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Kinetic Functions for Nonclassical Shocks, Entropy Stability, and Discrete Summation by Parts

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Abstract

We study nonlinear hyperbolic conservation laws with non-convex flux in one space dimension and, for a broad class of numerical methods based on summation by parts operators, we compute numerically the kinetic functions associated with each scheme. As established by LeFloch and collaborators, kinetic functions (for continuous or discrete models) uniquely characterize the macro-scale dynamics of small-scale dependent, undercompressive, nonclassical shock waves. We show here that various entropy-dissipative numerical schemes can yield nonclassical solutions containing classical shocks, including Fourier methods with (super-) spectral viscosity, finite difference schemes with artificial dissipation, discontinuous Galerkin schemes with or without modal filtering, and TeCNO schemes. We demonstrate numerically that entropy stability does not imply uniqueness of the limiting numerical solutions for scalar conservation laws in one space dimension, and we compute the associated kinetic functions in order to distinguish between these schemes. In addition, we design entropy-dissipative schemes for the Keyfitz–Kranzer system whose solutions are measures with delta shocks. This system illustrates the fact that entropy stability does not imply boundedness under grid refinement.

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Acknowledgements

The authors would like to thank David Ketcheson for very interesting discussions. This research work was supported by the King Abdullah University of Science and Technology (KAUST). The first author (PLF) was partially supported by the Innovative Training Network (ITN) Grant 642768 (ModCompShocks). The second author (HR) was partially supported by the German Research Foundation (DFG, Deutsche Forschungsgemeinschaft) under Grant SO 363/14-1.

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  1. Hendrik Ranocha

    Present address: Applied Mathematics Münster, University of Münster, Münster, Germany

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  1. Laboratoire Jacques-Louis Lions and Centre National de la Recherche Scientifique, Sorbonne Université, 4 Place Jussieu, 75252, Paris, France

    Philippe G. LeFloch

  2. Computer Electrical and Mathematical Science and Engineering Division (CEMSE), King Abdullah University of Science and Technology (KAUST), Thuwal, 23955-6900, Saudi Arabia

    Hendrik Ranocha

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LeFloch, P.G., Ranocha, H. Kinetic Functions for Nonclassical Shocks, Entropy Stability, and Discrete Summation by Parts. J Sci Comput 87, 55 (2021). https://doi.org/10.1007/s10915-021-01463-6

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