Section: New Results
Composing B-series of integrators and vector fields
Participants : Philippe Chartier, Gilles Vilmart.
This is a joint work with E. Hairer, from the University of Geneva.
Following the pioneering work of Butcher  ,  in the study of order conditions for Runge-Kutta methods applied to ordinary differential equations
Hairer and Wanner  introduced the concept of B-series. A B-series B(f, a)(y) is a formal expression of the form
where the index set T is a set of rooted trees, and a are real coefficients, and F(t) a derivative of f associated to t . B-series and extensions thereof are now exposed in various textbooks and lie at the core of several recent theoretical developments. B-Series owe their success to their ability to represent most numerical integrators, e.g. Runge-Kutta methods, splitting and composition methods, underlying one-step method of linear multistep formulae, as well as modified vector fields, i.e. vector fields built on derivatives of a given function. In some applications, B-series naturally combine with each other, according to two different laws. The composition law of Butcher and the substitution law of Chartier, Hairer and Vilmart.
The aim of the paper  is to explain the fundamental role in numerical analysis of these two laws and to explore their common algebraic structure and relationships. It complements, from a numerical analyst perspective, the work of Calaque, Ebraihimi-Fard & Manchon  , where more sophisticated algebra is used. We introduce into details the composition and substitution laws, as considered in the context of numerical analysis and relate each law to a Hopf algebra. Then we explore various relations between the two laws and consider a specific map related to the logarithm. Eventually, we mention the extension of the substitution law to P-series, which are of great use for partitionned or split systems of ordinary differential equations.