update: auto commit

report.tex

@@ -310,11 +310,20 @@
     \end{equation}
     The evolution of the constraints is given by
     \begin{equation}
-      \Partial{t} \mathcal{C}_i = \Partial{i} \pi - \Partial{i} \Partial{t} \phi = 0,
+      \Partial{t} \mathcal{C}_i = \Partial{t} \psi_i - \Partial{i} \Partial{t} \phi = \Partial{i} \pi - \Partial{i} \pi = 0,
     \end{equation}
     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}
+    For example, we can add a constraint damping term $-\gamma \mathcal{C}_i$ to the evolution equation of $\tensor{\psi}{_i}$, where $\gamma > 0$ is a constant. Then the evolution of the constraints becomes
+    \begin{equation}
+      \Partial{t} \mathcal{C}_i = -\gamma \mathcal{C}_i,
+    \end{equation}
+    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}{Existing first order formulations}
     \subsection{Existing first order formulations}
     \begin{itemize}