Abstract
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): \(u^{(m)}(t)=\sigma u(t),\, t\in (-1,1)\) with constant \(\sigma \not =0\) and usual initial conditions at t\(=-1,\) are associated with the generalised Bessel polynomials (GBPs). In particular, we derive analytical formulae for the eigenvalues and eigenvectors in the cases m\(=1,2\). 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.
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The code used in this work will be made available upon request to the authors.
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Funding
The work is supported in part by the National Natural Science Foundation of China (No. 12271528, No. 12001280, 11971407) and by Singapore MOE AcRF Tier 1 Grant: MOE2021-T1-RG15/21. The first author also acknowledges the support from the Fundamental Research Funds for the Central Universities (No. 2020zzts031) and the China Scholarship Council (CSC, No. 202106370101).
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Communicated by: Francesca Rapetti
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Kong, D., Shen, J., Wang, LL. et al. Eigenvalue analysis and applications of the Legendre dual-Petrov-Galerkin methods for initial value problems. Adv Comput Math 50, 97 (2024). https://doi.org/10.1007/s10444-024-10190-z
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DOI: https://doi.org/10.1007/s10444-024-10190-z
Keywords
- Legendre dual Petrov-Galerkin methods
- Bessel and generalised Bessel polynomials
- Spectral method in time
- Eigenvalue distributions
- Matrix diagonalisation
- QZ decomposition