Numerical solution of the two-dimensional shallow water equations by the application of relaxation methods

A generalization and extension of a finite difference method for calculating numerical solutions of the two dimensional shallow water system of equations is investigated. A previously developed non-oscillatory relaxation scheme is generalized as to included problems with source terms in two dimensions, with emphasis given to the bed topography, resulting to a class of methods of first- and second-order in space and time. The methods are based on classical relaxation models combined with TVD Runge-Kutta time stepping mechanisms where neither Riemann solvers nor characteristic decompositions are needed. Numerical results are presented for several test problems with or without the source term present. The wetting and drying process is also considered. The presented schemes are verified by comparing the results with documented ones.