The Lie Group Structure of the Butcher Group
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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 Fréchet 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 with the Butcher group by Connes and Kreimer.
KeywordsButcher group Infinite-dimensional Lie group Hopf algebra of rooted trees Regularity of Lie groups Symplectic methods
Mathematics Subject Classification22E65 (primary) 65L06 58A07 16T05 (secondary)
The research on this paper was partially supported by the projects Topology in Norway (Norwegian Research Council Project 213458) and Structure Preserving Integrators, Discrete Integrable Systems and Algebraic Combinatorics (Norwegian Research Council Project 231632). The second author would also like to thank Reiner Hermann for helpful discussions on Hopf algebras. Furthermore, we thank the anonymous referees for their insightful comments which led to substantial improvements of the paper.
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