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We introduce a family of fourth order two-step methods that preserve theenergy function of canonical polynomial Hamiltonian systems. Each method in thefamily may be viewed as a correction of a linear two-step method, where thecorrection term is O(h^5) (h is the stepsize of integration). The key tools thenew methods are based upon are the line integral associated with a conservativevector field (such as the one defined by a Hamiltonian dynamical system) andits discretization obtained by the aid of a quadrature formula. Energyconservation is equivalent to the requirement that the quadrature is exact,which turns out to be always the case in the event that the Hamiltonianfunction is a polynomial and the degree of precision of the quadrature formulais high enough. The non-polynomial case is also discussed and a number of testproblems are finally presented in order to compare the behavior of the newmethods to the theoretical results.

A Two Step, Fourth Order, Nearly-Linear Method with Energy Preserving Properties