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.
- 3.G. Bogfjellmo, R. Dahmen, and A. Schmeding. Character groups of Hopf algebras as infinite-dimensional Lie groups. arXiv:1501.05221v3, Apr. 2015.
- 6.M. P. Calvo, A. Murua, and J. M. Sanz-Serna. Modified equations for ODEs. In Chaotic numerics (Geelong, 1993), volume 172 of Contemp. Math., pages 63–74. Amer. Math. Soc., Providence, RI, 1994.Google Scholar
- 11.R. Dahmen. Direct Limit Constructions in Infinite Dimensional Lie Theory. PhD thesis, University of Paderborn, 2011. urn:nbn:de:hbz:466:2-239.Google Scholar
- 12.K. Deimling. Ordinary Differential Equations in Banach Spaces. Number 596 in Lecture Notes in Mathematics. Springer Verlag, Heidelberg, 1977.Google Scholar
- 16.H. Glöckner. Regularity properties of infinite-dimensional Lie groups, and semiregularity. arXiv:1208.0715v3, Jan. 2015.
- 17.H. Glöckner. Infinite-dimensional Lie groups without completeness restrictions. In A. Strasburger, J. Hilgert, K. Neeb, and W. Wojtyński, editors, Geometry and Analysis on Lie Groups, volume 55 of Banach Center Publication, pages 43–59. Warsaw, 2002.Google Scholar
- 20.E. Hairer, C. Lubich, and G. Wanner. Geometric Numerical Integration, volume 31 of Springer Series in Computational Mathematics. Springer Verlag, \(^2\)2006.Google Scholar
- 21.H. Jarchow. Locally Convex Spaces. Lecture Notes in Mathematics 417. Teubner, Stuttgart, 1981.Google Scholar
- 22.H. Keller. Differential Calculus in Locally Convex Spaces. Lecture Notes in Mathematics 417. Springer Verlag, Berlin, 1974.Google Scholar
- 24.A. Kriegl and P. W. Michor. The convenient setting of global analysis, volume 53 of Mathematical Surveys and Monographs. AMS, 1997.Google Scholar
- 25.R. I. McLachlan, K. Modin, H. Munthe-Kaas, and O. Verdier. B-series methods are exactly the local, affine equivariant methods. arXiv:1409.1019v3, Sept. 2014.
- 26.I. Mencattini. Structure of the insertion elimination Lie algebra in the ladder case. ProQuest LLC, Ann Arbor, MI, 2005. Thesis (Ph.D.)–Boston University.Google Scholar
- 27.P. Michor. Manifolds of Differentiable Mappings. Shiva Mathematics Series 3. Shiva Publishing Ltd., Orpington, 1980.Google Scholar
- 28.J. Milnor. Remarks on infinite-dimensional Lie groups. In B. DeWitt and R. Stora, editors, Relativity, Groups and Topology II, pages 1007–1057. North Holland, New York, 1983.Google Scholar
- 30.H. H. Schaefer. Topological vector spaces. Springer-Verlag, New York-Berlin, 1971. Third printing corrected, Graduate Texts in Mathematics, Vol. 3.Google Scholar