On the B-series composition theorem

TL;DR

This paper introduces a novel proof of the B-series composition theorem for ordinary differential equations that avoids the use of labelled trees, simplifying the theoretical framework by employing the concept of 'assignment' to handle pruning-related combinatorics.

Contribution

It provides a new proof of the B-series composition theorem using unlabelled trees, eliminating the need for labelled trees and introducing 'assignment' for combinatorial counting.

Findings

New proof of the B-series composition theorem without labelled trees

Introduction of 'assignment' to handle pruning combinatorics

Simplification of B-series analysis methods

Abstract

The B-series composition theorem has been an important topic in numerical analysis of ordinary differential equations for the past-half century. Traditional proofs of this theorem rely on labelled trees, whereas recent developments in B-series analysis favour the use of unlabelled trees. In this paper, we present a new proof of the B-series composition theorem that does not depend on labelled trees. A key challenge in this approach is accurately counting combinations related to ``pruning.'' This challenge is overcome by introducing the concept of ``assignment.''