Improvement and generalization of a finite element shallow-water solver to multi-layer systems

This paper improves and generalizes to multi-layer systems the shallow-water solver presented in [Bermudez et al., IMA J. Numer. Anal., 11, 79-97 (1991)]. The model equations are discretized in time using the method of characteristics and the Euler implicit method. The space discretization is performed using the first-order Raviart-Thomas mixed finite element. A formulation of the non-linear equations to solve at each time step that takes into account regions without water is given, and numerical results are presented in which this situation takes place for the one-dimensional case. These non-linear problems are solved by a duality technique with an automatic choice of parameters that greatly improves the convergence of the algorithm. A preconditioner has been designed for solving the linear problems that appear at each iteration of the duality method, which significantly reduces the computational cost. This is illustrated with some numerical examples. Finally, an application of the multi-layer model to a realistic geometry of the Alboran Sea is presented, giving good results from a qualitative point of view. Copyright (C) 1999 John Wiley & Sons, Ltd.