Numerical Integration of Lie--Poisson Systems While Preserving Coadjoint Orbits and Energy

Abstract

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 ${\mbox{\normalsize$\mathfrak{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 and a finite-dimensional truncation of the Euler equations for a two-dimensional (2D) incompressible fluid are used to illustrate the properties of the algorithm.