A Multigrid Solver for the Helmholtz Equation on a Semiregular Grid on the Sphere

Abstract

A multigrid finite-difference solver is developed for the Helmholtz equation on the sphere. The finite-difference grid resolution is constant in the latitudinal direction and variable in the longitudinal direction so as to keep the physical gridpoint spacing approximately uniform over the sphere. The cpu time per grid point required to reduce the residual by a given amount is independent of grid resolution. The discretization error is slightly worse than second order as a result of the variable grid spacing. The method should be applicable to general elliptic equations on the sphere and should be useful for problems where uniform grid spacing is disadvantageous.