Convergence Analysis of a Krylov Subspace Spectral Method for the 1-D Wave Equation in an Inhomogeneous Medium

TL;DR

This paper analyzes the convergence of a Krylov subspace spectral method for the 1-D wave equation in inhomogeneous media, demonstrating stability and spectral accuracy under certain conditions, supported by numerical experiments.

Contribution

It provides the first stability analysis of a KSS method without assuming a bandlimited reaction term, and extends the analysis to inhomogeneous media.

Findings

Unconditional stability for sufficiently regular initial data

Spectral accuracy in space and second-order in time

Effective performance in higher dimensions and nonlinear PDEs

Abstract

This paper presents a convergence analysis of a Krylov subspace spectral (KSS) method applied to a 1-D wave equation in an inhomogeneous medium. It will be shown that for sufficiently regular initial data, this KSS method yields unconditional stability, spectral accuracy in space, and second-order accuracy in time, in the case of constant wave speed and a bandlimited reaction term coefficient. Numerical experiments that corroborate the established theory are included, along with an investigation of generalizations, such as to higher space dimensions and nonlinear PDEs, that features performance comparisons with other Krylov subspace-based time-stepping methods. This paper also includes the first stability analysis of a KSS method that does not assume a bandlimited reaction term coefficient.