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Well-balanced mesh-based and meshless schemes for the shallow-water equations

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Abstract

We formulate a general criterion for the exact preservation of the “lake at rest” solution in general mesh-based and meshless numerical schemes for the strong form of the shallow-water equations with bottom topography. The main idea is a careful mimetic design for the spatial derivative operators in the momentum flux equation that is paired with a compatible averaging rule for the water column height arising in the bottom topography source term. We prove consistency of the mimetic difference operators analytically and demonstrate the well-balanced property numerically using finite difference and RBF-FD schemes in the one- and two-dimensional cases.

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Acknowledgements

This research was undertaken, in part, thanks to funding from the Canada Research Chairs program and the NSERC Discovery Grant program. The authors thank Grady Wright for helpful discussions, and the two anonymous referees for their helpful and considerate remarks.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C 5S7, Canada

    Alexander Bihlo & Scott MacLachlan

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  1. Alexander Bihlo
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  2. Scott MacLachlan
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Correspondence to Scott MacLachlan.

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Communicated by Elisabeth Larsson.

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Bihlo, A., MacLachlan, S. Well-balanced mesh-based and meshless schemes for the shallow-water equations. Bit Numer Math 58, 579–598 (2018). https://doi.org/10.1007/s10543-018-0696-y

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  • DOI: https://doi.org/10.1007/s10543-018-0696-y

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