An Explicit, Stable, High-Order Spectral Method for the Wave Equation Based on Block Gaussian Quadrature

This paper presents a modification of Krylov Subspace Spectral (KSS) Methods, which build on the work of Golub, Meurant and others pertaining to moments and Gaussian quadrature to produce high-order accurate approximate solutions to the variable-coefficient second-order wave equation. Whereas KSS methods currently use Lanczos iteration to compute the needed quadrature rules, the modification uses block Lanczos iteration in order to avoid the need to compute two quadrature rules for each component of the solution, or use perturbations of quadrature rules that tend to be sensitive in problems with oscillatory coefficients or data. It will be shown that under reasonable assumptions on the coefficients of the problem, a 1-node KSS method is second-order accurate and unconditionally stable, and methods with more than one node are shown to possess favorable stability properties as well, in addition to very high-order temporal accuracy. Numerical results demonstrate that block KSS methods are significantly more accurate than their non-block counterparts, especially for problems that feature oscillatory coefficients.