A Semi-Lagrangian Collocation Method for the Shallow Water Equations on the Sphere

Abstract

In this paper, we describe a numerical method for solving the shallow water equations (SWEs) in spherical coordinates. The most popular spatial discretization method used in global atmospheric models is currently the spectral transform method, which generates high-order numerical solutions and provides an elegant solution to the pole problems induced by a spherical coordinate system. However, the spectral transform method requires Legendre transforms, which have a computational complexity of $\mathcal O(N^3)$, where N is the number of subintervals in one spatial dimension. Thus, high-order finite element methods may be a viable alternative. In this implementation, the SWEs are discretized in time using the three-level semi-Lagrangian semi-implicit method and in space using the cubic spline collocation method. Numerical results for the standard SWEs test suite [D. L. Williamson et al., J. Comput. Phys., 102 (1992), pp. 211--224] are presented to demonstrate the stability and accuracy of the method. When compared to a previously applied Eulerian-based method, our method generates solutions with comparable accuracy while allowing larger timesteps and thus lower computational cost.