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Yingjie Wang
2026-04-09 23:19:09 -04:00
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\begin{frame}{Methods on evolving black holes: moving puncture} \begin{frame}{Methods on evolving black holes: moving puncture}
In the isotropic coordinate for Schwarzschild black hole, the spical metric In the isotropic coordinate for Schwarzschild black hole, the spical metric
\begin{equation*} \begin{equation}
\dd{l^2} = \left( 1+\frac{M}{2r} \right)^4 (\dd{r^2} + r^2 \dd{\Omega^2}) \dd{l^2} = \left( 1+\frac{M}{2r} \right)^4 (\dd{r^2} + r^2 \dd{\Omega^2})
\end{equation*} \end{equation}
is conformally flat, with the conformal factor is conformally flat, with the conformal factor
\begin{equation*} \begin{equation}
\psi = 1 + \frac{M}{2r}. \psi = 1 + \frac{M}{2r}.
\end{equation*} \end{equation}
General $N$ black hole puncture initial data: General $N$ black hole puncture initial data:
\begin{equation*} \begin{equation}
\psi = \sum_{i=1}^N \frac{M_i}{2|\myvec{r}-\myvec{r}_i|} + (\text{regular part}), \psi = \sum_{i=1}^N \frac{M_i}{2|\myvec{r}-\myvec{r}_i|} + (\text{regular part}),
\end{equation*} \end{equation}
again, if we factor out the conformal factor $\psi$, the remaining conformal metric is regular everywhere. again, if we factor out the conformal factor $\psi$, the remaining conformal metric is regular everywhere.
\end{frame} \end{frame}
@@ -168,13 +168,13 @@
\begin{frame}{Discontinuous Galerkin method and \texttt{nmesh}} \begin{frame}{Discontinuous Galerkin method and \texttt{nmesh}}
Core idea of DG method: if we have a first-order PDE system: Core idea of DG method: if we have a first-order PDE system:
\begin{equation*} \begin{equation}
\pdvt{u} + A^i \pdv{u}{x^i} = S, \pdvt{u} + A^i \pdv{u}{x^i} = S,
\end{equation*} \end{equation}
then for some test functions $\{v_a\}$, we have then for some test functions $\{v_a\}$, we have
\begin{equation*} \begin{equation}
\left(v_a, \pdvt{u}\right) + \left(v_a, A^i \pdv{u}{x^i}\right) = (v_a, S), \left(v_a, \pdvt{u}\right) + \left(v_a, A^i \pdv{u}{x^i}\right) = (v_a, S),
\end{equation*} \end{equation}
which reduce to a linear system of $\{ u_a = (v_a, u)\}$ after intergrating by parts. Boundary terms are replaced by numerical fluxes. which reduce to a linear system of $\{ u_a = (v_a, u)\}$ after intergrating by parts. Boundary terms are replaced by numerical fluxes.
In \texttt{nmesh}, we use Lagrange polynomials over Gauss-Legendre points on each element. In \texttt{nmesh}, we use Lagrange polynomials over Gauss-Legendre points on each element.
@@ -236,8 +236,12 @@
$\implies$ we need to find a first order, strongly hyperbolic formulation of Einstein's equations with constraint damping and compatible with the moving puncture gauge condition. $\implies$ we need to find a first order, strongly hyperbolic formulation of Einstein's equations with constraint damping and compatible with the moving puncture gauge condition.
\end{frame} \end{frame}
\begin{frame}{Searching for a first order Z4c}{Hyperbolicity of first order systems} \begin{frame}{Searching for a first order Z4c}{Why Z4c?}
\section{Searching for a first order Z4c} \section{Searching for a first order Z4c}
\subsection{Why Z4c?}
\end{frame}
\begin{frame}{Searching for a first order Z4c}{Hyperbolicity of first order systems}
\subsection{Hyperbolicity of first order systems} \subsection{Hyperbolicity of first order systems}
Say we have a list of variables $u^I$, and a first order PDE system Say we have a list of variables $u^I$, and a first order PDE system
\begin{equation} \begin{equation}