SUPERCONVERGENT GRIDS FOR 2-POINT BOUNDARY-VALUE-PROBLEMS

Consider a well-posed two-point boundary value problem on (0, 1) given by y" = F(x, y, y'), y(0) = y0, y(1) = y1. The solution may be approximated by the standard three-point finite difference scheme on a non-uniform grid with N subintervals given by x(i) = phi(i/N), where phi is a monotone grid function on [0, 1]. The grid adaption problem is to choose-phi so as to obtain minimal error. We show that all choices of phi, save at most one, yield mean local truncation errors which are O(N-2) but not o(N-2); and that for some problems there is a unique-phi, called the superconvergent grid function, which yields O(N-4) errors. Thus, no method of grid adaption, except that based on the superconvergent grid, can improve the order of convergence. A two-point boundary value problem solver based on the superconvergent grid is given, and numerical results are shown.