IJPAM: Volume 42, No. 4 (2008)


James V. Lambers
Department of Energy Resources Engineering
Stanford University
Stanford, CA 94305-2220, USA

Abstract.This paper describes a reformulation of Krylov subspace spectral (KSS) methods, which build on Gene Golub's many contributions pertaining to moments and Gaussian quadrature to produce high-order accurate approximate solutions to variable-coefficient time-dependent PDE. Because KSS methods rely on perturbations of Krylov subspaces in the direction of the data, they can be reformulated in terms of derivatives of nodes and weights of Gaussian quadrature rules, which can be computed analytically. Because these derivatives allow KSS methods to be described in terms of operator splittings, they facilitate stability analysis. Under reasonable assumptions on the coefficients of the problem, certain KSS methods are unconditionally stable.

Received: August 17, 2007

AMS Subject Classification: 65M12, 65M70, 65D32

Key Words and Phrases: spectral methods, Gaussian quadrature, variable-coefficient, Lanczos method, stability, heat equation, wave equation

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2008
Volume: 42
Issue: 4