On algebraic structures of numerical integration on vector spaces and manifolds

TL;DR

This paper surveys various algebraic structures like pre-Lie, post-Lie, and Hopf algebras that underpin numerical integration methods on vector spaces and manifolds, highlighting their applications in structure-preserving algorithms and geometric integration.

Contribution

It provides a comprehensive overview of algebraic frameworks used in numerical analysis of integration algorithms on manifolds and vector spaces, emphasizing recent developments.

Findings

Pre-Lie structures relate to flat, torsion-free connections.

Post-Lie and D-algebras analyze flows on manifolds with flat connections.

Non-commutative Bell polynomials appear in non-autonomous flow analysis.

Abstract

Numerical analysis of time-integration algorithms has been applying advanced 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 integration algorithms on manifolds have motivated the incorporation of other algebraic structures in numerical analysis. In this paper we will survey structures that have found applications within these areas. This includes pre-Lie structures for the geometry 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 Eulerian idempotents appear in the analysis of non-autonomous flows and in backward error analysis. Non-commutative Bell polynomials and a non-commutative Fa\`a di Bruno Hopf algebra are other examples of structures appearing naturally in the numerical analysis of integration on manifolds.