On energy stable discontinuous Galerkin spectral element approximations of the perfectly matched layer for the wave equation

Kenneth Duru, Alice Agnes Gabriel, Gunilla Kreiss

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


In this paper, we develop a provably energy stable discontinuous Galerkin spectral element method (DGSEM), approximating the perfectly matched layer (PML) for the three and two space dimensional (3D and 2D) linear acoustic wave equations, in first order form, subject to well-posed linear boundary conditions. First, using the well-known complex coordinate stretching, we derive an efficient un-split modal PML for the 3D acoustic wave equation, truncating a cuboidal computational domain. Second, we prove asymptotic stability of the continuous PML by deriving energy estimates in the Laplace space, for the 3D PML in a heterogeneous acoustic medium, assuming piece-wise constant PML damping. In the time-domain, the energy estimate translates to a bound for the solutions in terms of the initial data. Third, we develop a DGSEM for the wave equation using physically motivated numerical flux, with penalty weights, which are compatible with all well-posed, internal and external, boundary conditions. When the PML damping vanishes, by construction, our choice of penalty parameters yields an upwind scheme and a discrete energy estimate analogous to the continuous energy estimate. Fourth, to ensure numerical stability of the discretization when PML is present, it is necessary to systematically extend the numerical fluxes, and the inter-element and boundary procedures, to the PML auxiliary differential equations. This is critical for deriving discrete energy estimates analogous to the continuous energy estimates. Finally, we propose a procedure to compute PML damping coefficients such that the PML error converges to zero, at the optimal convergence rate of the underlying numerical method. Numerical solutions are evolved in time using the high order Taylor-type time stepping scheme of the same order of accuracy of the spatial discretization. By combining the DGSEM spatial approximation with the high order Taylor-type time stepping scheme and the accuracy of the PML we obtain an arbitrarily accurate wave propagation solver in the time domain. Numerical experiments are presented in 2D and 3D corroborating the theoretical results.
Original languageEnglish (US)
Pages (from-to)898-937
Number of pages40
JournalComputer Methods in Applied Mechanics and Engineering
StatePublished - Apr 16 2019
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2022-06-10
Acknowledged KAUST grant number(s): ORS-2016-CRG5-3027, ORS-2017-CRG6 3389.02
Acknowledgements: The work presented in this paper was enabled by funding from the European Union's Horizon 2020 research and innovation program under grant agreement No 671698 (ExaHyPE). [Figure presented], A.-A. G. acknowledges additional support by the German Research Foundation (DFG) (projects no. 1108 KA 2281/4-1, GA 2465/2-1, GA 2465/3-1), by BaCaTec 1109 (project no. A4) and BayLat, by KONWIHR- the Bavarian Competence Network for Technical and Scientific High Performance Computing (project NewWave), by KAUST-CRG (GAST, grant no. ORS-2016-CRG5-3027 and FRAGEN, grant no. ORS-2017-CRG6 3389.02), and by the European Union's Horizon research and innovation program (ChEESE, grant no. 823844). Computing resources were provided by the Institute of Geophysics of LMU Munich [34] and the Leibniz Supercomputing Centre (LRZ, projects no. h019z, 1103 pr63qo, and pr45fi on SuperMUC).
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

ASJC Scopus subject areas

  • Physics and Astronomy(all)
  • Mechanics of Materials
  • Mechanical Engineering
  • Computational Mechanics
  • Computer Science Applications


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