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Lie group methods

Hans Munthe-Kaas and Brynjulf Owren

University of Bergen and NTNU

For the basic Lie group theory, we refer to the first lecture of Peter Olver.

Part I Introduction

In the literature, one finds two different approaches for defining Lie group methods for nonlinear differential equations on manifolds. One is based on Lie group actions, and the other one on frames. We begin by considering these two notions and compare their uses. The Lie group methods also fall into two classes, those which can be interpreted as working in local coordinates (e.g. RKMK methods), and those which are based on composition of flows (e.g. Crouch-Grossman methods). We discuss the two types of methods, how to apply them to a given ODE on a manifold.

Part II Order theory

In this part, we discuss order conditions for Lie group methods, using Lie series and ordered rooted trees. Some extra attention will be given to a recent class of methods called ``commutator-free Lie group methods'' and we will briefly look at isotropy corrections as recently proposed in a paper by Lewis and Olver. We will introduce some useful algebraic structures which can be used to eliminate dependencies between order conditions as well as in backward error analysis. We may demonstrate some Maple programs which implement these structures. Finally, we will discuss some open problems that we are currently working on.

Part III Applications to ODEs and PDEs