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Finite Volume Evolution Galerkin Methods for the Shallow Water Equations with Dry Beds

Published online by Cambridge University Press:  20 August 2015

Andreas Bollermann ,
Sebastian Noelle  and
Maria Lukáčová-Medvid’ová
Andreas Bollermann*
Affiliation:
IGPM, RWTH Aachen, Templergraben 55, 52062 Aachen, Germany
Sebastian Noelle*
Affiliation:
IGPM, RWTH Aachen, Templergraben 55, 52062 Aachen, Germany
Maria Lukáčová-Medvid’ová*
Affiliation:
Institut fir Mathematik, Universität Mainz, Staudingerweg 9, 55099 Mainz, Germany
*
Corresponding author.Email:andreas.bollermann@rwth-aachen.de
Email:noelle@igpm.rwth-aachen.de
Email:lukacova@uni-mainz.de
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Abstract

We present a new Finite Volume Evolution Galerkin (FVEG) scheme for the solution of the shallow water equations (SWE) with the bottom topography as a source term. Our new scheme will be based on the FVEG methods presented in (Noelle and Kraft, J. Comp. Phys., 221 (2007)), but adds the possibility to handle dry boundaries. The most important aspect is to preserve the positivity of the water height. We present a general approach to ensure this for arbitrary finite volume schemes. The main idea is to limit the outgoing fluxes of a cell whenever they would create negative water height. Physically, this corresponds to the absence of fluxes in the presence of vacuum. Wellbalancing is then re-established by splitting gravitational and gravity driven parts of the flux. Moreover, a new entropy fix is introduced that improves the reproduction of sonic rarefaction waves.

Keywords

65M0876B1576M1235L5002.60.Cb47.11.Df92.10.SxWell-balanced schemesdry boundariesshallow water equationsevolution Galerkin schemessource terms
Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 2 , August 2011 , pp. 371 - 404
Copyright
Copyright © Global Science Press Limited 2011

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