Legendre--Gauss collocation method for initial value problems of second order ordinary differential equations

In this paper, we develop a new collocation method for solving initial value problems of second order ODEs. We approximate the solutions by the Legendre-Gauss interpolation directly. The numerical solutions possess the spectral accuracy. We also propose a multi-step version of Legendre-Gauss collocation method, which works well for long-time calculations. Numerical results demonstrate the effectiveness of proposed methods and coincide well with analysis.

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