Solution of time-dependent PDE through rapid estimation of block Gaussian quadrature nodes

Abstract

The stiffness of systems of ODEs that arise from spatial discretization of PDEs causes difficulties for both explicit and implicit time-stepping methods. Krylov Subspace Spectral (KSS) methods present a balance between the efficiency of explicit methods and the stability of implicit methods by computing each Fourier coefficient from an individualized approximation of the solution operator of the PDE. While KSS methods are explicit methods that exhibit a high order of accuracy and stability similar to that of implicit methods, their efficiency needs to be improved. A previous asymptotic study of block Lanczos iteration yielded estimates of extremal block Gaussian quadrature nodes for each Fourier component and led to an improvement in efficiency. In this paper, a more detailed asymptotic study is performed in order to rapidly estimate all nodes, thus drastically reducing computational expense without sacrificing accuracy. Numerical results demonstrate that the new node estimation scheme does in fact accomplish these aims.