Quadratic spline methods for the shallow water equations on the sphere: Collocation

In this study, we present numerical methods, based on the optimal quadratic spline collocation (OQSC) methods, for solving the shallow water equations (SWEs) in spherical coordinates. The error associated with quadratic spline interpolation is fourth order locally at certain points and third order globally, but the standard quadratic spline collocation methods generate only second-order approximations. In contrast, the OQSC methods generate approximations of the same order as quadratic spline interpolation. In the one-step OQSC method, the discrete differential operators are perturbed to eliminate low-order error terms, and a high-order approximation is computed using the perturbed operators. In the two-step OQSC method, a second-order approximation is generated first, using the standard formulation, and then a high-order approximation is computed in a second phase by perturbing the right sides of the equations appropriately. In this implementation, the SWEs are discretized in time using the semi-Lagrangian semi-implicit method, and in space using the OQSC methods. The resulting methods are efficient and yield stable and accurate representation of the meteorologically important Rossby waves. Moreover, by adopting the Arakawa C-type grid, the methods also faithfully capture the group velocity of inertia-gravity waves.