Eigenvalue analysis and applications of the Legendre dual-Petrov-Galerkin methods for initial value problems

In this paper, we show that the eigenvalues and eigenvectors of the spectral discretisation matrices resulting from the Legendre dual-Petrov-Galerkin (LDPG) method for the mth-order initial value problem (IVP): \(u^{(m)}(t)=\sigma u(t),\, t\in (-1,1)\) $u^{(m)}(t)=\sigma u(t),t\in (-1,1)$ with constant \(\sigma \not =0\) $\sigma \ne 0$ and usual initial conditions at t \(=-1,\) $=-1,$ are associated with the generalised Bessel polynomials (GBPs). In particular, we derive analytical formulae for the eigenvalues and eigenvectors in the cases m \(=1,2\) $=1,2$ . As a by-product, we are able to answer some open questions related to the collocation method at Legendre points (extensively studied in the 1980s) for the first-order IVP, by reformulating it into a Petrov-Galerkin formulation. Our results have direct bearing on the CFL conditions of time-stepping schemes with spectral or spectral-element discretisation in space. Moreover, we present two stable algorithms for computing zeros of the GBPs and develop a general space-time method for evolutionary PDEs. We provide ample numerical results to demonstrate the high accuracy and robustness of the space-time methods for some interesting examples of linear and nonlinear wave problems.