Optimized point shifts and poles in the linear rational pseudospectral method for boundary value problems

Abstract

Due to their rapid – often exponential – convergence as the number N of interpolation/collocation points is increased, polynomial pseudospectral methods are very efficient in solving smooth boundary value problems. However, when the solution displays boundary layers and/or interior fronts, this fast convergence will merely occur with very large N. To address this difficulty, we present a method which replaces the polynomial ansatz with a rational function r and considers the physical domain as the conformal map g of a computational domain. g shifts the interpolation points from their classical position in the computational domain to a problem-dependent position in the physical domain. Starting from a map by Bayliss and Turkel we have constructed a shift that can in principle accomodate an arbitrary number of fronts. Its parameters as well as the poles of r are optimized. Numerical results demonstrate how g best accomodates interior fronts while the poles also handle boundary layers.