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Entropy stable shock capturing space–time discontinuous Galerkin schemes for systems of conservation laws

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Abstract

We present a streamline diffusion shock capturing spacetime discontinuous Galerkin (DG) method to approximate nonlinear systems of conservation laws in several space dimensions. The degrees of freedom are in terms of the entropy variables and the numerical flux functions are the entropy stable finite volume fluxes. We show entropy stability of the (formally) arbitrarily high order accurate method for a general system of conservation laws. Furthermore, we prove that the approximate solutions converge to the entropy measure valued solutions for nonlinear systems of conservation laws. Convergence to entropy solutions for scalar conservation laws and for linear symmetrizable systems is also shown. Numerical experiments are presented to illustrate the robustness of the proposed schemes.

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Acknowledgments

SM thanks Dr. T. J. Barth, NASA, Ames, USA for interesting discussions on spacetime DG methods and for providing the particular forms of the shock capturing operators used by him.

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Authors and Affiliations

  1. Seminar for Applied Mathematics (SAM), Department of Mathematics, ETH Zürich, HG J 49, Zürich, 8092, Switzerland

    Andreas Hiltebrand

  2. Center of Mathematics for Applications (CMA), University of Oslo, P.O. Box 1053, Blindern, Oslo, 0316, Norway

    Siddhartha Mishra

  3. Seminar for Applied Mathematics (SAM), Department of Mathematics, ETH Zürich, HG G 57.2, Zürich, 8092, Switzerland

    Siddhartha Mishra

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  1. Andreas Hiltebrand
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  2. Siddhartha Mishra
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Correspondence to Siddhartha Mishra.

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Hiltebrand, A., Mishra, S. Entropy stable shock capturing space–time discontinuous Galerkin schemes for systems of conservation laws. Numer. Math. 126, 103–151 (2014). https://doi.org/10.1007/s00211-013-0558-0

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