nLab Butcher group

Butcher group

The Butcher group was introduced in Butcher’s seminal work on Runge-Kutta methods. 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.


  • The English wikipedia article is quite detailed

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

  • Ch. Brouder, Runge-Kutta methods and renormalization, Europ. Phys. J. C12 (2000) 512–534

  • Charles Brouder, Trees, renormalization and differential equations, BIT Numerical Mathematics, 44 (3): 425–438 doi

  • Alain Connes, Dirk Kreimer,_Lessons from quantum field theory: Hopf algebras and spacetime geometries_, Letters in Mathematical Physics, 48: 85–96, 1999 doi

  • G. Bogfjellmo, Alexander Schmeding: The Lie group structure of the Butcher group, Foundations of Computational Mathematics, (arXiv:1410.4761)

Last revised on June 13, 2018 at 15:18:10. See the history of this page for a list of all contributions to it.