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Lie Group Integrators

Conference paper
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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 267)

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.

Keywords

Lie group integrators Geometric integration Symplectic integrators Geometric mechanics Structure preserving integrators Energy preservation Numerical methods on manifolds Lie groups Homogeneous spaces 

MSC

65L05 34C40 34G20 37M15 

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Norwegian University of Science and TechnologyTrondheimNorway

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