Formal series and numerical integrators: some history and some new techniques

Resumen

This paper provides a brief history of B-series and the associated Butchergroup and presents the new theory of word series and extended word series. B-series(Hairer and Wanner 1976) are formal series of functions parameterized by rooted trees.They greatly simplify the study of Runge-Kutta schemes and other numerical integrators.We examine the problems that led to the introduction of B-series and survey a number ofmore recent developments, including applications outside numerical mathematics. Wordseries (series of functions parameterized by words from an alphabet) provide in somecases a very convenient alternative to B-series. Associated with word series is a group Gof coe cients with a composition rule simpler than the corresponding rule in the Butchergroup. From a more mathematical point of view, integrators, like Runge-Kutta schemes,that are a ne equivariant are represented by elements of the Butcher group, integratorsthat are equivariant with respect to arbitrary changes of variables are represented byelements of the word group G.