Energy-Based Discontinuous Galerkin Difference Methods for Second-Order Wave Equations

doi:10.1007/s42967-021-00149-y

Communications on Applied Mathematics and Computation ›› 2022, Vol. 4 ›› Issue (3): 855-879.doi: 10.1007/s42967-021-00149-y

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Energy-Based Discontinuous Galerkin Difference Methods for Second-Order Wave Equations

Lu Zhang¹, Daniel Appelö², Thomas Hagstrom³

1. Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA;
2. Department of Computational Mathematics, Science and Engineering and Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA;
3. Department of Mathematics, Southern Methodist University, Dallas, TX 75275, USA

Received:2020-11-23 Revised:2021-04-04 Online:2022-09-20 Published:2022-07-04
Contact: Thomas Hagstrom,E-mail:thagstrom@smu.edu;Lu Zhang,E-mail:lz2784@columbia.edu;Daniel Appelö,E-mail:appeloda@msu.edu E-mail:thagstrom@smu.edu;appeloda@msu.edu;appeloda@msu.edu
Supported by:
This work was supported by NSF Grants DMS-1913076 and DMS-2012296 and completed, while the third author was in residence at the Institute for Computational and Experimental Mathematics. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Abstract

Abstract: We combine the newly constructed Galerkin difference basis with the energy-based discontinuous Galerkin method for wave equations in second-order form. The approximation properties of the resulting method are excellent and the allowable time steps are large compared to traditional discontinuous Galerkin methods. The one drawback of the combined approach is the cost of inversion of the local mass matrix. We demonstrate that for constant coefficient problems on Cartesian meshes this bottleneck can be removed by the use of a modified Galerkin difference basis. For variable coefficients or non-Cartesian meshes this technique is not possible and we instead use the preconditioned conjugate gradient method to iteratively invert the mass matrices. With a careful choice of preconditioner we can demonstrate optimal complexity, albeit with a larger constant.

Key words: Discontinuous Galerkin, Galerkin difference, Simultaneous diagonalization

CLC Number:

65M60
65M06

Lu Zhang, Daniel Appelö, Thomas Hagstrom. Energy-Based Discontinuous Galerkin Difference Methods for Second-Order Wave Equations[J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 855-879.

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References

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Energy-Based Discontinuous Galerkin Difference Methods for Second-Order Wave Equations

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