Formal series and numerical integrators: some history and some new techniques
For researchers in numerical analysis, this work provides a new theoretical tool (word series) that simplifies the analysis of certain integrators, but it is primarily a survey and conceptual advance rather than a practical breakthrough.
This paper reviews the history of B-series and the Butcher group, and introduces word series and extended word series as a new framework for numerical integrators. Word series offer a simpler composition rule than B-series and represent integrators equivariant under arbitrary changes of variables.
This paper provides a brief history of B-series and the associated Butcher group 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 of more recent developments, including applications outside numerical mathematics. Word series (series of functions parameterized by words from an alphabet) provide in some cases a very convenient alternative to B-series. Associated with word series is a group G of coefficients with a composition rule simpler than the corresponding rule in the Butcher group. From a more mathematical point of view, integrators, like Runge-Kutta schemes, that are affine equivariant are represented by elements of the Butcher group, integrators that are equivariant with respect to arbitrary changes of variables are represented by elements of the word group G.