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
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.
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