Commun. Comput. Phys., 10 (2011), pp. 371-404.


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

Andreas Bollermann 1*, Sebastian Noelle 1, Maria Lukacova-Medvid'ova 2

1 IGPM, RWTH Aachen, Templergraben 55, 52062 Aachen, Germany.
2 Institut fur Mathematik, Universitat Mainz, Staudingerweg 9, 55099 Mainz, Germany.

Received 22 February 2010; Accepted (in revised version) 2 July 2010
Available online 27 April 2011
doi:10.4208/cicp.220210.020710a

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. Well-balancing 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.

AMS subject classifications: 65M08, 76B15, 76M12, 35L50
PACS: 02.60.Cb, 47.11.Df, 92.10.Sx
Key words: Well-balanced schemes, dry boundaries, shallow water equations, evolution Galerkin schemes, source terms.

*Corresponding author.
Email: andreas.bollermann@rwth-aachen.de (A. Bollermann), noelle@igpm.rwth-aachen.de (S. Noelle), lukacova@uni-mainz.de (M. Lukacova)
 

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