From e89ed5db20a099e3da9bdbfc688e44c76f04d11c Mon Sep 17 00:00:00 2001
From: Yingjie Wang <phywyj@gmail.com>
Date: Thu, 9 Apr 2026 23:19:09 -0400
Subject: [PATCH] update: auto commit

---
 report.tex | 26 +++++++++++++++-----------
 1 file changed, 15 insertions(+), 11 deletions(-)

diff --git a/report.tex b/report.tex
index 8dbb3db..9fd2158 100644
--- a/report.tex
+++ b/report.tex
@@ -117,18 +117,18 @@
 
   \begin{frame}{Methods on evolving black holes: moving puncture}
     In the isotropic coordinate for Schwarzschild black hole, the spical metric
-    \begin{equation*}
+    \begin{equation}
       \dd{l^2} = \left( 1+\frac{M}{2r} \right)^4 (\dd{r^2} + r^2 \dd{\Omega^2})
-    \end{equation*}
+    \end{equation}
     is conformally flat, with the conformal factor
-    \begin{equation*}
+    \begin{equation}
       \psi = 1 + \frac{M}{2r}.
-    \end{equation*}
+    \end{equation}
 
     General $N$ black hole puncture initial data:
-    \begin{equation*}
+    \begin{equation}
       \psi = \sum_{i=1}^N \frac{M_i}{2|\myvec{r}-\myvec{r}_i|} + (\text{regular part}),
-    \end{equation*}
+    \end{equation}
     again, if we factor out the conformal factor $\psi$, the remaining conformal metric is regular everywhere.
   \end{frame}
 
@@ -168,13 +168,13 @@
 
   \begin{frame}{Discontinuous Galerkin method and \texttt{nmesh}}
     Core idea of DG method: if we have a first-order PDE system:
-    \begin{equation*}
+    \begin{equation}
       \pdvt{u} + A^i \pdv{u}{x^i} = S,
-    \end{equation*}
+    \end{equation}
     then for some test functions $\{v_a\}$, we have
-    \begin{equation*}
+    \begin{equation}
       \left(v_a, \pdvt{u}\right) + \left(v_a, A^i \pdv{u}{x^i}\right) = (v_a, S),
-    \end{equation*}
+    \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.
 
     In \texttt{nmesh}, we use Lagrange polynomials over Gauss-Legendre points on each element.
@@ -236,8 +236,12 @@
     $\implies$ 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}{Searching for a first order Z4c}{Hyperbolicity of first order systems}
+  \begin{frame}{Searching for a first order Z4c}{Why Z4c?}
     \section{Searching for a first order Z4c}
+    \subsection{Why Z4c?}
+  \end{frame}
+
+  \begin{frame}{Searching for a first order Z4c}{Hyperbolicity of first order systems}
     \subsection{Hyperbolicity of first order systems}
     Say we have a list of variables $u^I$, and a first order PDE system
     \begin{equation}