Version 16 (modified by 11 years ago) ( diff ) | ,
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# Recap

The Euler equations are a set of non-linear PDE's, which comprise an eigenvalue problem. The eigenvalues for the equations are functions of the solution to the equations themselves. This means that the waves, which propagate with speeds = to their eigenvalues, distort the solution, and the solution distorts them over space and time. Thus the solution to the Riemann problem for the non-linear system does not consist of a closed form expression for the values of pstar and ustar like it does for a linear system of equations. To solve the Euler equations exactly then, we have developed the method of characteristics that describe the propagation of waves outside of the star region. To solve for values of the q-array inside of the star region, we used an iterative scheme and then sampled the solution in the different wave regions. We now are concerned with approximations to this exact solution. The method to be discussed here considers a 'linearized' version of the Euler equations, so analytical methods used for linear, constant coefficient systems of equations can be applied.

# The ROE Solver

The Roe solver is an approximation means for the numerical flux of the Godunov method, which is derived through linearizing a hyperbolic system of equations. For instance, the Euler equations in conservative form are written

, which using the chain rule is identical to

where, the Jacobian matrix A= is a NON-constant coefficient matrix, the Euler equations comprise a NON-linear set of equations. An easier system to solve would be one that is a linear, constant coefficient system of equations. We can transform the Euler equations into this simpler case, if we make transformations of variables in the matrix to be some average function of the left and right data state variables. This results in the set of equations:

Where now the Jacobian is an averaged matrix, representing constant coefficients for the Euler equations.

In considering an integral form of the conservation laws now that an exact solution to the 'approximate' Riemann problem is feasible, one can solve for the numerical flux in terms of 1) wave strengths, 2) eigenvalues, 3) right eigenvectors of the Jacobian. One possible algebraic form for the flux function is:

### Attachments (23)

- Test1Roe.png (11.5 KB ) - added by 11 years ago.
- waveStructures.png (30.2 KB ) - added by 11 years ago.
- RhoP.png (21.4 KB ) - added by 11 years ago.
- RhoU.png (16.5 KB ) - added by 11 years ago.
- RhoP.2.png (21.4 KB ) - added by 11 years ago.
- RoeRho.png (22.9 KB ) - added by 11 years ago.
- rhoSolver.png (118.5 KB ) - added by 11 years ago.
- CallRoe.png (14.1 KB ) - added by 11 years ago.
- RoePTest1.png (10.2 KB ) - added by 11 years ago.
- RoeUTest1.png (12.0 KB ) - added by 11 years ago.
- RoeRhoTest1.png (10.7 KB ) - added by 11 years ago.
- RoePTest3.png (12.2 KB ) - added by 11 years ago.
- RoeUTest3.png (12.4 KB ) - added by 11 years ago.
- RoeRhoTest3.png (12.7 KB ) - added by 11 years ago.
- RoePTest5.png (12.0 KB ) - added by 11 years ago.
- RoeUTest5.png (12.1 KB ) - added by 11 years ago.
- RoeRhoTest5.png (10.7 KB ) - added by 11 years ago.
- RoePTest4.png (13.5 KB ) - added by 11 years ago.
- RoeUTest4.png (10.5 KB ) - added by 11 years ago.
- RoeRhoTest4.png (11.0 KB ) - added by 11 years ago.
- Roe3.png (121.8 KB ) - added by 11 years ago.
- Roe2.png (127.3 KB ) - added by 11 years ago.
- Roe1.png (131.9 KB ) - added by 11 years ago.

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