Recent advances in Krylov subspace spectral methods

Abstract

AbstractThis paper reviews the main properties, and most recent developments, of Krylov subspace spectral (KSS) methods for time‐dependent variable‐coefficient PDE. These methods use techniques developed by Golub and Meurant for approximating elements of functions of matrices by Gaussian quadrature in the spectral domain in order to achieve high‐order accuracy in time and stability characteristic of implicit time‐stepping schemes, even though KSS methods themselves are explicit. In fact, for certain problems, 1‐node KSS methods are unconditionally stable. Furthermore, these methods are equivalent to high‐order operator splittings, thus offering another perspective for further analysis and enhancement. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)