A robust well-balanced scheme for multi-layer shallow water equations

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June 2010, 13(4): 739-758. doi: 10.3934/dcdsb.2010.13.739

A robust well-balanced scheme for multi-layer shallow water equations

François Bouchut ^1, and Vladimir Zeitlin ^2,

1.	DMA, CNRS & École Normale Supérieure, 45 Rue d’Ulm, F-75230 Paris cedex 05, France
2.	LMD, École Normale Supérieure, 24 Rue Lhomond, F-75231 Paris cedex 05, France

Received April 2009 Revised May 2009 Published March 2010

Abstract
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The numerical resolution of the multi-layer shallow water system encounters two additional difficulties with respect to the one-layer system. The first is that the system involves nonconservative terms, and the second is that it is not always hyperbolic. A splitting scheme has been proposed by Bouchut and Morales, that enables to ensure a discrete entropy inequality and the well-balanced property, without any theoretical difficulty related to the loss of hyperbolicity. However, this scheme has been shown to often give wrong solutions. We introduce here a variant of the splitting scheme, that has the overall property of being conservative in the total momentum. It is based on a source-centered hydrostatic scheme for the one-layer shallow water system, a variant of the hydrostatic scheme. The final method enables to treat an arbitrary number $m$ of layers, with arbitrary densities $\rho_1$,...,$\rho_m$, and arbitrary topography. It has no restriction concerning complex eigenvalues, it is well-balanced and it is able to treat vacuum, it satisfies a semi-discrete entropy inequality. The scheme is fast to execute, as is the one-layer hydrostatic method.

Keywords: nonhyperbolic system, hydrostatic reconstruction, discrete entropy inequality, splitting scheme, nonconservative system, well-balanced scheme, multi-layer shallow water.

Mathematics Subject Classification: Primary: 65M08; Secondary: 76B70, 76U0.

Citation: François Bouchut, Vladimir Zeitlin. A robust well-balanced scheme for multi-layer shallow water equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 739-758. doi: 10.3934/dcdsb.2010.13.739

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