Efficient and High-Order Time Spectral Methods for Initial Value Problems

In this paper, we revisit the Legendre (dual) Petrov–Galerkin spectral method in time for initial value problems (IVPs) and establish the relationship between the numerator of the Padé approximation to ez and the determinant of A^(z)=I-zM^, where M^ is the mass matrix from the Legendre dual Petrov–Galerkin scheme, for instance. Based upon this equivalence, we construct an efficient implementation of the spectral method in time with the aid of the zeros of the generalized reverse Bessel polynomials. As a by-product, we prove the equivalence between the Legendre (dual) Petrov–Galerkin spectral method and the spectral tau method, such that the implementation of the algorithm proposed in [Z. Chen and Y. Liu. SIAM J. Sci. Comput., 46(3): A2073–A2100, 2024] becomes more effective. Numerical experiments illustrate the accuracy and effectiveness of the proposed algorithms.