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An energy-based discontinuous Galerkin method with tame CFL numbers for the wave equation

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Abstract

We extend and analyze the energy-based discontinuous Galerkin method for second order wave equations on staggered and structured meshes. By combining spatial staggering with local time-stepping near boundaries, the method overcomes the typical numerical stiffness associated with high order piecewise polynomial approximations. In one space dimension with periodic boundary conditions and suitably chosen numerical fluxes, we prove bounds on the spatial operators that establish stability for CFL numbers \(c \frac{\Delta t}{h} < C\) independent of order when stability-enhanced explicit time-stepping schemes of matching order are used. For problems on bounded domains and in higher dimensions we demonstrate numerically that one can march explicitly with large time steps at high order temporal and spatial accuracy.

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Funding

This work was partially supported by NSF Grant Nos. DMS-1913076, DMS-2012296, DMS-1719942, DMS-1913072 and DMS-2208164. Any opinions, findings, conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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Authors and Affiliations

  1. Department of Computational Mathematics, Science and Engineering and Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Daniel Appelö

  2. Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY, 10027, USA

    Lu Zhang

  3. Department of Mathematics, Southern Methodist University, Dallas, TX, 75275, USA

    Thomas Hagstrom

  4. Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY, 12180, USA

    Fengyan Li

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  1. Daniel Appelö
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  2. Lu Zhang
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  4. Fengyan Li
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Correspondence to Daniel Appelö.

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Communicated by Jan Nordström.

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Appelö, D., Zhang, L., Hagstrom, T. et al. An energy-based discontinuous Galerkin method with tame CFL numbers for the wave equation. Bit Numer Math 63, 5 (2023). https://doi.org/10.1007/s10543-023-00954-2

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