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\begin{document}

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\section*{Butcher group}

\hypertarget{butcher_group}{}\section*{{Butcher group}}\label{butcher_group}

The Butcher group was introduced in Butcher’s seminal work on [[Runge-Kutta method]]s. The elements of the group are (generalised) numerical schemes which contain the Runge-Kutta schemes and the group product encodes composition of numerical schemes.

Later Brouder observed that the Butcher group can be viewed as the character group of the (Butcher-)Connes-Kreimer-Hopf algebra used in the combinatorial approach to renormalisation.

The group is used in numerical analysis to deal with order conditions of numerical methods. Further, it has been discovered that the Butcher group is an [[infinite-dimensional Lie group]].

References

\begin{itemize}%
\item The English \href{https://en.wikipedia.org/wiki/Butcher_group}{wikipedia} article is quite detailed


\item J. C. Butcher, An algebraic theory of integration methods, Math. Comp. 26 (1972), 79–106.


\item Ch. Brouder, \emph{Runge-Kutta methods and renormalization}, Europ. Phys. J. C12 (2000) 512–534


\item Charles Brouder, \emph{Trees, renormalization and differential equations}, BIT Numerical Mathematics, 44 (3): 425–438 \href{https://doi.org/10.1023%2FA%3A1007523409317}{doi}


\item Alain Connes, Dirk Kreimer,\_Lessons from quantum field theory: Hopf algebras and spacetime geometries\_, Letters in Mathematical Physics, 48: 85–96, 1999 \href{https://dx.doiorg/10.1023/A:1007523409317}{doi}


\item G. Bogfjellmo, [[Alexander Schmeding]]: The Lie group structure of the Butcher group, Foundations of Computational Mathematics, (\href{https://arxiv.org/abs/1410.4761}{arXiv:1410.4761})



\end{itemize}


\end{document}