Spectral methods for multi-dimensional diffusion problems

The numerical calculation of diffusion processes in complex geometries in multi-dimensions is of current interest in many applications involving the solution of the coupled equations of radiation transport or thermal conduction, as well as in applications involving unstable hydrodynamic flows. The application of pseudo-spectral techniques to the implicit solution of the diffusion equation in a distorted two-dimensional grid for both linear and nonlinear problems is considered. The implicit pseudo-spectral equations on a m x n mesh result in a full P x P matrix (where P equals the square of the product of m and n) which can be inverted by the methods introduced. The examples given illustrate the success of pseudo-spectral techniques applied to the solution of these problems. The computational and memory requirements for machine implementation of pseudo-spectral techniques for a given accuracy are shown to be superior to those standard finite difference techniques.