A Family of Implicit Higher Order Methods for the Numerical Integration of Second Order Differential Equations

Abstract

A family of higher order implicit methods with k steps is constructed, which exactly integrate the initial value problems of second order ordinary differential equations directly without reformulation to first order systems. Implicit methods with step numbers  are considered. For these methods, a study of local truncation error is made with their basic properties. Error and step length control based on Richardson extrapolation technique is carried out. Illustrative examples are solved with the aid of MATLAB package. Findings from the analysis of the basic properties of the methods show that they are consistent, symmetric and zero-stable. The results obtained from numerical examples show that these methods are much more efficient and accurate on comparison. These methods are preferable to some existing methods owing to the fact that they are efficient and simple in terms of derivation and computation

Keywords: Error constant, implicit methods, Order of accuracy, Zero-Stability, Symmetry