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<p>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.”</p>

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On the B-series composition theorem

  • 期刊论文

  • 发表日期：

    2025年5月4日

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

摘要

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.”