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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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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 HTML | PDF
Math. Comp. 90 (2021), 2603-2643 Request permission


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.
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Additional 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:
  • Simon Michel
  • Affiliation: Department of Mathematics and Computer Science, University of Basel, Spiegelgasse 1, 4051 Basel, Switzerland
  • Email:
  • Stefan A. Sauter
  • Affiliation: Institute for Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
  • MR Author ID: 313335
  • Email:
  • 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:
  • MathSciNet review: 4305363