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A New Discontinuous Galerkin Method for Elastic Waves with Physically Motivated Numerical Fluxes

Abstract

The discontinuous Galerkin (DG) method is an established method for computing approximate solutions of partial differential equations in many applications. Unlike continuous finite elements, in DG methods numerical fluxes are used to enforce inter-element conditions, and internal/external physical boundary conditions. For elastic wave propagation in complex media several wave types, including dissipative surface and interface waves, are simultaneously supported. The presence of multiple wave types and different physical phenomena pose a significant challenge for numerical fluxes. When modelling surface or interface waves an incompatibility of the numerical flux with the physical boundary condition leads to numerical artefacts. We present a stable and arbitrary order accurate DG method for elastic waves with a physically motivated numerical flux. Our numerical flux is compatible with all well-posed, internal and external, boundary conditions, including linear and nonlinear frictional constitutive equations for modelling spontaneously propagating shear ruptures in elastic solids and dynamic earthquake rupture processes. By construction our choice of penalty parameters yield an upwind scheme and a discrete energy estimate analogous to the continuous energy estimate. We derive a priori error estimate for the DG method proving optimal convergence to discontinuous and nearly singular exact solutions. The spectral radius of the resulting spatial operator has an upper bound which is independent of the boundary and interface conditions, thus it is suitable for efficient explicit time integration. We present numerical experiments in one and two space dimensions verifying high order accuracy and asymptotic numerical stability. We demonstrate the potential of the method for modelling complex nonlinear frictional problems in elastic solids with 2D dynamically adaptive meshes and non-planar topography with 2D curvilinear elements.

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Acknowledgements

We thank the anonymous reviewers and Editor-in-Chief Chi-Wang Shu for constructive criticism which significantly improved the clarity and quality of the manuscript. The work presented in this paper was enabled by funding from the European Union’s Horizon 2020 research and innovation program under Grant Agreements Nos. 671698 (ExaHyPE), 852992 (TEAR) and 823844 (ChEESE). A.-A.G. acknowledges additional support by the German Research Foundation (DFG) (Projects Nos., GA 2465/2-1, GA 2465/3-1), by KONWIHR—the Bavarian Competence Network for Technical and Scientific High Performance Computing (project NewWave), and by KAUST-CRG (FRAGEN, Grant No. ORS-2017-CRG6 3389.02).

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Correspondence to Kenneth Duru.

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Duru, K., Rannabauer, L., Gabriel, AA. et al. A New Discontinuous Galerkin Method for Elastic Waves with Physically Motivated Numerical Fluxes. J Sci Comput 88, 51 (2021). https://doi.org/10.1007/s10915-021-01565-1

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  • DOI: https://doi.org/10.1007/s10915-021-01565-1

Keywords

  • Elastic wave equation
  • First order systems
  • Boundary conditions
  • Interface conditions
  • Nonlinear friction laws
  • Stability
  • Discontinuous Galerkin method
  • Spectral accuracy
  • Penalty method

Mathematics Subject Classification

  • 35F55
  • 35F46
  • 65M60
  • 65M70
  • 65M12
  • 65M15