B-series and Algebraic Analysis

Abstract

One of the key questions in the analysis of numerical approximations to differential equations, and related problems, is in the comparison of two mappings. One mapping would be the exact flow through a specified time step and the other would be a numerical scheme, such as a Runge–Kutta method. The B-series approach to questions like this is to write the Taylor expansions of the two mappings in a special way, in terms of “elementary differentials”, and to then compare the coefficients in corresponding terms. The terms themselves can be indexed in terms of rooted trees and the theory of B-series hinges on this indexing. The chapter includes a discussion of multi-dimensional Taylor series and it is shown how this leads to the formulation of elementary differentials and B-series. Some sample problems, which are both easy and fundamental, are solved, first in a low order introduction and then in full generality. Special attention is given to the flow through a unit step, followed by an implicit variant of the Euler method. It is remarkable that the latter simple example is a direct path to the large family of Runge–Kutta methods. The composition rule for B-series is introduced and some of its wider ramifications are explored.