Stability of Discontinuous Galerkin Spectral Element Schemes for Wave\n Propagation when the Coefficient Matrices have Jumps

2020/11/23 by David A. Kopriva, Gregor J. Gassner, Kopriva, David A. +3 · 1 citation
Engineering · #Advanced Numerical Methods in Computational Mathematics #Computational Fluid Dynamics and Aerodynamics #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2011.11746

openalex publication_date 2020/11/23 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

We use the behavior of the L₂ norm of the solutions of linear hyperbolic\nequations with discontinuous coefficient matrices as a surrogate to infer\nstability of discontinuous Galerkin spectral element methods (DGSEM). Although\nthe L₂ norm is not bounded by the initial data for homogeneous and\ndissipative boundary conditions for such systems, the L₂ norm is easier to\nwork with than a norm that discounts growth due to the discontinuities. We show\nthat the DGSEM with an upwind numerical flux that satisfies the\nRankine-Hugoniot (or conservation) condition has the same energy bound as the\npartial differential equation does in the L₂ norm, plus an added\ndissipation that depends on how much the approximate solution fails to satisfy\nthe Rankine-Hugoniot jump.\n

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• Numerical boundary flux functions that give provable bounds for nonlinear initial boundary value problems with open boundaries

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