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On the B-series composition theorem

John C. Butcher, Taketomo Mitsui, Yuto Miyatake, Shun Sato

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.''

On the B-series composition theorem

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.''
Paper Structure (4 sections, 3 theorems, 33 equations)

This paper contains 4 sections, 3 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. B-series composition theorem and a key lemma
  3. Notation
  4. Proposition \ref{['prop1']} and the proof of Lemma \ref{['lemma1']}

Key Result

Theorem 1

Let $a\in\text{\sf B}$ and $b\in\text{\sf B}^*$. Then, $( {\bf B}_h ({\bf B}_hy_0)a ) b$ is also a B-series $({\bf B}_hy_0)(ab)$, and the map $ab \in \text{\sf B}^*$ is given as

Theorems & Definitions (6)

  • Theorem 1
  • Lemma 2
  • Proposition 3: cf. bu21
  • proof
  • proof
  • Remark 1