## Lie–Butcher series and geometric numerical integration on manifolds

##### Abstract

The thesis belongs to the field of “geometric numerical integration” (GNI), whose
aim it is to construct and study numerical integration methods for differential equations
that preserve some geometric structure of the underlying system. Many systems
have conserved quantities, e.g. the energy in a conservative mechanical system
or the symplectic structures of Hamiltonian systems, and numerical methods
that take this into account are often superior to those constructed with the more
classical goal of achieving high order.
An important tool in the study of numerical methods is the Butcher series (Bseries)
invented by John Butcher in the 1960s. These are formal series expansions
indexed by rooted trees and have been used extensively for order theory and the
study of structure preservation. The thesis puts particular emphasis on B-series
and their generalization to methods for equations evolving on manifolds, called
Lie–Butcher series (LB-series).
It has become apparent that algebra and combinatorics can bring a lot of insight
into this study. Many of the methods and concepts are inherently algebraic or
combinatoric, and the tools developed in these fields can often be used to great
effect. Several examples of this will be discussed throughout. The thesis is structured as follows: background material on geometric numerical
integration is collected in Part I. It consists of several chapters: in Chapter 1 we
look at some of the main ideas of geometric numerical integration. The emphasis
is put on B-series, and the analysis of these. Chapter 2 is devoted to differential
equations evolving on manifolds, and the series corresponding to B-series in this
setting. Chapter 3 consists of short summaries of the papers included in Part II.
Part II is the main scientific contribution of the thesis, consisting of reproductions
of three papers on material related to geometric numerical integration.

##### Has part(s)

Paper I: Lundervold, Alexander., Munthe-Kaas, Hans. Hopf algebras of formal diffeomorphisms and numerical integration on manifolds. Contemporary Mathematics, volume 539, 2011. Full text not available in BORA. The article is available at: http://arxiv.org/abs/0905.0087 Paper II: Lundervold, Alexander., Munthe-Kaas, Hans. Backward error analysis and the substitution law for Lie group integrators. Full text not available in BORA. The article is available at: http://arxiv.org/abs/1106.1071 Paper III: Lundervold, Alexander., Munthe-Kaas, Hans. On pre-Lie-type algebras with torsion. Full text not available in BORA.