Numerical comparisons between Gauss-Legendre methods and Hamiltonian BVMs defined over Gauss points. (arXiv:1002.2727v1 [math.NA])

Numerical comparisons between Gauss-Legendre methods and Hamiltonian BVMs defined over Gauss points. (arXiv:1002.2727v1 [math.NA]): "

Hamiltonian Boundary Value Methods are a new class of energy preserving one
step methods for the solution of polynomial Hamiltonian dynamical systems. They
can be thought of as a generalization of collocation methods in that they may
be defined by imposing a suitable set of extended collocation conditions. In
particular, in the way they are described in this note, they are related to
Gauss collocation methods with the difference that they are able to precisely
conserve the Hamiltonian function in the case where this is a polynomial of any
high degree in the momenta and in the generalized coordinates. A description of
these new formulas is followed by a few test problems showing how, in many
relevant situations, the precise conservation of the Hamiltonian is crucial to
simulate on a computer the correct behavior of the theoretical solutions.

"