Lie group methods in geometric integration

Abstract

The subject of geometric integration concerns the construction of numerical integrators which can successfully preserve the asymptotic behaviour or invariants of the approximate solutions, computed for certain dynamical systems. Whereas one of the aims when building a classical numerical integrator is to minimise the error produced by the method, the emphasis in the construction of a geomteric integrator lies in its ability to produce numerical solutions with the correct qualitative behaviour from the differential equations. This leads to the specific design of numerical integrators to solve certain types of dynamical systems. It turns out that the language of Lie groups and Lie algebras is particularly suited to the building of geometric integrators, based on their actions on manifolds. A geometric integrator which produces numerical solutions lying on the manifold of a dynamical sysem via the Lie group action, is an integrator which belongs to a class of numerical methods known as Lie group methods.