Cubic Spline Collocation Method for the Shallow Water Equations on the Sphere

Abstract

Spatial discretization schemes commonly used in global meteorological applications are currently limited to spectral methods or low-order finite-difference/finite-element methods. The spectral transform method, which yields high-order approximations, requires Legendre transforms, which have a computational complexity of O(N3), where N is the number of subintervals in one dimension. Thus, high-order finite-element methods may be a viable alternative to spectral methods. In this study, we present a new numerical method for solving the shallow water equations (SWE) in spherical coordinates. In this implementation, the SWE are discretized in time with the semi-implicit leapfrog method, and in space with the cubic spline collocation method on a skipped latitude–longitude grid. Numerical results for the Williamson et al. SWE test cases [D. L. Williamson, J. B. Blake, J. J. Hack, R. Jakob, and P. N. Swarztrauber, J. Comput. Phys.102, 211 (1992)] are presented to demonstrate the stability and accuracy of the method. Results are also shown for an efficiency comparison between this method and a similar method in which spatial discretization is done on a uniform latitude–longitude grid.