Eigenvalue analysis and applications of the Legendre dual-Petrov-Galerkin methods for initial value problems (Q6624465)

This paper addresses the significant scientific problem of developing and rigorously analysing spectral methods tailored for initial value problems (IVPs), particularly using the Legendre dual-Petrov-Galerkin (LDPG) method. Traditional spectral methods are well-established for spatial discretisation but often lack an analogous framework for time discretisation, typically relying on lower-order schemes. The study seeks to bridge this gap by extending the LDPG framework to time-dependent problems, providing a robust mathematical foundation for their application in evolutionary equations.\N\NTo tackle this challenge, the authors meticulously analyse the eigenvalue distributions of spectral discretisation matrices resulting from the LDPG method applied to \( m \)-th order IVPs, represented by \( u^{(m)}(t) = \sigma u(t) \) on the interval \( t \in (-1, 1) \). The methods are grounded in the properties of the generalised Bessel polynomials (GBPs), which are employed to characterise the eigenvalues and eigenvectors of the resulting discretisation matrices. Analytical formulas are derived explicitly for the first- and second-order cases. Additionally, the study reformulates classical collocation methods on Legendre points into the Petrov-Galerkin framework, resolving longstanding questions from earlier spectral studies. Two stable numerical algorithms are introduced for computing GBP zeros, and a general space-time spectral method for solving evolutionary partial differential equations (PDEs) is proposed and evaluated.\N\NThe main findings include an exact characterisation of eigenvalue distributions associated with the LDPG matrices, revealing their connection to the GBPs. The authors demonstrate that this connection informs the stability conditions of time-stepping schemes involving spectral or spectral-element discretisations. Furthermore, the numerical experiments highlight the high accuracy and stability of the proposed methods, showcasing their applicability to both linear and nonlinear wave problems. The study also addresses numerical stability challenges through diagonalisation and QZ decomposition techniques.\N\NThe significance of this research lies in its theoretical contributions and practical implications. By rigorously establishing the spectral properties of the LDPG methods and developing computational techniques for their application, the paper advances the use of spectral methods for space-time problems. This work paves the way for further innovations in high-resolution numerical simulations of time-dependent PDEs, enhancing the efficiency and accuracy of computational models in applied mathematics and physics. The integration of mathematical theory with computational practice makes this study a valuable resource for researchers in numerical analysis and applied mathematics.