Uniformly Accurate Numerical Solutions to Differential Equations Using Extrapolation and Interpolation.

In this work the author is concerned with numerical methods for solving ordinary differential equations. The author considers those methods that have asymptotic error expansions involving all powers of h sup q, where h is the steplength and q is a fixed integer. The process of extrapolation can be employed with such methods to obtain highly accurate solutions at grid points belonging to the coarsest mesh. In Chapter I the pullback interpolation method is developed. This method combines extrapolation with Hermite interpolation of the coefficient functions for the asymptotic error expansion to produce a highly accurate solution at all grid points of the finest mesh. In Chapter II the pullback interpolation method is modified so as to be applicable to boundary value problems. In Chapter III, difference differential equations with constant retardation are considered. Modified author abstract