Semi-implicit finite difference methods for the two-dimensional shallow water equations

Abstract

In this paper a semi-implicit finite difference method for the 2-dimensional shallow water equations is derived and discussed. A characteristic analysis of the governing equations is carried out first, in order to determine those terms to be discretized implicitly so that the stability of the method will not depend upon the celerity. Such terms are the gradient of the water surface elevation in the momentum equations and the velocity divergence in the continuity equation. The convective terms are discretized explicitly. The simpler explicit discretization for the convective terms is the upwind discretization which is conditionally stable and introduces some artificial viscosity. It is shown that the stability restriction is eliminated and the artificial viscosity is reduced when an Eulerian-Lagrangian approach with large time steps is used to discretize the convective terms. This method, at each time step, requires the solution of a linear, symmetric, 5-diagonal system. Such a system is diagonally dominant with positive elements on the main diagonal and negative ones elsewhere. Thus, existence and uniqueness of the numerical solution is assured. The resulting algorithm is mass conservative and fully vectorizable for an efficient implementation on modern vector computers. The performance of this method is further improved when used in combination with an ADI technique which results in two sets of simpler, linear 3-diagonal systems and maintains all the properties described above.