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Yingjie Wang
2026-04-10 00:35:11 -04:00
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@@ -412,7 +412,7 @@
where $\tensor{e}{^i}$ is the standard column vector basis of $\setR^3$, while $\tensor{e}{_i}$ is the standard row vector basis of $(\setR^3)^*$. 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} \end{frame}
\begin{frame}{Searching for a first order Z4c}{First order reduction} \begin{frame}{First order reduction: wave equation as an example}
The principal symbol matrix is The principal symbol matrix is
\begin{equation} \begin{equation}
\tensor{P}{^I_J}(\xi) = \begin{pmatrix} \tensor{P}{^I_J}(\xi) = \begin{pmatrix}
@@ -435,7 +435,7 @@
\end{equation} \end{equation}
\end{frame} \end{frame}
\begin{frame}{Searching for a first order Z4c}{constraints during the reduction} \begin{frame}{First order reduction: wave equation as an example}{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 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} \begin{equation}
\mathcal{C}_i \definedby \tensor{\psi}{_i} - \Partial{i} \phi = 0. \mathcal{C}_i \definedby \tensor{\psi}{_i} - \Partial{i} \phi = 0.
@@ -447,7 +447,7 @@
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. 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} \end{frame}
\begin{frame}{Searching for a first order Z4c}{constraints during the reduction} \begin{frame}{First order reduction: wave equation as an example}{constraint damping}
For example, we can add a constraint damping term $-\gamma \mathcal{C}_i$ to the evolution equation of $\tensor{\psi}{_i}$, For example, we can add a constraint damping term $-\gamma \mathcal{C}_i$ to the evolution equation of $\tensor{\psi}{_i}$,
\begin{equation} \begin{equation}
\begin{cases} \begin{cases}
@@ -463,7 +463,7 @@
which means that the constraint violation will decay exponentially with time, and thus the system is more stable for numerical simulations. which means that the constraint violation will decay exponentially with time, and thus the system is more stable for numerical simulations.
\end{frame} \end{frame}
\begin{frame}{Searching for a first order Z4c}{constraints during the reduction} \begin{frame}{First order reduction: wave equation as an example}{constraint damping and hyperbolicity recheck}
We have to check if the constraint damping term will change the hyperbolicity of the system. The new principal symbol matrix is We have to check if the constraint damping term will change the hyperbolicity of the system. The new principal symbol matrix is
\begin{equation} \begin{equation}
\tensor{P}{^I_J}(\xi) = \begin{pmatrix} \tensor{P}{^I_J}(\xi) = \begin{pmatrix}