update: auto commit

2026-04-09 19:59:14 -04:00
parent a387d26e2b
commit 1fdfad954e
1 changed files with 8 additions and 4 deletions
@@ -257,10 +257,14 @@
    \begin{equation}
      \pdvt[2]{\phi} - \pdv[2]{\phi}{x} = 0,
    \end{equation}
-  \end{frame}
-
-  \begin{frame}{Towards first order Z4c}{Example: wave equation}
-    
+    we can introduce $\pi := \pdvt{\phi}$ and $\psi := \pdv{\phi}{x}$, and rewrite the wave equation as
+    \begin{equation}
+      \begin{cases}
+        \pdvt{\phi} = \pi, \\
+        \pdvt{\psi} = \pdv{\pi}{x}, \\
+        \pdvt{\pi} = \pdv{\psi}{x}.
+      \end{cases}
+    \end{equation}
  \end{frame}

  \begin{frame}{Existing first order formulations}