Spectral Methods for Time-dependent Variable-coefficient PDE Based on Block Gaussian Quadrature

Block Krylov subspace spectral (KSS) methods are a “best-of-both-worlds” compromise between explicit and implicit time-stepping methods for variable-coefficient PDE, in that they combine the efficiency of explicit methods and the stability of implicit methods, while also achieving spectral accuracy in space and high-order accuracy in time. Block KSS methods compute each Fourier coefficient of the solution using techniques developed by Gene Golub and Gérard Meurant for approximating elements of functions of matrices by block Gaussian quadrature in the spectral, rather than physical, domain. This paper describes how block KSS methods can be applied to a variety of equations, and also demonstrates their superiority, in terms of accuracy and efficiency, to other Krylov subspace methods in the literature.