Approximate factorization in shallow water applications

Abstract

We consider the numerical integration of problems modelling phenomena in shallow water in 3 spatial dimensions. If the governing partial differential equations for such problems are spatially discretized, then the right-hand side of the resulting system of ordinary differential equations can be split into terms f 1, f 2, f 3 and f 4, respectively representing the spatial derivative terms with respect to the x, y and z directions, and the interaction terms. It is typical for shallow water applications that the interaction term f 4 is nonstiff and that the function f 3 corresponding with the vertical spatial direction is much more stiff than the functions f 1 and f 2 corresponding with the horizontal spatial directions. The reason is that in shallow seas the gridsize in the vertical direction is several orders of magnitude smaller than in the horizontal directions. In order to solve the initial value problem (IVP) for these systems numerically, we need a stiff IVP solver, which is necessarily implicit, requiring the iterative solution of large systems of implicit relations. The aim of this paper is the design of an efficient iteration process based on approximate factorization. Stability properties of the resulting integration method are compared with those of a number of integration methods from the literature. Finally, a performance test on a shallow water transport problem is reported.