Effectiveness of the operator splitting for solving the atmospherical shallow water equations

A semi-implicit semi-Lagrangian mixed finite-difference finite-volume model for the shallow water equations on a rotating sphere is considered. The main features of the model are the finite-volume approach for the continuity equation and the vectorial treatment of the momentum equation. Pressure and Coriolis terms in the momentum equation and velocity in the continuity equation are treated semi-implicitly. Discretization of this model led us to introduce, in a previous paper, a splitting technique which highly reduces the computational effort for the numerical solution. In this paper we solve the full set of equations, without splitting, introducing an 'ad hoc' algorithm. A von Neumann stability analysis of this scheme is performed to establish the unconditional stability of the new proposed method. Finally, we compare the efficiency of the two approaches by numerical experiments on a standard test problem. Our results show that, due to the devised algorithm, the solution of the full system of equations is much more accurate while slightly increases the computational cost.