315 lines
12 KiB
TeX
315 lines
12 KiB
TeX
\documentclass[10pt]{beamer}
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\input{mystyle_beamer.tex}
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\usetikzlibrary{arrows.meta}
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\usetikzlibrary{decorations.pathmorphing}
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\usetikzlibrary{calc}
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\usepackage{appendixnumberbeamer}
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\tikzset{zigzag/.style={decorate, decoration=zigzag}}
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\def \L {2.}
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\title{Towards Moving Picture Simulations in a Discontinuous Galerkin Framework}
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\author{Yingjie Wang}
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\institute{FAU}
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\addbibresource{ref.bib}
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\begin{document}
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\maketitle
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\begin{frame}{Contents}
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\tableofcontents
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\end{frame}
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\begin{frame}{Outline of the Project}
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\section{Introduction}
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\begin{enumerate}
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\item A first order evolution system, to be named FOZ4c
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\item An implementation in our discontinuous Galerkin code \texttt{nmesh}
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\item Simulations of binary black holes with large mass ratio with the moving puncture gauge
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\end{enumerate}
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\end{frame}
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\begin{frame}{Final Goal: binary black holes with large mass ratio}
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\subsection{binary black holes with large mass ratio}
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Why binary black holes?
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\begin{itemize}
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\item Main sources of gravitational waves
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\item Simplest two-body problem in general relativity
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\item Connecting strong field GR with post-Newtonian approximations
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\end{itemize}
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Why large mass ratio?
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\begin{itemize}
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\item LISA is able to detect extreme mass ratio inspirals (EMRIs)
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\item Compare with perturbation theory
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\end{itemize}
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\end{frame}
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\begin{frame}{Methods on evolving black holes}
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\subsection{Methods on evolving black holes}
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There are two main methods to evolve black holes in numerical relativity:
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\begin{itemize}
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\item Excision method: excise the black hole interior from the computational domain, and impose boundary conditions on the excision surface
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\item Moving puncture method: evolve the black hole as a puncture, and use a suitable gauge condition to avoid the singularity.
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\end{itemize}
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\end{frame}
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\begin{frame}{Methods on evolving black holes: excision}
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\begin{figure}
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\centering
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\begin{tikzpicture}[>=Latex, line cap=round, line join=round]
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% causal diamond
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\draw[thick,red,zigzag] (-\L,\L) coordinate(stl) -- (\L,\L) coordinate (str);
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\draw[thick,black] (\L,-\L) coordinate (sbr)
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-- (0,0) coordinate (bif) -- (stl);
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\draw[thick,black,fill=blue, fill opacity=0.2,text opacity=1]
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(bif) -- (str) -- (2*\L,0) node[right] (io) {$i^0$} -- (sbr);
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% null labels
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\draw[black] (1.4*\L,0.7*\L) node[right] (scrip) {$\mathcal{I}^+$}
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(1.5*\L,-0.6*\L) node[right] (scrip) {$\mathcal{I}^-$}
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(0.2*\L,-0.6*\L) node[right] (scrip) {$\mathcal{H}^-$}
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(0.5*\L,0.85*\L) node[right] (scrip) {$\mathcal{H}^+$};
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% singularity label
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\draw[thick,red,<-] (0,1.05*\L)
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-- (0,1.2*\L) node[above] {\color{red} singularity};
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% % Scwharzschild surface
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% \draw[thick,blue] (bif) .. controls (1.*\L,-0.35*\L) .. (2*\L,0);
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% \draw[thick,blue,<-] (1.75*\L,-0.1*\L) -- (1.9*\L,-0.5*\L)
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% -- (2*\L,-0.5*\L) node[right,align=left]
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% {$t=$ constant\\in Schwarzschild\\coordinates};
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% excision surface
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\draw[thick,dashed,red] (-0.3*\L,0.3*\L) -- (0.4*\L,\L);
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\draw[thick,red,<-] (-0.33*\L,0.3*\L)
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-- (-0.5*\L,0.26*\L) node[left,align=right] {excision\\surface};
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% Kerr-Schild surface
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\draw[green,thick] (0.325*\L,0.325*\L) .. controls (\L,0) .. (2*\L,0);
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\draw[green,dashed,thick] (0.325*\L,0.325*\L) -- (-0.051*\L,0.5*\L);
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% Kerr-Schild label
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\draw[green,thick,<-] (0.95*\L,0.15*\L) -- (1.2*\L,0.5*\L)
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-- (2*\L,0.5*\L) node[right,align=left]
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{time slice};
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\end{tikzpicture}
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\caption{excision on a single Schwarchild black hole}
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\end{figure}
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\end{frame}
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\begin{frame}{Methods on evolving black holes: excision}
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\begin{figure}
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\centering
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\includegraphics[width=0.7\textwidth]{imgs/black_hole_excision_mesh.png}
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\caption{An example mesh using excision method from \cite{Hemberger_2013}.}
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\end{figure}
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\end{frame}
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\begin{frame}{Methods on evolving black holes: moving puncture}
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In the isotropic coordinate for Schwarzschild black hole, the spical metric
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\begin{equation*}
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\dd{l^2} = \left( 1+\frac{M}{2r} \right)^4 (\dd{r^2} + r^2 \dd{\Omega^2})
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\end{equation*}
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is conformally flat, with the conformal factor
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\begin{equation*}
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\psi = 1 + \frac{M}{2r}.
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\end{equation*}
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General $N$ black hole puncture initial data:
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\begin{equation*}
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\psi = \sum_{i=1}^N \frac{M_i}{2|\vb*{r}-\vb*{r}_i|} + (\text{regular part}),
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\end{equation*}
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again, if we factor out the conformal factor $\psi$, the remaining conformal metric is regular everywhere.
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\end{frame}
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\begin{frame}{Methods on evolving black holes: moving puncture}
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In the moving puncture mothod, we won't cut out the black hole singularity, but choose a gauge condition that makes the singularity invisible to the numerical evolution.
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\begin{figure}
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\centering
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\includegraphics[width=0.7\textwidth, trim=4bp 4bp 4bp 4bp, clip]{imgs/moving_puncture_penrose_1.png}
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\caption{The numerical time slices in the moving puncture method in the Penrose diagram, from \cite{Hannam_2008}.}
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\end{figure}
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\end{frame}
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\begin{frame}{Methods on evolving black holes: moving puncture}
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\begin{figure}
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\centering
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\includegraphics[width=0.42\textwidth, trim=5bp 4bp 4bp 4bp, clip]{imgs/moving_puncture_init.png}
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\includegraphics[width=0.45\textwidth, trim=4bp 4bp 4bp 4bp, clip]{imgs/moving_puncture_stable.png}
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\caption{The embedding of the initial slice (left) and the final slice (right) in the moving puncture method, from \cite{Hannam_2008}.}
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\end{figure}
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\end{frame}
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\begin{frame}{Discontinuous Galerkin method and \texttt{nmesh}}
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\subsection{Discontinuous Galerkin method and \texttt{nmesh}}
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There are two main numerical methods in numerical relativity:
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\begin{itemize}
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\item Finite Difference / Finite Volume
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\begin{itemize}
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\item Easy to implement, and mature codes available
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\end{itemize}
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\item Spectral / Discontinuous Galerkin
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\begin{itemize}
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\item High accuracy for smooth solutions
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\end{itemize}
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\end{itemize}
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\end{frame}
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\begin{frame}{Discontinuous Galerkin method and \texttt{nmesh}}
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Core idea of DG method: if we have a first-order PDE system:
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\begin{equation*}
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\pdvt{u} + A^i \pdv{u}{x^i} = S,
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\end{equation*}
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then for some test functions $\{v_a\}$, we have
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\begin{equation*}
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\left(v_a, \pdvt{u}\right) + \left(v_a, A^i \pdv{u}{x^i}\right) = (v_a, S),
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\end{equation*}
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which reduce to a linear system of $\{ u_a = (v_a, u)\}$ after intergrating by parts. Boundary terms are replaced by numerical fluxes.
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In \texttt{nmesh}, we use Lagrange polynomials over Gauss-Legendre points on each element.
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\begin{center}
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\begin{tikzpicture}[scale=4]
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\draw[thick, blue] (0,0) -- (1,0);
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\draw[thick, red] (1,0) -- (2,0);
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\pgfmathsetmacro{\a}{0.5 - 0.5*sqrt(3/7)}
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\pgfmathsetmacro{\b}{0.5 + 0.5*sqrt(3/7)}
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\pgfmathsetmacro{\c}{1.5 - 0.5*sqrt(3/7)}
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\pgfmathsetmacro{\d}{1.5 + 0.5*sqrt(3/7)}
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\foreach \x in {0,\a,0.5,\b,1}
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\fill[blue] (\x,0) circle (0.018);
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\foreach \x in {1,\c,1.5,\d,2}
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\fill[red] (\x,0) circle (0.018);
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\fill[blue] (1,0) circle (0.022);
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\fill[red] (1,0) circle (0.013);
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\end{tikzpicture}
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\end{center}
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Information exchange only happens at the boundaries of elements through numerical fluxes, which makes it efficient for parallel computing.
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\end{frame}
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\begin{frame}{Discontinuous Galerkin method and \texttt{nmesh}}
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\texttt{nmesh} has two features that are useful for our project:
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\begin{itemize}
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\item Adaptive mesh refinement (AMR)
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\item DG + FD/FV dynamically switching
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\end{itemize}
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\begin{figure}
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\centering
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\includegraphics[width=0.65\textwidth]{imgs/MPA1_W-9sn12l5_GRHD_D_t0400.pdf}
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\caption{The mesh in a binary star emulation with \texttt{nmesh}}
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\end{figure}
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\end{frame}
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\begin{frame}{Moving puncture evolution in \texttt{nmesh}?}
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\subsection{Moving puncture evolution in \texttt{nmesh}?}
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There are many $3+1$ formulations of Einstein's equations,
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\begin{itemize}
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\item We want DG method:
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\begin{itemize}
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\item a {\color{blue}first order} formulation is needed, like the generalized harmonic (GH) formulation.
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\end{itemize}
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\item We want moving puncture evolution:
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\begin{itemize}
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\item the {\color{blue}moving puncture gauge condition} is needed, and not compatible with the GH formulation.
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\item People usually use BSSN, Z4c or CCZ4 formulation for moving puncture evolutions, but they are second order in space, and not directly suitable for DG method.
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\end{itemize}
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\item We want stable evolution:
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\begin{itemize}
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\item a {\color{blue}strongly hyperbolic} formulation is needed.
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\item {\color{blue} constraint damping} is needed.
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\end{itemize}
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\end{itemize}
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$\implies$ we need to find a first order, strongly hyperbolic formulation of Einstein's equations with constraint damping and compatible with the moving puncture gauge condition.
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\end{frame}
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\begin{frame}{Towards first order Z4c}{Hyperbolicity of first order systems}
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\section{Towards first order Z4c}
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\subsection{Hyperbolicity of first order systems}
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Say we have a list of variables $u^I$, and a first order PDE system
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\begin{equation}
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\pdvt{u^I} + \tensor{A}{^i^I_J} \pdv{u^J}{x^i} = S^I, \label{first-order-PDE}
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\end{equation}
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where both $A$ and $S$ can depend on $u^I$ but not their derivatives. For any unit covector $\xi_i$, the \emph{Principal symbol} matrix is defined as $\tensor{P}{^I_J}(\xi) := \tensor{A}{^i^I_J} \xi_i$.
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The system is said to be:
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\begin{itemize}
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\item \emph{weakly hyperbolic} if for any $\xi_i$, $\tensor{P}{^I_J}(\xi)$ has only real eigenvalues.
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\item \emph{strongly hyperbolic} if for any $\xi_i$, $\tensor{P}{^I_J}(\xi)$ has only real eigenvalues and a complete set of eigenvectors, i.e., it is diagonalizable.
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\item \emph{symmetrically hyperbolic} if there exists a positive definite symmetrizer matrix $H_{IJ}$ such that $H_{IK} \tensor{P}{^K_J}(\xi)$ is always symmetric.
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\end{itemize}
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\begin{theorem}
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System~\eqref{first-order-PDE} is well-posed in the $L^2$ sense if and only if it is strongly hyperbolic.
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\end{theorem}
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\end{frame}
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\begin{frame}{Towards first order Z4c}{First order reduction}
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We can introduce auxiliary variables to reduce a second order PDE system to a first order one. For example, for the wave equation on flat spacetime:
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\begin{equation}
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\pdvt[2]{\phi} - \tensor{\delta}{^i^j}\pdv{\phi}{x^i}{x^j} = 0,
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\end{equation}
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we can introduce $\pi := \pdvt{\phi}$ and $\psi := \pdv{\phi}{x}$, and rewrite the wave equation as
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\begin{equation}
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\begin{cases}
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\pdvt{\phi} = \pi, \\
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\pdvt{\psi} = \pdv{\pi}{x}, \\
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\pdvt{\pi} = \pdv{\psi}{x}.
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\end{cases}
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\end{equation}
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write $u^I = (\phi, \pi, \psi)$, then the system can be written in the form of~\eqref{first-order-PDE} with
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\begin{equation}
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\tensor{A}{^x^I_J} = \begin{pmatrix}
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0 & 0 & 0 \\
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0 & 0 & -1 \\
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0 & -1 & 0
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\end{pmatrix} \qc S=0.
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\end{equation}
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\end{frame}
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\begin{frame}{Existing first order formulations}
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\subsection{Existing first order formulations}
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\begin{itemize}
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\item GH
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\begin{itemize}
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\item Not compatible with moving puncture mothod
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\end{itemize}
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\item FOCCZ4
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\begin{itemize}
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\item No constraint damping
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\end{itemize}
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\item FOZ4
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\begin{itemize}
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\item No constraint damping
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\end{itemize}
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\item FOBSSN
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\begin{itemize}
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\item Probably works; not well tested in the community; WIP in \texttt{nmesh}
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\end{itemize}
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\end{itemize}
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\end{frame}
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\begin{frame}{References}
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\printbibliography
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\end{frame}
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\appendix
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\begin{frame}{Appendix}
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\section{Appendix}
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\begin{figure}
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\centering
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\includegraphics[width=0.7\textwidth, trim=4bp 4bp 4bp 4bp, clip]{imgs/moving_puncture_penrose.png}
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\caption{The numerical time slices in the moving puncture method in the Penrose diagram, from \cite{Hannam_2008}.}
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\end{figure}
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\end{frame}
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\end{document} |