def /isFixedPitch false def /UnderlinePosition -130 def /UnderlineThickness 90 def end readonly def /CharStrings 25 dict dup begin /space{318 0 0 0 0 0 _sc }_d /numbersign{{838 0 77 0 761 718 _sc 511 440 _m 369 440 _l 328 277 _l 471 277 _l 511 440 _l 438 718 _m 387 515 _l 530 515 _l 581 718 _l 659 718 _l 609 515 _l 761 515 _l 761 440 _l 590 440 _l 550 277 _l 705 277 _l 705 202 _l 531 202 _l 480 0 _l 402 0 _l 452 202 _l 309 202 _l 259 0 _l 180 0 _l 231 202 _l 77 202 _l 77 277 _l 249 277 _l 290 440 _l 133 440 _l }_e{133 515 _l 309 515 _l 359 718 _l 438 718 _l _cl}_e}_d /hyphen{361 0 49 234 312 314 _sc 49 314 _m 312 314 _l 312 234 _l 49 234 _l 49 314 _l _cl}_d /zero{636 0 66 -13 570 742 _sc 318 664 _m 267 664 229 639 203 589 _c 177 539 165 464 165 364 _c 165 264 177 189 203 139 _c 229 89 267 64 318 64 _c 369 64 407 89 433 139 _c 458 189 471 264 471 364 _c 471 464 458 539 433 589 _c 407 639 369 664 318 664 _c 318 742 _m 399 742 461 709 505 645 _c 548 580 570 486 570 364 _c 570 241 548 147 505 83 _c 461 19 399 -13 318 -13 _c 236 -13 173 19 130 83 _c 87 147 66 241 66 364 _c 66 486 87 580 130 645 _c 173 709 236 742 318 742 _c _cl}_d /one{636 0 110 0 544 729 _sc 124 83 _m 285 83 _l 285 639 _l 110 604 _l 110 694 _l 284 729 _l 383 729 _l 383 83 _l 544 83 _l 544 0 _l 124 0 _l 124 83 _l _cl}_d /two{{636 0 73 0 536 742 _sc 192 83 _m 536 83 _l 536 0 _l 73 0 _l 73 83 _l 110 121 161 173 226 239 _c 290 304 331 346 348 365 _c 380 400 402 430 414 455 _c 426 479 433 504 433 528 _c 433 566 419 598 392 622 _c 365 646 330 659 286 659 _c 255 659 222 653 188 643 _c 154 632 117 616 78 594 _c 78 694 _l 118 710 155 722 189 730 _c 223 738 255 742 284 742 _c }_e{359 742 419 723 464 685 _c 509 647 532 597 532 534 _c 532 504 526 475 515 449 _c 504 422 484 390 454 354 _c 446 344 420 317 376 272 _c 332 227 271 164 192 83 _c _cl}_e}_d /three{{636 0 76 -13 556 742 _sc 406 393 _m 453 383 490 362 516 330 _c 542 298 556 258 556 212 _c 556 140 531 84 482 45 _c 432 6 362 -13 271 -13 _c 240 -13 208 -10 176 -4 _c 144 1 110 10 76 22 _c 76 117 _l 103 101 133 89 166 81 _c 198 73 232 69 268 69 _c 330 69 377 81 409 105 _c 441 129 458 165 458 212 _c 458 254 443 288 413 312 _c 383 336 341 349 287 349 _c }_e{202 349 _l 202 430 _l 291 430 _l 339 430 376 439 402 459 _c 428 478 441 506 441 543 _c 441 580 427 609 401 629 _c 374 649 336 659 287 659 _c 260 659 231 656 200 650 _c 169 644 135 635 98 623 _c 98 711 _l 135 721 170 729 203 734 _c 235 739 266 742 296 742 _c 370 742 429 725 473 691 _c 517 657 539 611 539 553 _c 539 513 527 479 504 451 _c 481 423 448 403 406 393 _c _cl}_e}_d /four{636 0 49 0 580 729 _sc 378 643 _m 129 254 _l 378 254 _l 378 643 _l 352 729 _m 476 729 _l 476 254 _l 580 254 _l 580 172 _l 476 172 _l 476 0 _l 378 0 _l 378 172 _l 49 172 _l 49 267 _l 352 729 _l _cl}_d /five{{636 0 77 -13 549 729 _sc 108 729 _m 495 729 _l 495 646 _l 198 646 _l 198 467 _l 212 472 227 476 241 478 _c 255 480 270 482 284 482 _c 365 482 429 459 477 415 _c 525 370 549 310 549 234 _c 549 155 524 94 475 51 _c 426 8 357 -13 269 -13 _c 238 -13 207 -10 175 -6 _c 143 -1 111 6 77 17 _c 77 116 _l 106 100 136 88 168 80 _c 199 72 232 69 267 69 _c }_e{323 69 368 83 401 113 _c 433 143 450 183 450 234 _c 450 284 433 324 401 354 _c 368 384 323 399 267 399 _c 241 399 214 396 188 390 _c 162 384 135 375 108 363 _c 108 729 _l _cl}_e}_d /six{{636 0 70 -13 573 742 _sc 330 404 _m 286 404 251 388 225 358 _c 199 328 186 286 186 234 _c 186 181 199 139 225 109 _c 251 79 286 64 330 64 _c 374 64 409 79 435 109 _c 461 139 474 181 474 234 _c 474 286 461 328 435 358 _c 409 388 374 404 330 404 _c 526 713 _m 526 623 _l 501 635 476 644 451 650 _c 425 656 400 659 376 659 _c 310 659 260 637 226 593 _c }_e{192 549 172 482 168 394 _c 187 422 211 444 240 459 _c 269 474 301 482 336 482 _c 409 482 467 459 509 415 _c 551 371 573 310 573 234 _c 573 159 550 99 506 54 _c 462 9 403 -13 330 -13 _c 246 -13 181 19 137 83 _c 92 147 70 241 70 364 _c 70 479 97 571 152 639 _c 206 707 280 742 372 742 _c 396 742 421 739 447 735 _c 472 730 498 723 526 713 _c _cl}_e}_d /seven{636 0 82 0 551 729 _sc 82 729 _m 551 729 _l 551 687 _l 286 0 _l 183 0 _l 432 646 _l 82 646 _l 82 729 _l _cl}_d /eight{{636 0 68 -13 568 742 _sc 318 346 _m 271 346 234 333 207 308 _c 180 283 167 249 167 205 _c 167 161 180 126 207 101 _c 234 76 271 64 318 64 _c 364 64 401 76 428 102 _c 455 127 469 161 469 205 _c 469 249 455 283 429 308 _c 402 333 365 346 318 346 _c 219 388 _m 177 398 144 418 120 447 _c 96 476 85 511 85 553 _c 85 611 105 657 147 691 _c 188 725 245 742 318 742 _c }_e{390 742 447 725 489 691 _c 530 657 551 611 551 553 _c 551 511 539 476 515 447 _c 491 418 459 398 417 388 _c 464 377 501 355 528 323 _c 554 291 568 251 568 205 _c 568 134 546 80 503 43 _c 459 5 398 -13 318 -13 _c 237 -13 175 5 132 43 _c 89 80 68 134 68 205 _c 68 251 81 291 108 323 _c 134 355 171 377 219 388 _c 183 544 _m 183 506 194 476 218 455 _c }_e{242 434 275 424 318 424 _c 360 424 393 434 417 455 _c 441 476 453 506 453 544 _c 453 582 441 611 417 632 _c 393 653 360 664 318 664 _c 275 664 242 653 218 632 _c 194 611 183 582 183 544 _c _cl}_e}_d /nine{{636 0 63 -13 566 742 _sc 110 15 _m 110 105 _l 134 93 159 84 185 78 _c 210 72 235 69 260 69 _c 324 69 374 90 408 134 _c 442 178 462 244 468 334 _c 448 306 424 284 396 269 _c 367 254 335 247 300 247 _c 226 247 168 269 126 313 _c 84 357 63 417 63 494 _c 63 568 85 628 129 674 _c 173 719 232 742 306 742 _c 390 742 455 709 499 645 _c 543 580 566 486 566 364 _c }_e{566 248 538 157 484 89 _c 429 21 356 -13 264 -13 _c 239 -13 214 -10 189 -6 _c 163 -2 137 5 110 15 _c 306 324 _m 350 324 385 339 411 369 _c 437 399 450 441 450 494 _c 450 546 437 588 411 618 _c 385 648 350 664 306 664 _c 262 664 227 648 201 618 _c 175 588 162 546 162 494 _c 162 441 175 399 201 369 _c 227 339 262 324 306 324 _c _cl}_e}_d /L{557 0 98 0 552 729 _sc 98 729 _m 197 729 _l 197 83 _l 552 83 _l 552 0 _l 98 0 _l 98 729 _l _cl}_d /V{684 0 8 0 676 729 _sc 286 0 _m 8 729 _l 111 729 _l 342 115 _l 573 729 _l 676 729 _l 398 0 _l 286 0 _l _cl}_d /c{{550 0 55 -13 488 560 _sc 488 526 _m 488 442 _l 462 456 437 466 411 473 _c 385 480 360 484 334 484 _c 276 484 230 465 198 428 _c 166 391 150 339 150 273 _c 150 206 166 154 198 117 _c 230 80 276 62 334 62 _c 360 62 385 65 411 72 _c 437 79 462 90 488 104 _c 488 21 _l 462 9 436 0 410 -5 _c 383 -10 354 -13 324 -13 _c 242 -13 176 12 128 64 _c }_e{79 115 55 185 55 273 _c 55 362 79 432 128 483 _c 177 534 244 560 330 560 _c 358 560 385 557 411 551 _c 437 545 463 537 488 526 _c _cl}_e}_d /e{{615 0 55 -13 562 560 _sc 562 296 _m 562 252 _l 149 252 _l 153 190 171 142 205 110 _c 238 78 284 62 344 62 _c 378 62 412 66 444 74 _c 476 82 509 95 541 113 _c 541 28 _l 509 14 476 3 442 -3 _c 408 -9 373 -13 339 -13 _c 251 -13 182 12 131 62 _c 80 112 55 181 55 268 _c 55 357 79 428 127 481 _c 175 533 241 560 323 560 _c 397 560 455 536 498 489 _c }_e{540 441 562 377 562 296 _c 472 322 _m 471 371 457 410 431 440 _c 404 469 368 484 324 484 _c 274 484 234 469 204 441 _c 174 413 156 373 152 322 _c 472 322 _l _cl}_e}_d /f{352 0 23 0 371 760 _sc 371 760 _m 371 685 _l 285 685 _l 253 685 230 678 218 665 _c 205 652 199 629 199 595 _c 199 547 _l 347 547 _l 347 477 _l 199 477 _l 199 0 _l 109 0 _l 109 477 _l 23 477 _l 23 547 _l 109 547 _l 109 585 _l 109 645 123 690 151 718 _c 179 746 224 760 286 760 _c 371 760 _l _cl}_d /l{278 0 94 0 184 760 _sc 94 760 _m 184 760 _l 184 0 _l 94 0 _l 94 760 _l _cl}_d /m{{974 0 91 0 889 560 _sc 520 442 _m 542 482 569 511 600 531 _c 631 550 668 560 711 560 _c 767 560 811 540 842 500 _c 873 460 889 403 889 330 _c 889 0 _l 799 0 _l 799 327 _l 799 379 789 418 771 444 _c 752 469 724 482 686 482 _c 639 482 602 466 575 435 _c 548 404 535 362 535 309 _c 535 0 _l 445 0 _l 445 327 _l 445 379 435 418 417 444 _c 398 469 369 482 331 482 _c }_e{285 482 248 466 221 435 _c 194 404 181 362 181 309 _c 181 0 _l 91 0 _l 91 547 _l 181 547 _l 181 462 _l 201 495 226 520 255 536 _c 283 552 317 560 357 560 _c 397 560 430 550 458 530 _c 486 510 506 480 520 442 _c _cl}_e}_d /n{634 0 91 0 549 560 _sc 549 330 _m 549 0 _l 459 0 _l 459 327 _l 459 379 448 417 428 443 _c 408 469 378 482 338 482 _c 289 482 251 466 223 435 _c 195 404 181 362 181 309 _c 181 0 _l 91 0 _l 91 547 _l 181 547 _l 181 462 _l 202 494 227 519 257 535 _c 286 551 320 560 358 560 _c 420 560 468 540 500 501 _c 532 462 549 405 549 330 _c _cl}_d /o{612 0 55 -13 557 560 _sc 306 484 _m 258 484 220 465 192 427 _c 164 389 150 338 150 273 _c 150 207 163 156 191 118 _c 219 80 257 62 306 62 _c 354 62 392 80 420 118 _c 448 156 462 207 462 273 _c 462 337 448 389 420 427 _c 392 465 354 484 306 484 _c 306 560 _m 384 560 445 534 490 484 _c 534 433 557 363 557 273 _c 557 183 534 113 490 63 _c 445 12 384 -13 306 -13 _c 227 -13 165 12 121 63 _c 77 113 55 183 55 273 _c 55 363 77 433 121 484 _c 165 534 227 560 306 560 _c _cl}_d /r{411 0 91 0 411 560 _sc 411 463 _m 401 469 390 473 378 476 _c 366 478 353 480 339 480 _c 288 480 249 463 222 430 _c 194 397 181 350 181 288 _c 181 0 _l 91 0 _l 91 547 _l 181 547 _l 181 462 _l 199 495 224 520 254 536 _c 284 552 321 560 365 560 _c 371 560 378 559 386 559 _c 393 558 401 557 411 555 _c 411 463 _l _cl}_d /s{{521 0 54 -13 472 560 _sc 443 531 _m 443 446 _l 417 458 391 468 364 475 _c 336 481 308 485 279 485 _c 234 485 200 478 178 464 _c 156 450 145 430 145 403 _c 145 382 153 366 169 354 _c 185 342 217 330 265 320 _c 296 313 _l 360 299 405 279 432 255 _c 458 230 472 195 472 151 _c 472 100 452 60 412 31 _c 372 1 316 -13 246 -13 _c 216 -13 186 -10 154 -5 _c }_e{122 0 89 8 54 20 _c 54 113 _l 87 95 120 82 152 74 _c 184 65 216 61 248 61 _c 290 61 323 68 346 82 _c 368 96 380 117 380 144 _c 380 168 371 187 355 200 _c 339 213 303 226 247 238 _c 216 245 _l 160 257 119 275 95 299 _c 70 323 58 356 58 399 _c 58 450 76 490 112 518 _c 148 546 200 560 268 560 _c 301 560 332 557 362 552 _c 391 547 418 540 443 531 _c }_e{_cl}_e}_d /y{592 0 30 -207 562 547 _sc 322 -50 _m 296 -114 271 -157 247 -177 _c 223 -197 191 -207 151 -207 _c 79 -207 _l 79 -132 _l 132 -132 _l 156 -132 175 -126 189 -114 _c 203 -102 218 -75 235 -31 _c 251 9 _l 30 547 _l 125 547 _l 296 119 _l 467 547 _l 562 547 _l 322 -50 _l _cl}_d end readonly def /BuildGlyph {exch begin CharStrings exch 2 copy known not{pop /.notdef}if true 3 1 roll get exec end}_d /BuildChar { 1 index /Encoding get exch get 1 index /BuildGlyph get exec }_d FontName currentdict end definefont pop %!PS-Adobe-3.0 Resource-Font %%Title: cmmi10 %%Copyright: Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. %%Creator: Converted from TrueType to type 3 by PPR 25 dict begin /_d{bind def}bind def /_m{moveto}_d /_l{lineto}_d /_cl{closepath eofill}_d /_c{curveto}_d /_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d /_e{exec}_d /FontName /Cmmi10 def /PaintType 0 def /FontMatrix[.001 0 0 .001 0 0]def /FontBBox[-34 -250 1048 750]def /FontType 3 def /Encoding [ /e /r ] def /FontInfo 10 dict dup begin /FamilyName (cmmi10) def /FullName (cmmi10) def /Notice (Copyright (C) 1994, Basil K. Malyshev. All Rights Reserved.012BaKoMa Fonts Collection, Level-B. ) def /Weight (Regular) def /Version (1.1/12-Nov-94) def /ItalicAngle 0.0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def end readonly def /CharStrings 2 dict dup begin /e{{465 0 44 -10 430 442 _sc 201 -10 _m 169 -10 141 -2 117 14 _c 93 30 75 53 63 81 _c 50 109 44 138 44 170 _c 44 217 56 262 80 304 _c 104 346 136 379 177 404 _c 218 429 262 442 309 442 _c 337 442 362 434 384 419 _c 405 404 416 383 416 356 _c 416 317 400 288 370 270 _c 339 251 304 239 266 235 _c 228 231 185 229 139 229 _c 136 229 _l 125 189 120 155 120 127 _c }_e{120 97 126 71 140 49 _c 154 26 175 15 203 15 _c 243 15 281 24 318 42 _c 354 60 384 86 407 118 _c 409 120 411 121 415 121 _c 418 121 421 119 425 115 _c 428 111 430 108 430 105 _c 430 102 429 100 428 99 _c 404 65 371 38 330 19 _c 288 0 245 -10 201 -10 _c 142 255 _m 180 255 216 256 249 260 _c 282 264 312 273 339 287 _c 365 301 379 324 379 355 _c }_e{379 367 375 378 368 387 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0.000000 rotate /Cmsy10 findfont 14.4 scalefont setfont 0.000000 4.203125 moveto /bar glyphshow 3.984116 4.203125 moveto /bar glyphshow /Cmmi10 findfont 14.4 scalefont setfont 7.968231 4.203125 moveto /e glyphshow /Cmsy10 findfont 14.4 scalefont setfont 14.657623 4.203125 moveto /bar glyphshow 18.641739 4.203125 moveto /bar glyphshow grestore 0 setlinecap [6 6] 0 setdash gsave 483.092 367.525 m 503.252 367.525 l stroke grestore gsave 519.092500 358.485000 translate 0.000000 rotate /Cmsy10 findfont 14.4 scalefont setfont 0.000000 4.203125 moveto /bar glyphshow 3.984116 4.203125 moveto /bar glyphshow /Cmmi10 findfont 14.4 scalefont setfont 7.968231 4.203125 moveto /r glyphshow /Cmsy10 findfont 14.4 scalefont setfont 14.460876 4.203125 moveto /bar glyphshow 18.444992 4.203125 moveto /bar glyphshow grestore end showpage ================================================ FILE: multigrid/multigrid.tex ================================================ \label{ch:multigrid} \section{Elliptic equations} The simplest elliptic PDE is {\em Laplace's equation}: \begin{equation} \nabla^2 \phi = 0 \end{equation} Only slightly more complex is {\em Poisson's equation} (Laplace + a source term): \begin{equation} \nabla^2 \phi = f \end{equation} These equations can arise in electrostatics (for the electric potential), solving for the gravitational potential from a mass distribution, or enforcing a divergence constraint on a vector field (we'll see this when we consider incompressible flow). Another common elliptic equation is the {\em Helmholtz equation}: \begin{equation} (\alpha - \nabla \cdot \beta \nabla) \phi = f \end{equation} A Helmholtz equation can arise, for example, from a time-dependent equation (like diffusion) by discretizing in time. Notice that there is no time-dependence in any of these equations. The quantity $\phi$ is specified instantaneously in the domain subject to boundary conditions. This makes the solution methods very different then what we saw for hyperbolic problems. \section{\label{elliptic:sec:fft} Fourier Method} A direct way of solving a constant-coefficient elliptic equation is using Fourier transforms. Using a general Fourier transform (which we consider here) works only for periodic boundary conditions, but other basis functions (e.g., all sines or all cosines) can be used for other boundary conditions. \MarginPar{ref?} Consider the Poisson equation: \begin{equation} \nabla^2 \phi = f \end{equation} We will difference this in a second-order accurate fashion---see Figure~\ref{fig:mg:laplacian}. \begin{figure}[t] \centering \includegraphics[width=\linewidth]{laplacian} \caption[Data centerings for the discrete Laplacian]{\label{fig:mg:laplacian} The centerings of the first and second derivatives for a standard Laplacian discretization. Our data, $\phi$, is cell-centered. The first-derivatives, $d\phi/dx$, are edge-centered, and the second-derivative, $d^2\phi/dx^2$, is cell-centered.} \end{figure} % In 1-d, the Laplacian is just the second-derivative. If our solution is defined at cell-centers, then we first compute the first-derivative on cell edges: \begin{align} \left . \frac{d\phi}{dx} \right |_{i-\myhalf} &= \frac{\phi_i - \phi_{i-1}}{\dx}\\ \left . \frac{d\phi}{dx} \right |_{i+\myhalf} &= \frac{\phi_{i+1} - \phi_i}{\dx} \end{align} These are second-order accurate on the interface. We can then compute the second-derivative at the cell-center by differencing these edge values: \begin{equation} \left . \frac{d^2\phi}{dx^2} \right |_i = \frac{ d\phi/dx |_{i+\myhalf} - d\phi/dx |_{i-\myhalf}}{\dx} \end{equation} The extension to 2-d is straightforward. Thinking of the Laplacian as $\nabla^2 \phi$ = $\nabla \cdot \nabla \phi$, we first compute the gradient of $\phi$ on edges: \begin{align} [\nabla \phi \cdot \hat{x}]_{i+\myhalf,j} &= \frac{\phi_{i+1,j} - \phi_{i,j}}{\dx} \\ [\nabla \phi \cdot \hat{y}]_{i,j+\myhalf} &= \frac{\phi_{i,j+1} - \phi_{i,j}}{\dy} \end{align} Again, since this is defined on edges, this represents a centered difference, and is therefore second-order accurate. We then difference the edge-centered gradients to the center to get the Laplacian at cell-centers: \begin{align} [\nabla^2 \phi]_{i,j} &= \frac{[\nabla \phi \cdot \hat{x}]_{i+\myhalf,j} - [\nabla \phi \cdot \hat{x}]_{i-\myhalf,j}}{\dx} + \frac{[\nabla \phi \cdot \hat{y}]_{i,j+\myhalf} - [\nabla \phi \cdot \hat{y}]_{i,j-\myhalf}}{\dy} \nonumber\\ % &= \frac{\phi_{i+1,j} - 2\phi_{i,j} + \phi_{i-1,j}}{\dx^2} + \frac{\phi_{i,j+1} - 2\phi_{i,j} + \phi_{i,j-1}}{\dy^2} = f_{i,j} \end{align} Again, since we used a centered-difference of the edge values, this expression is second-order accurate. This is the standard {\em 5-point stencil} for the 2-d Laplacian\footnote{There are other possible second-order accurate stencils, including a 9-point stencil in 2-d, that are less commonly used.}. We now assume that we have an FFT subroutine (see \S~\ref{sec:intro:ffts}) that can take our discrete real-space data, $\phi_{i,j}$ and return the discrete Fourier coefficients, $\Phi_{k_x,k_y}$, and likewise for the source term: \begin{equation} \Phi_{k_x,k_y} = \mathcal{F}(\phi_{i,j}) \quad F_{k_x,k_y} = \mathcal{F}(f_{i,j}) \end{equation} The power of the Fourier method is that derivatives in real space are multiplications in Fourier space, which makes the solution process in Fourier space straightforward. We now express $\phi_{i,j}$ and $f_{i,j}$ as sums over their Fourier components. Here we define $M$ as the number of grid points in the $x$-direction and $N$ as the number of grid points in the $y$-direction. As before, we are using $i$ as the grid index, we will use $\imag$ as the imaginary unit: \begin{align} \phi_{i,j} &= \frac{1}{MN} \sum_{k_x = 0}^{M-1} \sum_{k_y = 0}^{N-1} \Phi_{k_x,k_y} e^{2\pi\imag i k_x/M} e^{2\pi\imag j k_y/N} \\ f_{i,j} &= \frac{1}{MN} \sum_{k_x = 0}^{M-1} \sum_{k_y = 0}^{N-1} F_{k_x,k_y} e^{2\pi\imag i k_x/M} e^{2\pi\imag j k_y/N} \end{align} Inserting these into the differenced equation, we have: \begin{align} \frac{1}{MN}\sum_{k_x=0}^{M-1}\sum_{k_y=0}^{N-1} \biggl \{ &\frac{\Phi_{k_x,k_y}}{\dx^2} e^{2\pi\imag j k_y/N} \left [ e^{2\pi\imag (i+1) k_x/M} -2 e^{2\pi\imag i k_x/M} + e^{2\pi\imag (i-1) k_x/M} \right ] + \nonumber \\ &\frac{\Phi_{k_x,k_y}}{\dy^2} e^{2\pi\imag i k_x/M} \left [ e^{2\pi\imag (j+1) k_y/N} -2 e^{2\pi\imag j k_y/N} + e^{2\pi\imag (j-1) k_y/N} \right ] \biggr \} = \nonumber \\ \frac{1}{MN}\sum_{k_x=0}^{M-1}\sum_{k_y=0}^{N-1} & F_{k_x,k_y} e^{2\pi\imag i k_x/M} e^{2\pi\imag j k_y/N} \end{align} We can bring the righthand side into the sums on the left, and we can then look at just a single $(k_x,k_y)$ term in the series: \begin{align} e^{2\pi\imag i k_x/M} e^{2\pi\imag j k_y/N} \biggl \{ &\frac{\Phi_{k_x,k_x}}{\dx^2} \left [e^{2\pi\imag k_x/M} + e^{-2\pi\imag k_x/M} - 2 \right ] + \nonumber \\ &\frac{\Phi_{k_x,k_x}}{\dy^2} \left [e^{2\pi\imag k_y/N} + e^{-2\pi\imag k_y/N} - 2 \right ] - F_{k_x,k_y} \biggr \} = 0 \end{align} Simplifying, we have: \begin{equation} \Phi_{k_x,k_y} = \frac{1}{2}\frac{F_{k_x,k_y}} {\left [\cos(2\pi k_x/M) - 1 \right ] \Delta x^{-2} + \left [\cos(2\pi k_y/N) - 1 \right ] \Delta y^{-2}} \label{eq:FFTsol} \end{equation} This is the algebraic solution to the Poisson equation in Fourier (frequency) space. Once we evaluate this, we can get the real-space solution by doing the inverse transform: \begin{equation} \phi_{i,j} = \mathcal{F}^{-1}(\Phi_{k_x,k_y}) \end{equation} We can test this technique with the source term: \begin{align} f =\, & 8\pi^2\cos(4\pi y) \left [\cos(4\pi x) - \sin(4\pi x) \right ] - \nonumber \\ & 16\pi^2 \left [ \sin(4\pi x)\cos(2\pi y)^2 + \sin(2\pi x)^2 \cos(4\pi y) \right ] \label{eq:mg:fftsource} \end{align} which has the analytic solution\footnote{Note: throughout this chapter, we devise test problems by picking a function that meets the desired boundary conditions and then inserting it into the analytic equation we are solving to find the righthand side}: \begin{equation} \phi = \sin(2\pi x)^2 \cos(4\pi y) + \sin(4\pi x)\cos(2\pi y)^2 \end{equation} Note that this solution has the required periodic behavior. Figure~\ref{fig:mg:fftpoisson} shows the solution. \begin{figure}[t] \centering \includegraphics[width=0.53\linewidth]{poisson_fft} \includegraphics[width=0.44\linewidth]{fft-poisson-converge} \caption[FFT solution to the Poisson equation]{\label{fig:mg:fftpoisson} (left) Solution to the Poisson equation on a $64^2$ grid with source from Eq.~\ref{eq:mg:fftsource}. (right) Error vs.\ the true solution as a function of resolution for the Fourier method, showing second-order convergence. \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/elliptic/poisson_fft.py}{poisson\_fft.py}}} \end{figure} The main downside of this approach is that, because we solve for a single component independently (Eq.~\ref{eq:FFTsol}), this only works for linear problems with constant coefficients. This makes it an excellent choice for cosmological problems solving the gravitational Poisson equation with periodic boundaries on all sides of the domain (see, e.g., \cite{heitmann:2008}). However, for a problem like: \begin{equation} \nabla \cdot (\beta \nabla \phi) = f \end{equation} there would be ``cross-talk'' between the Fourier modes of $\beta$ and $\phi$, and we would not be able to solve for a single mode of $\Phi_{k_x,k_y}$ independently. We discuss more general methods that work for these forms next. \section{Relaxation} Relaxation is an iterative technique, and as we will see shortly, it provides the basis for the multigrid technique. Consider Poisson's equation, again differenced as: \begin{equation} \frac{\phi_{i+1,j} - 2 \phi_{i,j} + \phi_{i-1,j}}{\dx^2} + \frac{\phi_{i,j+1} - 2 \phi_{i,j} + \phi_{i,j-1}}{\dy^2} = f_{i,j} \end{equation} For each zone $(i,j)$, we couple in the zones $\pm 1$ in $x$ and $\pm 1$ in $y$. For the moment, consider the case where $\dx = \dy$. If we solve this discretized equation for $\phi_{i,j}$, then we have: \begin{equation} \phi_{i,j} = \frac{1}{4} (\phi_{i+1,j} + \phi_{i-1,j} + \phi_{i,j+1} + \phi_{i,j-1} - \dx^2 f_{i,j} ) \end{equation} A similar expression exists for every zone in our domain, coupling all the zones together. We can't separate the solution of $\phi_{i,j}$ for the neighboring zones, but instead can apply an iterative technique called {\em relaxation} (also sometimes called {\em smoothing} because generally speaking the solution to elliptic equations is a smooth function) to find the solution for $\phi$ everywhere. Imagine an initial guess to $\phi$: $\phi_{i,j}^{(0)}$. We can improve that guess by using our difference equation to define a new value of $\phi$, $\phi_{i,j}^{(1)}$: \begin{equation} \phi_{i,j}^{(1)} = \frac{1}{4} (\phi_{i+1,j}^{(0)} + \phi_{i-1,j}^{(0)} + \phi_{i,j+1}^{(0)} + \phi_{i,j-1}^{(0)} - \dx^2 f_{i,j} ) \end{equation} or generally, the $k+1$ iteration will see: \begin{equation} \phi_{i,j}^{(k+1)} = \frac{1}{4} (\phi_{i+1,j}^{(k)} + \phi_{i-1,j}^{(k)} + \phi_{i,j+1}^{(k)} + \phi_{i,j-1}^{(k)} - \dx^2 f_{i,j} ) \end{equation} This will (slowly) converge to the true solution\footnote{Formally, convergence is only guaranteed if the matrix in our linear system is {\em diagonally dominant}. The Laplacian used here is not quite diagonally domainant, but these methods still converge.}, since each zone is coupled to each other zone (and to the boundary values that we need to specify---more on that in a moment). This form of relaxation is called {\em Jacobi iteration}. To implement this, you need two copies of $\phi$---the old iteration value and the new iteration value. An alternate way to do the relaxation is to update $\phi_{i,j}$ in place, as soon as the new value is known. Thus the neighboring cells will see a mix of the old and new solutions. We can express this in-place updating as: \begin{equation} \phi_{i,j} \leftarrow \frac{1}{4} (\phi_{i+1,j} + \phi_{i-1,j} + \phi_{i,j+1} + \phi_{i,j-1} - \dx^2 f_{i,j} ) \end{equation} This only requires a single copy of $\phi$ to be stored. This technique is called {\em Gauss-Seidel iteration}. A host of other relaxation methods exist, including linear combinations of these two. An excellent discussion of these approaches, and their strengths and weaknesses is given in \cite{multigridtutorial}. Next consider the Helmholz equation with constant coefficients: \begin{equation} (\alpha - \beta \nabla^2) \phi = f \end{equation} We can discretize this as: \begin{equation} \alpha \phi_{i,j} - \beta \left ( \frac{\phi_{i+1,j} - 2 \phi_{i,j} + \phi_{i-1,j}}{\dx^2} + \frac{\phi_{i,j+1} - 2 \phi_{i,j} + \phi_{i,j-1}}{\dy^2} \right ) = f_{i,j} \end{equation} and the update of $\phi_{i,j}$ through relaxation is: \begin{equation} \phi_{i,j} \leftarrow \left . \left ( f_{i,j} + \frac{\beta}{\dx^2} \phi_{i+1,j} + \frac{\beta}{\dx^2} \phi_{i-1,j} + \frac{\beta}{\dy^2} \phi_{i,j+1} + \frac{\beta}{\dy^2} \phi_{i,j-1} \right ) \middle / \left ( \alpha + \frac{2 \beta}{\dx^2} + \frac{2 \beta}{\dy^2} \right ) \right . \end{equation} Notice that if $\alpha = 0$, $\beta = -1$, and $\dx = \dy$, we recover the relaxation expression for Poisson's equation from above. \subsection{Boundary conditions} When using a cell-centered grid, no points fall exactly on the boundary, so we need to use ghost cells to specify boundary conditions. A single ghost cell is sufficient for the 5-point stencil used here. The common types of boundary conditions are {\em Dirichlet} (specified value on the boundary), {\em Neumann} (specified first derivative on the boundary), and periodic. Some restrictions apply (see discuss this later, in \S~\ref{sec:multigrid:solvability}). % created by figures/multigrid/fv-fd_bnd.py \begin{figure}[h] \centering \includegraphics[width=\linewidth]{fv-fd_grid_bc} \caption[Node-centered vs.\ cell-centered data at boundaries]{\label{mg:fig:bcs} A node-centered (top) and cell-centered (bottom) finite difference grid showing the data and domain boundaries. Notice that for the cell-centered grid, there is no data point precisely at the boundary.} \end{figure} Consider Dirichlet boundary conditions, specifying values $\phi_l$ on the left and $\phi_r$ on the right boundaries.\footnote{If the value, $\phi_l$ or $\phi_r$ is zero, we call this a {\em homogeneous boundary condition}. Otherwise we call it an {\em inhomogeneous boundary condition}} We'll label the first zone inside the domain, at the left boundary, $\mathrm{lo}$, and the last zone inside the domain, at the right boundary, $\mathrm{hi}$---see Figure~\ref{mg:fig:bcs}. To second order, we can average the zone values on either side of the interface to get the boundary condition: \begin{eqnarray} \phi_l &=& \frac{1}{2} ( \phi_\mathrm{lo} + \phi_\mathrm{lo-1} ) \\ \phi_r &=& \frac{1}{2} ( \phi_\mathrm{hi} + \phi_\mathrm{hi+1} ) \end{eqnarray} This then tells us that the values we need to assign to the ghost cells are: \begin{eqnarray} \label{eq:bc_inhomo_dir} \phi_\mathrm{lo-1} &=& 2 \phi_l - \phi_\mathrm{lo} \\ \phi_\mathrm{hi+1} &=& 2 \phi_r - \phi_\mathrm{hi} \end{eqnarray} If we instead consider Neumann boundary conditions, we specify values of the derivative on the boundaries: $\phi_x |_l$ on the left and $\phi_x |_r$ on the right. We note that a single difference across the boundary is second-order accurate on the boundary (it is a centered-difference there), so to second-order: \begin{eqnarray} \phi_x |_l &=& \frac{\phi_\mathrm{lo} - \phi_\mathrm{lo-1}}{\dx} \\ \phi_x |_r &=& \frac{\phi_\mathrm{hi+1} - \phi_\mathrm{hi}}{\dx} \end{eqnarray} This then tells us that the ghost cells are filled as: \begin{eqnarray} \label{eq:bc_inhomo_neum} \phi_\mathrm{lo-1} &=& \phi_\mathrm{lo} - \dx \, \phi_x |_l \\ \phi_\mathrm{hi+1} &=& \phi_\mathrm{hi} + \dx \, \phi_x |_r \end{eqnarray} \subsection{Residual and true error} The {\em residual error} is a measure of how well our discrete solution satisfies the discretized equation. For the Poisson equation, we can the residual as: \begin{equation} r_{i,j} = f_{i,j} - (L \phi)_{i,j} \end{equation} and the residual error as: \begin{equation} \epsilon^{(r)} = \| r \| \end{equation} where $L$ represents our discretized Laplacian. Note that $r$ is the error with respect to the discrete form of the equation. The true error is the measure of how well our discrete solution approximates the true solution. If $\phi^\mathrm{true}$ satisfies $\nabla^2 \phi^\mathrm{true} = f$, then the true error in each zone is \begin{equation} e_{i,j} = \phi^\mathrm{true}(x_i,y_j) - \phi_{i,j} \end{equation} and \begin{equation} \epsilon^\mathrm{true} = \| e_{i,j} \| \end{equation} We can make $\epsilon^{(r)}$ approach machine precision by performing more and more relaxation iterations, but after some point, this will no longer improve $\epsilon^\mathrm{true}$. The only way to improve $\epsilon^\mathrm{true}$ is to make $\dx$ and $\dy$ smaller. In practice we do not know the true solution so we cannot compute $\epsilon^\mathrm{true}$ and will instead have to rely on $\epsilon^{(r)}$ to monitor our error. Note that since our operator is linear, \begin{equation} L e = L\phi^\mathrm{true} - L\phi = f - L\phi = r \end{equation} so the error in our solution obeys a Poisson equation with the residual as the source---we'll see this in the next section. \subsection{Norms} There are several different norms that are typically used in defining errors on the grid. The $L_\infty$ norm (or `inf'-norm) is just the maximum error on the grid: \begin{equation} \|e\|_\infty = \max_{i,j} \{ |e_{i,j}| \} \end{equation} This will pick up on local errors. The $L_1$ norm and $L_2$ norms are more global. \begin{eqnarray} \|e\|_1 &=& \frac{1}{N} \sum_{i,j} |e_{i,j} | \\ \|e\|_2 &=& \left ( \frac{1}{N} \sum_{i,j} |e_{i,j} |^2 \right )^{1/2} \end{eqnarray} Generally, the measure in $L_2$ falls between $L_\infty$ and $L_1$. Regardless of the norm used, if the problem converges, it should converge in all norms. For reference, the AMReX library\footnote{\url{https://github.com/amrex-codes}} uses $L_\infty$ in its multigrid solvers. \subsection{Performance} % this figure can be created with figures/multigrid/smooth-separate.py \begin{figure} \centering \includegraphics[width=\linewidth]{smooth-error} \caption[Convergence as a function of number of iterations using Gauss-Seidel relaxation]{\label{fig:smootherror} Gauss-Seidel relaxation applied to $\phi_{xx} = \sin(x)$ with $\phi(0) = \phi(1) = 0$. Shown are the L2 norm of the error compared with the true solution (solid lines) and the L2 norm of the residual (dotted lines) for 3 different resolutions (16, 32, and 64 zones). \\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/multigrid/smooth.py}{smooth.py}}} \end{figure} Consider the simple Poisson problem\footnote{This is the test problem used throughout {\em A Multigrid Tutorial} \cite{multigridtutorial}. We use the same problem here to allow for easy comparison to the discussions in that text.} on $x \in [0,1]$: \begin{equation} \phi_{xx} = \sin(x), \qquad \phi(0) = \phi(1) = 0 \end{equation} The analytic solution to this is simply \begin{equation} \phi^a(x) = -\sin(x) + x \sin(1) \end{equation} We can perform smoothing and compute both the error against the analytic solution (the `true' error), $e \equiv \| \phi^a(x_i) - \phi_i \|_2$ and the residual error, $\| r_i \|_2$. Figure~\ref{fig:smootherror} shows these errors as a function of the number of smoothing iterations for 3 different resolutions. Notice that the true error stalls at a relatively high value---this is the truncation error of the method. From one resolution to the next, the true error changes as $\dx^2$, indicating that we are converging as our method should. No additional amount of smoothing will change this---we are getting the best answer to the problem we can with our choice of discretization. In contrast, the residual error decreases to machine precision levels---this is indicating that our solution is an exact solution to the discrete equation (to roundoff-error). In practice, we can only monitor the residual error, not the true error, and we hope that small residual error implies a small true error. Figure~\ref{fig:smoothnorms} shows the error with respect to the true solution and of the residual for pure smoothing in three different norms. The overall behavior is qualitative similar regardless of the choice of norm. % this figure can be created with figures/multigrid/smooth-separate.py \begin{figure} \centering \includegraphics[width=\linewidth]{smooth-error-norms} \caption[Convergence of smoothing in different norms]{\label{fig:smoothnorms} Gauss-Seidel relaxation applied to $\phi_{xx} = \sin(x)$ with $\phi(0) = \phi(1) = 0$. This is like figure \ref{fig:smootherror}, but now we show the error in 3 different norms: $L_1$, $L_2$, and $L_\infty$. \\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/multigrid/smooth-norms.py}{smooth-norms.py}}} \end{figure} To demonstrate the influence of the boundary conditions, Figure~\ref{fig:smooth-badbcs} shows the norm of the true error for the same problem, but this time with a naive implementation of the boundary conditions---simply initializing the ghost cell to the boundary value, instead of averaging to the interface: \begin{align} \phi_\mathrm{lo-1} &= \phi_\mathrm{lo} \\ \phi_\mathrm{hi+1} &= \phi_\mathrm{hi} \end{align} We see that with this mistake at the boundaries, the error of the entire solution is affected, and we get only first-order convergence with resolution (this can be seen by looking at the spacing of the curves). The previous solution, with the correct BCs is shown for reference, and shows second-order convergence. It is important to note that because every zone is linked to every other zone (and the boundary) in elliptic problems, an error at the boundary can pollute the global solution. \begin{figure} \centering \includegraphics[width=\linewidth]{smooth-badBCs} \caption[Convergence of smoothing in first-order BCs]{\label{fig:smooth-badbcs} The same problem as in figure \ref{fig:smootherror}, but now we done with a naive first-order boundary conditions---just initializing the ghost cell to the boundary value (solid lines). We see that this achieves only first-order convergence in the true error. The correct second-order implmentation is shown as the dotted lines. \\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/multigrid/smooth-badbcs.py}{smooth-badbcs.py}}} \end{figure} \begin{figure}[t] \centering \includegraphics[width=0.8\linewidth]{smooth-multimode} \caption[Smoothing of different wavenumbers]{\label{fig:mg:smooth} Error in the solution to $\phi'' = 0$ given an initial guess with 3 different wavenumbers of noise. A 128 zone grid was used. The different curves are different numbers of smoothing iterations. \\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/multigrid/smooth-modes.py}{smooth-modes.py}}} \end{figure} \subsection{Frequency/wavelength-dependent error} We can think of the error in the solution as a superposition of high (short) and low (long) frequency (wavelength) modes. Smoothing works really well to eliminate the short wavelength noise quickly, but many iterations are needed to remove the long wavelength noise (see Figure~\ref{fig:smooth}). A very important concept to understand here is that when we talk about long wavelength error, we are expressing it in terms of the number of zones across the feature, not as physical length. We can get an intuitive feel for this behavior by thinking about how smoothing works. Every zone is coupled to every other zone, and we can think about each zone seeing one more zone away for each iteration. When the error is short wavelength, that means that there are only a few zones across it, and after a few iterations, all of the zones have seen the short wavelength error, and can eliminate it. For a very long wavelength error, many iterations will be needed until the smoothing couples one zone to another that is a wavelength away. This behavior suggests that if we could represent our problem on a coarser grid, the error will now be of shorter wavelength, and smoothing will be more efficient. This is the core idea behind multigrid, which we see next. \begin{exercise}[Smoothing the 1-d Laplace equation] {Implement 1-d smoothing for the Laplace equation on cc-grid. Use an initial guess for the solution: \begin{equation} \label{eq:mg:phimodes} \phi_0(x) = \frac{1}{3} ( \sin(2\pi x) + \sin(2\pi \, 8 x) + \sin(2\pi \, 16 x) ) \end{equation} on a 128 zone grid with Dirichlet boundary conditions. This initial guess has both high-frequency and low-frequency noise. Observe that the high-frequency stuff goes after only a few smoothing iterations, but many iterations are needed to remove the low-frequency noise. You should see something like Figure~\ref{fig:smooth}. } \end{exercise} Figure~\ref{fig:mg:smooth} illustrates this behavior. We are solving the Laplace equation, $\phi^{\prime\prime} = 0$ in 1-d on $[0,1]$ with homogeneous Dirichlet boundary conditions. The solution is simply $\phi = 0$. We use Eq.~\ref{eq:mg:phimodes} as an initial guess---this is a superposition of 3 different modes. We see that the after just a few iterations, the shortest wavelength mode, $\sin(2\pi 16x)$ is no longer apparent in the error, and just the two longer wavelength modes dominate. By 100 iterations, the error appears to be only due to the longest wavelength mode, $\sin(2\pi x)$. Even after 1000 iterations, we still see this mode in the error. This demonstrates that the longest wavelength modes in the error take the longest to smooth away. \section{Multigrid} The text {\em A Multigrid Tutorial}~\cite{multigridtutorial} provides an excellent introduction to the mechanics of multigrid. The discussion here differs mainly in that we are dealing with cell-centered/finite-volume data. We already saw that the treatment of boundary conditions is more complicated because we do not have a point directly on the boundary. The other complication comes in transferring the data from a fine grid to a coarser grid, and back. Before we discuss the multigrid technique, we'll understand how the operations between grids work. \subsection{Prolongation and restriction on cell-centered grids} Multigrid relies on transferring the problem up and down a hierarchy of grids. The process of moving the data from the fine grid to the coarse grid is called {\em restriction}. The reverse process, moving data from the coarse grid to the fine grid is called {\em prolongation}. If the data on our grid is a conserved quantity, we want restriction and prolongation to conserve the amount of stuff when transitioning data between the fine and coarse grids. \subsubsection{1-d} Consider a 1-d finite-volume/cell-centered finite-difference grid shown in Figure~\ref{fig:mg:1dgrid-prolong}---we see a fine grid and the corresponding coarser grid. % % this figure can be created by figures/multigrid/fvrestrict.py \begin{figure}[t] \centering \includegraphics[width=0.8\linewidth]{fvrestrict} \caption[The geometry for 1-d prolongation and restriction]{\label{fig:mg:1dgrid-prolong} A fine grid and corresponding underlying coarse grid.} \end{figure} % If $\phi$ represents a density, then conservation requires: \begin{equation} \phi^c_j = \frac{1}{\dx^c} (\dx^f \phi^f_i + \dx^f \phi^f_{i+1} ) \end{equation} or, for a jump in 2 in resolution ($\dx^c = 2 \dx^f$), \begin{equation} \phi^c_j = \frac{1}{2} (\phi^f_i + \phi^f_{i+1} ) \end{equation} This latter form appears as a simple average to the interface of the two fine cells / center of the corresponding coarse cell. The simplest type of prolongation is simply {\em direct injection}: \begin{align} \phi^f_i &= \phi^c_j \\ \phi^f_{i+1} &= \phi^c_j \end{align} A higher-order method is to do linear reconstruction of the coarse data and then average under the profile, e.g., \begin{equation} \phi(x) = \frac{\phi^c_{j+1} - \phi^c_{j-1}}{2\dx^c} (x - x_j^c) + \phi_j^c \end{equation} To second-order, we can find the values of $\phi^f_i$ and $\phi^f_{i+1}$ by evaluating $\phi(x)$ at the $x$-coordinate corresponding to their cell-centers, \begin{align} x^f_i &= x^c_j - \frac{\dx^c}{4} \\ x^f_{i+1} &= x^c_j + \frac{\dx^c}{4} \end{align} giving \begin{align} \phi^f_i &= \phi^c_j - \frac{1}{8} (\phi^c_{j+1} - \phi^c_{j-1}) \\ \phi^f_{i+1} &= \phi^c_j + \frac{1}{8} (\phi^c_{j+1} - \phi^c_{j-1}) \end{align} Notice that this is conservative, since $\dx^f (\phi^f_i + \phi^f_{i+1}) = \dx^c \phi^c_j$. \subsubsection{2-d} % this figure can be created by figures/multigrid/2dgrid-mg.py \begin{figure}[t] \centering \includegraphics[width=0.8\linewidth]{2dgrid-prolong} \caption[The geometry for 2-d prolongation and restriction]{\label{fig:2dgrid-prolong} Four fine cells and the underlying coarse grid. For prolongation, the fine cells in red are initialized from a coarse parent. The gray coarse cells are used in the reconstruction of the coarse data. For restriction, the fine cells are averaged to the underlying coarse cell.} \end{figure} Restriction from the fine grid to the coarse grid is straightforward. Since the fine cells are perfectly enclosed by a single coarse cell, we simply average: \begin{equation} \phi_{i,j}^c = \frac{1}{4} ( \phi_{--}^f + \phi_{+-}^f + \phi_{-+}^f + \phi_{++}^f ) \end{equation} Prolongation requires us to reconstruct the coarse data and use this reconstruction to determine what the fine cell values are. For instance, a linear reconstruction of the coarse data in $x$ and $y$ is: \begin{equation} \phi(x,y) = \frac{m_x}{\dx^c} (x - x_i^c) + \frac{m_y}{\dy^c} (y - y_j^c) + \phi_{i,j}^c \end{equation} with slopes: \begin{eqnarray} m_x &=& \frac{1}{2}({\phi_{i+1,j}^c - \phi_{i-1,j}^c}) \\ m_y &=& \frac{1}{2}({\phi_{i,j+1}^c - \phi_{i,j-1}^c}) \end{eqnarray} % When averaged over the coarse cell, $\phi(x,y)$ recovers the average, $\phi_{i,j}^c$ in that cell (this means that our interpolant is conservative). We can evaluate the value in the fine cells by evaluating $\phi(x,y)$ at the center of the fine cells, \begin{eqnarray} x_\pm^f &=& x_i^c \pm \frac{\dx^c}{4} \\ y_\pm^f &=& y_j^c \pm \frac{\dy^c}{4} \\ \end{eqnarray} This gives \begin{equation} \phi_{\pm\pm}^f = \phi_{i,j}^c \pm \frac{1}{4}m_x \pm \frac{1}{4}m_y \end{equation} (Note: you would get the same expression if you averaged $\phi(x,y)$ over the fine cell.) There are other options for prolongation and restriction, both of higher and lower order accuracy. However, the methods above seem to work well. \subsection{Multigrid cycles} The basic idea of multigrid\footnote{ In these discussions, we use {\em multigrid} to mean {\em geometric multigrid}, where the coarsening is done to the grid geometry directly. The alternative is {\em algebraic multigrid}, where it is the structure of the matrix in the linear system itself that is coarsened.} is to smooth a little on the current grid solving $L\phi = f$, compute the residual, $r$, then {\em restrict} $r$ to a coarser grid and smooth on that grid solving $Le = r$, restrict again, $\ldots$. Once you reach a sufficiently coarse grid, the problem solved exactly. Then the data is moved up to the finer grids, a process called {\em prolongation}. The error on the coarse grid, $e$, is prolonged to the finer grid. This error is then used to correct the solution on the finer grid, some smoothing is done, and then the data is prolonged up again. Note: on the coarse grids, you are not solving the original system, but rather an error equation. If the boundary conditions in the original system are inhomogeneous, the boundary conditions for the error equations are now homogeneous. This must be understood by any ghost cell filling routines. There are many different forms of the multigrid process. The simplest is called the {\em V-cycle}. Here you start of the fine grid, restrict down to the coarsest, solve, and then prolong back up to the finest. The flow looks like a `V'. You continue with additional V-cycles until the residual error is smaller than your tolerance. \subsection{Bottom solver} % this figure can be created with figures/multigrid/mgtower.py \begin{figure}[t] \centering \includegraphics[width=\linewidth]{mgtower} \caption[A multigrid hierarchy]{\label{fig:mgtower} Illustration of the hierarchy of grids leading to the coarsest 2-zone grid (in one-dimension). Each grid has a single ghost cell to accommodate boundary conditions.} \end{figure} Once the grid is sufficiently coarse, the linear system is small enough to be solved directly. This is the bottom solver operation. In the most ideal case, where the finest grid is some power of 2, $N_x = N_y = 2^n$, then the multigrid procedure can continue down until a $2\times 2$ grid is created (Figure~\ref{fig:mgtower} illustrates this idea for a one-dimensional grid). This is the coarsest grid upon which one can still impose boundary conditions. With this small grid, just doing additional smoothing is sufficient enough to `solve' the problem. No fancy bottom solver is needed. For a general rectangular grid or one that is not a power of 2, the coarsest grid will likely be larger. For the general case, a linear system solver like conjugate gradient (or a variant) is used on the coarsest grid. \subsection{Boundary conditions throughout the hierarchy} The general inhomogeneous boundary conditions from Eqs.~\ref{eq:bc_inhomo_dir} and \ref{eq:bc_inhomo_neum} apply to the finest level. But because we are solving the residual equation of the coarsest levels in the multigrid hierarchy, the boundary conditions on $Le = r$ are all homogeneous (but of the same type, Dirichlet, Neumann, or periodic, as the fine level). Implementing these boundary conditions in your multigrid solver means that you will have separate actions for the fine level (where inhomogeneous boundaries may apply) and the coarser levels (where you will always be homogeneous). An alternate way to enforce boundary conditions is via {\em boundary charges}. For inhomogeneous boundary conditions, boundary charges can be used to convert the BCs to homogeneous BCs. This has the advantage of allowing the ghost cell filling routines only deal with the homogeneous case. Consider the one-dimensional Poisson equation, near the left boundary our discretized equation appears as: \begin{equation} \frac{\phi_\mathrm{lo-1} - 2\phi_\mathrm{lo} + \phi_\mathrm{lo+1}}{\dx^2} = f_\mathrm{lo} \end{equation} Inhomogeneous BCs at the left boundary would give the condition: \begin{equation} \phi_\mathrm{lo-1} = 2 \phi_l - \phi_\mathrm{lo} \end{equation} Substituting this into the discrete equation, we have: \begin{equation} \frac{2 \phi_l - \phi_\mathrm{lo} - 2\phi_\mathrm{lo} + \phi_\mathrm{lo+1}}{\dx^2} = f_\mathrm{lo} \end{equation} Bringing the boundary condition value over to the RHS, we see \begin{equation} \frac{- 3\phi_\mathrm{lo} + \phi_\mathrm{lo+1}}{\dx^2} = f_\mathrm{lo} - \frac{2\phi_l}{\dx^2} \end{equation} Now the left side looks precisely like the differenced Poisson equation with homogeneous Dirichlet BCs. The RHS has an additional `charge' that captures the boundary value. By modifying the source term, $f$, in the multigrid solver to include this charge, we can use the homogeneous ghost cell filling routines throughout the multigrid algorithm. This technique is discussed a bit in~\cite{colellanotes}. Note that the form of the boundary charge will depend on the form of the elliptic equation---the expressions derived above apply only for $\nabla^2 \phi = f$. \subsection{Stopping criteria} Repeated V-cycles are done until: \begin{equation} \| r \| < \epsilon \|f\| \end{equation} on the finest grid, for some user-input tolerance, $\epsilon$. Here, $\|f\|$ is called the {\em source norm}. If $\|f\| = 0$, then we stop when \begin{equation} \| r \| < \epsilon \end{equation} Picking the tolerance $\epsilon$ is sometimes problem-dependent, and generally speaking, a problem with a large number of zones will require a looser tolerance. The general rule-of-thumb is that each V-cycle should reduce your residual by about 1 order of magnitude. It is important that your bottom solver solves the coarse problem to a tolerance of $10^{-3}$ or $10^{-4}$ in order for the solver to converge. Figure~\ref{fig:mgerror} shows the true and residual errors for $\phi_{xx} = \sin(x)$ as a function of V-cycle number, illustrating the expected performance. \begin{figure} \centering \includegraphics[width=0.85\linewidth]{mg_error_vs_cycle} \caption[Error in solution as a function of multigrid V-cycle number]{\label{fig:mgerror} Error in the multigrid solution to our model problem ($\phi_{xx} = \sin(x)$) as a function of V-cycle. We see that the true error, $\|e\|$ stalls at truncation error while the residual error, $\|r\|$ reaches roundoff error, the same behavior as seen with smoothing alone (as expected). \\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/multigrid/mg_test.py}{mg\_test.py}}} \end{figure} The overall convergence of the multigrid algorithm is limited by the discretization of the Laplacian operator used and the implementation of the boundary conditions. Figure~\ref{fig:mg:convergence} shows the error in the solution as the number of zones is increased---demonstrating second-order convergence for our implementation. \begin{figure}[t] \centering \includegraphics[width=0.85\linewidth]{mg-converge} \caption[Convergence of the multigrid algorithm]{\label{fig:mg:convergence} Convergence of the multigrid algorithm. \\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/multigrid/mg_converge.py}{mg\_converge.py}}} \end{figure} \section{Solvability} \label{sec:multigrid:solvability} For $\nabla^2 \phi = f$ with periodic or Neumann boundaries all around, the sum of $f$ must equal $0$ otherwise the solution will not converge. Instead, we will simply find the solution increase each V-cycle. This is seen as follows: \begin{equation} \int_\Omega f d\Omega = \int_\Omega \nabla^2 \phi d\Omega = \int_{\partial \Omega} \nabla \phi \cdot n dS = 0 \end{equation} For all homogeneous Neumann boundaries, we have $\nabla \phi \cdot dS = 0$ by construction, so that integral is zero, requiring that the source integrate to zero. If the Neumann boundaries are inhomogeneous, there is still a solvability condition on $f$ based on the sum on the boundary values. A simple example of solvability is: \begin{equation} \phi^{\prime\prime} = 0 \end{equation} on $[a, b]$ with \begin{align} \phi^\prime(a) &= A \\ \phi^\prime(b) &= B \end{align} We can integrate $\phi^{\prime\prime} = 0$ to get $\phi(x) = \alpha x + \beta$ where $\alpha$ and $\beta$ are integration constants. But note that this is just a straight line, with a single slope, $\alpha$, so it is not possible to specify two unique slopes, $A$ and $B$ at the boundary unless there is a non-zero source term. For all periodic boundaries, we have $\nabla \phi |_\mathrm{left} = -\nabla \phi |_\mathrm{right}$ on the left and right boundaries by definition of the periodicity (and similarly for the top and bottom). Again this implies that $f$ must integrate to zero. Sometimes, with periodic boundary conditions all around, you need to enforce that $f$ integrate to zero numerically to test convergence. This is discussed in \S~\ref{sec:lm:periodicbcs}. \section{Going Further} \label{sec:multigrid:other} \subsection{Red-black Ordering} When using domain decomposition to spread the problem across parallel processors, the smoothing is often done as {\em red-black Gauss-Seidel}. In this ordering, you imagine the grid to be a checkerboard (see Figure~\ref{fig:rb}). In the first Gauss-Seidel pass you update the red squares and in the second, the black squares. The advantage is that when updating the red, you can be sure that none of the zones you depend on (the neighboring black zones) will change. This makes the decomposition parallel. Note: this works for the standard 5-point Laplacian. If you are doing some other operator with a different stencil, then this decomposition may no longer hold. % this figure can be created by figures/multigrid/red_black.py \begin{figure}[t] \centering \includegraphics[width=0.6\linewidth]{rb} \caption[Red-black ordering of zones] {\label{fig:rb} The red-black ordering of zones.} \end{figure} \subsection{More General Elliptic Equations} The most general {\em second-order} elliptic equation takes the form: \begin{equation} \alpha \phi + \nabla \cdot (\beta \nabla \phi) + \gamma \cdot \nabla \phi + \nabla \cdot (\zeta \phi) = f \end{equation} Here, $\gamma$ and $\zeta$ are vectors. Solving a general elliptic equation of this form can be accomplished with multigrid using the same basic ideas here. The main change is that the smoothing algorithm and the construction of the residual will need to discretize the more general operator, and these coefficients will need to be restricted to the coarser grids (some on edges). 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331.478 l 225.743 303.087 l cl gsave 1.000 0.000 0.000 setrgbcolor fill grestore stroke grestore gsave 388.9 390.4 17.17 15.62 clipbox 268.164 303.087 m 268.164 331.478 l 296.445 331.478 l 296.445 303.087 l cl gsave 1.000 0.000 0.000 setrgbcolor fill grestore stroke grestore gsave 388.9 390.4 17.17 15.62 clipbox 338.866 303.087 m 338.866 331.478 l 367.147 331.478 l 367.147 303.087 l cl gsave 1.000 0.000 0.000 setrgbcolor fill grestore stroke grestore gsave 388.9 390.4 17.17 15.62 clipbox 20.707 338.576 m 20.707 366.967 l 48.9878 366.967 l 48.9878 338.576 l cl gsave 1.000 0.000 0.000 setrgbcolor fill grestore stroke grestore gsave 388.9 390.4 17.17 15.62 clipbox 91.409 338.576 m 91.409 366.967 l 119.69 366.967 l 119.69 338.576 l cl gsave 1.000 0.000 0.000 setrgbcolor fill grestore stroke grestore gsave 388.9 390.4 17.17 15.62 clipbox 162.111 338.576 m 162.111 366.967 l 190.392 366.967 l 190.392 338.576 l cl gsave 1.000 0.000 0.000 setrgbcolor fill grestore stroke grestore gsave 388.9 390.4 17.17 15.62 clipbox 232.813 338.576 m 232.813 366.967 l 261.094 366.967 l 261.094 338.576 l cl gsave 1.000 0.000 0.000 setrgbcolor fill grestore stroke grestore gsave 388.9 390.4 17.17 15.62 clipbox 303.515 338.576 m 303.515 366.967 l 331.796 366.967 l 331.796 338.576 l cl gsave 1.000 0.000 0.000 setrgbcolor fill grestore stroke grestore end showpage ================================================ FILE: multiphysics/multiphysics.tex ================================================ \section{Integrating Multiphysics} Consider a system whose evolution depends on several different physical processes, represented by the operators $A$, $D$, $R$ (advection, diffusion, and reactions, respectively). \begin{equation} \phi_t = -A(\phi) + D(\phi) + R(\phi) \end{equation} One way to solve this system is to discretize each of the operators in space. For instance the discrete advection operator, $[A(\phi)]_i$ might use the ideas on piecewise linear reconstruction techniques discussed in chapter~\ref{ch:advection}, $[D(\phi)]_i$ can use the discrete form of the Laplacian from chapter~\ref{ch:diffusion}, and $[R(\phi)]_i$ may be an algebraic relation. This leaves us with an ordinary differential equation for the time-evolution of $\phi$, \begin{equation} \frac{d\phi_i}{dt} = -[A(\phi)]_i + [D(\phi)]_i + [R(\phi)]_i \end{equation} which can be solve using standard ODE techniques. This is the {\em method of lines} technique we saw with advection (\S~\ref{adv:sec:mol_2d}) and compressible hydrodynamics (\S~\ref{sec:comp:mol}), and can be a powerful technique to solve PDEs or systems of PDEs with multiple physics operators A difficulty arises if these processes each have different timescales associated with them. For instance, reactions may be vigorous and require a small timestep to accurately capture the changes, but the advection is slow. Or, recall that the timestep limiter for explicit diffusion scales as $\Delta x^2$ while explicit advection scales as $\Delta x$, so these processes could demand very different timescale for evolution. Therefore, we don't want to use the same timestep for all the processes, since that will needlessly make things computationally expensive. {\em Operator splitting} solves for the effects of each operator separately, using whichever timestep (and time-discretization, e.g., explicit or implicit) is most suited to the operation. The result of one operation is used as the input to the next\footnote{This directly parallels the dimensional splitting approach we saw for advection in \S~\ref{adv:sec:dimensionalsplit}}. The downside of this approach is that the operations may not be well coupled. \section{Ex: diffusion-reaction} \label{sec:multiphys:diffreact} Consider a diffusion-reaction equation: \begin{equation} \phi_t = \kappa \phi_{xx} + \frac{1}{\tau} R(\phi) \end{equation} This can be thought of as a simple model for a combustion flame, and can propagate a front. It is often the case that the reactions are stiff, and require a smaller timestep then the diffusion part. In fact, we may want to use an implicit integration method designed for stiff ODEs for the reaction part, but use a standard explicit method for the diffusion part. This requires operator splitting. We can use Strang splitting \cite{strang} to make the integration second-order accurate overall: \begin{equation} \phi^{n+1} = R_{\Delta t/2} D_{\Delta t} R_{\Delta t/2} \phi^n \end{equation} where $R_{\Delta t/2}$ represents reacting for a step of $\Delta t/2$ and $D_{\Delta t}$ represents diffusing for a step of $\Delta t$. In each case, these operators act as if the other were not present, but they see the effect of the previous operation on the input $\phi$. Note that no explicit source terms describing one process appear in the other process's update. The procedure for updating appears as: \begin{enumerate} \item {\em Evolve reaction ODE system for $\Delta t/2$} Define $\phi^\star$ as the the solution to the ODE: \begin{equation} \frac{d\phi}{dt} = \frac{1}{\tau} R(\phi), \quad \phi(t^n) = \phi^n, \quad t \in [t^n, t^{n+\myhalf}] \end{equation} \item {\em Solve the diffusion equation for $\Delta t$ with an implicit Crank-Nicolson discretization} % \begin{equation} \frac{\phi^{\star\star} - \phi^\star}{\Delta t} = \frac{1}{2} (D(\phi^\star) + D(\phi^{\star\star})) \end{equation} Note that the starting point is $\phi^\star$. \item {\em Evolve reaction ODE system for $\Delta t/2$} Define $\phi^{n+1}$ as the the solution to the ODE: \begin{equation} \text{define } \phi^{n+1}: \, \frac{d\phi}{dt} = \frac{1}{\tau} R(\phi), \quad \phi(t^{n+\myhalf}) = \phi^{\star\star}, \quad t \in [t^{n+\myhalf}, t^{n+1}] \end{equation} \end{enumerate} Consider a simple reaction source \begin{equation} \label{eq:multiphysics:react} R(\phi) = \frac{1}{4} \phi (1 - \phi) \end{equation} This is called a KPP reaction source. Here $\phi$ can be thought of as a progress variable that varies between pure ash ($\phi = 0$) and pure fuel ($\phi = 1$). Figure~\ref{fig:diffreact} shows the solution to our diffusion-reaction equation with 256 zones, $\kappa = 0.1$, $\tau = 1.0$ at several times. \begin{figure}[t] \centering \includegraphics[width=5.0in]{flame_seq} \caption[Solution to the diffusion-reaction equation] {\label{fig:diffreact} Solution to the diffusion-reaction equation with 256 zones, and $\kappa = 0.1$, $\tau = 1.0$. The lines shown are spaced 8.0 time-units apart. We see the initial smoothed tophat profile giving rise to a traveling front. \\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/multiphysics/diffusion-reaction.py}{diffusion-reaction.py}} } \end{figure} The solution in this case is a wave with speed $S = \sqrt{\kappa/\tau}$ and thickness $\delta = \sqrt{\kappa \tau}$ (see \cite{vladimirova2006} for some details of this system). \begin{exercise}[Diffusion-reaction system] Solve the the difusion reaction equation with the source given in Eq.~\ref{eq:multiphyiscs:react}. Note that you should begin with some smooth initial conditions---if they are too sharp than the C-N discretization will cause jagged features to appear. Compare your results to Figure~\ref{fig:diffreact}. \end{exercise} \section{Ex: advection-diffusion} \label{ch:multiphysics:sec:adburgers} The viscous Burgers' equation appears as: \begin{equation} u_t + u u_x = \epsilon u_{xx} \end{equation} This admits shocks and rarefactions just like the inviscid form we studied in Chapter~\ref{ch:burgers}, but now the viscosity can act to smooth out the shock---instead of being infinitely thin, it will have a physical width. As we saw earlier, there are efficient, accurate methods for handling the advective parts explicitly, but for diffusion, we often want to solve it implicitly. We can split the solution up, but couple the two processes together to make a method that is overall second-order accurate in time. We write our equation as: \begin{equation} u_t + A(u) = D(u) \end{equation} with $A(u) = [\frac{1}{2} u^2]_x$ and $D(u) = \epsilon u_{xx}$. Then our update appears in two steps. \begin{enumerate} \item {\em Find the advective update over the timestep}: We use an approximation of the diffusion term at time-level $n$, $D(u^n)$ as a source in the construction of the interface states for the advective part. Once the interface states, $u_{i+\myhalf}^{n+\myhalf}$ are known, we construct the advective update term as: \begin{equation} A_i^{n+\myhalf} = \frac{\left [ \frac{1}{2} \left (u_{i+\myhalf}^{n+\myhalf}\right)^2\right ] - \left [ \frac{1}{2} \left (u_{i-\myhalf}^{n+\myhalf}\right)^2\right ]} {\Delta x} \end{equation} \item {\em Solve the diffusion equation with the advective source}: We use a Crank-Nicolson discretization of the diffusion part of our equation, with the advective update term appearing as a source. \begin{equation} \frac{u^{n+1} - u^n}{\Delta t} = \frac{1}{2}D(u^n) + \frac{1}{2}D(u^{n+1}) - A^{n+\myhalf} \end{equation} This is a linear system that can be solved as a tridiagonal matrix or with multigrid. The result of this step is that $u^{n+1}$ is updated with both the advection and diffusion terms. \end{enumerate} Because the diffusion is done implicitly, the timestep constraint (for stability) for this solve is due to the advective portion only. \begin{figure}[t] \centering \includegraphics[width=5in]{burgersvisc} \caption[Viscous Burgers' equation solution] {\label{fig:viscburger} Solution to the viscous Burgers' equation with a variety of different viscosities. The initial conditions was a single wavelength of a sine wave for $x \in [1/3,2/3]$, and $u = 1$ otherwise. \\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/multiphysics/burgersvisc.py}{burgersvisc.py}}} \end{figure} For step 1, the addition of the explicit diffusion source requires a small change to the method we used to predict the interface states. \begin{eqnarray} u^{n+1}_{i+\myhalf,L} &=& u^n_i + \frac{\Delta x}{2} \frac{\partial u}{\partial x} + \frac{\Delta t}{2} \frac{\partial u}{\partial t} + \ldots \\ &=& u^n_i + \frac{\Delta x}{2} \frac{\partial u}{\partial x} + \frac{\Delta t}{2} \left (-u_i \frac{\partial u}{\partial x} + D(u^n_i) \right ) + \ldots \\ &=& 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 \end{eqnarray} here the source term (shown in red) incorporates the effects of the diffusion on the prediction of the states for advection. This entered into our states when we replaced $\partial u/\partial t$ with our PDE equation. The spatial derivative, $\partial u/\partial x$ is replaced by a monotonized difference, and the method then proceeds as with the regular Burgers' equation. The Riemann problem is unchanged from the inviscid case. Figure~\ref{fig:viscburger} shows the solution of the viscous Burgers' equation for shock initial conditions with different amounts of viscosity. Notice that the effect of the viscosity is to smooth the shock profile, but the shock position itself agrees between the cases. \subsection{Convergence without an analytic solution} Assessing the convergence of this is more difficult than the tests we looked at before, since there is no analytic solution to compare to. In this case, we can compare a very high-resolution solution and take this to be the {\em right} answer / benchmark solution, and then compare coarser simulations to a coarsened version of the benchmark solution. This is done in Figure~\ref{fig:multiphysics:burgersconverge}. The fine zones of the benchmark are averaged into a corresponding coarse grid zone to produce a coarsened representation of the benchmark, and the norm of the error with the coarse simulation is computed. \begin{figure}[t] \centering \includegraphics[width=5.0in]{burgersvisc_converge} \caption[Convergence of the viscous Burgers' equation] {\label{fig:multiphysics:burgersconverge} Convergence of the Burgers' solution with $\epsilon = 0.005$. To test convergence a run with 4096 zones was done as a benchmark solution. We see second-order convergence for the higher-resolution runs in this sample.\\ \hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/multiphysics/burgersvisc_converge.py}{burgersvisc\_converge.py}} } \end{figure} ================================================ FILE: mysymbols.tex ================================================ \newcommand{\cfl}{{\mathcal{C}}} % velocity \newcommand{\Ub}{{\bf U}} \newcommand{\gb}{{\bf g}} \newcommand{\Gb}{{\bf G}} \newcommand{\Hb}{{\bf H}} % systems \newcommand{\Ab}{{\bf A}} \newcommand{\Uc}{{\,\boldsymbol{\mathcal{U}}}} \newcommand{\Rc}{{\,\boldsymbol{\mathcal{R}}}} \newcommand{\Fb}{{\bf F}} \newcommand{\qb}{{\bf q}} % reactions \newcommand{\omegadot}{{\dot{\omega}}} \newcommand{\Hnuc}{{H_\mathrm{nuc}}} % diffusion \newcommand{\kth}{{k_\mathrm{th}}} % eigenvectors \newcommand{\evm}{{(-)}} \newcommand{\evz}{{(\circ)}} \newcommand{\evzs}[1]{{(\circ,#1)}} \newcommand{\evp}{{(+)}} \newcommand{\enu}{{(\nu)}} \newcommand{\lb}{{\bf l}} \newcommand{\rb}{{\bf r}} \newcommand{\Lb}{{\bf L}} \newcommand{\Rb}{{\bf R}} \newcommand{\Lambdab}{{\boldsymbol{\Lambda}}} \newcommand{\wb}{{\bf w}} % directional exponents \newcommand{\exx}{{(x)}} \newcommand{\eyy}{{(y)}} % Riemann and compressible \newcommand{\riemann}{{\mathcal{R}}} \newcommand{\Ic}{{\mathcal{I}}} % derivative shortcuts \newcommand{\ddx}[1]{{\frac{{\partial#1}}{\partial x}}} \newcommand{\ddy}[1]{{\frac{{\partial#1}}{\partial y}}} \newcommand{\ddz}[1]{{\frac{{\partial#1}}{\partial z}}} \newcommand{\ddr}[1]{{\frac{{\partial#1}}{\partial r}}} \newcommand{\ddt}[1]{{\frac{{\partial#1}}{\partial t}}} \newcommand{\DDt}[1]{{\frac{{D#1}}{D t}}} % F-V shortcuts \newcommand{\favg}[1]{{\langle #1\rangle}} \newcommand{\dx}{\Delta x} \newcommand{\dy}{\Delta y} \newcommand{\dt}{\Delta t} % limited slopes \newcommand{\Dux}{\overline{\Delta u}^{(x)}} \newcommand{\Duy}{\overline{\Delta u}^{(y)}} \newcommand{\Dvx}{\overline{\Delta v}^{(x)}} \newcommand{\Dvy}{\overline{\Delta v}^{(y)}} \newcommand{\imag}{I} ================================================ FILE: pde-classes/pde-classes.tex ================================================ \section{Introduction} Partial differential equations (PDEs) are usually grouped into one of three different classes: {\em hyperbolic}, {\em parabolic}, or {\em elliptic}. You can find the precise mathematical definition of these classifications in most books on PDEs, but this formal definition is not very intuitive or useful. Instead, it is helpful to look at some prototypical examples of each type of PDE. When we are solving multiphysics problems, we will see that our system of PDEs spans these different types. Nevertheless, we will look at solutions methods for each type separately first, and then use what we learn to solve more complex systems of equations. \section{Hyperbolic PDEs} The canonical hyperbolic PDE is the wave equation: \begin{equation} \frac{\partial^2 \phi}{\partial t^2} = c^2 \frac{\partial^2 \phi}{\partial x^2} \end{equation} The general solution to this is traveling waves in either direction: \begin{equation} \label{eq:pde:wavesol} \phi(x,t) = \alpha f_0(x - ct) + \beta g_0(x + ct) \end{equation} Here $f_0$ and $g_0$ are set by the initial conditions, and the solution propagates $f_0$ to the right and $g_0$ to the left at a speed $c$. \begin{exercise}[Wave equation] {Show by substitution that Eq.~\ref{eq:pde:wavesol} is a solution to the wave equation} \end{exercise} A simple first-order hyperbolic PDE is the linear advection equation: \begin{equation} a_t + u a_x = 0 \end{equation} This simply propagates any initial profile to the right at the speed $u$. We will use linear advection as our model equation for numerical methods for hyperbolic PDEs. A system of first-order hyperbolic PDEs takes the form: \begin{equation} {\bf a}_t + \Ab {\bf a}_x = 0 \end{equation} where ${\bf a} = (a_0, a_1, \ldots a_{N-1})^\intercal$ and $\Ab$ is a matrix. This system is hyperbolic if the eigenvalues of $A$ are real (see \cite{leveque:2002} for an excellent introduction). An important concept for hyperbolic PDEs are {\em characteristics}---these are curves in a space-time diagram along which the solution is constant. Associated with these curves is a speed---this is the wave speed at which information on how the solution changes is communicated. For a linear PDE (or system of PDEs), these will tell you everything you need to know about the solution. \section{Elliptic PDEs} The canonical elliptic PDE is the Poisson equation: \begin{equation} \nabla^2 \phi = f \end{equation} Note that there is no time-variable here. This is a pure boundary value problem. The solution, $\phi$ is determined completely by the source, $f$, and the boundary conditions. In contrast to the hyperbolic case, there is no propagation of information here. The potential, $\phi$, is known instantaneously everywhere in the domain. For astrophysical flows, this commonly arises as the Poisson equation describing the gravitational potential. \section{Parabolic PDEs} The canonical parabolic PDE is the heat equation: \begin{equation} \label{eq:pde:heat} \frac{\partial \phi}{\partial t} = k \frac{\partial^2 f}{\partial x^2} \end{equation} This has aspects of both hyperbolic and elliptic PDEs. The heat equation represents diffusion---an initially sharp feature will spread out into a smoother profile on a timescale that depends on the coefficient $k$. We'll encounter parabolic equations for thermal diffusion and other types of diffusion (like species, mass), and with viscosity. \begin{exercise}[Diffusion timescale] {Using dimensional analysis, estimate the characteristic timescale for diffusion from Eq.~\ref{eq:pde:heat}.} \end{exercise} ================================================ FILE: preface/preface.tex ================================================ This text started as a set of notes to help new students at Stony Brook University working on projects in computational astrophysics. They focus on the common methods used in computational hydrodynamics for astrophysical flows and are written at a level appropriate for upper-level undergraduates. Problems integrated in the text help demonstrate the core ideas. An underlying principle is that source code is provided for all the methods described here (including all the figures). This allows the reader to explore the routines themselves. These notes are very much a work in progress, and new chapters will be added with time. The page size is formatted for easy reading on a tablet or for 2-up printing in a landscape orientation on letter-sized paper. This text is part of the Open Astrophysics Bookshelf. Contributions to these notes are welcomed. The \LaTeX\ source for these notes is available online on github at: \\[0.25em] % \url{https://github.com/Open-Astrophysics-Bookshelf/numerical_exercises} \\[0.25em] % Simply fork the notes, hack away, and submit a pull-request to add your contributions. All contributions will be acknowledged in the text. A PDF version of the notes is always available at: \\[0.25em] % \url{https://open-astrophysics-bookshelf.github.io/numerical_exercises/CompHydroTutorial.pdf} \\[0.25em] % These notes are updated at irregular intervals, usually when I have a new student working with me, or if I am using them for a course. The source (usually python) for all the figures is also contained in the main git repo. The line drawings of the grids are done using the classes in \href{https://github.com/Open-Astrophysics-Bookshelf/numerical_exercises/blob/master/grid_plot.py}{{\tt grid\_plot.py}}. This needs to be in your {\tt PYTHONPATH} if you wish to run the scripts. The best way to understand the methods described here is to run them for yourself. There are several sets of example codes that go along with these notes: \begin{enumerate} \item \hydroex\ is a set of simple 1-d, standalone python scripts that illustrate some of the basic solvers. Many of the figures in these notes were created using these codes---the relevant script will be noted in the figure caption. You can get this set of scripts from github at:\\ \url{https://github.com/zingale/hydro_examples/} References to the scripts in \hydroex\ are shown throughout the text as: \\[0.5em] \hydroexdoit{scriptname} \\[0.5em] Clicking on the name of the script will bring up the source code to the script (on github) in your web browser. More details on the codes available in \hydroex\ are described in Appendix~\ref{app:hydroex}. \item The \pyro\ code~\cite{pyro} is a 2-d simulation code with solvers for advection, diffusion, compressible and incompressible hydrodynamics, as well as multigrid. A gray flux-limited diffusion radiation hydrodynamics solver is in development. \pyro\ is designed with clarity in mind and to make experimentation easy. You can download \pyro\ at: \\ \url{https://github.com/python-hydro/pyro2/} A brief overview of \pyro\ is given in Appendix~\ref{app:pyro}, and more information can be found at: \\ \url{http://python-hydro.github.io/pyro2/} \item \hydrooned\ is a simple one-dimensional compressible hydrodynamics code that implements the piecewise parabolic method from Chapter~\ref{ch:compressible}. It can be obtained from github at:\\ \url{https://github.com/zingale/hydro1d/} Details on it are given in Appendix~\ref{app:hydro1d}. \end{enumerate} Wherever possible we try to use standardized notation for physical quantities, as listed in Table~\ref{table:sym}. These notes benefited {\em immensely} from numerous conversations and an ongoing collaboration with Ann Almgren, John Bell, Andy Nonaka, \& Weiqun Zhang---pretty much everything I know about projection methods comes from working with them. Discussions with Alan Calder, Sean Couch, Max Katz, Chris Malone, and Doug Swesty have also been influential in the presentation of these notes. If you find errors, please e-mail me at michael.zingale@stonybrook.edu, or issue a pull request to the git repo noted above. \begin{flushright} Michael Zingale \\ Stony Brook University \end{flushright} \clearpage \input symbols/symbols \clearpage \section*{Authorship} \subsection*{Primary Author} Michael Zingale (Stony Brook) \subsection*{Contributions} Thank you to the following people for pointing out typos or confusing remarks in the text: \begin{itemize} \item Chen-Hung \item Rixin Li (Arizona) \item Zhi Li (Shanghai Astronomical Observatory) \item Chris Malone \item Sai Praneeth (Waterloo) \item Donald Willcox (Stony Brook) \end{itemize} Material on WENO schemes was contributed by Ian Hawke (Southampton). See the git log for full details on contributions. All contributions via pull-requests will be acknowledged here. ================================================ FILE: public-codes/NOTES ================================================ Describe the features of public codes: Athena Castro Enzo Flash Maestro pyro pluto give details on: -- solver -- griding (including subcycling?) -- language -- EOS -- other physics -- species -- particles -- visualization tools -- gravity -- other details -- diffusion -- radiation ================================================ FILE: pyro/pyro.tex ================================================ \label{app:pyro} %% \begin{center} %% \includegraphics[width=0.125\linewidth]{pyro-sm} %% \end{center} \section{Introduction} \pyro\ is a simple two-dimensional code that implements the core solvers described in these notes. It is written primarily in python, with some kernels in Fortran. Clarity of the methods is emphasized over performance. \section{Getting \pyro} \pyro\ can be downloaded from its github repository, \url{https://github.com/python-hydro/pyro2} as: \begin{verbatim} git clone https://github.com/python-hydro/pyro2 \end{verbatim} \pyro\ uses the {\tt h5py}, {\sf matplotlib}, and {\sf numpy} libraries. Several routines are written in Fortran and need to be compiled. The script {\tt mk.sh} will build the packages. This requires that {\tt f2py} is installed. By default, \pyro\ will use python 3. The following sequence will setup \pyro: \begin{itemize} \item Set the {\tt PYRO\_HOME} environment variable to point to the main {\tt pyro2/} directory, e.g., \begin{verbatim} export PYRO_HOME=/path/to/pyro2 \end{verbatim} \item Set the {\tt PYTHONPATH} environment variable to point to the main {\tt pyro2/} directory, e.g., \begin{verbatim} export PYTHONPATH="$PYTHONPATH:$PYRO_HOME" \end{verbatim} \item Build the Fortran source using the {\tt mk.sh} script. In the {\tt pyro2/} directory, simply do \begin{verbatim} ./mk.sh \end{verbatim} \end{itemize} \section{\pyro\ solvers} \pyro\ offers the following 2-d solvers: \begin{itemize} \item {\em advection}: an unsplit, second-order method for linear advection, following the ideas from Chapter~\ref{ch:advection}. \item {\em advection\_rk}: an alternate advection algorithm using method-of-lines time integration, again following the ideas from Chapter~\ref{ch:advection}. \item {\em compressible}: an unsplit, second-order compressible hydrodynamics solver using the piecewise linear reconstruction discussed in Chapter~\ref{ch:compressible}. \item {\em compressible\_rk}: an alternate compressible algorithm using method-of-lines time integration, using the piecewise linear reconstruction discussed in Chapter~\ref{ch:compressible}. \item {\em diffusion}: a second-order implicit diffusion solver, based on the ideas from Chapter~\ref{ch:diffusion}. \item {\em incompressible}: a second-order incompressible hydrodynamics solver using a cell-centered approximate projection, as discussed in Chapter~\ref{ch:incompressible}. \item {\em lm\_atm}: a low-Mach number hydrodynamics solver for atmospheric flows, as discussed in S~\ref{sec:lm:atm} \item {\em multigrid}: a multigrid solver for constant-coefficient Helmholtz elliptic equations. This follows the ideas from Chapter~\ref{ch:multigrid}, and is used by the diffusion and incompressible solvers. \end{itemize} \section{\pyro 's structure} The structure of the code and descriptions of the various runtime parameters is found on the pyro webpage, \url{http://python-hydro.github.io/pyro2/}, and described in \cite{pyro}. Here we provide a summary. The grid structure is managed by the {\tt patch.Grid2d} class. Data is that lives on the grid is contained in a {\tt patch.CellCenterData2d} object. Methods are available to provide access to the data and fill the ghost cells. When accessing the data on the grid, a {\tt array\_indexer.ArrayIndexer} object is returned. This is a subclass of the normal NumPy {\tt ndarray} that provides additional methods that are useful for interacting with finite-volume data. In particular, for an object {\tt a}, you can do {\tt a.ip(1)} to mimic $a_{i+1,j}$. The Jupyter notebook \href{https://github.com/python-hydro/pyro2/blob/master/mesh/mesh-examples.ipynb}{mesh-examples.ipynb} walks through these classes to demonstrate how they work. Each \pyro\ solver is its own python module. All but the multigrid solver represent time-dependent problems. Each of these provide a {\tt Simulation} class that provides the routines necessary to initialize the solver, determine the timestep, and evolve the solution in time. Each solver has one or more sets of initial conditions defined in the solver's {\tt problems/} directory. All time-dependent problems are run through the {\tt pyro.py} script. The general form is: \begin{verbatim} ./pyro.py solver problem inputs \end{verbatim} where {\em solver} is one of \{{\tt advection}, {\tt advection\_rk}, {\tt compressible}, {\tt compressible\_rk}, {\tt diffusion}, {\tt incompressible}, {\tt lm\_atm}\}, {\em problem} is one of the problems defined in the solver's {\tt problems/} directory, and {\em inputs} is the name of an input file that defines the values of runtime parameter. The possible runtime parameters and their defaults are defined in the {\tt \_defaults} files in the main directory and each solver and problem directory. Note that the inputs file need not be in the {\tt pyro2/} directory. The solver's {\tt problems/} directory will also be checked. \section{Running \pyro} A simple Gaussian advection simulation is provided by the advection {\em smooth} problem. This is run as: \begin{verbatim} ./pyro.py advection smooth inputs.smooth \end{verbatim} As this is run, the solution will be visualized at each step, showing the progression of the simulation. A list of the problems available for each solver is given in Table~\ref{tab:problems}. For the multigrid solver, there are scripts available in the {\tt multigrid/} directory that illustrate its use. \section{Output and visualization} \pyro\ uses HDF5 for I/O\footnote{Previously a simple python {\tt pickle} was done, but this was not very portable.} The simple script {\tt plot.py} in the {\tt pyro/} directory can read in the data and plot the output. It will determine which solver was used for the run and do the visualization using that solver's methods. \section{Testing} The script {\tt test.py} provides an interface to \pyro's unit and regression tests. Simply running it was {\tt ./test.py} will executate all of the tests. The unit tests are enabled through the {\sf pytest} framework. \begin{sidewaystable*} \centering \renewcommand{\arraystretch}{1.2} \begin{tabular}{lp{1in}lp{4.0in}} \hline {\bf solver} & {\bf section} & {\bf problem} & {\bf problem description} \\ \hline % \multirow{2}{*}{\tt advection} & \multirow{2}{*}{\S~\ref{adv:sec:unsplit_2d}} & {\tt smooth} & advect a smooth Gaussian profile \\ & & {\tt tophat} & advect a discontinuous tophat profile \\ \hline % \multirow{1}{*}{\tt advection\_rk} & \multirow{1}{*}{\S~\ref{adv:sec:mol_2d}} & \multicolumn{2}{l}{same as {\tt advection}} \\ \hline % \multirow{7}{*}{\tt compressible} & \multirow{7}{*}{Ch.~\ref{ch:compressible}} & {\tt advect} & a Gaussian profile in the density field advected diagonally (in constant pressure background) \\ & & {\tt bubble} & a buoyant bubble in a stratified atmosphere \\ & & {\tt kh} & setup a shear layer to drive Kelvin-Helmholtz instabilities \\ & & {\tt quad} & 2-d Riemann problem based on \cite{schulz-rinne:1993} \\ & & {\tt rt} & a simple Rayleigh-Taylor instability\\ & & {\tt sedov} & the classic Sedov-Taylor blast wave \\ & & {\tt sod} & the Sod shock tube problem \\ \hline % \multirow{1}{*}{\tt compressible\_rk} & \multirow{1}{*}{Ch.~\ref{ch:compressible}} & \multicolumn{2}{l}{same as {\tt compressible}} \\ \hline % {\tt diffusion} & \S~\ref{diff:sec:implicit_mg} & {\tt gaussian} & diffuse an initial Gaussian profile \\ \hline % \multirow{2}{*}{\tt incompressible} & \multirow{2}{*}{Ch.~\ref{ch:incompressible}} & {\tt converge} & A simple incompressible problem with known analytic solution.\\ & & {\tt shear} & a doubly-periodic shear layer \\ \hline % \multirow{1}{*}{\tt lm\_atm} & \multirow{1}{*}{Ch.~\ref{sec:lm:atm}} & {\tt bubble} & a buoyant bubble in a stratified atmosphere \\ \hline \end{tabular} \caption{\label{tab:problems} \pyro\ solvers and their distributed problems. 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/p0_2b { newpath translate 293.116 73.5111 m 304.837 73.5111 l 304.837 74.4268 l 293.116 74.4268 l 293.116 73.5111 l } bind def /p0_2c { newpath translate 293.116 74.4268 m 304.837 74.4268 l 304.837 75.3426 l 293.116 75.3426 l 293.116 74.4268 l } bind def /p0_2d { newpath translate 293.116 75.3426 m 304.837 75.3426 l 304.837 76.2583 l 293.116 76.2583 l 293.116 75.3426 l } bind def /p0_2e { newpath translate 293.116 76.2583 m 304.837 76.2583 l 304.837 77.174 l 293.116 77.174 l 293.116 76.2583 l } bind def /p0_2f { newpath translate 293.116 77.174 m 304.837 77.174 l 304.837 78.0897 l 293.116 78.0897 l 293.116 77.174 l } bind def /p0_30 { newpath translate 293.116 78.0897 m 304.837 78.0897 l 304.837 79.0054 l 293.116 79.0054 l 293.116 78.0897 l } bind def /p0_31 { newpath translate 293.116 79.0054 m 304.837 79.0054 l 304.837 79.9212 l 293.116 79.9212 l 293.116 79.0054 l } bind def /p0_32 { newpath translate 293.116 79.9212 m 304.837 79.9212 l 304.837 80.8369 l 293.116 80.8369 l 293.116 79.9212 l } bind def /p0_33 { newpath translate 293.116 80.8369 m 304.837 80.8369 l 304.837 81.7526 l 293.116 81.7526 l 293.116 80.8369 l } bind def /p0_34 { newpath translate 293.116 81.7526 m 304.837 81.7526 l 304.837 82.6683 l 293.116 82.6683 l 293.116 81.7526 l } bind def /p0_35 { newpath translate 293.116 82.6683 m 304.837 82.6683 l 304.837 83.584 l 293.116 83.584 l 293.116 82.6683 l } bind def /p0_36 { newpath translate 293.116 83.584 m 304.837 83.584 l 304.837 84.4998 l 293.116 84.4998 l 293.116 83.584 l } bind def /p0_37 { newpath translate 293.116 84.4998 m 304.837 84.4998 l 304.837 85.4155 l 293.116 85.4155 l 293.116 84.4998 l } bind def /p0_38 { newpath translate 293.116 85.4155 m 304.837 85.4155 l 304.837 86.3312 l 293.116 86.3312 l 293.116 85.4155 l } bind def /p0_39 { newpath translate 293.116 86.3312 m 304.837 86.3312 l 304.837 87.2469 l 293.116 87.2469 l 293.116 86.3312 l } bind def /p0_3a { newpath translate 293.116 87.2469 m 304.837 87.2469 l 304.837 88.1626 l 293.116 88.1626 l 293.116 87.2469 l } bind def /p0_3b { newpath translate 293.116 88.1626 m 304.837 88.1626 l 304.837 89.0784 l 293.116 89.0784 l 293.116 88.1626 l } bind def /p0_3c { newpath translate 293.116 89.0784 m 304.837 89.0784 l 304.837 89.9941 l 293.116 89.9941 l 293.116 89.0784 l } bind def /p0_3d { newpath translate 293.116 89.9941 m 304.837 89.9941 l 304.837 90.9098 l 293.116 90.9098 l 293.116 89.9941 l } bind def /p0_3e { newpath translate 293.116 90.9098 m 304.837 90.9098 l 304.837 91.8255 l 293.116 91.8255 l 293.116 90.9098 l } bind def /p0_3f { newpath translate 293.116 91.8255 m 304.837 91.8255 l 304.837 92.7413 l 293.116 92.7413 l 293.116 91.8255 l } bind def /p0_40 { newpath translate 293.116 92.7413 m 304.837 92.7413 l 304.837 93.657 l 293.116 93.657 l 293.116 92.7413 l } bind def /p0_41 { newpath translate 293.116 93.657 m 304.837 93.657 l 304.837 94.5727 l 293.116 94.5727 l 293.116 93.657 l } bind def /p0_42 { newpath translate 293.116 94.5727 m 304.837 94.5727 l 304.837 95.4884 l 293.116 95.4884 l 293.116 94.5727 l } bind def /p0_43 { newpath translate 293.116 95.4884 m 304.837 95.4884 l 304.837 96.4041 l 293.116 96.4041 l 293.116 95.4884 l } bind def /p0_44 { newpath translate 293.116 96.4041 m 304.837 96.4041 l 304.837 97.3199 l 293.116 97.3199 l 293.116 96.4041 l } bind def /p0_45 { newpath translate 293.116 97.3199 m 304.837 97.3199 l 304.837 98.2356 l 293.116 98.2356 l 293.116 97.3199 l } bind def /p0_46 { newpath translate 293.116 98.2356 m 304.837 98.2356 l 304.837 99.1513 l 293.116 99.1513 l 293.116 98.2356 l } bind def /p0_47 { newpath translate 293.116 99.1513 m 304.837 99.1513 l 304.837 100.067 l 293.116 100.067 l 293.116 99.1513 l } bind def /p0_48 { newpath translate 293.116 100.067 m 304.837 100.067 l 304.837 100.983 l 293.116 100.983 l 293.116 100.067 l } bind def /p0_49 { newpath translate 293.116 100.983 m 304.837 100.983 l 304.837 101.898 l 293.116 101.898 l 293.116 100.983 l } bind def /p0_4a { newpath translate 293.116 101.898 m 304.837 101.898 l 304.837 102.814 l 293.116 102.814 l 293.116 101.898 l } bind def /p0_4b { newpath translate 293.116 102.814 m 304.837 102.814 l 304.837 103.73 l 293.116 103.73 l 293.116 102.814 l } bind def /p0_4c { newpath translate 293.116 103.73 m 304.837 103.73 l 304.837 104.646 l 293.116 104.646 l 293.116 103.73 l } bind def /p0_4d { newpath translate 293.116 104.646 m 304.837 104.646 l 304.837 105.561 l 293.116 105.561 l 293.116 104.646 l } bind def /p0_4e { newpath translate 293.116 105.561 m 304.837 105.561 l 304.837 106.477 l 293.116 106.477 l 293.116 105.561 l } bind def /p0_4f { newpath translate 293.116 106.477 m 304.837 106.477 l 304.837 107.393 l 293.116 107.393 l 293.116 106.477 l } bind def /p0_50 { newpath translate 293.116 107.393 m 304.837 107.393 l 304.837 108.309 l 293.116 108.309 l 293.116 107.393 l } bind def /p0_51 { newpath translate 293.116 108.309 m 304.837 108.309 l 304.837 109.224 l 293.116 109.224 l 293.116 108.309 l } bind def /p0_52 { newpath translate 293.116 109.224 m 304.837 109.224 l 304.837 110.14 l 293.116 110.14 l 293.116 109.224 l } bind def /p0_53 { newpath translate 293.116 110.14 m 304.837 110.14 l 304.837 111.056 l 293.116 111.056 l 293.116 110.14 l } bind def /p0_54 { newpath translate 293.116 111.056 m 304.837 111.056 l 304.837 111.971 l 293.116 111.971 l 293.116 111.056 l } bind def /p0_55 { newpath translate 293.116 111.971 m 304.837 111.971 l 304.837 112.887 l 293.116 112.887 l 293.116 111.971 l } bind def /p0_56 { newpath translate 293.116 112.887 m 304.837 112.887 l 304.837 113.803 l 293.116 113.803 l 293.116 112.887 l } bind def /p0_57 { newpath translate 293.116 113.803 m 304.837 113.803 l 304.837 114.719 l 293.116 114.719 l 293.116 113.803 l } bind def /p0_58 { newpath translate 293.116 114.719 m 304.837 114.719 l 304.837 115.634 l 293.116 115.634 l 293.116 114.719 l } bind def /p0_59 { newpath translate 293.116 115.634 m 304.837 115.634 l 304.837 116.55 l 293.116 116.55 l 293.116 115.634 l } bind def /p0_5a { newpath translate 293.116 116.55 m 304.837 116.55 l 304.837 117.466 l 293.116 117.466 l 293.116 116.55 l } bind def /p0_5b { newpath translate 293.116 117.466 m 304.837 117.466 l 304.837 118.381 l 293.116 118.381 l 293.116 117.466 l } bind def /p0_5c { newpath translate 293.116 118.381 m 304.837 118.381 l 304.837 119.297 l 293.116 119.297 l 293.116 118.381 l } bind def /p0_5d { newpath translate 293.116 119.297 m 304.837 119.297 l 304.837 120.213 l 293.116 120.213 l 293.116 119.297 l } bind def /p0_5e { newpath translate 293.116 120.213 m 304.837 120.213 l 304.837 121.129 l 293.116 121.129 l 293.116 120.213 l } bind def /p0_5f { newpath translate 293.116 121.129 m 304.837 121.129 l 304.837 122.044 l 293.116 122.044 l 293.116 121.129 l } bind def /p0_60 { newpath translate 293.116 122.044 m 304.837 122.044 l 304.837 122.96 l 293.116 122.96 l 293.116 122.044 l } bind def /p0_61 { newpath translate 293.116 122.96 m 304.837 122.96 l 304.837 123.876 l 293.116 123.876 l 293.116 122.96 l } bind def /p0_62 { newpath translate 293.116 123.876 m 304.837 123.876 l 304.837 124.791 l 293.116 124.791 l 293.116 123.876 l } bind def /p0_63 { newpath translate 293.116 124.791 m 304.837 124.791 l 304.837 125.707 l 293.116 125.707 l 293.116 124.791 l } bind def /p0_64 { newpath translate 293.116 125.707 m 304.837 125.707 l 304.837 126.623 l 293.116 126.623 l 293.116 125.707 l } bind def /p0_65 { newpath translate 293.116 126.623 m 304.837 126.623 l 304.837 127.539 l 293.116 127.539 l 293.116 126.623 l } bind def /p0_66 { newpath translate 293.116 127.539 m 304.837 127.539 l 304.837 128.454 l 293.116 128.454 l 293.116 127.539 l } bind def /p0_67 { newpath translate 293.116 128.454 m 304.837 128.454 l 304.837 129.37 l 293.116 129.37 l 293.116 128.454 l } bind def /p0_68 { newpath translate 293.116 129.37 m 304.837 129.37 l 304.837 130.286 l 293.116 130.286 l 293.116 129.37 l } bind def /p0_69 { newpath translate 293.116 130.286 m 304.837 130.286 l 304.837 131.202 l 293.116 131.202 l 293.116 130.286 l } bind def /p0_6a { newpath translate 293.116 131.202 m 304.837 131.202 l 304.837 132.117 l 293.116 132.117 l 293.116 131.202 l } bind def /p0_6b { newpath translate 293.116 132.117 m 304.837 132.117 l 304.837 133.033 l 293.116 133.033 l 293.116 132.117 l } bind def /p0_6c { newpath translate 293.116 133.033 m 304.837 133.033 l 304.837 133.949 l 293.116 133.949 l 293.116 133.033 l } bind def /p0_6d { newpath translate 293.116 133.949 m 304.837 133.949 l 304.837 134.864 l 293.116 134.864 l 293.116 133.949 l } bind def /p0_6e { newpath translate 293.116 134.864 m 304.837 134.864 l 304.837 135.78 l 293.116 135.78 l 293.116 134.864 l } bind def /p0_6f { newpath translate 293.116 135.78 m 304.837 135.78 l 304.837 136.696 l 293.116 136.696 l 293.116 135.78 l } bind def /p0_70 { newpath translate 293.116 136.696 m 304.837 136.696 l 304.837 137.612 l 293.116 137.612 l 293.116 136.696 l } bind def /p0_71 { newpath translate 293.116 137.612 m 304.837 137.612 l 304.837 138.527 l 293.116 138.527 l 293.116 137.612 l } bind def /p0_72 { newpath translate 293.116 138.527 m 304.837 138.527 l 304.837 139.443 l 293.116 139.443 l 293.116 138.527 l } bind def /p0_73 { newpath translate 293.116 139.443 m 304.837 139.443 l 304.837 140.359 l 293.116 140.359 l 293.116 139.443 l } bind def /p0_74 { newpath translate 293.116 140.359 m 304.837 140.359 l 304.837 141.274 l 293.116 141.274 l 293.116 140.359 l } bind def /p0_75 { newpath translate 293.116 141.274 m 304.837 141.274 l 304.837 142.19 l 293.116 142.19 l 293.116 141.274 l } bind def /p0_76 { newpath translate 293.116 142.19 m 304.837 142.19 l 304.837 143.106 l 293.116 143.106 l 293.116 142.19 l } bind def /p0_77 { newpath translate 293.116 143.106 m 304.837 143.106 l 304.837 144.022 l 293.116 144.022 l 293.116 143.106 l } bind def /p0_78 { newpath translate 293.116 144.022 m 304.837 144.022 l 304.837 144.937 l 293.116 144.937 l 293.116 144.022 l } bind def /p0_79 { newpath translate 293.116 144.937 m 304.837 144.937 l 304.837 145.853 l 293.116 145.853 l 293.116 144.937 l } bind def /p0_7a { newpath translate 293.116 145.853 m 304.837 145.853 l 304.837 146.769 l 293.116 146.769 l 293.116 145.853 l } bind def /p0_7b { newpath translate 293.116 146.769 m 304.837 146.769 l 304.837 147.685 l 293.116 147.685 l 293.116 146.769 l } bind def /p0_7c { newpath translate 293.116 147.685 m 304.837 147.685 l 304.837 148.6 l 293.116 148.6 l 293.116 147.685 l } bind def /p0_7d { newpath translate 293.116 148.6 m 304.837 148.6 l 304.837 149.516 l 293.116 149.516 l 293.116 148.6 l } bind def /p0_7e { newpath translate 293.116 149.516 m 304.837 149.516 l 304.837 150.432 l 293.116 150.432 l 293.116 149.516 l } bind def /p0_7f { newpath translate 293.116 150.432 m 304.837 150.432 l 304.837 151.347 l 293.116 151.347 l 293.116 150.432 l } bind def /p0_80 { newpath translate 293.116 151.347 m 304.837 151.347 l 304.837 152.263 l 293.116 152.263 l 293.116 151.347 l } bind def /p0_81 { newpath translate 293.116 152.263 m 304.837 152.263 l 304.837 153.179 l 293.116 153.179 l 293.116 152.263 l } bind def /p0_82 { newpath translate 293.116 153.179 m 304.837 153.179 l 304.837 154.095 l 293.116 154.095 l 293.116 153.179 l } bind def /p0_83 { newpath translate 293.116 154.095 m 304.837 154.095 l 304.837 155.01 l 293.116 155.01 l 293.116 154.095 l } bind def /p0_84 { newpath translate 293.116 155.01 m 304.837 155.01 l 304.837 155.926 l 293.116 155.926 l 293.116 155.01 l } bind def /p0_85 { newpath translate 293.116 155.926 m 304.837 155.926 l 304.837 156.842 l 293.116 156.842 l 293.116 155.926 l } bind def /p0_86 { newpath translate 293.116 156.842 m 304.837 156.842 l 304.837 157.757 l 293.116 157.757 l 293.116 156.842 l } bind def /p0_87 { newpath translate 293.116 157.757 m 304.837 157.757 l 304.837 158.673 l 293.116 158.673 l 293.116 157.757 l } bind def /p0_88 { newpath translate 293.116 158.673 m 304.837 158.673 l 304.837 159.589 l 293.116 159.589 l 293.116 158.673 l } bind def /p0_89 { newpath translate 293.116 159.589 m 304.837 159.589 l 304.837 160.505 l 293.116 160.505 l 293.116 159.589 l } bind def /p0_8a { newpath translate 293.116 160.505 m 304.837 160.505 l 304.837 161.42 l 293.116 161.42 l 293.116 160.505 l } bind def /p0_8b { newpath translate 293.116 161.42 m 304.837 161.42 l 304.837 162.336 l 293.116 162.336 l 293.116 161.42 l } bind def /p0_8c { newpath translate 293.116 162.336 m 304.837 162.336 l 304.837 163.252 l 293.116 163.252 l 293.116 162.336 l } bind def /p0_8d { newpath translate 293.116 163.252 m 304.837 163.252 l 304.837 164.168 l 293.116 164.168 l 293.116 163.252 l } bind def /p0_8e { newpath translate 293.116 164.168 m 304.837 164.168 l 304.837 165.083 l 293.116 165.083 l 293.116 164.168 l } bind def /p0_8f { newpath translate 293.116 165.083 m 304.837 165.083 l 304.837 165.999 l 293.116 165.999 l 293.116 165.083 l } bind def /p0_90 { newpath translate 293.116 165.999 m 304.837 165.999 l 304.837 166.915 l 293.116 166.915 l 293.116 165.999 l } bind def /p0_91 { newpath translate 293.116 166.915 m 304.837 166.915 l 304.837 167.83 l 293.116 167.83 l 293.116 166.915 l } bind def /p0_92 { newpath translate 293.116 167.83 m 304.837 167.83 l 304.837 168.746 l 293.116 168.746 l 293.116 167.83 l } bind def /p0_93 { newpath translate 293.116 168.746 m 304.837 168.746 l 304.837 169.662 l 293.116 169.662 l 293.116 168.746 l } bind def /p0_94 { newpath translate 293.116 169.662 m 304.837 169.662 l 304.837 170.578 l 293.116 170.578 l 293.116 169.662 l } bind def /p0_95 { newpath translate 293.116 170.578 m 304.837 170.578 l 304.837 171.493 l 293.116 171.493 l 293.116 170.578 l } bind def /p0_96 { newpath translate 293.116 171.493 m 304.837 171.493 l 304.837 172.409 l 293.116 172.409 l 293.116 171.493 l } bind def /p0_97 { newpath translate 293.116 172.409 m 304.837 172.409 l 304.837 173.325 l 293.116 173.325 l 293.116 172.409 l } bind def /p0_98 { newpath translate 293.116 173.325 m 304.837 173.325 l 304.837 174.24 l 293.116 174.24 l 293.116 173.325 l } bind def /p0_99 { newpath translate 293.116 174.24 m 304.837 174.24 l 304.837 175.156 l 293.116 175.156 l 293.116 174.24 l } bind def /p0_9a { newpath translate 293.116 175.156 m 304.837 175.156 l 304.837 176.072 l 293.116 176.072 l 293.116 175.156 l } bind def /p0_9b { newpath translate 293.116 176.072 m 304.837 176.072 l 304.837 176.988 l 293.116 176.988 l 293.116 176.072 l } bind def /p0_9c { newpath translate 293.116 176.988 m 304.837 176.988 l 304.837 177.903 l 293.116 177.903 l 293.116 176.988 l } bind def /p0_9d { newpath translate 293.116 177.903 m 304.837 177.903 l 304.837 178.819 l 293.116 178.819 l 293.116 177.903 l } bind def /p0_9e { newpath translate 293.116 178.819 m 304.837 178.819 l 304.837 179.735 l 293.116 179.735 l 293.116 178.819 l } bind def /p0_9f { newpath translate 293.116 179.735 m 304.837 179.735 l 304.837 180.65 l 293.116 180.65 l 293.116 179.735 l } bind def /p0_a0 { newpath translate 293.116 180.65 m 304.837 180.65 l 304.837 181.566 l 293.116 181.566 l 293.116 180.65 l } bind def /p0_a1 { newpath translate 293.116 181.566 m 304.837 181.566 l 304.837 182.482 l 293.116 182.482 l 293.116 181.566 l } bind def /p0_a2 { newpath translate 293.116 182.482 m 304.837 182.482 l 304.837 183.398 l 293.116 183.398 l 293.116 182.482 l } bind def /p0_a3 { newpath translate 293.116 183.398 m 304.837 183.398 l 304.837 184.313 l 293.116 184.313 l 293.116 183.398 l } bind def /p0_a4 { newpath translate 293.116 184.313 m 304.837 184.313 l 304.837 185.229 l 293.116 185.229 l 293.116 184.313 l } bind def /p0_a5 { newpath translate 293.116 185.229 m 304.837 185.229 l 304.837 186.145 l 293.116 186.145 l 293.116 185.229 l } bind def /p0_a6 { newpath translate 293.116 186.145 m 304.837 186.145 l 304.837 187.061 l 293.116 187.061 l 293.116 186.145 l } bind def /p0_a7 { newpath translate 293.116 187.061 m 304.837 187.061 l 304.837 187.976 l 293.116 187.976 l 293.116 187.061 l } bind def /p0_a8 { newpath translate 293.116 187.976 m 304.837 187.976 l 304.837 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def 635.407 254.127 o grestore gsave 639.406906 250.814135 translate 0.000000 rotate 0.000000 0.000000 m /zero glyphshow 7.634766 0.000000 m /period glyphshow 11.449219 0.000000 m /zero glyphshow 19.083984 0.000000 m /zero glyphshow 26.718750 0.000000 m /zero glyphshow 34.353516 0.000000 m /zero glyphshow 41.988281 0.000000 m /zero glyphshow 49.623047 0.000000 m /one glyphshow 57.257812 0.000000 m /six glyphshow grestore end showpage ================================================ FILE: radiation/radiation.tex ================================================ \label{ch:radiation} \section{Equations of Radiation Hydrodynamics} \subsection{Reference Frames} \subsection{Angular Approximations / Moments} \subsection{Closures} \section{Hyperbolic System} \section{Parabolic System} \subsection{General Elliptic Solver} \label{sec:rad:generalelliptic} For the gray radiation hydrodynamics system, we need to solve a linear system that takes the form: \begin{equation} \alpha \phi + \nabla \cdot \beta \nabla \phi + \gamma \cdot \nabla \phi = f \end{equation} (see Zhang et al., Eq. 45). We can discretize this for a cell-centered $\phi$ to second-order as: \begin{align} \alpha_{i,j} \phi_{i,j} &+ \frac{(\beta \nabla \phi)_{i+1/2,j} - (\beta \nabla \phi)_{i-1/2,j}}{\Delta x} + \frac{(\beta \nabla \phi)_{i,j+1/2} - (\beta \nabla \phi)_{i,j-1/2}}{\Delta y} \nonumber \\ &+ \gamma^{(x)}_{i,j} \frac{\phi_{i+1,j} - \phi_{i-1,j}}{2\Delta x} + \gamma^{(y)}_{i,j} \frac{\phi_{i,j+1} - \phi_{i,j-1}}{2\Delta y} = f_{i,j} \end{align} where we decompose the vector, $\gamma$ as $\gamma = \gamma^{(x)}\hat{x} + \gamma^{(y)}\hat{y}$. Expanding the gradients: \begin{align} \label{eq:general_L} \alpha_{i,j} \phi_{i,j} &+ \frac{\beta_{i+1/2,j} (\phi_{i+1,j} - \phi_{i,j}) - \beta_{i-1/2,j} (\phi_{i,j} - \phi_{i-1,j})}{\Delta x^2} \nonumber \\ &+ \frac{\beta_{i,j+1/2} (\phi_{i,j+1} - \phi_{i,j}) - \beta_{i,j-1/2} (\phi_{i,j} - \phi_{i,j-1})}{\Delta y^2} \nonumber \\ &+ \gamma^{(x)}_{i,j} \frac{\phi_{i+1,j} - \phi_{i-1,j}}{2\Delta x} + \gamma^{(y)}_{i,j} \frac{\phi_{i,j+1} - \phi_{i,j-1}}{2\Delta y} = f_{i,j} \end{align} There are several different linear system algorithms people use to solve these types of systems. Since we have already developed a multigrid solver, we will add equations of this type to the multigrid framework. Defining $\tilde{\beta}_{i\pm1/2,j} \equiv \beta_{i\pm1/2,j}/\Delta x^2$, $\tilde{\beta}_{i,j\pm1/2} \equiv \beta_{i,j\pm1/2}/\Delta y^2$, $\tilde{\gamma}^{(x)}_{i,j} = \gamma^{(x)}_{i,j}/(2\Delta x)$, and $\tilde{\gamma}^{(y)}_{i,j} = \gamma^{(y)}_{i,j}/(2\Delta y)$, we have as an update to $\phi_{i,j}$, \begin{align} \label{eq:general_mg_smooth} \phi_{i,j} = \frac{1}{D_{i,j}} \bigg [ f_{i,j} &-(\tilde{\beta}_{i+1/2,j} + \tilde{\gamma}^{(x)}_{i,j}) \phi_{i+1,j} -(\tilde{\beta}_{i-1/2,j} - \tilde{\gamma}^{(x)}_{i,j}) \phi_{i-1,j} \nonumber \\ &-(\tilde{\beta}_{i,j+1/2} + \tilde{\gamma}^{(y)}_{i,j}) \phi_{i,j+1} -(\tilde{\beta}_{i,j-1/2} - \tilde{\gamma}^{(y)}_{i,j}) \phi_{i,j-1} \bigg ] \end{align} with \begin{equation} D_{i,j} = \alpha_{i,j} - \tilde{\beta}_{i+1/2,j} - \tilde{\beta}_{i-1/2,j} - \tilde{\beta}_{i,j+1/2} - \tilde{\beta}_{i,j-1/2} \end{equation} The remaining details of the multigrid solver are unchanged. The boundary conditions are implemented in the same way as done for Poisson's equation. For radiation, we will need to implement inhomogeneous boundary conditions, which we can do via the ghost cell filling using Eqs.~\ref{eq:bc_inhomo_dir} and \ref{eq:bc_inhomo_neum}. Note that because of this general form of the equation, it is no longer simple to do the boundary charge method described in \S~\ref{sec:multigrid:other}, so we must modify the ghost cell fill routines explicitly. The only changes to the core solver are in the smoothing function, which implements Eq.~\ref{eq:general_mg_smooth} and in the residual, which constructs our operator as discretized in Eq.~\ref{eq:general_L}. To test this solver out, we can define a test problem by picking functional forms of $\alpha$, $\beta$, and $\gamma$, and a solution, $\phi$, with desired boundary conditions and find the resulting $f$ matching: \begin{equation} \label{eq:general_elliptic} \alpha \phi + \nabla \cdot (\beta \nabla \phi) + \gamma \cdot \nabla \phi = f \end{equation} Let's find a set of coefficients and righthand side for which \begin{equation} \phi = \cos(\pi x/2) \cos(\pi y/2) \end{equation} is a solution on $[0,1]\times[0,1]$. This would satisfy the Dirichlet boundary conditions: \begin{align} \phi |_{x=0} &= \cos(\pi y/2) \\ \phi |_{x=1} &= 0 \\ \phi |_{y=0} &= \cos(\pi x/2) \\ \phi |_{y=1} &= 0 \end{align} For the coefficients, we choose: on with homogeneous Dirichlet boundary conditions. We use \begin{align} \alpha &= 10 \\ \beta &= xy+1 \\ \gamma &= \hat{x} + \hat{y} \end{align} This gives \begin{align} \label{eq:general_elliptic_rhs} f = &- \frac{\pi}{2} (x + 1) \sin{\left (\frac{\pi y}{2} \right )} \cos{\left (\frac{\pi x}{2} \right )} - \frac{\pi}{2} (y + 1) \sin{\left (\frac{\pi x}{2} \right )} \cos{\left (\frac{\pi y}{2} \right )} \nonumber \\ &+ \left(10- \frac{x y+1}{2} \pi^{2}\right) \cos{\left (\frac{\pi x}{2} \right )} \cos{\left (\frac{\pi y}{2} \right )} \end{align} Figure~\ref{fig:general_mg_converge} shows the convergence of multigrid for this problem, and figure~\ref{fig:general_mg_solution} shows the solution. We see second order convergence, as expected with this discretization. \begin{figure} \centering \includegraphics[width=0.6\linewidth]{mg_general_inhomogeneous_converge} \caption[Convergence of the general elliptic multigrid solver]{\label{fig:general_mg_converge} Convergence of the multigrid solver on our test problem $\alpha \phi + \nabla \cdot (\beta \nabla \phi) + \gamma \cdot \nabla \phi = f$ with inhomogeneous Dirichlet boundary conditions.} \end{figure} \begin{figure} \centering \includegraphics[width=\linewidth]{mg_general_inhomogeneous_test} \caption[Solution of a general elliptic equation] {\label{fig:general_mg_solution} The solution to our general elliptic text problem, Eqs.~\ref{eq:general_elliptic} to \ref{eq:general_elliptic_rhs} with inhomogeneous boundary conditions on a $512^2$ grid. This test can be run in \pyro\ {\tt multigrid/test\_mg\_general\_inhomogeneous.py}.} \end{figure} ================================================ FILE: reactive_flow/notes.txt ================================================ -- mass fractions, partial density -- ODEs for reaction network -- number density -- coupling (Strang splitting vs. SDC) -- flames -- detonations ================================================ FILE: reactive_flow/reactive_flow.tex ================================================ \section{Introduction} Many astrophysical problems involve modeling nuclear (or chemical) reactions coupled with hydrodynamics. Astrophysical reacting flows share a lot of similarities with terrestrial combustion, including both subsonic (deflagrations) and supersonic (detonations) burning fronts. Nice discussions of the physics of reacting flows can be found in, e.g., \cite{oranboris, peters:2000} For multifluid flows, the Euler equations are augmented with continuity equations for each of the nuclear (or chemical) species. If we denote the density of species $k$ as $\rho_k$, then conservation of mass for each species individually implies: \begin{equation} \label{react:eq:partialdensitycont} \ddt{\rho_k} + \nabla \cdot (\rho_k \Ub) = 0 \end{equation} Summing Eq.~\ref{react:eq:partialdensitycont} over $k$ gives us back the mass contunity equation, since $\sum_k \rho_k = \rho$. The $\rho_k$ are sometimes called {\em partial densities}. Note that using both Eq.~\ref{react:eq:partialdensitycont} and the normal mass continuity equation for $\rho$ overspecifies the system. It is often easier to define a mass fraction of species $k$, $X_k$, as \begin{equation} X_k = \frac{\rho_k}{\rho} \end{equation} and it is easy to see that $\sum_k X_k = 1$, and \begin{equation} \label{react:eq:partialrho} \frac{\partial (\rho X_k)}{\partial t} + \nabla \cdot (\rho X_k \Ub) = 0 \end{equation} \begin{exercise}[Species advection] Using the continuity equation, show that we can write Eq.~\ref{react:eq:partialrho} as an advection equation: \begin{equation} \frac{\partial X_k}{\partial t} + \Ub \cdot \nabla X_k = 0 \end{equation} or \begin{equation} \DDt{X_k} = 0 \end{equation} \end{exercise} This latter form says that as a fluid element advects with the flow, its composition does not change. However, reactions can turn one species into another, so for reacting flow, $DX_k/Dt \ne 0$. If we define the creation rate for species $k$ as $\omegadot_k$, then we have: \begin{equation} \DDt{X_k} = \omegadot_k \end{equation} Since $\sum_k X_k = 1$, it must be that $\sum_k \omegadot_k = 0$. Reactions will also release (or consume) energy, due to the change in nuclear or chemical binding energy. This energy will be a source to the internal energy. In particular, our first law of thermodynamics now has a source:\MarginPar{this assumes chemical equilibrium} \begin{equation} T ds = de + p d \left( \frac{1}{\rho}\right ) = q_\mathrm{react} \end{equation} where $q_\mathrm{react}$ is the specific energy release due to reactions (energy / mass). Following a fluid element, this takes the form: \begin{equation} T\DDt{s} = \DDt{e} + p \DDt{(1/\rho)} = \Hnuc \end{equation} Where now $\Hnuc$ is the time-rate of energy release per unit mass ($dq_\mathrm{react}/dt$). An additional energy source that we often consider along with reactions is thermal diffusion. Fick's law says that the heat flux is proportional to $\nabla T$, so we write: \begin{equation} F_\mathrm{cond} = -\kth \nabla T \end{equation} as the conductive heat flux. Here the `$-$' indicates that heat flows from hot to cold and $\kth$ is the thermal conductivity. See \S~\ref{sec:diffusion} for more discussion. The conservative Euler equations with reactive and diffusive source terms appear as: \begin{align} \ddt{\rho} + \nabla \cdot (\rho \Ub) &= 0 \\ \ddt{\rho X_k} + \nabla \cdot (\rho \Ub X_k) &= \rho \omegadot_k \\ \ddt{(\rho \Ub)} + \nabla \cdot (\rho \Ub \Ub) + \nabla p &= 0 \\ \ddt{(\rho E)} + \nabla \cdot (\rho E \Ub + p \Ub ) &= \nabla \cdot (\kth \nabla T) + \rho \Hnuc \end{align} We'll write the heat sources simply as $H$, \begin{equation} H = \Hnuc + \frac{1}{\rho} \nabla \cdot \kth \nabla T \end{equation} In primitive form, the derivation of the pressure equation is a complex. Previously, we found the pressure evolution in the absence of sources by differentiating pressure along streamlines (Eq.~\ref{eq:euler:presnosource}): \begin{equation} \frac{Dp}{Dt} = \left . \frac{\partial p}{\partial \rho} \right |_s \DDt{\rho} + \left . \frac{\partial p}{\partial s} \right |_\rho \DDt{s} \end{equation} where, as before, we recognize that $\Gamma_1 \equiv \partial \log p/\partial \log \rho |_s$. Now, however, we have an entropy source, so $Ds/Dt \ne 0$. The Maxwell relations\footnote{see, for instance, \cite{shu}, for some discussion on these relations} tell us that \begin{equation} \partial p/\partial s |_\rho = \rho^2 \partial T/\partial \rho |_s \end{equation} This gives us the form: \begin{equation} \frac{Dp}{Dt} = \frac{p}{\rho} \Gamma_1 \frac{D\rho}{Dt} + \left . \rho^2 \frac{\partial T}{\partial \rho} \right |_s \frac{H}{T} \end{equation} We need an expression for $\partial T/\partial \rho |_s$ in terms of derivatives of $\rho$ and $T$ (since that is what our equations of state typically provide). Consider $\Gamma_1$, with $p = p(\rho, T)$: \begin{equation} \Gamma_1 = \frac{\rho}{p} \left . \frac{dp}{d\rho} \right |_s = \frac{\rho}{p} \left [ p_\rho + p_T \left . \frac{dT}{d\rho} \right |_s \right ] \end{equation} where we use the shorthand $p_\rho \equiv \partial p/\partial \rho |_T$ and $p_T \equiv \partial p/\partial T |_\rho$. This tells us that \begin{equation} \left . \frac{\partial T}{\partial \rho} \right |_s = \frac{p}{\rho p_T} (\Gamma_1 - \chi_\rho) \end{equation} where $\chi_\rho \equiv \partial \log p / \partial \log \rho |_T$. To further simplify this, we use some standard relations for general equations of state \begin{equation} \frac{\Gamma_1}{\chi_\rho} = \frac{c_p}{c_v} \end{equation} and \begin{equation} c_p - c_v = \frac{p}{\rho T} \frac{\chi_T^2}{\chi_\rho} \end{equation} with $\chi_T \equiv \partial \log p / \partial \log T |_\rho$ (see, e.g., \cite{HKT} for both of these relations). These allow us to write \begin{equation} \left . \frac{\partial T}{\partial \rho} \right |_s = \frac{p\chi_T}{\rho^2 c_v} \end{equation} Putting this all together, we have: \begin{equation} \frac{Dp}{Dt} = \frac{p}{\rho}\Gamma_1 \frac{D\rho}{Dt} + \frac{p \chi_T}{c_v T} H \end{equation} and using the continuity equation, $D\rho/Dt = -\rho \nabla \cdot \Ub$, we finally have: \begin{equation} \label{eq:reacting:p} \ddt{p} + \Ub \cdot \nabla p + \Gamma_1 p \nabla \cdot \Ub = \Gamma_1 p \sigma H \end{equation} where we defined \begin{equation} \sigma = \frac{\partial p/\partial T |_\rho}{\rho c_p \partial p/\partial \rho |_T} \end{equation} The $\sigma$ notation follows from \cite{ABRZ:II}. The primitive variable is then \begin{align} \ddt{\rho} + \Ub \cdot \nabla \rho + \rho \nabla \cdot \Ub &= 0\\ \ddt{\Ub} + \Ub \cdot\nabla \Ub + \frac{1}{\rho} \nabla p &= 0\\ \ddt{p} + \Ub \cdot \nabla p + \Gamma_1 p \nabla \cdot \Ub &= \Gamma_1 p \sigma H \\ \ddt{X_k} + \Ub \cdot \nabla {X_k} &= \omegadot_k \end{align} Finally, we can derive a temperature evolution equation following the method from \S~\ref{sec:euler:otherthermo}. We find \begin{equation} c_v \DDt{T} = \left ( \left . \frac{\partial e}{\partial \rho} \right |_T - \frac{p}{\rho} \right ) \nabla \cdot \Ub + H \end{equation} or \begin{equation} c_p \DDt{T} = \left ( \frac{1}{\rho} - \left . \frac{\partial h}{\partial p} \right |_T \right ) \DDt{p} + H \end{equation} depending on whether we start with $e$ or $h$. \section{Operator splitting approach} Using operator splitting, we separate the equations into hydrodynamics and reactive parts and we do these operations in turn. Following the ideas from \S~\ref{sec:multiphys:diffreact}, we perform the update in stages. If we denote the reaction update operator as $R_{\Delta t}$ and the hydrodynamics operator as $A_{\Delta t}$, then our update appears as: \begin{equation} \Uc^{n+1} = R_{\Delta t/2} A_{\Delta t} R_{\Delta t/2} \Uc^n \end{equation} To make it more explicit, we write out system as: \begin{equation} \ddt{\Uc} + \nabla \cdot \Fb(\Uc) = \Rc(\Uc) \end{equation} where $\Rc = (0, \omegadot_k, 0, \rho \Hnuc)^\intercal$ is the vector of reactive sources. The advance through $\Delta t$ in time is: \begin{enumerate} \item {\em Evolve reaction part for $\Delta t/2$}. We define $\Uc_i^\star$ as the result of integrating \begin{equation} \frac{d\Uc_i}{dt} = \Rc(\Uc_i) \end{equation} over $t \in [t^n, t^{n+\myhalf}]$ with initial conditions, $\Uc_i(t^n) = \Uc_i^n$. \item {\em Solve the hydrodynamics portion for $\Delta t$, beginning with the state $\Uc^\star$} % \begin{equation} \frac{\Uc^{\star\star}_i - \Uc^\star_i}{\Delta t} = \frac{\Fb^\exx(\Uc^{\star,n+\myhalf}_{i-\myhalf}) - \Fb^\exx(\Uc^{\star,n+\myhalf}_{i+\myhalf})}{\Delta x} \end{equation} where $\Uc^{\star,n+\myhalf}_{i-\myhalf}$ is the predict interface state (using the ideas from \S~\ref{sec:onedrecon}), beginning with the state $\Uc^\star$. Note that $\Rc$ does not appear here explicitly. Alternately, we could use the MOL hydrodynamics update here, but the basic idea will be the same---the hydrodynamics does not explicitly see the reaction terms. \item {\em Evolve reaction part for $\Delta t/2$} We reach the final update, $\Uc^{n+1}$, by integrating \begin{equation} \frac{d\Uc}{dt} = \Rc(\Uc) \end{equation} over $t \in [t^{\myhalf}, t^{n+1}]$ with initial conditions $\Uc(t^{n+\myhalf}) = \Uc^{\star\star}$. \end{enumerate} \subsection{Adding species to hydrodynamics} In the hydrodynamics algorithm, we neglect the diffusion and reaction terms. Their effects are incorporated implicitly by performing the hydrodynamics update starting with the state that has already experienced burning. \begin{align} \ddt{\rho} + \nabla \cdot (\rho \Ub) &= 0 \\ \ddt{(\rho X_k)} + \nabla \cdot (\rho \Ub X_k) &= 0 \\ \ddt{(\rho \Ub)} + \nabla \cdot (\rho \Ub \Ub) + \nabla p &= 0 \\ \ddt{(\rho E)} + \nabla \cdot (\rho E \Ub + p \Ub ) &= 0 \end{align} When we now consider our primitive variables: $\qb = (\rho, u, p, X_k)$, we find \begin{equation} \Ab(\qb) = \left ( \begin{array}{cccc} u & \rho & 0 & 0\\ 0 & u & 1/\rho & 0\\ 0 & \Gamma_1 p & u & 0\\ 0 & 0 & 0 & u \end{array} \right ) \end{equation} There are now 4 eigenvalues, with the new one also being simply $u$. This says that the species simply advect with the flow. The right eigenvectors are now: \begin{equation} \rb^\evm = \left ( \begin{array}{c} 1 \\ -c/\rho \\ c^2 \\ 0\end{array} \right ) % \qquad \rb^\evz = \left ( \begin{array}{c} 1 \\ 0 \\ 0 \\ 0\end{array} \right ) % \qquad \rb^\evzs{X} = \left ( \begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \end{array} \right ) % \qquad \rb^\evp = \left ( \begin{array}{c} 1 \\ c/\rho \\ c^2 \\ 0 \end{array} \right ) \end{equation} corresponding to $\lambda^\evm = u -c$, $\lambda^\evz = u$, $\lambda^\evzs{X} = u$, and $\lambda^\evp = u + c$. We see that for the species, the only non-zero element is for one of the $u$ eigenvectors ($\rb^\evzs{X}$). This means that $X_k$ only jumps over this middle wave. In the Riemann solver then, there is no `star' state for the species, it just jumps across the contact wave. To add species into the solver, you simply need to reconstruct $X_k$ as described in Chapter~\ref{ch:compressible} to find the interface values. If we are doing characteristic tracing, then we use this new $\Ab(\qb)$ and associated eigenvectors. We then solve the Riemann problem, with $X_k$ on the interface taken as simply the left or right state depending on the sign of the contact wave speed, and do the conservative update for $\rho X_k$ using the species flux. One issue that can arise with species is that even if $\sum_k X_k = 1$ initially, after the update, that may no longer be true. There are a variety of ways to handle this: \begin{itemize} \item You can update the species, $(\rho X_k)$ to the new time and then define the density to be $\rho = \sum_k (\rho X_k)$---this means that you are not relying on the value of the density from the mass continuity equation itself. \item You can force the interface states of $X_k$ to sum to 1. Because the limiting is non-linear, this is where problems can arise. If the interface values of $X_k$ are forced to sum to 1 (by renormalizing), then the updated cell-centered value of $X_k$ will as well. This is the approach discussed in \cite{plewamuller:1999}. \item You can design the limiting procedure to preserve the summation property. This approach is sometimes taken in the combustion field. For piecewise linear reconstruction, this can be obtained by computing the limited slopes of all the species, and taking the most restrictive slope and applying this same slope to all the species. \end{itemize} \subsection{Integrating the reaction network} The reactive portion of our operator split system is: \begin{align} \frac{dX_k}{dt} &= \omegadot_k \\ \frac{d(\rho e)}{dt} &= \rho \Hnuc \end{align} As written, this system is not closed without the equation of state, since $\omegadot_k = \omegadot_k(\rho, T, X_k)$, and likewise for $\Hnuc$. Our temperature evolution equation reduces to \begin{equation} c_x \frac{dT}{dt} = \Hnuc \end{equation} where $c_x$ is $c_v$ or $c_p$, depending on the physical effects we are trying to capture. \if DEBUG \footnote{formally, there should be a composition term here too, of the form \[ \sum_k \left . \frac{\partial e}{\partial X_k} \right |_{\rho,T} \frac{dX_k}{dt} = \sum_k \mu_k \frac{dX_k}{dt} \] where $\mu_k$ is the chemical potential. But this term is zero if we are in chemical equilibrium, which is usually assumed (but perhaps should be checked) for astrophysical flows. } \fi This difference only arises because we are neglecting the hydrodynamic portions of the temperature evolution during the reaction step. Many simulation codes neglect the evolution of temperature during the reaction step (see, e.g., \cite{flash}). An excellent introduction to integrating astrophysical nuclear reaction networks is found in \cite{timmes_nets}. The main issue with reaction networks is that they tend to be stiff, which means that implicit integration techniques are needed. There are a number of well-tested, freely available implicit ODE integrators (e.g.~\cite{vode}). For stiff systems, you need to supply both a righthand side routine and a Jacobian routine\footnote{although this could be calculated via finite-differences if needed}. \subsection{Incorporating explicit diffusion} \section{Burning modes} \subsection{Convective burning} \subsection{Deflagrations} A deflagration or flame is a subsonic burning front that propagates via a balance of diffusion and reactions. Thermal diffusion raises the temperature of fuel ahead of the flame to ignition and the energy release heats the ash, keeping the thermal gradient needed for diffusion. In terrestrial flames, species diffusion can be important as well, but this is usually negligible in astrophysical flames. Because the burning front is subsonic, sound waves can communicate across the flame and therefore the fuel and ash are at the same pressure. The temperature jump across the flame means that the density will drop behind the flame to preserve the constant pressure. Our temperature evolution equation, taking $Dp/Dt = 0$, is \begin{equation} \rho c_p \DDt{T} = \nabla \cdot \kth \nabla T + \rho \Hnuc \end{equation} We can estimate the timescale for diffusion by neglecting the reactions and writing \begin{equation} \DDt{T} = \frac{1}{\rho c_p} \nabla \cdot \kth \nabla T \approx \mathcal{D} \nabla^2 T \end{equation} where $\mathcal{D} = \kth/(\rho c_p)$ is the diffusion coefficient (assuming $\rho c_p$ is spatially constant). Dimensional analysis suggests that the diffusion timescale is then \begin{equation} t_\mathrm{diff} \approx \frac{\delta^2}{\mathcal{D}} \end{equation} where $\delta$ is the width of the region where diffusion takes place---we'll identify this as the flame thickness. Similarly, we can estimate the reaction timescale by considering \begin{equation} \rho c_p \DDt{T} = \rho \Hnuc \end{equation} and writing \begin{equation} t_\mathrm{burn} \approx \frac{c_p T}{\Hnuc} \end{equation} We can alternately expand $\Hnuc$ in terms of temperature if we wish to capture some of the temperature sensitivity. Equating these timescales gives \begin{equation} \delta \sim \left ( \frac{\kth T}{\rho \Hnuc} \right )^{1/2} \end{equation} which is an estimate of the flame thickness. We can define the flame speed as the time it takes to burn through this thickness: \begin{equation} v_f \sim \frac{\delta}{t_\mathrm{burn}} \sim \left ( \frac{\kth \Hnuc}{\rho c_p T} \right )^{1/2} \end{equation} We note the main dependences are: \begin{equation} v_f \sim \sqrt{\kth \Hnuc} \qquad \delta \sim \sqrt{\frac{\kth}{\Hnuc}} \end{equation} \subsection{Detonations} ================================================ FILE: refs.bib ================================================ @String{ JCP = "J Comput Phys"} @String{ JAS = "J Atmos Sci"} @String{ APJ = "Astrophys J"} @String{ APJ:sup = "Astrophys J Supplement"} @String{ AAP = "Astron Astrophys"} @String{ SJSSC = "SIAM J Sci Statist Comput"} @String{ CAMCOS = "Communications in Applied Mathematics and Computational Science"} @String{ MNRAS = "Mon Not R Astron Soc"} @ARTICLE{durran:1989, author={D. R. 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The first comes with the choice of equation set. As discussed earlier, for pure hydrodynamics, this often means solving the Euler rather than the full Navier-Stokes equations. The next approximation is made when discretizing the equations, using the methods we described in the earlier chapters. Astrophysical flows involve a range of length and timescales. Often it is not possible to capture all these scales in a single simulation, so some comprimises must be made. If you are interested in the dynamics on the full scale of the system, $L$, then you will resolve down to some cutoff scale $l$, where $L/l \sim O(10^3\mbox{--}10^4)$. At scales smaller than $l$, you neglect the flow features. The smallest scale of importance is the disipation scale $\lambda \ll L$. If you are not resolving down to the dissipation scales in our problem, then you are working in the realm of {\em large eddy simulations} (LES; see \cite{schmidt:2014}). Proper LES requires a subgrid model to treat the unresolve scales and their feedback onto your grid. If you instead rely on numerical diffusion of your method to do the dissipation, you are doing {\em implicit large eddy simulation} or ILES---for many flows, this be sufficient to capture the dynamics (including turbulence) that are important~\cite{aspden:2008,MargolinRider:2002}. \section{Dimensionality and picking your resolution} Nature is three-dimensional, so realistic simulaitons generally require full 3-d modeling. Nevertheless, 2-d simulations can be useful to scope out the problem space, and in particular, to get a feel for the timescales and resolution needed to resolve smallscale features. The major place where dimensionality affects results is for instabilities and turbulence. The trend in 2-d is for small scale vortices to merge and build larger and larger structures (see, e.g., \cite{twodturbulence}), while in 3-d motions at the largest scale cascade down to smaller scales, driving turbulent flow down to the smallest scales. Turbulence can be very important in astrophysical flows, and properly capturing it requires modeling a large range of lengthscales (physically, the objective is to get a large {\em Reynolds number}: the ratio of the advective force to the dissipative force). Generally, you'll just start to get a resolved inertial range over a decade in wavenumber in your power spectrum at around 512 zones on a side of your domain (using the methods that we discussed here). With fewer points than this, you are unlikely to capture turbulence. If nuclear burning is modeled, the steepness of the reaction rate will set a lengthscale. Burning can also wash out instabilities, a process sometimes called {\em fire polishing} \cite{timmeswoosley,SNrt}. Here, if the timescale for reactions is faster than the growth rate of an instability, the instability is suppressed. This can set a small scale cutoff that is well above the viscous dissipation scale. Hydrostatic equilibrium also puts demands on the resolution. Recall that the momentum equation is: \begin{equation} \rho \DDt{\Ub} + \nabla p = \rho \gb \end{equation} If hydrostatic equilibrium, $\nabla p = \rho \gb$, is not exactly satisfied, then the residual will appear as a force that generates an acceleration and non-zero velocities will build up. A good rule of thumb is that you need $10$ points per pressure scale height to get a reasonable hydrostatic balance (this is discussed at bit in \cite{hse}). Spatial resolution is often in contention with improved physics---both can add to the computational cost of a simulation, so for fixed cost, you have often have to choose between more physics (e.g., larger reaction network) or more resolution. An important part of the simulation process is to ensure that your results are converged. In a {\em convergence study}, you run the same simulation at various resolutions and look to see that the major results (e.g., integral quantities) do not change significantly with resolution. Ideally you would show that any results you are interested are moving toward an asymptotic value. Even if you are not converged however, you could use the trends observed with resolution to help understand the possible error in your simulation. \section{Boundary conditions} Another approximation you make is how to treat the boundaries. Usually one chooses symmetry / reflecting, outflow / zero-gradient, inflow (specifying the value on the boundary), or periodic. In the case of symmetry, the fluid state is reflected and the normal velocity is reflected with a sign change. The transverse velocity can be left as is (a slip wall) or set to zero at the boundary (no-slip wall). Generally speaking, you should put your boundaries as far away from the place where the action is occuring as possible. For example, if you are modeling a convective region bounded by stably stratified flow, then you should have a large buffer of stable atmosphere bounding it. Periodic boundary conditions don't magically fix everything either. With a triply-periodic box (periodic on all boundaries), then you are confining the flow. If energy is injected into the box, then there is no place for it to expand, and this can lead to artifical heating or compression. Symmetry / reflecting boundaries provide a simple way to reduce a problem size, for example, modeling only an octant of a star in 3-d instead of the full star. But they also introduce artifacts, since they force the normal velocity to zero right on the boundary. This means that flow through the center of a star (e.g., if we model only an octant) can be restricted, resulting in artifical flow / convective features. Wall heating can also be an issue---forcing the normal velocity to zero at the symmetry boundary can lead to stagnation of the flow there, allowing hot spots to develop and not be transported away. \MarginPar{characteristic BCs?} \section{Timestep considerations} The timestep constraint we discussed (the CFL condition) is required for stability, but that is different than accuracy. In particular, accurately capturing some physical processes (that may appear as source terms) might require more restrictive timesteps to ensure that they are properly coupled with the hydrodynamics. An example where this can commonly arise is reactive flows---combining hydrodynamics with nuclear or chemical reactions. If the reactions are very vigorous, then you may run into a situation where you exaust your fuel in a zone in a single step, without giving the hydrodynamics a chance to advect new fuel in. This represents a breakdown of the coupling of the hydrodynamics and reactions. A common way to improve the accuracy here is to monitor the change in mass fractions of a fuel and set a timestep limit such that: \begin{equation} \Delta t_\mathrm{react} = C_\mathrm{react} \frac{X_\mathrm{fuel}}{|\omegadot_\mathrm{fuel}|} \end{equation} where $C_\mathrm{react} < 1$. The species creation rate, $\dot{X}_\mathrm{react}$, is typically evaluated using the value from the previous step. \section{Convergence and multiphysics} For a full simulation, with hydrodynamics coupled to other physics, you should test your code to ensure that it is converging at the rate that you expect. Usually in this case there is no analytic solution, so you need to measure the convergence numerically. A standard way to do this is to run pairs of simulations that differ by a factor of two in resolution, we'll denote the coarser simulation with the subscript $c$ and the finer simulation with the subscript $f$. The error is computed for a field $\phi$ by coarsening the finer simulation by a factor of two, and computing the norm of the difference: \begin{equation} \epsilon \equiv \| \phi_c - \mathcal{C}(\phi_f) \| \end{equation} Here, we denote the coarsening operator as $\mathcal{C}(.)$. In this case, two simulations give us a single error. We then repeat this, taking the fine simulation from this comparison as the coarse for the next, and using an even finer simulation to generate a new error. We can then measure the convergence of a multiphysics simulation by comparing the errors from the two pairs of the simulations. This procedure was used in, e.g., \cite{ABNZ:III,mccorquodalecolella}. Ideally, a smooth problem should be used, since discontinuities by definition do not converge with resolution (and limiters, if used, would kick in there as well). \section{Computational cost} The main limitation to your choice of resolution and timestep is computational cost. For a three-dimensional simulation in a box with $N$ zones on a side, the work scales as $O(N^4)$. The cost to advance a single timestep is based on the volume, $N^3$, and the number of timesteps to get you to a fixed simulation time, $t_\mathrm{max}$, is $t_\mathrm{max}/\Delta t$, but $\Delta t \sim \Delta x / s \sim L/(N s)$, where $s$ is the fastest information speed. So the number of timesteps increases with $N$, making the total cost $N^4$. Think about this for a minute---if you double the number of zones, then the amount of work you need to do increases by 16$\times$. A common conflict that arises is: Do you do one ``hero'' calculation or many smaller calculations to understand the sensitivity of your results to input parameters? Computing centers that grant allocations usually setup their queues to favor big jobs that use as much of the machine as possible (as that's what their funders use as a metric for their success). It is important to understand that no single algorithm will offer everything you need. \section{I/O} Another challenge with simulations is data storage. For a calculation on a $1024^3$ grid, using double precision, each number you store takes 8~bytes / zone, so a single variable on the grid will require 8~GB. For compressible hydro, you will have atleast 5 variables (in 3-d), $\rho$, $(\rho {\bf U})$, and $(\rho E)$, so this is a minimum of 40~GB per file. Often you will want some derived quantities (like temperature) and composition quantities in your output, which will greatly increase your storage requirements. For this reason, simulation codes often have different types of output. Checkpoint files store all the variables that are needed to restart the calculation from where you left off---these should be stored in a (portable) binary format at machine precision. You don't need to keep these around forever, perhaps only saving the last few as necessary to restart your calculation as it works through a computing center's queue. Plotfiles store a lot of variables---basically anything that will be needed for the analysis of the simulation. Since you don't need these to restart, you can probably afford to store the variables at single precision. Sometimes you might have multiple levels of plotfiles, storing a few variables (like density) very frequently to allow for the production of smooth movies, and storing a lot of variables and derived quantities at a much longer cadence. A perpetual challenge with the analysis and visualization of simulations is that sometimes you don't know what you are looking for until the simulation is complete, so you cannot simply do runtime visualization and not save the raw data as the simulation progresses. This leads to collections of plotfiles that can easily top 100~TB per simulation. Runtime diagnostics can help a little---if you know any global quantities that you will be interested in over the course of the simulation (like peak temperature, or total energy) you can compute these on the fly and simply store a single number per step (or every $N$ steps) into a file. ================================================ FILE: software-engineering/software-engineering.tex ================================================ \section{Version control} Version control gives you the ability to keep track of changes to your source, interact with other developers on the same project, and to roll back changes when bugs are introduced. It is easy to get started and is perhaps the single most important part of code development for everyone. There are several choices for version control, split loosely between centralized systems and distributed version control systems. In the former category are CVS and subversion---these have been around for a long time and are still in wide use, but not recommedned for starting a new project. Mercurial and git are the two most popular distributed version control systems, each actively used. There are some subtle differences between the two, but generally the workflow in each system follows the same ideas. Diagram showing a distributed workflow. We generally want to pull bitbucket and github \section{Testing} There are several types of testing that are important in computational hydrodynamics. When concerning source code quality, {\em unit testing} and {\em regression testing} are invaluable in making sure that your code works as designed and that you don't change the results as you continue to develop. \section{Reproducibility} ================================================ FILE: software-engineering/topics.txt ================================================ -- git & github -- diff -- gnuplot -- awk -- grep / ack -- basic scripting (for i in ... [ seq, *.png ] . basename -- screen ================================================ FILE: symbols/symbols.tex ================================================ \renewcommand{\arraystretch}{1.5} % \begin{center} \begin{longtable}{|l|p{3.25in}|l|} \caption[Definition of symbols.]{Definition of symbols.} \label{table:sym} \\ % \hline \multicolumn{1}{|c|}{\textbf{symbol}} & \multicolumn{1}{ c|}{\textbf{meaning}} & \multicolumn{1}{ c|}{\textbf{units}} \\ \hline \endfirsthead \multicolumn{3}{c}% {{\tablename\ \thetable{}---continued}} \\ \hline \multicolumn{1}{|c|}{\textbf{symbol}} & \multicolumn{1}{ c|}{\textbf{meaning}} & \multicolumn{1}{ c|}{\textbf{units}} \\ \hline \endhead \multicolumn{3}{|r|}{{\em continued on next page}} \\ \hline \endfoot \hline \endlastfoot $\Ab$ & Jacobian matrix & N/A \\ % $C$ & Lagrangian sound speed, $C = \sqrt{\Gamma_1 p \rho}$ & g~m$^{-2}$~s$^{-1}$\\ % $\cfl$ & CFL number & --\\ % $c$ & sound speed, $c = \sqrt{\Gamma_1 p/\rho}$ & m~s$^{-1}$ \\ % $c_p$ & specific heat at constant pressure ($c_p \equiv \left . \partial h / \partial T \right |_{p,X_k}$) & erg~g$^{-1}$~K$^{-1}$ \\ % $c_v$ & specific heat at constant density ($c_v \equiv \left . \partial e / \partial T \right |_{\rho,X_k}$) & erg~g$^{-1}$~K$^{-1}$ \\ % $E$ & specific total energy & erg~g$^{-1}$ \\ % $e$ & specific internal energy & erg~g$^{-1}$ \\ % $\Fb$ & flux vector & N/A \\ % $\gb$ & gravitational acceleration & cm~s$^{-2}$ \\ % $\Gamma_1$ & first adiabatic exponent ($\Gamma_1 \equiv \left . d \log p/d \log \rho \right |_s$) & -- \\ % $\gamma$ & ratio of specific heats, $\gamma = c_p/c_v$ & -- \\ % $\gamma_e$ & the quantity $p/(\rho e) + 1$ & -- \\ % $H$ & heat sources & erg~g$^{-1}$~s$^{-1}$ \\ % $\Hnuc$ & nuclear energy source & erg~g$^{-1}$~s$^{-1}$ \\ % $h$ & specific enthalpy & erg~g$^{-1}$ \\ % $\Ic$ & integral under a (piecewise) parabolic polynomial reconstruction & N/A \\ % $\kth$ & thermal conductivity & erg~cm$^{-1}$~s$^{-1}$~K$^-1$ \\ % ${\bf L}$ & matrix of left eigenvectors & -- \\ % ${\boldsymbol{\Lambda}}$ & diagonal matrix of eigenvalue & -- \\ % $\lb$ & left eigenvector & N/A \\ % $\lambda$ & eigenvalue & N/A \\ % $M$ & Mach number, $M = |\Ub|/c$ & -- \\ % $\omegadot_k$ & species creation rate & s$^{-1}$ \\ % $p$ & pressure & erg~cm$^{-3}$ \\ % $\qb$ & primitive variable vector & N/A \\ % ${\bf R}$ & matrix of right eigenvectors & -- \\ % $\riemann(q_L, q_R)$ & Riemann problem between states $q_L$ and $q_R$ & N/A \\ % $\rb$ & right eigenvector & N/A \\ % $\rho$ & mass density & g~cm$^{-3}$ \\ % $S$ & source term to the divergence constraint & s$^{-1}$ \\ % $s$ & specific entropy & erg~g$^{-1}$~K$^{-1}$ \\ % $T$ & temperature & K \\ % $t$ & time & s \\ % $\tau$ & specific volume ($\tau = 1/\rho$) & cm$^3$~g$^{-1}$ \\ % $\Ub$ & total velocity vector, $\Ub = (u, v)^\intercal$ in 2-d & cm~s$^{-1}$ \\ % $\Uc$ & conserved variable vector & N/A \\ % $u$ & x-velocity & cm~s$^{-1}$ \\ % $v$ & y-velocity & cm~s$^{-1}$ \\ % $\wb$ & characteristic variables vector & N/A \\ % $X_k$ & mass fraction of the species ($\sum_k X_k = 1$) & -- \\ \end{longtable} \end{center} \renewcommand{\arraystretch}{1.0}