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/
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File [added]: Makefile
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--- numerical_relativity/Makefile (rev 0)
+++ numerical_relativity/Makefile 2012-04-02 01:52:31 UTC (rev 21)
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+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
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--- numerical_relativity/numerical_relativity.tex (rev 0)
+++ numerical_relativity/numerical_relativity.tex 2012-04-02 01:52:31 UTC (rev 21)
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+\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
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===================================================================
--- numerical_relativity/preamble.tex (rev 0)
+++ numerical_relativity/preamble.tex 2012-04-02 01:52:31 UTC (rev 21)
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+\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}
