Solution of Time-Dependent PDE Through Component-wise Approximation of Matrix Functions

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 demonstrates the superiority of block KSS methods, in terms of accuracy and efficiency, to other Krylov subspace methods in the literature. It is also described how the ideas behind block KSS methods can be applied to a variety of equations, including problems for which Fourier spectral methods are not normally feasible. In particular, the versatility of the approach behind block KSS methods is demonstrated through application to nonlinear diffusion equations for signal and image processing, and adaptation to finite element discretization.