A shock-capturing finite volume scheme to solve 1D multi-layer shallow water equations

The multi-layer shallow water approach can be regarded as a development of De Saint Venant equations in the direction of a more accurate description of the physical problem, keeping as far as possible the efficiency of classical De Saint Venant numerical models. From this point of view, in the present paper, the one dimensional multi-layer De Saint Venant equations are briefly developed, marking the fact that the stresses due to the presence of neighboring layers can be treated as the effect of a virtual topography. In this way, continuity and momentum equation on each layer furnish a system of equations that is very similar to classic single-layer De Saint Venant equations. This similitude suggests the possibility to solve the resulting differential equations by means of the techniques originally developed for the solution of De Saint Venant equations. Following this idea, the 1D multi-layer De Saint Venant equations are solved numerically by means of a shock-capturing finite volume technique applied to each layer separately. The resulting numerical scheme is applied to some benchmark test, and the results are presented and discussed.