We consider the system of partial differential equations governing
the one-dimensional flow of two superposed immiscible layers of
shallow water. The difficulty in this system comes
from the coupling terms involving some derivatives of the unknowns
that make the system nonconservative, and eventually nonhyperbolic.
Due to these terms, a numerical scheme obtained by performing an
arbitrary scheme to each layer, and using time-splitting or
other similar techniques leads to instabilities in general.
Here we use entropy inequalities in order to control
the stability. We introduce a stable well-balanced time-splitting scheme
for the two-layer shallow water system that satisfies a fully discrete
entropy inequality. In contrast with Roe type solvers,
it does not need the computation of eigenvalues, which is
not simple for the two-layer shallow water system.
The solver has the property to keep the water heights nonnegative,
and to be able to treat vanishing values.
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