Spectral Methods for Time-Dependent Variable-Coefficient PDE Based on Block Gaussian Quadrature

Abstract

Krylov subspace spectral (KSS) methods have previously been applied to the variable-coefficient heat equation and wave equation, as well as systems of coupled equations such as Maxwell’s equations, and have demonstrated high-order accuracy, as well as stability characteristic of implicit time-stepping schemes, even though KSS methods are explicit. KSS methods compute each Fourier coefficient of the solution using techniques developed by Gene Golub and Gerard Meurant for approximating elements of functions of matrices by Gaussian quadrature in the spectral, rather than physical, domain. In this paper, we review the most effective type of KSS method, that relies on block Gaussian quadrature, and compare its performance to that of Krylov subspace methods from the literature.