High-order compact finite difference schemes for the vorticity-divergence representation of the spherical shallow water equations

This work is devoted to the application of the super compact finite difference method (SCFDM) and the combined compact finite difference method (CCFDM) for spatial differencing of the spherical shallow water equations in terms of vorticity, divergence, and height. The fourth-order compact, the sixth-order and eighth-order SCFDM, and the sixth-order and eighth-order CCFDM schemes are used for the spatial differencing. To advance the solution in time, a semi-implicit Runge-Kutta method is used. In addition, to control the nonlinear instability, an eighth-order compact spatial filter is employed. For the numerical solution of the elliptic equations in the problem, a direct hybrid method, which consists of a high-order compact scheme for spatial differencing in the latitude coordinate and a fast Fourier transform in longitude coordinate, is utilized. The accuracy and convergence rate for all methods are verified against exact analytical solutions. Qualitative and quantitative assessments of the results for an unstable barotropic mid-latitude zonal jet employed as an initial condition are addressed. It is revealed that the sixth-order and eighth-order CCFDMs and SCFDMs lead to a remarkable improvement of the solution over the fourth-order compact method.