update: auto commit

report.tex

@@ -301,7 +301,18 @@
         \myvec{e}_5 = \mqty(0,-1,\xi_1,\xi_2,\xi_3)^T.
       \end{gathered}
     \end{equation}
+  \end{frame}
 
+  \begin{frame}{Towards 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.
+    \end{equation}
+    The evolution of the constraints is given by
+    \begin{equation}
+      \Partial{t} \mathcal{C}_i = 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}{Existing first order formulations}