HBVMs Homepage
Hamiltonian Boundary Value Methods 

Energy Preserving Discrete Line Integral Methods

A few examples taken from:

[12] L.Brugnano, F.Iavernaro, T.Susca. Numerical comparisons between Gauss-Legendre methods and Hamiltonian BVMs defined over Gauss points. Monografias de la Real Academia de Ciencias de Zaragoza, Special Issue devoted to the 65th birthday of Manuel Calvo,  33 (2010) 95-112 (arXiv:1002.2727).

Another example taken from:

[9] L.Brugnano, F.Iavernaro, D.Trigiante. Analisys of Hamiltonian Boundary Value Methods (HBVMs): a class of energy-preserving methods for the numerical solution of polynomial Hamiltonian dynamical systems.   (2009) (arXiv:0909.5659).

[10] L.Brugnano, F.Iavernaro, D.Trigiante. Hamiltonian Boundary Value Methods (Energy Conserving Discrete Line Integral Methods). Jour. Numer. Anal., Industrial and Appl. Math., 5,1-2 (2010) 17-37 (arXiv:0910.3621).