update: makefile and vscode latex recipy

.vscode/settings.json

@@ -0,0 +1,20 @@
+{
+  "latex-workshop.view.pdf.trim": 0,
+  "latex-workshop.latex.recipes": [
+    {
+      "name": "make",
+      "tools": ["make"]
+    }
+  ],
+  "latex-workshop.latex.tools": [
+    {
+      "name": "make",
+      "command": "make",
+      "args": [
+          "autopush"
+      ],
+      "env": {}
+    }
+  ],
+  "latex-workshop.view.pdf.zoom": "page-fit"
+}

Makefile

@@ -1,7 +1,8 @@
 main = report
 latexmk = latexmk -pdflatex
+git_add_files = *.tex imgs/ ref.bib
 
-.PHONY : main clean cleanall
+.PHONY : main clean cleanall autopush
 
 main: $(main).tex
 	$(latexmk) $<
@@ -11,3 +12,7 @@ clean:
 
 cleanall:
 	$(latexmk) -C $(main).tex
+
+autopush: main
+	git add $(git_add_files)
+	git diff --cached --quiet || (git commit -m "update: auto commit" && git push)

report.tex

@@ -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}