wiki:u/erica/RoeSolver

Version 5 (modified by Erica Kaminski, 11 years ago) ( diff )

The ROE Solver

The ROE solver combines a few different concepts and techniques from my previous studies on approximation methods. The ROE solver attempts to solve the Riemann Problem using an approximation to the numerical flux, albeit with a slightly different spin — linearizing the hyperbolic system of equations. That is, given the Euler equations

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can be written in matrix form as:

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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:

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