User: bmundim Date: 2012/04/04 11:13 AM

Modified: /numerical_relativity/ numerical_relativity.tex

Log: approximate riemann solvers, an example.

File Changes:

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

File [modified]: numerical_relativity.tex Delta lines: +24 -0 =================================================================== --- numerical_relativity/numerical_relativity.tex 2012-04-04 15:55:15 UTC (rev 45) +++ numerical_relativity/numerical_relativity.tex 2012-04-04 16:13:46 UTC (rev 46) @@ -609,7 +609,31 @@ \includegraphics[width=6cm]{rarefaction.pdf} \end{figure}

+}

+\frame{\frametitle{Approximate Riemann Solvers} +Usually the exact solution of the Riemann problem is computationally +very expensive. \pause + +Fortunately we can obtain very good approximation for the solutions +by approximating the conservation law as a quasi-linear system: +\begin{equation} +q_t + A q_x = 0 +\end{equation} \pause +where $A$ is a diagonalizable matrix given by: +\begin{equation} +A(q_l,q_r) = \left. \frac{\partial f}{\partial q} \right|_{q=1/2(q_l+q_R)} +\end{equation} \pause +Roe solver: +\begin{equation} +F^{\rm Roe}_{i+1/2}=\frac{1}{2} \left[f(q^r_{i+1/2})+f(q^l_{i-1/2}) +-\sum_{\alpha} |\lambda_{\alpha}| \omega_{\alpha} r_{\alpha} \right] +\end{equation} +where $\lambda_{\alpha}$ are the characteristics speeds, $\omega_{\alpha}$ +the jumps in the characteristics and $r_{\alpha}$ the right eigenvector +of $A$. }

+ + \end{document}