Numerical integration of Lie-Poisson systems while preserving coadjoint orbits and energy
Reports in Informatics No. 179, October 1999, Department of
Informatics, University of Bergen, Norway.
In this paper we apply geometric integrators of the RKMK type to the
problem of integrating Lie--Poisson systems numerically. By using the
coadjoint action of the Lie group $G$ on the dual Lie algebra $\g^*$
to advance the numerical flow, we devise methods of arbitrary order
that automatically stay on the coadjoint orbits. First integrals known
as Casimirs are retained to machine accuracy by the numerical
algorithm. Within the proposed class of methods we find integrators
that also conserve the energy. These schemes are implicit and of
second order. Nonlinear iteration in the Lie algebra and linear error
growth of the global error are discussed. Numerical experiments with
the rigid body, the heavy top and a finite--dimensional truncation of
the Euler equations for a $2$D incompressible fluid are used to
illustrate the properties of the algorithm.