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Volume 42, Issue 1
Legendre-Gauss-Radau Spectral Collocation Method for Nonlinear Second-Order Initial Value Problems with Applications to Wave Equations

Lina Wang, Qian Tong, Lijun Yi & Mingzhu Zhang

DOI: 10.4208/jcm.2203-m2021-0244

J. Comp. Math., 42 (2024), pp. 217-247.

Published online: 2023-12

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  • Abstract

We propose and analyze a single-interval Legendre-Gauss-Radau (LGR) spectral collocation method for nonlinear second-order initial value problems of ordinary differential equations. We design an efficient iterative algorithm and prove spectral convergence for the single-interval LGR collocation method. For more effective implementation, we propose a multi-interval LGR spectral collocation scheme, which provides us great flexibility with respect to the local time steps and local approximation degrees. Moreover, we combine the multi-interval LGR collocation method in time with the Legendre-Gauss-Lobatto collocation method in space to obtain a space-time spectral collocation approximation for nonlinear second-order evolution equations. Numerical results show that the proposed methods have high accuracy and excellent long-time stability. Numerical comparison between our methods and several commonly used methods are also provided.

  • Keywords

Legendre-Gauss-Radau collocation method, Second-order initial value problem, Spectral convergence, Wave equation.

  • AMS Subject Headings

65M70, 41A10, 65L05, 35L05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-42-217, author = {Wang , LinaTong , QianYi , Lijun and Zhang , Mingzhu}, title = {Legendre-Gauss-Radau Spectral Collocation Method for Nonlinear Second-Order Initial Value Problems with Applications to Wave Equations}, journal = {Journal of Computational Mathematics}, year = {2023}, volume = {42}, number = {1}, pages = {217--247}, abstract = {

We propose and analyze a single-interval Legendre-Gauss-Radau (LGR) spectral collocation method for nonlinear second-order initial value problems of ordinary differential equations. We design an efficient iterative algorithm and prove spectral convergence for the single-interval LGR collocation method. For more effective implementation, we propose a multi-interval LGR spectral collocation scheme, which provides us great flexibility with respect to the local time steps and local approximation degrees. Moreover, we combine the multi-interval LGR collocation method in time with the Legendre-Gauss-Lobatto collocation method in space to obtain a space-time spectral collocation approximation for nonlinear second-order evolution equations. Numerical results show that the proposed methods have high accuracy and excellent long-time stability. Numerical comparison between our methods and several commonly used methods are also provided.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2203-m2021-0244}, url = {http://global-sci.org/intro/article_detail/jcm/22158.html} }
TY - JOUR T1 - Legendre-Gauss-Radau Spectral Collocation Method for Nonlinear Second-Order Initial Value Problems with Applications to Wave Equations AU - Wang , Lina AU - Tong , Qian AU - Yi , Lijun AU - Zhang , Mingzhu JO - Journal of Computational Mathematics VL - 1 SP - 217 EP - 247 PY - 2023 DA - 2023/12 SN - 42 DO - http://doi.org/10.4208/jcm.2203-m2021-0244 UR - https://global-sci.org/intro/article_detail/jcm/22158.html KW - Legendre-Gauss-Radau collocation method, Second-order initial value problem, Spectral convergence, Wave equation. AB -

We propose and analyze a single-interval Legendre-Gauss-Radau (LGR) spectral collocation method for nonlinear second-order initial value problems of ordinary differential equations. We design an efficient iterative algorithm and prove spectral convergence for the single-interval LGR collocation method. For more effective implementation, we propose a multi-interval LGR spectral collocation scheme, which provides us great flexibility with respect to the local time steps and local approximation degrees. Moreover, we combine the multi-interval LGR collocation method in time with the Legendre-Gauss-Lobatto collocation method in space to obtain a space-time spectral collocation approximation for nonlinear second-order evolution equations. Numerical results show that the proposed methods have high accuracy and excellent long-time stability. Numerical comparison between our methods and several commonly used methods are also provided.

Wang , LinaTong , QianYi , Lijun and Zhang , Mingzhu. (2023). Legendre-Gauss-Radau Spectral Collocation Method for Nonlinear Second-Order Initial Value Problems with Applications to Wave Equations. Journal of Computational Mathematics. 42 (1). 217-247. doi:10.4208/jcm.2203-m2021-0244
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