Algebraic Structures of B-series

Philippe Chartier (1) , Ernst Hairer (2) , Gilles Vilmart (3)

1 IPSO - Invariant Preserving SOlvers
2 Section de mathématiques [Genève]
3 Section de Mathematiques [Lausanne]

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.

Mots clés: B-series; Rooted trees; Composition law; Substitution law; Butcher group; Hopf algebra of trees; Coproduct; Antipode; P-series; S-series

Domaines: Autre [cs.OH]; Analyse numérique [math.NA]

Philippe Chartier, Ernst Hairer, Gilles Vilmart. Algebraic Structures of B-series. Foundations of Computational Mathematics, 2010, 10 (4), pp.407-420. ⟨10.1007/s10208-010-9065-1⟩. ⟨inria-00598369⟩