A Finite Difference Approach to Pseudospectral Methods

University Of Colorado At Boulder, Boulder CO

Investigators

Abstract

Pseudospectral (PS) methods are high-accuracy alternatives to finite difference (FD) and finite element methods (FEM) for the numerical solution of PDEs. They are particularly effective for solving convection-dominated equations over long times and in relatively simple geometries. Both the algorithms themselves and the analysis of them have traditionally been closely tied to expansions in different classes of orthogonal functions. The recent book "A Practical Guide to Pseudospectral Methods" (by the present investigator) notes that a large body of generalizations, enhancements and insights can be gained by viewing PS methods instead as special cases of FD methods. Several such opportunities were explored under the grant DMS-9706916. Building on these experiences, we will now introduce radial basis functions (RBFs) as an additional component to be integrated with PS schemes. Although computationally quite costly, the spectral accuracy of RBFs, combined with their extreme geometric flexibility, would seem to make them ideally suited to complement PS methods in the vicinity of irregular boundaries. To realize this requires much research regarding the basic features of RBF approximations, as well as research on issues related to interfacing them with PS and FD methods. Pseudospectral (PS) methods were first proposed in the early 1970's (in connection with meteorology and turbulence modeling). They have since been shown to be extraordinary effective for high-accuracy calculations in numerous other fields, such as computational electromagnetics and nonlinear waves. Strategically important application areas include simulating the radar scattering from airplanes, the electrical interference between components on computer chips, and the transmissions of signals in optical fibres. The main weakness with PS methods has been the difficulty to apply them in complex geometries. Some of this has been overcome in recent years (usually by splitting a complicated domain in simpler parts, and then applying some suitable computational means for coupling of the subdomains). A quite different and highly promising approach will be introduced and then explored under this grant. This involves combining 'classical' PS methods with the geometrically extremely flexible approach known as radial basis functions (RBFs). This has never been attempted, and the research falls in the 'high risk, high gain' category. Success is by no means certain, but if it happens, it could have a major impact, particularly on computational electromagnetics in applications such as simulation of radar scattering / stealth properties of objects.