Semi‐implicit finite difference methods for three‐dimensional shallow water flow

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Abstract
A semi‐implicit finite difference method for the numerical solution of three‐dimensional shallow water flows is presented and discussed. The governing equations are the primitive three‐dimensional turbulent mean flow equations where the pressure distribution in the vertical has been assumed to be hydrostatic. In the method of solution a minimal degree of implicitness has been adopted in such a fashion that the resulting algorithm is stable and gives a maximal computational efficiency at a minimal computational cost. At each time step the numerical method requires the solution of one large linear system which can be formally decomposed into a set of small three‐diagonal systems coupled with one five‐diagonal system. All these linear systems are symmetric and positive definite. Thus the existence and uniquencess of the numerical solution are assured. When only one vertical layer is specified, this method reduces as a special case to a semi‐implicit scheme for solving the corresponding two‐dimensional shallow water equations. The resulting two‐ and three‐dimensional algorithm has been shown to be fast, accurate and mass‐conservative and can also be applied to simulate flooding and drying of tidal mud‐flats in conjunction with three‐dimensional flows. Furthermore, the resulting algorithm is fully vectorizable for an efficient implementation on modern vector computers.