Comparison of several methods for the solution of the Helmholtz equation

Summary A series of Helmholtz equations has to be solved in short-range weather forecast models which use a splitting scheme of integration. For these purposes the successive overrelaxation, the Gauss-Seidel relaxation, the conjugate gradient method, the steepest descent method, the full-multigrid method and the direct method based on the minimum degree algorithm were used and their efficiencies were compared. It was found that the full-multigrid method was the most efficient among the iterative methods in terms of computational time, and that the effect rapidly increased with the grid size. The direct method may be an appropriate approach if the solution is repeated for various right-hand sides, but it requires large auxiliary computer memory. The selection of the optimum method depends on the concrete problem being solved and on the computer memory available.