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

A form of collocation is presented, analysed, and illustratedfor the approximate solution of second-order two-point boundary-valueproblems for ordinary differential equations by use of smoothcubic splines. By collocating to a perturbed differential equationwhich is satisfied by an accurate spline interpolant of thetrue solution, we achieve the desired O(h^4-j) global accuracyfor the jth derivative of the solution, together with enhancedaccuracy for the derivatives at certain points; this shouldbe compared with the O(h^2-j) bounds given by standard collocationat the joints.