High-order quasi-uniform approximation on the sphere using fourier-finite-elements

Solving transport equations on the whole sphere using an explicit time stepping and a Eulerian formulation on a latitude-longitude grid is relatively straightforward but suffers from the pole problem: due to the increased zonal resolution near the pole, numerical stability requires unacceptably small time steps. Commonly used workarounds such as near-pole zonal filters affect the qualitative properties of the numerical method. Rigorous solutions based on spherical harmonics have a high computational cost. The numerical method we propose to avoid this problem is based on a Galerkin formulation in a subspace of a Fourier-finite element spatial discretization, providing quasi-uniform resolution and high-order accuracy. For N ² degrees of freedom, the computational cost is O(N ²logN), intermediate between finite-difference or finite-volume methods and spherical harmonics methods. We present experimental results and standard benchmarks demonstrating the accuracy and stability of the method applied to the rotating shallow-water equations.