Optimal Quadrati Spline Collo ation Methods for the Shallow Water Equations on the Sphere

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. A quadratic spline collocation method approximates the solution of a differential problem by a quadratic spline. In the standard formulation, the quadratic spline is computed by making the residual of the differential equations zero at a set of collocation points; the resulting error is second order, while the error associated with quadratic spline interpolation is fourth order locally at certain points and third order globally. 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 semiimplicit scheme, which allows large timesteps while maintaining numerical stability, 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 Part of this work was done while the author was at the Department of Computer Science, University of Toronto. yPartly supported by Natural Sciences and Engineering Research Council (NSERC) of Canada.