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Stability of a Cartesian grid projection method for zero Froude number shallow water flows

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Abstract

In this paper a Godunov-type projection method for computing approximate solutions of the zero Froude number (incompressible) shallow water equations is presented. It is second-order accurate and locally conserves height (mass) and momentum. To enforce the underlying divergence constraint on the velocity field, the predicted numerical fluxes, computed with a standard second order method for hyperbolic conservation laws and applied to an auxiliary system, are corrected in two steps. First, a MAC-type projection adjusts the advective velocity divergence. In a second projection step, additional momentum flux corrections are computed to obtain new time level cell-centered velocities, which satisfy another discrete version of the divergence constraint. The scheme features an exact and stable second projection. It is obtained by a Petrov–Galerkin finite element ansatz with piecewise bilinear trial functions for the unknown height and piecewise constant test functions. The key innovation compared to existing finite volume projection methods is a correction of the in-cell slopes of the momentum by the second projection. The stability of the projection is proved using a generalized theory for mixed finite elements. In order to do so, the validity of three different inf-sup conditions has to be shown. The results of preliminary numerical test cases demonstrate the method’s applicability. On fixed grids the accuracy is improved by a factor four compared to a previous version of the scheme.

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  1. Numerische Mathematik/Scientific Computing, FB Mathematik und Informatik, Institut  für Mathematik, Freie Universität (FU) Berlin, Arnimallee 6, 14195, Berlin, Germany

    Stefan Vater & Rupert Klein

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  1. Stefan Vater
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  2. Rupert Klein
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Correspondence to Stefan Vater.

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Vater, S., Klein, R. Stability of a Cartesian grid projection method for zero Froude number shallow water flows. Numer. Math. 113, 123–161 (2009). https://doi.org/10.1007/s00211-009-0224-8

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  • DOI: https://doi.org/10.1007/s00211-009-0224-8

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