9 research outputs found
Analysis of Energy and QUadratic Invariant Preserving (EQUIP) methods
In this paper we are concerned with the analysis of a class of geometric
integrators, at first devised in [14, 18], which can be regarded as an
energy-conserving variant of Gauss collocation methods. With these latter they
share the property of conserving quadratic first integrals but, in addition,
they also conserve the Hamiltonian function itself. We here reformulate the
methods in a more convenient way, and propose a more refined analysis than that
given in [18] also providing, as a by-product, a practical procedure for their
implementation. A thorough comparison with the original Gauss methods is
carried out by means of a few numerical tests solving Hamiltonian and Poisson
problems.Comment: 28 pages, 2 figures, 4 table
Energy-preserving methods for Poisson systems
AbstractWe present and analyze energy-conserving methods for the numerical integration of IVPs of Poisson type that are able to preserve some Casimirs. Their derivation and analysis is done following the ideas of Hamiltonian BVMs (HBVMs) (see Brugnano et al. [10] and references therein). It is seen that the proposed approach allows us to obtain the methods recently derived in Cohen and Hairer (2011) [17], giving an alternative derivation of such methods and a new proof of their order. 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. In particular, order and preservation properties when the involved integrals are approximated by means of a quadrature formula, are derived