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This paper summarizes a Riemann-solver-free spacetime discontinuous Galerkin method developed for general conservation laws. The method integrates the best features of the spacetime Conservation Element-Solution Element CE-SE method and the discontinuous Galerkin DG method. The core idea is to construct a staggered spacetime mesh through alternate cell-centered CEs and vertex-centered CEs within each time step. Inside each SE, the solution is approximated using high-order spacetime DG basis polynomials. The spacetime flux conservation is enforced inside each CE using the DG concept. The unknowns are stored at both vertices and cell centroids of the spatial mesh. However, the solutions at vertices and cell centroids are updated at different time levels within each time step in an alternate fashion. Thanks to the staggered spacetime formulation, there are no left and right states for the solution at the spacetime interface. Instead, the solution available to evaluate the flux is continuous across the interface. Therefore, no approximate Riemann solvers are needed to provide a unique numerical flux. The current method can be used to solve arbitrary conservation laws including the compressible Euler equations, shallow water equations and magnetohydrodynamics MHD equations without the need of any form of Riemann solvers. A set of benchmark problems of various conservation laws are presented to demonstrate the accuracy of the method.

KEYWORDS

Riemann-Solver Free, Spacetime, Discontinuous Galerkin Method, Conservation Laws

Cite this paper

Tu, S. 2015 A Riemann-Solver Free Spacetime Discontinuous Galerkin Method for General Conservation Laws. American Journal of Computational Mathematics, 5, 55-74. doi: 10.4236-ajcm.2015.52004.





Autor: Shuang Z. Tu

Fuente: http://www.scirp.org/



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