From 1f024e5ea16f79826f801b36b9ce52db858e432d Mon Sep 17 00:00:00 2001
From: Yingjie Wang <phywyj@gmail.com>
Date: Wed, 8 Apr 2026 07:24:01 -0400
Subject: [PATCH] update

---
 report.tex | 63 ++++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 63 insertions(+)

diff --git a/report.tex b/report.tex
index 38478cf..04c5b10 100644
--- a/report.tex
+++ b/report.tex
@@ -4,6 +4,7 @@
 
 \usetikzlibrary{arrows.meta}
 \usetikzlibrary{decorations.pathmorphing}
+\usetikzlibrary{calc}
 
 \tikzset{zigzag/.style={decorate, decoration=zigzag}}
 \def \L {2.}
@@ -104,6 +105,22 @@
   \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 $\psi = 1 + \frac{M}{2r}$.
+
+    General $N$ black hole puncture initial data:
+    \begin{equation*}
+      \psi = \sum_{i=1}^N \frac{M_i}{2|\vb*{r}-\vb*{r}_i|} + (\text{regular part}),
+    \end{equation*}
+    again, if we factor out the conformal factor $\psi$, the remaining conformal metric is regular everywhere.
+  \end{frame}
+
+  \begin{frame}{Methods on evolving black holes: moving puncture}
+
+    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}
@@ -122,6 +139,52 @@
 
   \begin{frame}{Discontinuous Galerkin method and \texttt{nmesh}}
     \subsection{Discontinuous Galerkin method and \texttt{nmesh}}
+    There are two main numerical methods in numerical relativity:
+    \begin{itemize}
+      \item Finite Difference / Finite Volume
+      \begin{itemize}
+        \item Easy to implement, and mature codes available
+      \end{itemize}
+      \item Spectral / Discontinuous Galerkin
+      \begin{itemize}
+        \item High accuracy for smooth solutions
+      \end{itemize}
+    \end{itemize}
+  \end{frame}
+
+  \begin{frame}{Discontinuous Galerkin method and \texttt{nmesh}}
+    Core idea of DG method: if we have a first-order PDE system:
+    \begin{equation*}
+      \pdvt{u} + A^i \pdv{u}{x^i} = S,
+    \end{equation*}
+    then for some test functions $\{v_a\}$, we have
+    \begin{equation*}
+      \left(v_a, \pdvt{u}\right) + \left(v_a, A^i \pdv{u}{x^i}\right) = (v_a, S),
+    \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.
+    \begin{center}
+      \begin{tikzpicture}[scale=4]
+        \draw[thick, blue] (0,0) -- (1,0);
+        \draw[thick, red] (1,0) -- (2,0);
+
+        \pgfmathsetmacro{\a}{0.5 - 0.5*sqrt(3/7)}
+        \pgfmathsetmacro{\b}{0.5 + 0.5*sqrt(3/7)}
+        \pgfmathsetmacro{\c}{1.5 - 0.5*sqrt(3/7)}
+        \pgfmathsetmacro{\d}{1.5 + 0.5*sqrt(3/7)}
+
+        \foreach \x in {0,\a,0.5,\b,1}
+          \fill[blue] (\x,0) circle (0.018);
+
+        \foreach \x in {1,\c,1.5,\d,2}
+          \fill[red] (\x,0) circle (0.018);
+
+        \fill[blue] (1,0) circle (0.022);
+        \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.
   \end{frame}
 
   \begin{frame}{References}