DOI 10.1007/s100529900235
Runge-Kutta methods and renormalization
Ch. Brouder
Laboratoire de Minéralogie-Cristallographie, CNRS UMR7590,
Universités Paris 6, Paris 7, IPGP, Case 115, 4 place Jussieu,
75252 Paris Cedex 05, France (e-mail: brouder@lmcp.jussieu.fr)
Received: 6 July 1999 / Published online: 10 December 1999
Abstract
Rooted trees have been used to calculate the solution of
nonlinear flow equations and Runge-Kutta methods. More recently,
rooted trees have helped systematizing the algebra underlying
renormalization in quantum field theories. The Butcher group and
B-series establish a link
between these two approaches to rooted trees. On the one hand,
this link allows for an alternative representation of the algebra
of renormalization, leading to nonperturbative results. On the
other hand, it helps to renormalize singular flow equations.
The usual approach is extended here to nonlinear partial differential
equations. A nonlinear Born expansion is given, and renormalization is
used to partly remove the secular terms of the perturbative expansion.
Copyright Springer-Verlag 2000
