An algebraic theory of integration methods

Abstract

A class of integration methods which includes Runge-Kutta methods, as well as the Picard successive approximation method, is shown to be related to a certain group which can be represented as the family of real-valued functions on the set of rooted trees. For each integration method, a group element is defined corresponding to it and it is shown that the numerical result obtained using the method is characterised by this group element. If two methods are given, then a new method may be defined in such a way that when it is applied to a given initial-value problem the result is the same as for the successive application of the given methods. It is shown that the group element for this new method is the product of the group elements corresponding to the given methods. Various properties of the group and certain of its subgroups are examined. The concept of order is defined as a relationship between group elements.