Lie–Butcher series and geometric numerical integration on manifolds
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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.
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/0905.0087Paper 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/1106.1071Paper III: Lundervold, Alexander., Munthe-Kaas, Hans. On pre-Lie-type algebras with torsion. Full text not available in BORA.