Mathematics – Numerical Analysis
Scientific paper
2011-12-19
Mathematics
Numerical Analysis
42 pages
Scientific paper
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.
Lundervold Alexander
Munthe-Kaas Hans Z.
No associations
LandOfFree
On algebraic structures of numerical integration on vector spaces and manifolds does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On algebraic structures of numerical integration on vector spaces and manifolds, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On algebraic structures of numerical integration on vector spaces and manifolds will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-53588