[svn:einsteintoolkit] Workshop_Summer_2013/et_intro_details/ (Rev. 20) - Commits

26 Jul 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/
=============================
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)
@@ -0,0 +1,83 @@
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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:

    