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}