update: auto commit

report.tex

@@ -6,6 +6,8 @@
 \usetikzlibrary{decorations.pathmorphing}
 \usetikzlibrary{calc}
 
+\usepackage{appendixnumberbeamer}
+
 \tikzset{zigzag/.style={decorate, decoration=zigzag}}
 \def \L {2.}
 
@@ -62,12 +64,6 @@
   \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);
@@ -102,14 +98,28 @@
             -- (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
+      \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: 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=0.7\textwidth, trim=4bp 4bp 4bp 4bp, clip]{imgs/moving_puncture_penrose_1.png}
-      \includegraphics[width=0.45\textwidth]{imgs/moving_puncture_stable.png}
+      \caption{The numerical time slices in the moving puncture method in the Penrose diagram, from \cite{Hannam_2008}.}
-      \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}
+      \includegraphics[width=0.42\textwidth, trim=5bp 4bp 4bp 4bp, clip]{imgs/moving_puncture_init.png}
-      \caption{The numerical time slices in the moving puncture method in the Penrose diagram, from \cite{Hannam_2008}.}
+      \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}