194 lines
7.4 KiB
TeX
194 lines
7.4 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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\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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\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}
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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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\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 $\psi = 1 + \frac{M}{2r}$.
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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=0.42\textwidth]{imgs/moving_puncture_init.png}
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\includegraphics[width=0.45\textwidth]{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}{Methods on evolving black holes: moving puncture}
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\begin{figure}
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\centering
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\includegraphics[width=0.7\textwidth]{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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\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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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 easy to parallelize.
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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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\end{document} |