Efficient Space-Time Spectral Methods for Second-Order Problems on Unbounded Domains

In this paper, we propose efficient space-time spectral methods for problems on unbounded domains. For this purpose, we first introduce two series of new basis functions on the half/whole line by matrix decomposition techniques. The new basis functions are mutually orthogonal in both and inner products, and lead to diagonal systems for second order problems with constant coefficients. Then we construct efficient space-time spectral methods based on Laguerre/Hermite-Galerkin methods in space and dual-Petrov-Galerkin formulations in time for problems defined on unbounded domains. Using these suggested methods, higher accuracy can be obtained. We also demonstrate that the use of simultaneously orthogonal basis functions in space may greatly simplify the implementation of the space-time spectral methods.