From c4c74f98a660cdf081328e21b2e99ba6cb88b4fc Mon Sep 17 00:00:00 2001 From: bangerth Date: Mon, 12 May 2008 20:49:43 +0000 Subject: [PATCH] Fix matrix. git-svn-id: https://svn.dealii.org/trunk@16081 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-33/doc/intro.dox | 13 +++++++++---- 1 file changed, 9 insertions(+), 4 deletions(-) diff --git a/deal.II/examples/step-33/doc/intro.dox b/deal.II/examples/step-33/doc/intro.dox index 30cce446e6..f0e0cc8812 100644 --- a/deal.II/examples/step-33/doc/intro.dox +++ b/deal.II/examples/step-33/doc/intro.dox @@ -34,16 +34,21 @@ with the solution $\mathbf{w}=(\rho v_1,\ldots,\rho v_d,\rho, E)^{\top}$ consisting of $\rho$ the fluid density, ${\mathbf v}=(v_1,\ldots v_d)^T$ the flow velocity (and thus $\rho\mathbf v$ being the linear momentum density), and -$E$ the energy density of the gas. The flux matrix $\mathbf F$ (or system of flux functions) +$E$ the energy density of the gas. We interpret the equations above as +$\partial_t \mathbf{w}_i + \nabla \cdot \mathbf{F}_i(\mathbf{w}) = 0$, $i=1,\ldots,dim+2$. + +For the Euler equations, the flux matrix $\mathbf F$ (or system of flux functions) is defined as (shown here for the case $d=3$) @f{eqnarray*} \mathbf F = \left( \begin{array}{ccc} - \rho v_1^2+p & \rho v_2v_1 & \rho v_3v_1 & \rho v_1 & (E+p)v_1 \\ - \rho v_1v_2 & \rho v_2^2+p & \rho v_3v_2 & \rho v_2 & (E+p)v_2 \\ - \rho v_1v_3 & \rho v_2v_3 & \rho v_3^2+p & \rho v_3 & (E+p)v_3 + \rho v_1^2+p & \rho v_2v_1 & \rho v_3v_1 \\ + \rho v_1v_2 & \rho v_2^2+p & \rho v_3v_2 \\ + \rho v_1v_3 & \rho v_2v_3 & \rho v_3^2+p \\ + \rho v_1 & \rho v_2 & \rho v_3 \\ + (E+p) v_1 & (E+p) v_2 & (E+p) v_3 \end{array} \right), @f} -- 2.39.5