update: auto commit

report.tex

@@ -236,8 +236,8 @@
     $\implies$ 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}{Towards first order Z4c}{Hyperbolicity of first order systems}
-    \section{Towards first order Z4c}
+  \begin{frame}{Searching for a first order Z4c}{Hyperbolicity of first order systems}
+    \section{Searching for a first order Z4c}
     \subsection{Hyperbolicity of first order systems}
     Say we have a list of variables $u^I$, and a first order PDE system
     \begin{equation}
@@ -256,7 +256,7 @@
     \end{theorem}
   \end{frame}
 
-  \begin{frame}{Towards first order Z4c}{First order reduction}
+  \begin{frame}{Searching for a first order Z4c}{First order reduction}
     \subsection{First order reduction: wave equation as an example}
     We can introduce auxiliary variables to reduce a second order PDE system to a first order one. For example, for the wave equation on flat spacetime:
     \begin{equation}
@@ -282,7 +282,7 @@
     where $\tensor{e}{^i}$ is the standard column vector basis of $\setR^3$, while $\tensor{e}{_i}$ is the standard row vector basis of $(\setR^3)^*$.
   \end{frame}
 
-    \begin{frame}{Towards first order Z4c}{First order reduction}
+    \begin{frame}{Searching for a first order Z4c}{First order reduction}
     The principal symbol matrix is
     \begin{equation}
       \tensor{P}{^I_J}(\xi) = \begin{pmatrix}
@@ -305,7 +305,7 @@
     \end{equation}
   \end{frame}
 
-  \begin{frame}{Towards first order Z4c}{constraints during the reduction}
+  \begin{frame}{Searching for a first order Z4c}{constraints during the reduction}
     During the first order reduction, we introduce new variables. A solution to the new system is a solution to the original system if and only if the new variables satisfy some constraints. For example, in the wave equation case, we have the constraints
     \begin{equation}
       \mathcal{C}_i \definedby \tensor{\psi}{_i} - \Partial{i} \phi = 0.
@@ -317,7 +317,7 @@
     which means that if the constraints are satisfied initially, they will be satisfied for all time. However, in numerical simulations, there will always be some constraint violation due to numerical errors, and thus it's better to add some constraint damping terms to the evolution equations to suppress the growth of constraint violation.
   \end{frame}
 
-  \begin{frame}{Towards first order Z4c}{constraints during the reduction}
+  \begin{frame}{Searching for a first order Z4c}{constraints during the reduction}
     For example, we can add a constraint damping term $-\gamma \mathcal{C}_i$ to the evolution equation of $\tensor{\psi}{_i}$,
     \begin{equation}
       \begin{cases}
@@ -333,7 +333,7 @@
     which means that the constraint violation will decay exponentially with time, and thus the system is more stable for numerical simulations.
   \end{frame}
 
-  \begin{frame}{Towards first order Z4c}{constraints during the reduction}
+  \begin{frame}{Searching for a first order Z4c}{constraints during the reduction}
     We have to check if the constraint damping term will change the hyperbolicity of the system. The new principal symbol matrix is
     \begin{equation}
       \tensor{P}{^I_J}(\xi) = \begin{pmatrix}