Spectral collocation in space and time for linear PDEs

Abstract

Spectral methods solve elliptic partial differential equations (PDEs) numerically with errors bounded by an exponentially decaying function of the number of modes when the solution is analytic. For time dependent problems, almost all focus has been on low-order finite difference schemes for the time derivative and spectral schemes for spatial derivatives. Spectral methods that converge spectrally in both space and time have appeared recently. This paper is a continuation of the authors' previous works on Legendre and Chebyshev space-time methods for the heat equation. Here space-time spectral collocation methods for the Schrodinger, wave, Airy and beam equations are proposed and analyzed. In particular, a condition number estimate of each global Chebyshev space-time operator is shown. The analysis requires new estimates of eigenvalues of some spectral derivative matrices. In particular, it is shown that the real part of every eigenvalue of the third-order Chebyshev derivative matrix is positive and bounded away from zero, settling a twenty-year-old conjecture. Similarly, the real part of every eigenvalue of the fourth-order Chebyshev derivative matrix with Dirichlet boundary conditions is shown to be also positive and bounded away from zero. Numerical results verify the theoretical results, and demonstrate that the space-time methods also work well for some common nonlinear PDEs.