update

report.tex

@@ -2,10 +2,19 @@
 
 \input{mystyle_beamer.tex}
 
+\usetikzlibrary{arrows.meta}
+\usetikzlibrary{decorations.pathmorphing}
+
+\tikzset{zigzag/.style={decorate, decoration=zigzag}}
+\def \L {2.}
+
+
 \title{Towards Moving Picture Simulations in a Discontinuous Galerkin Framework}
 \author{Yingjie Wang}
 \institute{FAU}
 
+\addbibresource{ref.bib}
+
 \begin{document}
 	\maketitle
 
@@ -40,8 +49,83 @@
     \end{itemize}
   \end{frame}
 
+  \begin{frame}{Methods on evolving black holes}
+    \subsection{Methods on evolving black holes}
+    There are two main methods to evolve black holes in numerical relativity:
+    \begin{itemize}
+      \item Excision method: excise the black hole interior from the computational domain, and impose boundary conditions on the excision surface
+      \item Moving puncture method: evolve the black hole as a puncture, and use a suitable gauge condition to avoid the singularity.
+    \end{itemize}
+  \end{frame}
+
+  \begin{frame}{Methods on evolving black holes: excision}
+    \begin{figure}
+      \centering
+      \includegraphics[width=0.7\textwidth]{imgs/black_hole_excision_mesh.png}
+      \caption{An example mesh using excision method from \cite{Hemberger_2013}.}
+    \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}
+    \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}.}
+    \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}.}
+    \end{figure}
+  \end{frame}
+
   \begin{frame}{Discontinuous Galerkin method and \texttt{nmesh}}
     \subsection{Discontinuous Galerkin method and \texttt{nmesh}}
   \end{frame}
 
+  \begin{frame}{References}
+    \printbibliography
+  \end{frame}
+
 \end{document}