Recent Advances on Preserving Methods for Poisson Systems

We present and analyze energy-conserving methods for the numerical integration of IVPs of Poisson type systems. They are also able to preserve some Casimirs. Their derivation and analysis is done following the ideas of Hamiltonian BVMs (HBVMs) (see [3] and references therein). The proposed methods turn out to be equivalent to those recently derived in [8], giving therefore an alternative point of view that provides additional insight on the methods. Sufficient conditions that ensure the existence of a unique solution of the implicit equations defining the formulae are given. A study of the implementation of the methods is provided.