Parallel geometric multigrid for global weather prediction

The subject of this work is an optimal and scalable parallel geometric multigrid solver for elliptic problems on the sphere, crucial to the forecasting and the data assimilation tools used at the U.K. Met office. The optimality of multilevel techniques for elliptic problems makes them a suitable choice for these applications. The Met office uses spherical polar grids which, although structured, have the drawback of creating strong anisotropies near the poles. Moreover, a higher resolution in the radial direction introduces further anisotropies, and so modifications to the standard multigrid relaxation and the coarsening procedures are necessary to retain optimal efficiency. As the strength of anisotropy varies, we propose a non-uniform strategy, coarsening the grid only in regions that are sufficiently isotropic. This is combined with line relaxation in the radial direction. The success of non-uniform coarsening strategies has been demonstrated with algebraic multigrid (AMG) methods. Without the large setup costs required by AMG, however, we aim to surpass them with the geometric approach. We demonstrate the advantages of the method with experiments on model problems, both sequentially and in parallel, and show robustness and optimal efficiency of the method with constant convergence factors of less than 0.1. It substantially outperforms Krylov subspace methods with one-level preconditioners and the BoomerAMG implementation of AMG on typical grid resolutions. The parallel implementation scales almost optimally on up to 256 processors, so that a global solve of the quasi-geostrophic omega-equation with a maximum horizontal resolution of about 10 km and 3 x 109 unknowns takes about 60s.