Extrapolated Collocation for Two-point Boundary-value Problems using Cubic Splines

A form of collocation is presented, analysed, and illustrated for the approximate solution of second-order two-point boundary-value problems for ordinary differential equations by use of smooth cubic splines. By collocating to a perturbed differential equation which is satisfied by an accurate spline interpolant of the true solution, we achieve the desired O(h4-j) global accuracy for the jth derivative of the solution, together with enhanced accuracy for the derivatives at certain points; this should be compared with the O(h2-j) bounds given by standard collocation at the joints.