Algebraic structures of B-series
P. Chartier, E. Hairer, and G. Vilmart
Abstract. B-series are a fundamental tool in practical and theoretical
aspects of numerical integrators for ordinary differential equations. A composition
law for B-series permits an elegant derivation of order conditions, and a substitution
law gives much insight into modified differential equations of backward error
analysis. These two laws give rise to algebraic structures (groups and Hopf algebras
of trees) that have recently received much attention also in the non-numerical
literature. This article emphasizes these algebraic structures and presents interesting
relationships among them.
Key Words. B-series, rooted trees, composition law, substitution law,
Butcher group, Hopf algebra of trees, coproduct, antipode, P-series, S-series.