[Commits] [svn:einsteintoolkit] Workshop_Summer_2013/et_intro_details/ (Rev. 20)
rhaas at tapir.caltech.edu
rhaas at tapir.caltech.edu
Fri Jul 26 10:52:55 CDT 2013
User: rhaas
Date: 2013/07/26 10:52 AM
Added:
/et_intro_details/
WaveToy.tar.gz
Modified:
/et_intro_details/
et_intro_details.tex
Log:
revert WaveToy example to old non-MoL method
File Changes:
Directory: /et_intro_details/
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File [added]: WaveToy.tar.gz
Delta lines: +45 -0
===================================================================
--- et_intro_details/WaveToy.tar.gz (rev 0)
+++ et_intro_details/WaveToy.tar.gz 2013-07-26 15:52:55 UTC (rev 20)
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File [modified]: et_intro_details.tex
Delta lines: +68 -21
===================================================================
--- et_intro_details/et_intro_details.tex 2013-07-26 14:39:05 UTC (rev 19)
+++ et_intro_details/et_intro_details.tex 2013-07-26 15:52:55 UTC (rev 20)
@@ -264,29 +264,20 @@
\begin{center}
\begin{tikzpicture}[scale=1.1]
\draw (-1,0) -- (1,0);
- \draw (-1,1) -- (1,1);
- \draw (-1,-1) -- (1,-1);
+ \draw (0,-1) -- (0,1);
\scope[fill=green!50!blue, draw=black]
\filldraw (0,0) circle (0.07cm);
\filldraw (1,0) circle (0.07cm);
+ \filldraw (0,1) circle (0.07cm);
\filldraw (-1,0) circle (0.07cm);
- \filldraw (0,1) circle (0.07cm);
- \filldraw (1,1) circle (0.07cm);
- \filldraw (-1,1) circle (0.07cm);
\filldraw (0,-1) circle (0.07cm);
- \filldraw (1,-1) circle (0.07cm);
- \filldraw (-1,-1) circle (0.07cm);
\endscope
\scope[xshift=0.45cm,yshift=0.25cm]
\draw (0,0) node {{\tiny $\varphi^{(n)}_{i}$}};
\draw (1,0) node {{\tiny $\varphi^{(n)}_{i+1}$}};
- \draw (-1,0) node {{\tiny $\varphi^{(n)}_{i}$}};
\draw (0,1) node {{\tiny $\varphi^{(n+1)}_{i}$}};
- \draw (1,1) node {{\tiny $\varphi^{(n+1)}_{i+1}$}};
- \draw (-1,1) node {{\tiny $\varphi^{(n+1)}_{i}$}};
- \draw (0,-1) node {{\tiny $\varphi^{(n+1)}_{i}$}};
- \draw (1,-1) node {{\tiny $\varphi^{(n+1)}_{i+1}$}};
- \draw (-1,-1) node {{\tiny $\varphi^{(n+1)}_{i}$}};
+ \draw (-1,0) node {{\tiny $\varphi^{(n)}_{i-1}$}};
+ \draw (0,-1) node {{\tiny $\varphi^{(n-1)}_{i}$}};
\endscope
\end{tikzpicture}
\end{center}
@@ -299,19 +290,75 @@
c^2 (\partial_x^2\varphi + \partial_y^2\varphi)
+ S \\
\Downarrow (c\equiv1)\\
- \begin{aligned}
- %\text{\makebox[0pt][l]{\framebox{\parbox{4cm}{Semi-descretization for Method of Lines}}}}
- \frac{\partial \phi_{i,j}}{\partial t} = & \psi_{i,j} \\
- \frac{\partial \psi_{i,j}}{\partial t} = &
- \frac{\varphi_{i+1,j} - 2\varphi_{i,j}+\varphi_{i-1,j}}
+ \begin{split}
+ \frac{\varphi^{(n+1)}_{i,j} - 2\varphi^{(n)}_{i,j}+\varphi^{(n-1)}_{i,j}}
+ {2\Delta t^2} = &
+ \frac{\varphi^{(n)}_{i+1,j} - 2\varphi^{(n)}_{i,j}+\varphi^{(n)}_{i-1,j}}
{2\Delta x^2} + \\
- & \frac{\varphi_{i,j+1} - 2\varphi_{i,j}+\varphi_{i,j-1}}
- {2\Delta y^2} + S_{i,j}
- \end{aligned}
+ & \frac{\varphi^{(n)}_{i,j+1} - 2\varphi^{(n)}_{i,j}+\varphi^{(n)}_{i,j-1}}
+ {2\Delta y^2} + S^{(n)}_{i,j}
+ \end{split}
\end{gather*}
\end{columns}
}
+%\frame{ \frametitle{WaveToy Thorn: Discretization}
+% Discretization: \\
+% approximating continuous function $\varphi(x,y,t)$
+% with a grid function $\varphi^{(n)}_{i,j}$:
+% \begin{columns}[t]
+% \column{.2\textwidth}
+% \begin{figure}[!htp]
+% \begin{center}
+% \begin{tikzpicture}[scale=1.1]
+% \draw (-1,0) -- (1,0);
+% \draw (-1,1) -- (1,1);
+% \draw (-1,-1) -- (1,-1);
+% \scope[fill=green!50!blue, draw=black]
+% \filldraw (0,0) circle (0.07cm);
+% \filldraw (1,0) circle (0.07cm);
+% \filldraw (-1,0) circle (0.07cm);
+% \filldraw (0,1) circle (0.07cm);
+% \filldraw (1,1) circle (0.07cm);
+% \filldraw (-1,1) circle (0.07cm);
+% \filldraw (0,-1) circle (0.07cm);
+% \filldraw (1,-1) circle (0.07cm);
+% \filldraw (-1,-1) circle (0.07cm);
+% \endscope
+% \scope[xshift=0.45cm,yshift=0.25cm]
+% \draw (0,0) node {{\tiny $\varphi^{(n)}_{i}$}};
+% \draw (1,0) node {{\tiny $\varphi^{(n)}_{i+1}$}};
+% \draw (-1,0) node {{\tiny $\varphi^{(n)}_{i}$}};
+% \draw (0,1) node {{\tiny $\varphi^{(n+1)}_{i}$}};
+% \draw (1,1) node {{\tiny $\varphi^{(n+1)}_{i+1}$}};
+% \draw (-1,1) node {{\tiny $\varphi^{(n+1)}_{i}$}};
+% \draw (0,-1) node {{\tiny $\varphi^{(n+1)}_{i}$}};
+% \draw (1,-1) node {{\tiny $\varphi^{(n+1)}_{i+1}$}};
+% \draw (-1,-1) node {{\tiny $\varphi^{(n+1)}_{i}$}};
+% \endscope
+% \end{tikzpicture}
+% \end{center}
+% \end{figure}
+%
+% \column{.8\textwidth}
+% \begin{gather*}
+% \\
+% \frac{\partial^2\varphi}{\partial t^2} =
+% c^2 (\partial_x^2\varphi + \partial_y^2\varphi)
+% + S \\
+% \Downarrow (c\equiv1)\\
+% \begin{aligned}
+% \frac{\partial \phi_{i,j}}{\partial t} = & \psi_{i,j} \\
+% \frac{\partial \psi_{i,j}}{\partial t} = &
+% \frac{\varphi_{i+1,j} - 2\varphi_{i,j}+\varphi_{i-1,j}}
+% {2\Delta x^2} + \\
+% & \frac{\varphi_{i,j+1} - 2\varphi_{i,j}+\varphi_{i,j-1}}
+% {2\Delta y^2} + S_{i,j}
+% \end{aligned}
+% \end{gather*}
+% \end{columns}
+%}
+
\frame{ \frametitle{WaveToyMoL Thorn}
Thorn structure:
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