Abstract: The subject of this paper is the analysis of evolution Galerkin schemes applied to two dimensional Riemann problems with finitely many constant states. We solve a linear first order system for the wave equation and test evolution Galerkin methods as well as other commonly used schemes with respect to their accuracy and the ability to capture important structural phenomena of the solution. For the two-dimensional Riemann problems with finitely many constant states parts of the exact solution are constructed and the analytical formulae are derived in the following way. Using a self-similar transformation $(x/t, y/t)$ and solving the resulting 1D Riemann problems we are able to describe propagation of finitely many planar waves and give analytical descriptions of the discontinous solution up to the region where the waves start to interact. Then a Goursant-type problem has to be solved to describe the interaction of waves up to the sonic circle. Inside it the system changes type being partially elliptic. Friedrichs theory for symmetric positive first order system and the classical elliptic theory is used to show the existence, uniqueness and the maximum principle for the solution in the subsonic region.
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