Flooding and drying in discontinuous Galerking finite element discretizations of shallow-water equations. Part I: One dimension.

Onno Bokhove

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    52 Citations (Scopus)
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    Abstract

    Free boundaries in shallow-water equations demarcate the time-dependent water line between ‘‘flooded’’ and ‘‘dry’’ regions. We present a novel numerical algorithm to treat flooding and drying in a formally second-order explicit space discontinuous Galerkin finite-element discretization of the one-dimensional or symmetric shallow-water equations. The algorithm uses fixed Eulerian flooded elements and a mixed Eulerian–Lagrangian element at each free boundary. When the time step is suitably restricted, we show that the mean water depth is positive. This time-step restriction is based on an analysis of the discretized continuity equation while using the HLLC flux. The algorithm and its implementation are tested in comparison with a large and relevant suite of known exact solutions. The essence of the flooding and drying algorithm pivots around the analysis of a continuity equation with a fluid velocity and a pseudodensity (in the shallow water case the depth). It therefore also applies, for example, to space discontinuous Galerkin finite-element discretizations of the compressible Euler equations in which vacuum regions emerge, in analogy of the above dry regions. We believe that the approach presented can be extended to finite-volume discretizations with similar mean level and slope reconstruction.
    Original languageUndefined
    Article number10.1007/s10915-004-4136-6
    Pages (from-to)47-82
    Number of pages36
    JournalJournal of scientific computing
    Volume22-23
    Issue numberplease ass/1-3
    DOIs
    Publication statusPublished - 2005

    Keywords

    • IR-61963
    • EWI-11248
    • Shallow-water equations - flooding and drying - free-boundary dynamics - discontinuous Galerkin finite-element method - positivity of mean water depth
    • METIS-248157

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