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Analysis and Entropy Stability of the Line-Based Discontinuous Galerkin Method

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Abstract

We develop a discretely entropy-stable line-based discontinuous Galerkin method for hyperbolic conservation laws based on a flux differencing technique. By using standard entropy-stable and entropy-conservative numerical flux functions, this method guarantees that the discrete integral of the entropy is non-increasing. This nonlinear entropy stability property is important for the robustness of the method, in particular when applied to problems with discontinuous solutions or when the mesh is under-resolved. This line-based method is significantly less computationally expensive than a standard DG method. Numerical results are shown demonstrating the effectiveness of the method on a variety of test cases, including Burgers’ equation and the Euler equations, in one, two, and three spatial dimensions.

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Acknowledgements

This work was supported by the National Aeronautics and Space Administration (NASA) under Grant Number NNX16AP15A, by the Director, Office of Science, Office of Advanced Scientific Computing Research, U.S. Department of Energy under Contract No. DE-AC02-05CH11231 and by the AFOSR Computational Mathematics program under Grant Number FA9550-15-1-0010. Lawrence Livermore National Laboratory is operated by Lawrence Livermore National Security, LLC, for the U.S. Department of Energy, National Nuclear Security Administration under Contract DE-AC52-07NA27344.

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  1. Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Livermore, USA

    Will Pazner

  2. Department of Mathematics, University of California, Berkeley, USA

    Per-Olof Persson

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Pazner, W., Persson, PO. Analysis and Entropy Stability of the Line-Based Discontinuous Galerkin Method. J Sci Comput 80, 376–402 (2019). https://doi.org/10.1007/s10915-019-00942-1

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