On algebraic structures of numerical integration on vector spaces and manifolds

Abstract. Numerical analysis of time-integration algorithms has been applying ad-vanced algebraic techniques for more than fourty years. An explicit description of the group of characters in the Butcher–Connes–Kreimer Hopf algebra first appeared in Butcher’s work on composition of integration methods in 1972. In more recent years, the analysis of structure preserving algorithms, geometric integration techniques and in-tegration algorithms on manifolds have motivated the incorporation of other algebraic structures in numerical analysis. In this paper we will survey algebraic structures that have found applications within these areas. This includes pre-Lie structures for the ge-ometry of flat and torsion free connections appearing in the analysis of numerical flows on vector spaces. The much more recent post-Lie and D-algebras appear in the analysis of flows on manifolds with flat connections with constant torsion. Dynkin and Eule-rian idempotents appear in the analysis of non-autonomous flows and in backward error analysis. Non-commutative Bell polynomials and a non-commutative Faa ̀ di Bruno Hopf algebra are other examples of structures appearing naturally in the numerical analysis of integration on manifolds