The numerical treatment of wet/dry fronts in shallow flows:: Application to one-layer and two-layer systems

This paper deals with the numerical difficulties related to the appearance of the so-called wet/dry fronts that may occur during the simulation of free-surface waves in shallow fluids and internal waves in stratified fluids composed by two shallow layers of immiscible liquids. In the one-layer case, the fluid is supposed to be governed by the shallow water equations. In the case of two layers, the system to solve is formulated under the form of two-coupled system of shallow water equations. In both cases, the discretization of the equations are performed by means of the Q-schemes of Roe upwinding the source term developed in [1,2] for the one-layer system, and [3] for the two-layer system. These schemes satisfy the so-called C-property: they solve exactly steady solutions corresponding to water at rest. In order to handle properly with wet/dry fronts, a numerical scheme has to verify also an extended C-property: it has to solve exactly steady solutions corresponding to water at rest including wet/dry transitions. In [4-6], some numerical schemes satisfying this property has been obtained. In this paper, we propose an improvement of these schemes and its extension to two-layer systems. We present some numerical results: for one-layer fluids, we compare the numerical results with some measurements corresponding to a physical model. For two-layer systems, we use the numerical scheme to perform a lock-exchange experiment, and we verify its validity by comparing the steady state reached with an approximate solution obtained by Armi and Farmer in [7,8] by using a simplified model. (c) 2005 Elsevier Ltd. All rights reserved.