update: auto commit

report.tex

@@ -238,15 +238,17 @@
   \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 
       \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}
+    \subsection{Methods on evolving black holes}
+    Excision method: excise the black hole interior from the computational domain, and impose boundary conditions on the excision surface.
     \begin{figure}
       \centering
       \begin{tikzpicture}[>=Latex, line cap=round, line join=round]
@@ -288,6 +290,7 @@
   \end{frame}
 
   \begin{frame}{Methods on evolving black holes: excision}
+    Excision method: excise the black hole interior from the computational domain, and impose boundary conditions on the excision surface
     \begin{figure}
       \centering
       \includegraphics[width=0.7\textwidth]{imgs/black_hole_excision_mesh.png}
@@ -297,6 +300,8 @@
 
 
   \begin{frame}{Methods on evolving black holes: moving puncture}
+    Moving puncture method: use a suitable gauge condition to make sure the singularity is not on our time slices at all.
+
     In the isotropic coordinate for Schwarzschild black hole, the spatial metric
     \begin{equation}
       \dd{l^2} = \left( 1+\frac{M}{2r} \right)^4 (\dd{r^2} + r^2 \dd{\Omega^2})