A fourth-order-accurate Fourier method for the Helmholtz equation in three dimensions

Abstract

We present fourth-order-accurate compact discretizations of the Helmholtz equation on rectangular domains in two and three dimensions with any combination of Dirichlet, Neumann, or periodic boundary conditions. The resulting systems of linear algebraic equations have the same block-tridiagonal structure as traditional central differences and hence may be solved efficiently using the Fourier method. The performance of the method for a variety of test cases, including problems with nonsmooth solutions, is presented. The method is seen to be roughly twice as fast as deferred corrections and, in many cases, results in a smaller discretization error.