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

François Bouchut (1) , Vladimir Zeitlin (2)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 LMD - Laboratoire de Météorologie Dynamique (UMR 8539)

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 of layers, with arbitrary densities 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.

Mots clés: Multi-layer shallow water; Nonconservative system; Non-hyperbolic system; Well-balanced scheme; Splitting scheme; Hydrostatic reconstruction; Discrete entropy inequality

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

François Bouchut, Vladimir Zeitlin. A robust well-balanced scheme for multi-layer shallow water equations. Discrete and Continuous Dynamical Systems - Series B, 2010, 13 (4), pp.739-758. ⟨10.3934/dcdsb.2010.13.739⟩. ⟨hal-00538338⟩