The goal of this paper is to construct a first-order upwind scheme
for solving the system of partial differential equations governing the
one-dimensional flow of two superposed immiscible layers of shallow water
This is done by generalizing a numerical scheme presented by
Bermúdez and Vázquez-Cendón [3, 6, 27] for solving one-layer shallow water equations, consisting
in a Q-scheme with a suitable treatment of the source terms.
The difficulty in the two layer system comes from the coupling terms
involving some derivatives of the unknowns.
Due to these terms, a numerical scheme obtained by performing the
upwinding of each layer, independently from the other one, can be
In order to define a suitable numerical scheme with global upwinding,
we first consider an abstract system that generalizes the problem under study.
This system is not a system of conservation laws but, nevertheless,
Roe's method can be applied to obtain an upwind scheme based on
Approximate Riemann State Solvers.
Following this, we present some numerical tests to validate the resulting
schemes and to highlight the fact that, in general, numerical schemes
obtained by applying a Q-scheme to each separate conservation law of the
system do not yield good results.
First, a simple system of coupled Burgers' equations is considered.
Then, the Q-scheme obtained is applied to the two-layer shallow water system.