Lie group integrators

TL;DR

This survey reviews the development, classes, applications, and theoretical aspects of Lie group integrators, which are numerical methods for solving differential equations on manifolds with a focus on structure preservation.

Contribution

It provides a comprehensive overview of Lie group integrators, including their theoretical foundations, practical implementations, and various subclasses, highlighting recent advances and open issues.

Findings

Various classes of Lie group integrators are introduced and compared.

Applications include systems with natural Lie group actions on manifolds.

Discussion of structure-preserving properties like symplecticity and first integrals.

Abstract

In this survey we discuss a wide variety of aspects related to Lie group integrators. These numerical integration schemes for differential equations on manifolds have been studied in a general and systematic manner since the 1990s and the activity has since then branched out in several different subareas, focussing both on theoretical and practical issues. From two alternative setups, using either frames or Lie group actions on a manifold, we here introduce the most important classes of schemes used to integrate nonlinear ordinary differential equations on Lie groups and manifolds. We describe a number of different applications where there is a natural action by a Lie group on a manifold such that our integrators can be implemented. An issue which is not well understood is the role of isotropy and how it affects the behaviour of the numerical methods. The order theory of numerical Lie group integrators has become an advanced subtopic in its own right, and here we give a brief introduction on a somewhat elementary level. Finally, we shall discuss Lie group integrators having the property that they preserve a symplectic structure or a first integral.