The Lie Group Structure of the Butcher Group

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• 作者：Geir Bogfjellmo ; Alexander Schmeding
• 关键词：Butcher group ; Infinite ; dimensional Lie group ; Hopf algebra of rooted trees ; Regularity of Lie groups ; Symplectic methods
• 刊名：Foundations of Computational Mathematics
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：17
• 期：1
• 页码：127-159
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Numerical Analysis; Economics, general; Applications of Mathematics; Linear and Multilinear Algebras, Matrix Theory; Math Applications in Computer Science; Computer Science, general;
• 出版者：Springer US
• ISSN：1615-3383
• 卷排序：17

文摘

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.