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May  2003, 9(3): 559-576. doi: 10.3934/dcds.2003.9.559

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

Jiequan Li 1, , Mária Lukáčová - MedviĎová 2, and  Gerald Warnecke 3,

1. 

School of Mathematical Sciences, Capital Normal University, 100037, Beijing, China
2. 

Arbeitsbereich Mathematik, Technische Universität Hamburg-Harburg, Hamburg, Germany
3. 

Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg, Magdeburg, Germany

Received  January 2001 Revised  October 2002 Published  February 2003

The subject of this paper is a demonstration of the accuracy and robustness of evolution Galerkin schemes applied to two-dimensional Riemann problems with finitely many constant states. In order to have a test case with known exact solution we consider 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 in capturing important structural phenomena of the solution. For the two-dimensional Riemann problems with finitely many constant states some parts of the exact solution are constructed in the following three steps. Using a self-similar transformation we solve the Riemann problem outside a neighborhood of the origin and then work inwards. Next a Goursant-type problem has to be solved to describe the interaction of waves up to the sonic circle. Inside it a system of composite elliptic-hyperbolic type is obtained, which may not always be solvable exactly. There an interesting local maximum principle can be shown. Finally, an exact partial solution is used for numerical comparisons.
Keywords: Riemann problem., evolution Galerkin schemes, Genuinely multidimensional schemes, hyperbolic systems, wave equation, Euler equations.
Mathematics Subject Classification: 35L05, 65M06, 35L45, 35L65, 65M25, 65M1.
Citation: Jiequan Li, Mária Lukáčová - MedviĎová, Gerald Warnecke. Evolution Galerkin schemes applied to two-dimensional Riemann problems for the wave equation system. Discrete & Continuous Dynamical Systems, 2003, 9 (3) : 559-576. doi: 10.3934/dcds.2003.9.559
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