Efficient space-time Legendre rational spectral method for parabolic problems in unbounded domains

Abstract

In this paper, we first introduce three series of Legendre rational basis functions on the half/whole line by the diagonalization technique and the matrix decomposition technique. The new basis functions are mutually orthogonal in both L2- and H1-inner products, and lead to diagonal systems for second order problems with constant coefficients. Then we construct efficient space-time spectral methods for parabolic problems in unbounded domains using Legendre rational approximation in space and Legendre-Gauss collocation method in time, which can be implemented in a synchronous parallel fashion. Numerical results demonstrate that the use of simultaneously orthogonal basis functions in space may greatly simplify the implementation of the space-time spectral methods. Using these suggested methods, higher accuracy can also be obtained.