Volume preservation of Butcher series methods from the operad viewpoint

TL;DR

This paper explores a generalized operad structure related to rooted trees, providing new proofs for key properties of Butcher series methods in numerical ODE solutions, including volume preservation and aromatic bicomplex acyclicity.

Contribution

It introduces a new operad framework that generalizes rooted trees and offers conceptual proofs for volume preservation and aromatic bicomplex properties in Butcher series methods.

Findings

Proves absence of volume-preserving schemes in certain contexts.

Shows aromatic bicomplex is acyclic.

Provides a classification of volume-preserving integrators.

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.