The method of particular solutions for the Helmholtz equation

Abstract

We have previously showed that the advection-diffusion equation in steady hill-slope seepage problems can be reduced to the solution of the Helmholtz equation in two dimensions. Initially, solutions were found using an analytic series method (or the method of particular solutions). However, the accuracy of these solutions is limited by ill-conditioning in the set of basis functions as the number of basis functions increases. Here we show that these problems are overcome by choosing a different set of basis functions and modifying the method of particular solutions as suggested by Betcke and Trefethen. The different methods are tested on a number of simple domains. In most cases spectral convergence is obtained for the eigenvalue.