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Yingjie Wang
2026-04-09 15:52:16 -04:00
parent 1f024e5ea1
commit ea93697fd9
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report.tex
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@@ -6,6 +6,8 @@
\usetikzlibrary{decorations.pathmorphing}
\usetikzlibrary{calc}
\usepackage{appendixnumberbeamer}
\tikzset{zigzag/.style={decorate, decoration=zigzag}}
\def \L {2.}
@@ -59,6 +61,47 @@
\end{itemize}
\end{frame}
\begin{frame}{Methods on evolving black holes: excision}
\begin{figure}
\centering
\begin{tikzpicture}[>=Latex, line cap=round, line join=round]
% causal diamond
\draw[thick,red,zigzag] (-\L,\L) coordinate(stl) -- (\L,\L) coordinate (str);
\draw[thick,black] (\L,-\L) coordinate (sbr)
-- (0,0) coordinate (bif) -- (stl);
\draw[thick,black,fill=blue, fill opacity=0.2,text opacity=1]
(bif) -- (str) -- (2*\L,0) node[right] (io) {$i^0$} -- (sbr);
% null labels
\draw[black] (1.4*\L,0.7*\L) node[right] (scrip) {$\mathcal{I}^+$}
(1.5*\L,-0.6*\L) node[right] (scrip) {$\mathcal{I}^-$}
(0.2*\L,-0.6*\L) node[right] (scrip) {$\mathcal{H}^-$}
(0.5*\L,0.85*\L) node[right] (scrip) {$\mathcal{H}^+$};
% singularity label
\draw[thick,red,<-] (0,1.05*\L)
-- (0,1.2*\L) node[above] {\color{red} singularity};
% % Scwharzschild surface
% \draw[thick,blue] (bif) .. controls (1.*\L,-0.35*\L) .. (2*\L,0);
% \draw[thick,blue,<-] (1.75*\L,-0.1*\L) -- (1.9*\L,-0.5*\L)
% -- (2*\L,-0.5*\L) node[right,align=left]
% {$t=$ constant\\in Schwarzschild\\coordinates};
% excision surface
\draw[thick,dashed,red] (-0.3*\L,0.3*\L) -- (0.4*\L,\L);
\draw[thick,red,<-] (-0.33*\L,0.3*\L)
-- (-0.5*\L,0.26*\L) node[left,align=right] {excision\\surface};
% Kerr-Schild surface
\draw[green,thick] (0.325*\L,0.325*\L) .. controls (\L,0) .. (2*\L,0);
\draw[green,dashed,thick] (0.325*\L,0.325*\L) -- (-0.051*\L,0.5*\L);
% Kerr-Schild label
\draw[green,thick,<-] (0.95*\L,0.15*\L) -- (1.2*\L,0.5*\L)
-- (2*\L,0.5*\L) node[right,align=left]
{time slice};
\end{tikzpicture}
\caption{excision on a single Schwarchild black hole}
\end{figure}
\end{frame}
\begin{frame}{Methods on evolving black holes: excision}
\begin{figure}
\centering
@@ -67,49 +110,16 @@
\end{figure}
\end{frame}
\begin{frame}{Methods on evolving black holes}
\begin{tikzpicture}[>=Latex, line cap=round, line join=round]
% causal diamond
\draw[thick,red,zigzag] (-\L,\L) coordinate(stl) -- (\L,\L) coordinate (str);
\draw[thick,black] (\L,-\L) coordinate (sbr)
-- (0,0) coordinate (bif) -- (stl);
\draw[thick,black,fill=blue, fill opacity=0.2,text opacity=1]
(bif) -- (str) -- (2*\L,0) node[right] (io) {$i^0$} -- (sbr);
% null labels
\draw[black] (1.4*\L,0.7*\L) node[right] (scrip) {$\mathcal{I}^+$}
(1.5*\L,-0.6*\L) node[right] (scrip) {$\mathcal{I}^-$}
(0.2*\L,-0.6*\L) node[right] (scrip) {$\mathcal{H}^-$}
(0.5*\L,0.85*\L) node[right] (scrip) {$\mathcal{H}^+$};
% singularity label
\draw[thick,red,<-] (0,1.05*\L)
-- (0,1.2*\L) node[above] {\color{red} singularity};
% % Scwharzschild surface
% \draw[thick,blue] (bif) .. controls (1.*\L,-0.35*\L) .. (2*\L,0);
% \draw[thick,blue,<-] (1.75*\L,-0.1*\L) -- (1.9*\L,-0.5*\L)
% -- (2*\L,-0.5*\L) node[right,align=left]
% {$t=$ constant\\in Schwarzschild\\coordinates};
% excision surface
\draw[thick,dashed,red] (-0.3*\L,0.3*\L) -- (0.4*\L,\L);
\draw[thick,red,<-] (-0.33*\L,0.3*\L)
-- (-0.5*\L,0.26*\L) node[left,align=right] {excision\\surface};
% Kerr-Schild surface
\draw[green,thick] (0.325*\L,0.325*\L) .. controls (\L,0) .. (2*\L,0);
\draw[green,dashed,thick] (0.325*\L,0.325*\L) -- (-0.051*\L,0.5*\L);
% Kerr-Schild label
\draw[green,thick,<-] (0.95*\L,0.15*\L) -- (1.2*\L,0.5*\L)
-- (2*\L,0.5*\L) node[right,align=left]
{time slice};
\end{tikzpicture}
\end{frame}
\begin{frame}{Methods on evolving black holes: moving puncture}
In the isotropic coordinate for Schwarzschild black hole, the spical metric
\begin{equation*}
\dd{l^2} = \left( 1+\frac{M}{2r} \right)^4 (\dd{r^2} + r^2 \dd{\Omega^2})
\end{equation*}
is conformally flat, with the conformal factor $\psi = 1 + \frac{M}{2r}$.
is conformally flat, with the conformal factor
\begin{equation*}
\psi = 1 + \frac{M}{2r}.
\end{equation*}
General $N$ black hole puncture initial data:
\begin{equation*}
@@ -123,17 +133,17 @@
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.
\begin{figure}
\centering
\includegraphics[width=0.42\textwidth]{imgs/moving_puncture_init.png}
\includegraphics[width=0.45\textwidth]{imgs/moving_puncture_stable.png}
\caption{The embedding of the initial slice (left) and the final slice (right) in the moving puncture method, from \cite{Hannam_2008}.}
\includegraphics[width=0.7\textwidth, trim=4bp 4bp 4bp 4bp, clip]{imgs/moving_puncture_penrose_1.png}
\caption{The numerical time slices in the moving puncture method in the Penrose diagram, from \cite{Hannam_2008}.}
\end{figure}
\end{frame}
\begin{frame}{Methods on evolving black holes: moving puncture}
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{imgs/moving_puncture_penrose.png}
\caption{The numerical time slices in the moving puncture method in the Penrose diagram, from \cite{Hannam_2008}.}
\includegraphics[width=0.42\textwidth, trim=5bp 4bp 4bp 4bp, clip]{imgs/moving_puncture_init.png}
\includegraphics[width=0.45\textwidth, trim=4bp 4bp 4bp 4bp, clip]{imgs/moving_puncture_stable.png}
\caption{The embedding of the initial slice (left) and the final slice (right) in the moving puncture method, from \cite{Hannam_2008}.}
\end{figure}
\end{frame}
@@ -163,7 +173,7 @@
\end{equation*}
which reduce to a linear system of $\{ u_a = (v_a, u)\}$ after intergrating by parts. Boundary terms are replaced by numerical fluxes.
We use Lagrange polynomials over Gauss-Legendre points on each element.
In \texttt{nmesh}, we use Lagrange polynomials over Gauss-Legendre points on each element.
\begin{center}
\begin{tikzpicture}[scale=4]
\draw[thick, blue] (0,0) -- (1,0);
@@ -184,11 +194,86 @@
\fill[red] (1,0) circle (0.013);
\end{tikzpicture}
\end{center}
Information exchange only happens at the boundaries of elements through numerical fluxes, which makes it easy to parallelize.
Information exchange only happens at the boundaries of elements through numerical fluxes, which makes it efficient for parallel computing.
\end{frame}
\begin{frame}{Discontinuous Galerkin method and \texttt{nmesh}}
\texttt{nmesh} has two features that are useful for our project:
\begin{itemize}
\item Adaptive mesh refinement (AMR)
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{imgs/MPA1_W-9sn12l5_GRHD_D_t0400.pdf}
\caption{The mesh in a binery star emulation with \texttt{nmesh}}
\end{figure}
\end{itemize}
\end{frame}
\begin{frame}{Discontinuous Galerkin method and \texttt{nmesh}}
\texttt{nmesh} has two features that are useful for our project:
\begin{itemize}
\item DG/FV dynamically switching
\end{itemize}
\end{frame}
\begin{frame}{Moving puncture evolution in \texttt{nmesh}?}
\subsection{Moving puncture evolution in \texttt{nmesh}?}
There are many $3+1$ formulations of Einstein's equations,
\begin{itemize}
\item We want DG method:
\begin{itemize}
\item a {\color{blue}first order} formulation is needed, like the generalized harmonic (GH) formulation.
\end{itemize}
\item We want moving puncture evolution:
\begin{itemize}
\item the {\color{blue}moving puncture gauge condition} is needed, and not compatible with the GH formulation.
\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.
\end{itemize}
\item We want stable evolution:
\begin{itemize}
\item a {\color{blue}strongly hyperbolic} formulation is needed.
\item {\color{blue} constraint damping} is needed.
\end{itemize}
\end{itemize}
$\implies$ 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.
\end{frame}
\begin{frame}{Existing first order formulations}
\section{Towards first order Z4c}
\subsection{Existing first order formulations}
\begin{itemize}
\item GH
\begin{itemize}
\item Not compatible with moving puncture mothod
\end{itemize}
\item FOCCZ4
\begin{itemize}
\item No constraint damping
\end{itemize}
\item FOZ4
\begin{itemize}
\item No constraint damping
\end{itemize}
\item FOBSSN
\begin{itemize}
\item Probably works; not well tested in the community; WIP in \texttt{nmesh}
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame}{References}
\printbibliography
\end{frame}
\appendix
\begin{frame}{Appendix}
\section{Appendix}
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth, trim=4bp 4bp 4bp 4bp, clip]{imgs/moving_puncture_penrose.png}
\caption{The numerical time slices in the moving puncture method in the Penrose diagram, from \cite{Hannam_2008}.}
\end{figure}
\end{frame}
\end{document}