Roe solver

From Wikipedia, the free encyclopedia

This article or section is in need of attention from an expert on the subject.
Please help recruit one or improve this article yourself. See the talk page for details.
Please consider using {{Expert-subject}} to associate this request with a WikiProject

This article does not cite any references or sources. (March 2008)
Please help improve this article by adding citations to reliable sources. Unverifiable material may be challenged and removed.

This article or section needs to be wikified to meet Wikipedia's quality standards.
Please help improve this article with relevant internal links. (March 2008)

The Roe approximate Riemann solver uses the flux Jacobian to express the Euler equations, in this case the homogenous 1D Euler equation:

$\frac{\partial Q}{\partial t} + A(Q)\frac{\partial Q}{\partial x} = 0$

Where A(Q) is given as:

$\frac{\partial E(Q)}{\partial Q}$

The x-direction flux Jacobian is given as:

$A= \left[ \begin{array}{c c c c c} 0 & 1 & 0 & 0 & 0 \\ \hat{\gamma}H-\tilde{u}^2- \tilde{a}^2 & (3-\gamma)\tilde{u} & -\hat{\gamma}v & -\hat{\gamma}w & \hat{\gamma} \\ \tilde{u} \tilde{v}& \tilde{v} & \tilde{u} & 0 & 0\\ \tilde{u}\tilde{w} & \tilde{w} & 0 & u & 0 \\ (\frac{1}{2}\tilde{u}[(\gamma-3)\tilde{H}-\tilde{a}^2] & H-\hat{\gamma}\tilde{u}^2 &\hat{\gamma}\tilde{u}\tilde{v} & \hat{\gamma}\tilde{u}\tilde{w} & \gamma \tilde{u} \end{array} \right]$

Where the tilde represents the Roe-averaged quantities. (todo: Roe averages)

The eigenvalues of the Jacobian are:

$\tilde{\lambda}_1 = \tilde{u}-\tilde{a}, \quad \tilde{\lambda}_2 = \tilde{\lambda}_3 = \tilde{\lambda}_4 =\tilde{u}, \quad \tilde{\lambda}_5=\tilde{u}+\tilde{a}$

And the right eigenvectors:

$P= \left[ \begin{array}{c c c c c} 1 & 1 & 0 & 0 & 1 \\ \tilde{u} - \tilde{a} & \tilde{u} & 0 & 0 & \tilde{u}+\tilde{a} \\ \tilde{v} & \tilde{v} & 1 & 0 & \tilde{v}\\ \tilde{w} & \tilde{w} & 0 & 1 & \tilde{w} \\ \tilde{H}-\tilde{u}\tilde{a} & \frac{1}{2}\vec{\tilde{U}}^2 &\tilde{v} & \tilde{w} & \tilde{H}+\tilde{u}\tilde{a} \end{array} \right]$