Numerical solution of the Helmholtz equation in 2D and 3D using a high-order Nystrom discretization

We show how to solve time-harmonic scattering problems by means of a high-order Nystrom discretization of the boundary integral equations of wave scattering in 2D and 3D. The novel aspect of our new method is its use of local corrections to the discretized kernel in the vicinity of the kernel singularity. Enhanced by local corrections, the new algorithm has the simplicity and speed advantages of the traditional Nystrom method, but also enjoys the advantages of high-order convergence for controlling solution error. We explain the practical details of implementing a scattering code based on a high-order Nystrom discretization and demonstrate by numerical example that a scattering code based on this algorithm can achieve high-order convergence to the correct answer. We also demonstrate its performance advantages over a high-order Galerkin code. (C) 1998 Academic Press