From a387d26e2bef4e352919db3489c48a88f876e7f0 Mon Sep 17 00:00:00 2001
From: Yingjie Wang <phywyj@gmail.com>
Date: Thu, 9 Apr 2026 19:56:52 -0400
Subject: [PATCH] update: auto commit

---
 report.tex | 11 +++++++++++
 1 file changed, 11 insertions(+)

diff --git a/report.tex b/report.tex
index 5efee02..0e8a4ce 100644
--- a/report.tex
+++ b/report.tex
@@ -252,6 +252,17 @@
     \end{theorem}
   \end{frame}
 
+  \begin{frame}{Towards first order Z4c}{First order reduction}
+    We can introduce auxiliary variables to reduce a second order PDE system to a first order one. For example, for the wave equation on $1+1$-dimensional flat spacetime:
+    \begin{equation}
+      \pdvt[2]{\phi} - \pdv[2]{\phi}{x} = 0,
+    \end{equation}
+  \end{frame}
+
+  \begin{frame}{Towards first order Z4c}{Example: wave equation}
+    
+  \end{frame}
+
   \begin{frame}{Existing first order formulations}
     \subsection{Existing first order formulations}
     \begin{itemize}