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Volume preservation of Butcher series methods from the operad viewpoint

Vladimir Dotsenko, Paul Laubie

Abstract

We study a coloured operad involving rooted trees and directed cycles of rooted trees that generalizes the operad of rooted trees of Chapoton and Livernet. We describe all the relations between the generators of a certain suboperad of that operad, and compute the Chevalley-Eilenberg homology of two naturally arising differential graded Lie algebras. This allows us to give short and conceptual new proofs of two important results on Butcher series methods of numerical solution of ODEs: absence of volume-preserving integration schemes and the acyclicity of the aromatic bicomplex, the key step in a complete classification of volume-preserving integration schemes using the so called aromatic Butcher series.

Volume preservation of Butcher series methods from the operad viewpoint

Abstract

We study a coloured operad involving rooted trees and directed cycles of rooted trees that generalizes the operad of rooted trees of Chapoton and Livernet. We describe all the relations between the generators of a certain suboperad of that operad, and compute the Chevalley-Eilenberg homology of two naturally arising differential graded Lie algebras. This allows us to give short and conceptual new proofs of two important results on Butcher series methods of numerical solution of ODEs: absence of volume-preserving integration schemes and the acyclicity of the aromatic bicomplex, the key step in a complete classification of volume-preserving integration schemes using the so called aromatic Butcher series.

Paper Structure

This paper contains 19 sections, 14 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Conventions
  4. Recollections on volume preservation
  5. Butcher series methods
  6. Aromatic Butcher series methods
  7. Rooted trees, cycles of rooted trees, and related coloured operads
  8. The embedding theorem
  9. Operadic approach to divergence
  10. The action of the tadpole graph on trees
  11. Lack of volume-preserving B-series methods
  12. Differential graded Lie algebras and the aromatic bicomplex
  13. Two differential graded Lie algebras in species
  14. The aromatic bicomplex and the Chevalley--Eilenberg complex
  15. Further consequences of Theorem \ref{['th:div0']}
  16. ...and 4 more sections

Key Result

Proposition 4.1

The operad $\mathop{\mathrm{PLMC}}\nolimits$ admits the following combinatorial description. Here one considers the pre-Lie algebra $\mathop{\mathrm{RT}}\nolimits$ in species as a Lie algebra, and then forms the corresponding universal enveloping algebra and the module of Kähler differentials. The coloured operad structure corresponds to the following combinatorially define structure. The s

Theorems & Definitions (29)

  • Proposition 4.1
  • proof
  • Theorem 4.2
  • proof
  • Proposition 5.1
  • proof
  • Theorem 5.2
  • proof
  • Proposition 5.3
  • proof
  • ...and 19 more