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 N 2 degrees of freedom, the computational cost is ( N 2 log N ), dominated by the zonal Fourier transforms. This is more than with a finite‐difference or finite‐volume method, which costs ( N 2 ), and less than with a spherical harmonics method, which costs ( N 3 ). Differential operators with latitude‐dependent coefficients are inverted at a cost of ( N 2 ). 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. Copyright © 2009 Royal Meteorological Society