Volume 28, pp. 114-135, 2007-2008.

Derivation of high-order spectral methods for time-dependent PDE using modified moments

James V. Lambers


This paper presents 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. This reformulation serves two useful purposes. First, it more clearly illustrates the distinction between KSS methods and existing Krylov subspace methods for solving stiff systems of ODE arising from discretizions of PDE. KSS methods rely on perturbations of Krylov subspaces in the direction of the data, and therefore rely on directional derivatives of nodes and weights of Gaussian quadrature rules. Second, because these directional derivatives allow KSS methods to be described in terms of operator splittings, they facilitate stability analysis. It will be shown that under reasonable assumptions on the coefficients of the problem, certain KSS methods are unconditionally stable. This paper also discusses preconditioning similarity transformations that allow more general problems to benefit from this property.

Full Text (PDF) [252 KB], BibTeX

Key words

spectral methods, Gaussian quadrature, variable-coefficient, Lanczos method, stability, heat equation, wave equation

AMS subject classifications

65M12, 65M70, 65D32

Links to the cited ETNA articles

[19]Vol. 20 (2005), pp. 212-234 James V. Lambers: Krylov subspace spectral methods for variable-coefficient initial-boundary value problems

ETNA articles which cite this article

Vol. 31 (2008), pp. 86-109 James V. Lambers: Enhancement of Krylov subspace spectral methods by block Lanczos iteration
Vol. 60 (2024), pp. 136-168 Bailey Rester, Anzhelika Vasilyeva, and James V. Lambers: Convergence analysis of a Krylov subspace spectral method for the 1D wave equation in an inhomogeneous medium

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