# American Institute of Mathematical Sciences

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

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

 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

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.
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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