Elena Celledoni, Håkon Marthinsen, Brynjulf Owren
We give a short and elementary introduction to Lie group methods. A selection of applications of Lie group integrators are discussed. Finally, a family of symplectic integrators on cotangent bundles of Lie groups is presented and the notion of discrete gradient methods is generalised to Lie groups.
Håkon Marthinsen, Brynjulf Owren
Symplectic integration of autonomous Hamiltonian systems is a well-known field of study in geometric numerical integration, but for non-autonomous systems the situation is less clear, since symplectic structure requires an even number of dimensions. We show that one possible extension of symplectic methods in the autonomous setting to the non-autonomous setting is obtained by using canonical transformations. Many existing methods fit into this framework. We also perform experiments which indicate that for exponential integrators, the canonical and symmetric properties are important for good long time behaviour. In particular, the theoretical and numerical results support the well documented fact from the literature that exponential integrators for non-autonomous linear problems have superior accuracy compared to general ODE schemes.
Geir Bogfjellmo, Håkon Marthinsen
In this article, a unified approach to obtain symplectic integrators on T*G from Lie group integrators on a Lie group G is presented. The approach is worked out in detail for symplectic integrators based on Runge--Kutta--Munthe-Kaas methods and Crouch--Grossman methods. These methods can be interpreted as symplectic partitioned Runge--Kutta methods extended to the Lie group setting in two different ways. In both cases, we show that it is possible to obtain symplectic integrators of arbitrarily high order by this approach.