[Commits] [svn:einsteintoolkit] Workshop_Spring_2012/numerical_relativity/ (Rev. 31)

[diener at cct.lsu.edu]

User: diener
Date: 2012/04/03 10:41 PM

Modified:
 /numerical_relativity/
  numerical_relativity.tex

Log:
 First slide on MoL.

File Changes:

Directory: /numerical_relativity/
=================================

File [modified]: numerical_relativity.tex
Delta lines: +30 -5
===================================================================
--- numerical_relativity/numerical_relativity.tex	2012-04-04 01:40:55 UTC (rev 30)
+++ numerical_relativity/numerical_relativity.tex	2012-04-04 03:41:18 UTC (rev 31)
@@ -15,8 +15,8 @@
 %  \item 3+1 spacetime decomposition.
   \item The BSSN formulation.
   \item Puncture data.
-  \item Finite Differencing.
-%  \item The method of lines.
+  \item Finite differencing.
+  \item The method of lines.
  \end{itemize}
 }
 
@@ -214,7 +214,7 @@
 that $\psi$ has a unique solution.
 }
 
-\frame{\frametitle{Finite Differencing}
+\frame{\frametitle{Finite differencing}
 With finite differencing we discretize a function by sampling it at a 
 collection of grid points.\pause
 
@@ -231,7 +231,7 @@
   \left. \frac{df}{dx} \right |_{x_i}\approx \frac{1}{\Delta x}\sum_{j=-1}^{j=1} a_j f_{i+j}.
 \end{equation}
 }
-\frame{\frametitle{Finite Differencing (continued)}
+\frame{\frametitle{Finite differencing (continued)}
 The coefficients $a_j$ can be found be expanding $f$ in a Taylor series around
 $x_i$ for the grid points in the stencil
 \begin{eqnarray}
@@ -253,7 +253,7 @@
 with the solution $a_{-1}=-1/2, a_0=0, a_1=1/2$.
 }
 
-\frame{\frametitle{Finite Differencing (continued)}
+\frame{\frametitle{Finite differencing (continued)}
 Thus we find that
 \begin{equation}
 \left. \frac{df}{dx} \right |_{x_i} = \frac{f_{i+1}-f_{i-1}}{2\Delta x}
@@ -276,4 +276,29 @@
 It is also clear from either approach that the error estimates are only
 correct if $f$ is smooth enough.
 }
+
+\frame{\frametitle{The method of lines}
+Consider the set of hyperbolic PDE's
+\begin{equation}
+\partial_t \mathbf{q}+\mathbf{A}^i(\mathbf{q})\partial_i 
+\mathbf{B}(\mathbf{q}) = \mathbf{S}(\mathbf{q}).
+\end{equation}\pause
+The idea then is to discretize in space first, i.e.\ write the equations as
+\begin{equation}
+\partial_t \mathbf{q} = \mathbf{L}(\mathbf{q}),
+\end{equation}
+where $\mathbf{L}(\mathbf{q})$ is a discrete approximation to the equations
+(e.g.\ using finite differencing).\pause
+
+This then turns the equations into a set of coupled ODE's with respect to
+time.\pause
+
+If the spatial discretization (including boundary conditions) is stable
+we we can then evolve the system of equations using any stable ODE time
+integrator.\pause
+
+Often Runge-Kutta integration schemes are the scheme of choice.
+
+
+}
 \end{document}