Evolution Galerkin schemes applied to two-dimensional Riemann problems for the wave equation system

Abstract

The subject of this paper is ademonstration of the accuracy and robustness of evolutionGalerkin schemes applied to two-dimensional Riemann problems withfinitely many constant states.In order to have a test case with known exact solution weconsider a linear first ordersystem for the wave equation and test evolution Galerkin methodsas well as other commonly used schemes with respect to theiraccuracyin capturing important structural phenomenaof the solution. For the two-dimensional Riemann problems withfinitely many constant states some parts of the exact solution areconstructed in thefollowing three steps.Using a self-similar transformationwe solve the Riemann problem outside a neighborhoodof the origin and then work inwards.Next a Goursant-typeproblem has to be solved to describe the interaction of waves upto the sonic circle. Inside it a systemof composite elliptic-hyperbolic type is obtained, whichmay not always be solvable exactly.There an interesting local maximum principle can be shown.Finally, an exact partial solution is used for numericalcomparisons.