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Stabilized leapfrog based local time-stepping method for the wave equation


Grote, Marcus J; Michel, Simone; Sauter, Stefan A (2021). Stabilized leapfrog based local time-stepping method for the wave equation. Mathematics of Computation, 90(332):2603-2643.

Abstract

Local time-stepping methods permit to overcome the severe stability constraint on explicit methods caused by local mesh refinement without sacrificing explicitness. Diaz and Grote [SIAM J. Sci. Comput. 31 (2009), pp. 1985–2014] proposed a leapfrog based explicit local time-stepping (LF-LTS) method for the time integration of second-order wave equations. Recently, optimal convergence rates were proved for a conforming FEM discretization, albeit under a CFL stability condition where the global time-step, Δt, depends on the smallest elements in the mesh (see M. J. Grote, M. Mehlin, and S. A. Sauter [SIAM J. Numer. Anal. 56 (2018), pp. 994–1021]). In general one cannot improve upon that stability constraint, as the LF-LTS method may become unstable at certain discrete values of Δt. To remove those critical values of Δt, we apply a slight modification (as in recent work on LF-Chebyshev methods by Carle, Hochbruck, and Sturm [SIAM J. Numer. Anal. 58 (2020), pp. 2404–2433]) to the original LF-LTS method which nonetheless preserves its desirable properties: it is fully explicit, second-order accurate, satisfies a three-term (leapfrog like) recurrence relation, and conserves the energy. The new stabilized LF-LTS method also yields optimal convergence rates for a standard conforming FE discretization, yet under a CFL condition where Δt no longer depends on the mesh size inside the locally refined region.

Abstract

Local time-stepping methods permit to overcome the severe stability constraint on explicit methods caused by local mesh refinement without sacrificing explicitness. Diaz and Grote [SIAM J. Sci. Comput. 31 (2009), pp. 1985–2014] proposed a leapfrog based explicit local time-stepping (LF-LTS) method for the time integration of second-order wave equations. Recently, optimal convergence rates were proved for a conforming FEM discretization, albeit under a CFL stability condition where the global time-step, Δt, depends on the smallest elements in the mesh (see M. J. Grote, M. Mehlin, and S. A. Sauter [SIAM J. Numer. Anal. 56 (2018), pp. 994–1021]). In general one cannot improve upon that stability constraint, as the LF-LTS method may become unstable at certain discrete values of Δt. To remove those critical values of Δt, we apply a slight modification (as in recent work on LF-Chebyshev methods by Carle, Hochbruck, and Sturm [SIAM J. Numer. Anal. 58 (2020), pp. 2404–2433]) to the original LF-LTS method which nonetheless preserves its desirable properties: it is fully explicit, second-order accurate, satisfies a three-term (leapfrog like) recurrence relation, and conserves the energy. The new stabilized LF-LTS method also yields optimal convergence rates for a standard conforming FE discretization, yet under a CFL condition where Δt no longer depends on the mesh size inside the locally refined region.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:340 Law
610 Medicine & health
510 Mathematics
Scopus Subject Areas:Physical Sciences > Algebra and Number Theory
Physical Sciences > Computational Mathematics
Physical Sciences > Applied Mathematics
Uncontrolled Keywords:Applied Mathematics, Computational Mathematics, Algebra and Number Theory
Language:English
Date:18 June 2021
Deposited On:17 Nov 2021 13:35
Last Modified:26 Apr 2024 01:36
Publisher:American Mathematical Society
ISSN:0025-5718
OA Status:Closed
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.1090/mcom/3650