A center compact scheme for the Shallow Water equations on the sphere

Abstract

We consider the Shallow Water equations (SW) on a rotating sphere and their approximation by a finite difference scheme. The discrete unknowns are located at the vertices of the equiangular Cubed Sphere grid, (Croisille, 2013; 2015). The standard fourth order Hermitian difference derivative Lele (1991) is used along a set of suitable great circles. No one sided difference formula is used at any point. All differential operators on the sphere (gradient, divergence and curl) are approximated in a centered fashion. The approximation procedure is close in spirit to the one of compact schemes used in Computational Aeroacoustics. Numerical results on a series of numerical test cases for SW on the sphere are presented. A particular attention is devoted to the temporal scheme. In particular two Rosenbrock exponential time schemes are used and compared to the RK4 scheme.The results demonstrate the interest of the present approach in a variety of situations of interest in numerical climatology.