A nonlinear multigrid solver for a semi-Lagrangian potential vorticity-based shallow-water model on the sphere

In an earlier paper, we developed a new formulation of the shallow-water eq uations on the sphere, based on the semi-Lagrangian advection of potential vorticity. An important feature of our two-time-level model is that forward extrapolation of the nonlinear terms is avoided. In this paper, we develop a multigrid algorithm for solving the resulting system of three coupled no nlinear equations. We include a discussion of basic multigrid methods, alon g with some more advanced techniques required for the development of the mu ltigrid scheme for this system. Results are presented showing that usual op timal multigrid efficiency is obtained.