Efficient space-time Jacobi rational spectral methods for second order time-dependent problems on unbounded domains

Abstract

We propose some efficient space-time Jacobi rational spectral methods for solving second order time-dependent problems on unbounded domains in this paper. Based on the Jacobi rational functions on the whole line, we first construct a set of Fourier-like basis functions for the spatial discretization which are simultaneously orthogonal in both L2- and H1-inner products. Meanwhile, composite (multi-domain) Legendre-Gauss collocation schemes with the knowns being time function values are developed for time integration. Owing to the simultaneously diagonal mass and stiffness matrices in space, the resulted linear system is eventually decoupled into a system of discrete ordinary differential equations stemmed from the multi-domain Legendre-Gauss collocation in time. Next, rigorous error estimates are carried out for the one-dimensional parabolic equations. Finally, some numerical results are presented to illustrate the spectral accuracy and the high efficiency of our space-time spectral methods, and to validate our main theory.