Quadratic-Spline Collocation Methods for Two-Point Boundary Value Problems.

摘要

A new collocation method based on quadratic splines is presented for second order two-point boundary value problems. First, O (h to the 4th power) approximations to the first and second derivative of a function are derived using a quadratic-spline interpolant of u. Then these approximations are used to define an O(h to the 4th power) perturbation of the given boundary value problem. Second, the perturbed problem is used to define a collocation approximation at interval midpoints for which an optimal O(h to the 3-jth power) global estimate for the jth derivative of the error is derived. Further, O(h to the 4-jth power) error bounds for the jth derivative are obtained for certain superconvergence points. It should be observed that standard collocation at midpoints gives O(h to 2-jth power) bounds. Results from numerical experiments are reported that verify the theoretical behaviour of the method. Reprints. (JHD)