A conservative Fourier-finite-element method for solving partial differential equations on the whole sphere

Solving transport equations on the whole sphere using an explicit time stepping and an 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. The functional space we construct provides quasi-uniform resolution and high-order accuracy, while the Galerkin formalism guarantees the conservation of linear and quadratic invariants. For N² degrees of freedom, the computational cost is O(N² log N), dominated by the zonal Fourier transforms. This is more than with a finite-difference or finite-volume method, which costs O(N²), and less than with a spherical harmonics method, which costs O(N³). Differential operators with latitude-dependent coefficients are inverted at a cost of O(N²). We present experimental results and standard benchmarks demonstrating the accuracy, stability and efficiency of the method applied to the advection of a scalar field by a prescribed velocity field and to the incompressible rotating Navier-Stokes equations. The steps required to extend the method towards compressible flows and the Saint-Venant equations are described.