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 [34] , [35] in the study of order conditions for Runge-Kutta methods applied to ordinary differential equations
Hairer and Wanner [45] 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 [15] 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 [36] , 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.
