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In this paper, we show that the eigenvalues and eigenvectors of the spectraldiscretisation matrices resulted from the Legendre dual-Petrov-Galerkin (LDPG)method for the $m$th-order initial value problem (IVP): $u^{(m)}(t)=\sigmau(t),\, t\in (-1,1)$ with constant $\sigma\not=0$ and usual initial conditionsat $t=-1,$ are associated with the generalised Bessel polynomials (GBPs). Theessential idea of the analysis is to properly construct the basis functions forthe solution and its dual spaces so that the matrix of the $m$th derivative isan identity matrix, and the mass matrix is then identical or approximatelyequals to the Jacobi matrix of the three-term recurrence of GBPs with specificinteger parameters. This allows us to characterise the eigenvalue distributionsand identify the eigenvectors. As a by-product, we are able to answer some openquestions related to the very limited known results on the collocation methodat Legendre points (studied in 1980s) for the first-order IVP, by reformulatingit into a Petrov-Galerkin formulation. Moreover, we present two stablealgorithms for computing zeros of the GBPs, and develop a general space-timespectral method for evolutionary PDEs using either the matrix diagonalisation,which is restricted to a small number of unknowns in time due to theill-conditioning but is fully parallel, or the QZ decomposition which isnumerically stable for a large number of unknowns in time but involvessequential computations. We provide ample numerical results to demonstrate thehigh accuracy and robustness of the space-time spectral methods for someinteresting examples of linear and nonlinear wave problems.

Eigenvalue Analysis and Applications of the Legendre Dual-Petrov-Galerkin Methods for Initial Value Problems