Legendre-Gauss-Radau Spectral Collocation Method for Nonlinear Second-Order Initial Value Problems with Applications to Wave Equations (Journal of Computational Mathematics, 2023)
We propose and analyze a single-interval Legendre-Gauss-Radau (LGR) spectral collocation method for nonlinear second-order initial value problems of ordinary differentialequations. We design an efficient iterative algorithm and prove spectral convergence forthe single-interval LGR collocation method. For more effective implementation, we propose a multi-interval LGR spectral collocation scheme, which provides us great flexibilitywith 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-Lobattocollocation method in space to obtain a space-time spectral collocation approximation fornonlinear second-order evolution equations. Numerical results show that the proposedmethods 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.