Spectral methods in space and time for parabolic problems on semi-infinite domains

In this paper, we introduce three series of Jacobi rational basis functions on the half line by using the matrix decomposition technique. The new basis functions are simultaneously orthogonal in both \(L^2\)- and \(H^1\)-inner products, and lead to diagonal systems for second order problems with constant coefficients. We construct efficient space-time spectral methods for parabolic problems on semi-infinite domains using Jacobi rational approximation in space and multi-domain Legendre–Gauss collocation approximation in time, which can be implemented in a synchronous parallel fashion. Some rigorous error estimates are carried out for one-dimensional parabolic equations. Numerical results demonstrate that the suggested approaches possess high-order accuracy and greatly improve the efficiency.