User: diener Date: 2012/04/01 08:52 PM

Added: /numerical_relativity/ Makefile, numerical_relativity.tex, preamble.tex

Log: The beginning of material for introduction to numerical methods & numerical relativity.

File Changes:

Directory: /numerical_relativity/ =================================

File [added]: Makefile Delta lines: +10 -0 =================================================================== --- numerical_relativity/Makefile (rev 0) +++ numerical_relativity/Makefile 2012-04-02 01:52:31 UTC (rev 21) @@ -0,0 +1,10 @@ +all: numerical_relativity.pdf + +%.pdf: %.tex + pdflatex $* + +clean: + /bin/bash -exec "rm -f numerical_relativity.{aux,log,nav,out,snm,pdf,toc}" + +.PHONY: all clean +

File [added]: numerical_relativity.tex Delta lines: +217 -0 =================================================================== --- numerical_relativity/numerical_relativity.tex (rev 0) +++ numerical_relativity/numerical_relativity.tex 2012-04-02 01:52:31 UTC (rev 21) @@ -0,0 +1,217 @@ +\input{preamble} +\usepackage{ragged2e} +\usecolortheme[RGB={200,200,200}]{structure} + +\subtitle[Introduction to numerical methods & numerical Relativity]{{\large Einstein Toolkit Workshop}\*[0.3em] Numerical methods for spacetime and matter evolution} +\author[\mbox{Peter Diener & Bruno Mundim}]{Dr Peter Diener & Dr Bruno Mundim} +\date{Apr 05 2012} + +\begin{document} + +\frame{\titlepage} + +\frame{\frametitle{Outline} + \begin{itemize} +% \item 3+1 spacetime decomposition. + \item The BSSN formulation. + \item Puncture data. +% \item Finite Differencing. +% \item The method of lines. + \end{itemize} +} + +\frame{\frametitle{The BSSN formulation} +The 3+1 ADM evolution equations are +\begin{align} +\dt \gamma_{ij} &= - 2 \alpha K_{ij}, \label{dgdt} \ +\dt K_{ij} &= - D_i D_j \alpha + \alpha (R_{ij} + K K_{ij} +- 2 K_{ik} {K^k}{}_j), \label{dKdt} +\end{align} +and the constraints are +\begin{align} +{\cal H} & \equiv R + K^2 - K_{ij} K^{ij} = 0, \ +{\cal M}^i & \equiv D_j (K^{ij} - \gamma^{ij} K) = 0. +\end{align} +These equations were used a lot in the early years of numerical relativity. + +However, it was later discovered that the ADM evolution equations are only +weakly hyperbolic. + +In the mid to late 90's a new formulation was introduced that proved to be +much more robust and stable: The BSSN formulation. +} + +\frame{\frametitle{The BSSN formulation (continued)} +\setlength{\vs}{-1.5ex} +Introduce a conformal rescaling of the three metric\vspace{\vs} +\begin{equation} +\gamma_{ij} = \psi^4 \tg_{ij}.\vspace{\vs} +\end{equation} +We choose $\psi = \gamma^{1/12}$ such that the determinant of $\tg_{ij}$ is 1\vspace{\vs} +\begin{equation} +\mathrm{det}(\tg_{ij}) = \mathrm{det}(\psi^{-4}\gamma_{ij}) = +\mathrm{det}(\gamma^{-1/3}\gamma_{ij}) = +\gamma^{-1} \mathrm{det}(\gamma_{ij}) = 1. \vspace{\vs} +\end{equation} +In addition we introduce a trace decomposition of the extrinsic curvature.\vspace{\vs} +\begin{align} +K & = \gamma^{ij} K_{ij}, \ +A_{ij} & = K_{ij} - \frac{1}{3} \gamma_{ij} K.\vspace{\vs} +\end{align} +We then promote the following variables to evolution variables\vspace{\vs} +\begin{align} + \phi &= \ln \psi = \frac{1}{12} \ln \gamma, +\ + K &= \gamma_{ij} K^{ij}, +\ + \tg_{ij} &= e^{-4\phi} \gamma_{ij}, +\ + \tA_{ij} &= e^{-4\phi} A_{ij}. +\end{align} +} + +\frame{\frametitle{The BSSN formulation (continued)} +We finally additionally promote the conformal connection functions +\begin{equation} + \tG^i = \tg^{jk} \tG^i{}_{jk} = - \partial_j \tg^{ij}, +\end{equation} +to evolved variables as well. +The final set of evolution variables are $\phi$, $K$, $\tg_{ij}$, $\tA_{ij}$ +and $\tG^i$. + +The BSSN evolution equations can be derived from the ADM equations. As an +example take the equation for $\phi$. +\begin{equation} +\dt \phi = \partial_0\phi = \partial_0 \left (\frac{1}{12}\ln\gamma\right) += \frac{1}{12}\frac{1}{\gamma}\partial_0\gamma. +\end{equation} +Using the expression for the derivative of the determinant of the metric +in terms of the derivatives of the metric ($\partial_0\gamma = \gamma +\gamma^{ij} \partial_0\gamma_{ij}$) we find +\begin{equation} +\dt \phi = \partial_0\phi = \frac{1}{12}\gamma^{ij}\partial_0\gamma_{ij} = +\frac{1}{12}\gamma^{ij}(-2\alpha K_{ij}) = -\frac{1}{6}\alpha K. +\end{equation} +} + +\frame{\frametitle{The BSSN formulation (continued)} +The evolution equation for all the BSSN variables are +\begin{align} + \partial_t \tg_{ij} = {} & -2 \alpha \tA_{ij} + \lbt{\tg}{i}{j}{k}, \ + \partial_t \phi = {} & - \frac{1}{6} \alpha K + \beta^{k}\partial_{k}\phi + + \frac{1}{6}\partial_{k}\beta^{k} , \ + \partial_t \tA_{ij} = {} & e^{-4\phi} [ - D_i D_j \alpha + + \alpha R_{ij}]^{TF} + + \alpha (K \tA_{ij} - 2 \tA_{ik} \tA^k{}_j) \nonumber \ + & + \lbt{\tA}{i}{j}{k}, \ + \partial_t K = {} & - D^i D_i \alpha + ;\alpha (\tA_{ij} \tA^{ij} + + \frac{1}{3} K^2) + \beta^{k}\partial_{k} K, \ + \partial_t \tG^i = {} & \tilde\gamma^{jk} \partial_j\partial_k \beta^i ++ \frac{1}{3} \tilde\gamma^{ij} \partial_j\partial_k\beta^k ++ \beta^j\partial_j \tilde\Gamma^i - \tilde\Gamma^j \partial_j \beta^i ++ \frac{2}{3} \tilde\Gamma^i \partial_j\beta^j +\nonumber \ +& - 2 \tilde{A}^{ij} \partial_j\alpha ++ 2 \alpha ( \tilde{\Gamma}^i{}_{jk} \tilde{A}^{jk} + 6 \tilde{A}^{ij} +\partial_j \phi - \frac{2}{3} \tg^{ij} \partial_j K). +\end{align}%\vspace{\vs} +} + +\frame{\frametitle{The BSSN formulation (continued)} +%\raisebox{-1.2\vs}[0pt][0pt]{ +Here $R_{ij} = \tilde{R}_{ij} + R^{\phi}_{ij}$, where\vspace{1.0\vs} +%}\vspace{\vs} +\begin{align} +R^{\phi}_{ij} = {} & - 2 \tilde{D}_i +\tilde{D}_j \phi - 2 \tilde{\gamma}_{ij} +\tilde{D}^k \tilde{D}_k \phi + 4 \tilde{D}_i \phi ; \tilde{D}_j \phi +- 4 \tilde{\gamma}_{ij} \tilde{D}^k \phi ; \tilde{D}_k \phi , \ +\tilde R_{ij} = {} & - \frac{1}{2} \tg^{lm} \partial_l \partial_m +\tg_{ij} + \tg_{k(i} \partial_{j)} \tG^k + \tG^k \tG_{(ij)k} \nonumber \ + & + \tg^{lm} \left( 2 \tG^k{}_{l(i} \tG_{j)km} + \tG^k{}_{im} \tG_{klj} \right). +\end{align} +} + +\frame{\frametitle{The BSSN formulation (continued)} +The constraints are +\setlength{\jot}{3pt} +\begin{align} +\tilde{{\cal H}} & \equiv R + \frac{2}{3} K^2 - \tA_{ij} \tA^{ij} = 0, \ +\tilde{{\cal M}}^i & \equiv \tilde{D}_j\tA^{ij}+6 \tA^{ij}\partial_j\phi - +\frac{2}{3}\tg^{ij}\partial_j K = 0, \ +\tilde{{\cal G}} & \equiv \tg-1 = 0, \ +\tilde{{\cal A}} & \equiv \tg^{ij}\tA_{ij} = 0, \ +\tilde{{\cal L}}^{i} & \equiv \tG^{i}+\partial_j\tg^{ij} = 0. +\end{align} +The constraints $\tilde{{\cal G}}$ and $\tilde{{\cal A}}$ are enforced actively +at each timestep. + +The other constraints ($\tilde{{\cal H}}$, $\tilde{{\cal M}}^i$ and +$\tilde{{\cal L}}^{i}$) are not enforced. + +To improve stability and to help maintain $\tilde{{\cal L}}^{i}$ at a low level +the following rule is employed in an implementation +\begin{itemize} +\item Where derivatives of $\tG^{i}$ are needed the evolved $\tG^{i}$ are used + directly. +\item Where $\tG^{i}$ are needed without taking derivatives + $\tg^{jk}\tG^{i}{}_{jk}$ are used instead. +\end{itemize} +} + +\frame{\frametitle{The BSSN formulation (continued)} +In order to evolve a spacetime with the BSSN equations, you have to specify +the gauges $\alpha$ and $\beta^{i}$. + +Most codes use the ``moving puncture'' gauges +\begin{align} +\partial_t \alpha & = - 2\alpha K +\beta^i \partial_i\alpha, \ +\partial_t\beta^i & = \frac{3}{4} B^i + \beta^j\partial_j\beta^i, \ +\partial_t B^i & = \partial_t\tG^i - \eta B^i + \beta^j\partial_j(B^i-\tG^i) +\end{align} +or slight variations thereof for black hole spacetime evolutions. +} + +\frame{\frametitle{Puncture data} +\setlength{\vs}{-0.8ex} +The puncture approach to binary black hole initial data is\vspace{\vs} +\begin{equation} +\gamma^{\mathrm{ph}}_{ij} = \psi^4\gamma_{ab}, \mbox{\hspace{2em}} K^{\mathrm{ph}}_{ij} = \psi^{-2} K_{ij}, +\end{equation} +where $\gamma_{ij}$ is chosen to be the flat metric and $K_{ij}$ is assumed +to be tracefree. \ +The constraint equations become\vspace{\vs} +\begin{align} +0 & = \Delta\psi +\frac{1}{8}K^{ij}K_{ij}\psi^{-7} \ +0 & = D_j K^{ij}. +\end{align} +The momentum constraint has an analytic solution\vspace{\vs} +\begin{align} +K^{ij}_{\mathrm{BY}} = {} & \frac{3}{2r^2}\left ( P^i n^j + P^j n^i - + (\gamma^{ij}-n^i n^j) P^k n_k\right ) \nonumber \ +& +\frac{3}{r^3}\left (\epsilon^{ikl}S_k n_l n^j+ + \epsilon^{jkl}S_k n_l n^i\right). +\end{align} +$P^i$ and $S^i$ are the linear momentum and spin of the black hole. +} + +\frame{\frametitle{Puncture data (continued)} +\setlength{\vs}{-0.8ex} +For $K_{ij} = 0$ the Hamiltonian constraint has a simple solution for $N$ black holes. +\begin{equation} +\psi = 1+\sum_{i=1}^N\frac{m_i}{2|\vec{r}-\vec{r}_i|}. +\end{equation} +Inspired by this, for $K_{ij}\ne 0$ we make the ansatz +\begin{equation} +\psi=\frac{1}{\alpha}+u, \mbox{\hspace{2em}} +\frac{1}{\alpha} = \sum_{i=1}^N\frac{m_i}{2|\vec{r}-\vec{r}_i|}, +\end{equation} +In which case the Hamiltonian constraint becomes an equation for $u$ +\begin{equation} +\Delta u+\frac{1}{8}\alpha^7 K^{ij}K_{ij}(1+\alpha u)^{-7} = 0. +\end{equation} +It can be shown that $u$ will be $C^2$ at the location of the `punctures' and +that $\psi$ has a unique solution. +} + +\end{document}

File [added]: preamble.tex Delta lines: +73 -0 =================================================================== --- numerical_relativity/preamble.tex (rev 0) +++ numerical_relativity/preamble.tex 2012-04-02 01:52:31 UTC (rev 21) @@ -0,0 +1,73 @@ +\documentclass{beamer} + +\usepackage[utf8]{inputenc} +%\usepackage{beamerthemesplit} +\usepackage{url} +\usepackage{hyperref} +\usepackage{tikz} +\usepackage{alltt} + +\usetheme{Madrid} +\definecolor{cactusgreen}{RGB}{132,186,75} +\graphicspath{{pics/}} + +\newcommand{\ilshell}[1]{\colorbox{lightgray}{\small{\texttt{#1}}}} +\newcommand{\shell}[1]{\hspace{1cm}\colorbox{lightgray}{\small{\texttt{#1}}}} + +\hyphenation{Cac-tus-Ein-stein Schwarz-schild vacuum in-fra-struc-ture} + +% \logo{\includegraphics[height=1cm]{pics/cactuslogo.png}} + +% We want to use the infolines outer theme because it uses so less space, but +% it also tries to print an institution (which we do not have) and the slide +% numbers (which we might not want to show). Therefore, we here redefine the +% footline ourselfes - mostly a copy & paste from +% /usr/share/texmf/tex/latex/beamer/themes/outer/beamerouterthemeinfolines.sty +\defbeamertemplate*{footline}{infolines theme without institution and slide numbers} +{ + \leavevmode% + \hbox{% + \begin{beamercolorbox}[wd=.333333\paperwidth,ht=2.25ex,dp=1ex,center]{author in head/foot}% + \usebeamerfont{author in head/foot}\insertshortauthor + \end{beamercolorbox}% + \begin{beamercolorbox}[wd=.333333\paperwidth,ht=2.25ex,dp=1ex,center]{title in head/foot}% + \usebeamerfont{title in head/foot}\insertshorttitle + \end{beamercolorbox}% + \begin{beamercolorbox}[wd=.333333\paperwidth,ht=2.25ex,dp=1ex,center]{date in head/foot}% + \usebeamerfont{date in head/foot}\insertshortdate{} + \end{beamercolorbox}}% + \vskip0pt% +} + +% For some reason this is not displayed by default. This is just a copy & paste +% from the same file as the footline +\defbeamertemplate*{headline}{infolines theme with sections/subsections} +{ + \leavevmode% + \hbox{% + \begin{beamercolorbox}[wd=.5\paperwidth,ht=2.25ex,dp=1ex,right]{section in head/foot}% + \usebeamerfont{section in head/foot}\insertsectionhead\hspace*{2ex} + \end{beamercolorbox}% + \begin{beamercolorbox}[wd=.5\paperwidth,ht=2.25ex,dp=1ex,left]{subsection in head/foot}% + \usebeamerfont{subsection in head/foot}\hspace*{2ex}\insertsubsectionhead + \end{beamercolorbox}}% + \vskip0pt% +} + +\newcommand{\abspic}[4] + {\vspace{ #2\paperheight}\hspace{ #3\paperwidth}\includegraphics[height=#4\paperheight]{#1}\ + \vspace{-#2\paperheight}\vspace{-#4\paperheight}\vspace{-0.0038\paperheight}} + +% Make a todo comment stand out visually +\newcommand{\todo}[1]{{~\textbf{\color{red}[TODO: #1]}}} + +\newcommand{\tg}{\tilde\gamma} +\newcommand{\tG}{\tilde\Gamma} +\newcommand{\tA}{\tilde A} +\newcommand{\tK}{\tilde K} +\newcommand{\lb}{{\cal L}_\beta} +\newcommand{\dt}{(\partial_t - {\cal L}_\beta);} +\newcommand{\tr}{\mbox{tr}} +\newcommand{\lbt}[4]{\beta^{#4}\partial_{#4}#1_{#2#3}+#1_{#2#4}\partial_{#3}\beta^{#4}+#1_{#3#4}\partial_{#2}\beta^{#4}-\frac{2}{3}#1_{#2#3}\partial_{#4}\beta^{#4}} + +\newlength{\vs}