Comparative Test of Direct and Iterative Methods for Solving Helmholtz-Type Equations

Abstract

The Helmholtz-type equation arises in many areas of fluid dynamics, and, in recent years, there has been a rapid increase in the numerical procedures available for solving the equation. In this note, the various methods currently available are discussed, and representatives from the main categories are compared. We suggest that for certain problems, the most important of which is Poisson's equation on a rectangle, direct methods are now available that are far superior to widely used iterative methods. For problems involving irregular domains, mixed boundary conditions, and variable Helmholtz coefficients, however, existing direct methods often cannot be used with the same flexibility as iterative methods; there is a continuing need to extend direct methods to these more general cases.