The Lie group structure of the Butcher group

TL;DR

This paper establishes an infinite-dimensional Lie group structure on the Butcher group, enhancing its algebraic framework and connecting it to symplectic maps and existing Lie algebra models for better analysis of numerical methods.

Contribution

It introduces a natural Lie group structure on the Butcher group, making it a real analytic Baker--Campbell--Hausdorff Lie group modeled on a Fréchet space, and explores its algebraic and subgroup properties.

Findings

The Butcher group is a regular Lie group in the sense of Milnor.

Contains the subgroup of symplectic tree maps as a closed Lie subgroup.

Computes the Lie algebra of the Butcher group and relates it to existing models.

Abstract

The Butcher group is a powerful tool to analyse integration methods for ordinary differential equations, in particular Runge--Kutta methods. In the present paper, we complement the algebraic treatment of the Butcher group with a natural infinite-dimensional Lie group structure. This structure turns the Butcher group into a real analytic Baker--Campbell--Hausdorff Lie group modelled on a Fr\'echet space. In addition, the Butcher group is a regular Lie group in the sense of Milnor and contains the subgroup of symplectic tree maps as a closed Lie subgroup. Finally, we also compute the Lie algebra of the Butcher group and discuss its relation to the Lie algebra associated to the Butcher group by Connes and Kreimer.