Spectral methods using rational basis functions on an infinite interval

Abstract

By using the map y = L cot( t) where L is a constant, differential equations on the interval y ϵ [− ∞, ∞] can be transformed into t ϵ [0, π] and solved by an ordinary Fourier series. In this article, earlier work by Grosch and Orszag ( J. Comput. Phys. 25, 273 (1977)), Cain, Ferziger, and Reynolds ( J. Comput. Phys. 56, 272 (1984)), and Boyd ( J. Comput. Phys. 25, 43 (1982); 57, 454 (1985); SIAM J. Numer. Anal. (1987)) is extended in several ways. First, the series of orthogonal rational functions converge on the exterior of bipolar coordinate surfaces in the complex y-plane. Second, Galerkin's method will convert differential equations with polynomial or rational coefficients into banded matrix problems. Third, with orthogonal rational functions it is possible to obtain exponential convergence even for u( y) that asymptote to a constant although this behavior would wreck alternatives such as Hermite or sinc expansions. Fourth, boundary conditions are usually “natural” rather than “essential” in the sense that the singularities of the differential equation will force the numerical solution to have the correct behavior at infinity even if no constraints are imposed on the basis functions. Fifth, mapping a finite interval to an infinite one and then applying the rational Chebyshev functions gives an exponentially convergent method for functions with bounded endpoint singularities. These concepts are illustrated by five numerical examples.