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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} }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.