Stabilized leapfrog based local time-stepping method for the wave equation
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- by Marcus J. Grote, Simon Michel and Stefan A. Sauter;
- Math. Comp. 90 (2021), 2603-2643
- DOI: https://doi.org/10.1090/mcom/3650
- Published electronically: June 18, 2021
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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, $\Delta t$, 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 $\Delta t$. To remove those critical values of $\Delta t$, 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 $\Delta t$ no longer depends on the mesh size inside the locally refined region.References
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Bibliographic Information
- Marcus J. Grote
- Affiliation: Department of Mathematics and Computer Science, University of Basel, Spiegelgasse 1, 4051 Basel, Switzerland
- MR Author ID: 360720
- ORCID: 0000-0001-8129-0799
- Email: marcus.grote@unibas.ch
- Simon Michel
- Affiliation: Department of Mathematics and Computer Science, University of Basel, Spiegelgasse 1, 4051 Basel, Switzerland
- Email: simon.michel@unibas.ch
- Stefan A. Sauter
- Affiliation: Institute for Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
- MR Author ID: 313335
- Email: stas@math.uzh.ch
- Received by editor(s): May 29, 2020
- Received by editor(s) in revised form: February 5, 2021, and March 9, 2021
- Published electronically: June 18, 2021
- Additional Notes: This work was supported by the Swiss National Science Foundation under grant SNF 200020-188583
- © Copyright 2021 American Mathematical Society
- Journal: Math. Comp. 90 (2021), 2603-2643
- MSC (2020): Primary 65M12, 65M20, 65M60, 65L06, 65L20
- DOI: https://doi.org/10.1090/mcom/3650
- MathSciNet review: 4305363