A multigrid solver for semi‐implicit global shallow‐water models

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Abstract
The multigrid principle produces fast solvers for systems of algebraic equations, particularly those that arise from discretizing elliptic boundary‐value problems. A multigrid solver is developed for the discretized two‐dimensional elliptic equation on the sphere that arises from a semi‐implicit time discretization of the global shallow‐water equations. We experiment with different formulations of the semi‐implicit scheme that result in variable‐coefficient Helmholtz‐type equations for which no fast direct solvers are available. The efficiency of the multigrid solver is optimal, in the sense that the total operation count is proportional to the number of unknowns. Numerical experiments using initial data derived from actual 300‐mb height and wind velocity fields indicate that our semi‐implicit global shallow‐water model has very good accuracy and stability properties.