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

  1. Murua, Ander 2
  2. Sanz Serna, Jesús María 1
  1. 1 Universidad de Valladolid
    info

    Universidad de Valladolid

    Valladolid, España

    ROR https://ror.org/01fvbaw18

  2. 2 Universidad del País Vasco/Euskal Herriko Unibertsitatea
    info

    Universidad del País Vasco/Euskal Herriko Unibertsitatea

    Lejona, España

    ROR https://ror.org/000xsnr85

Book:
Proceedings of the 8th International Congress on Industrial and Applied Mathematics (ICIAM 2015)

Publisher: Higher Education Press

ISBN: 978-7-04-043453-8

Year of publication: 2015

Type: Book chapter

Abstract

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.

Funding information

Ministerio de Economía, Industria y Competitividad, proyectos MTM2013-46553-C3-2-P y MTM2013-46553-C3-1-P

Funders