[Commits] [svn:einsteintoolkit] Workshop_Spring_2012/numerical_relativity/ (Rev. 46)

[bcmsma at astro.rit.edu]

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/
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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}