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.