Explicit high-order time stepping based on componentwise application of asymptotic block Lanczos iteration

This paper describes the development of explicit time-stepping methods for linear partial differential equations that are specifically designed to cope with the stiffness of the system of ordinary differential equations that results from spatial discretization. As stiffness is caused by the contrasting behavior of coupled components of the solution, it is proposed to adopt a componentwise approach in which each coefficient of the solution in an appropriate basis is computed using an individualized approximation of the solution operator. This has been accomplished by Krylov subspace spectral (KSS) methods, which use techniques from “matrices, moments and quadrature” to approximate bilinear forms involving functions of matrices via block Gaussian quadrature rules. These forms correspond to coefficients with respect to the chosen basis of the application of the solution operator of the PDE to the solution at an earlier time. In this paper it is proposed to substantially enhance the efficiency of KSS methods through the prescription of quadrature nodes based on asymptotic analysis of the recursion coefficients produced by block Lanczos iteration for each Fourier coefficient as a function of the frequency. The potential of this idea is illustrated through numerical results obtained from the application of the modified KSS methods to diffusion equations and wave equations. Copyright c © 2010 John Wiley & Sons, Ltd.