Efficient Space-time Rational Spectral Method On The Whole Line

Space-time spectral/finite element method has important research value in theory and application.In this paper,an efficient space-time Jacobi rational spectral method for solving timedependent second-order partial differential equations in unbounded domains is proposed.Firstly,we introduce the classical Jacobi polynomials and their recursive relations.We study the properties of Jacobi rational functions on the whole line obtained by rational transformation.We also calculate the explicit expressions of elements for the stiffness matrix and mass matrix.Since the stiffness matrix is symmetric and positive definite and the mass matrix is an identity matrix,they have exactly the same real eigenvectors and can be diagonalized at the same time.On this basis,a set of Fourier-like basis functions for spatial discretization are constructed.They are orthogonal at the same time under-and-inner product.The corresponding stiffness matrix and mass matrix are diagonal.Therefore,the semi-discrete scheme after spatial discretization is decoupled into mutually independent ordinary differential equations.In terms of time integration,we have developed a composite(multi-interval)Legendre Gauss collocation scheme and a composite Hermite-Legendre-Gauss collocation scheme for the first and second-order time derivatives,so as to achieve the overall high accuracy of space-time matching.The fully discrete scheme after time discretization can be calculated in parallel.In addition,taking a one-dimensional parabolic equation as an example,we strictly estimate the error of its space-time spectral approximation scheme.Finally,in order to verify our main conclusions,some numerical results are shown,which validate the spectral accuracy and efficiency of our method.