Energy-preserving methods for Poisson systems

We present and analyze energy-conserving methods for the numerical inte-gration of IVPs of Poisson type systems that are able to preserve some Casimirs.Their derivation and analysis is done within the framework following the ideasof Hamiltonian BVMs (HBVMs) (see [1] and references therein). The pro-posed methods turn out to be equivalent to those recently derived in [2], givingtherefore an alternative point of view that provides additional insight on themethods. Sufficient conditions that ensure the existence of a unique solutionof the implicit equations defining the formulae are given. A study of the im-plementation of the methods is provided. In particular, order and preservationproperties when the involved integrals are approximated by means of a quadra-ture formula, are derived. References[1] L. Brugnano, F. Iavernaro and D. Trigiante. The Hamiltonian BVMs (HBVMs)Homepage, arXiv:1002.2757.(URL: http://www.math.unifi.it/~brugnano/HBVM/).[2] D. Cohen, E. Hairer. Linear energy-preserving integrators for Poisson systems.BIT Numer. math. 51, 1 (2011) 91–101.