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Relaxing the CFL Condition for the Wave Equation on Adaptive Meshes

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Abstract

The Courant–Friedrichs–Lewy (CFL) condition guarantees the stability of the popular explicit leapfrog method for the wave equation. However, it limits the choice of the time step size to be bounded by the minimal mesh size in the spatial finite element mesh. This essentially prohibits any sort of adaptive mesh refinement that would be required to reveal optimal convergence rates on domains with re-entrant corners. This paper shows how a simple subspace projection step inspired by numerical homogenisation can remove the critical time step restriction so that the CFL condition and approximation properties are balanced in an optimal way, even in the presence of spatial singularities.

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Acknowledgements

The authors would like to thank Andreas Longva for pointing out that mass lumping indeed works. Parts of this paper were written while the authors enjoyed the kind hospitality of the Hausdorff Institute for Mathematics (Bonn).

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  1. Institut für Numerische Simulation, Universität Bonn, Wegelerstraße 6, 53115, Bonn, Germany

    Daniel Peterseim & Mira Schedensack

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  1. Daniel Peterseim
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  2. Mira Schedensack
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Correspondence to Mira Schedensack.

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D. Peterseim gratefully acknowledges support by the Hausdorff Center for Mathematics Bonn and by Deutsche Forschungsgemeinschaft in the Priority Program 1748 “Reliable simulation techniques in solid mechanics: Development of non-standard discretization methods, mechanical and mathematical analysis” under the project “Adaptive isogeometric modeling of propagating strong discontinuities in heterogeneous materials”.

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Peterseim, D., Schedensack, M. Relaxing the CFL Condition for the Wave Equation on Adaptive Meshes. J Sci Comput 72, 1196–1213 (2017). https://doi.org/10.1007/s10915-017-0394-y

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  • DOI: https://doi.org/10.1007/s10915-017-0394-y

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