Sixth order C²-spline collocation method for integrating second order ordinary initial value problems

A new procedure based on sixth degree (Hexic) C²-Spline for the numerical integration of the second order initial value problems (IVPs) y″ = f(x, y), including those possessing oscillatory solutions, is presented. The proposed method is essentially an implicit sixth order one-step method. Stability analysis shows that the method possesses (0, 75.3)∪(030.2, 201.9) as interval of periodicity and/or absolute stability. In addition, the method has phase-lag (dispersion) of order six with actual phase-lag H ⁶/774144. Convergence results yield error bounds ∥ s ^(r) - y^(r) ∥= O(h⁶), r = 0, 1, in the uniform norm, provided y ∈ C⁸[0, b]. Furthermore, it turns out that the method is a continuous extension of a sixth order four-stage Runge-Kutta (-Nyström) method. Numerical experiments will also be considered.