Energy preservation in separable Hamiltonian systems by splitting schemes

Splitting and composition schemes for the numerical integration of separable Hamiltonian problems, in general, fail to yield conservation of generic nonlinear Hamiltonians. In the proposed approach we compose, at each step, symplectic maps which depend on a parameter chosen in order to have methods which minimize the error on the Hamiltonian function. Second and fourth order symmetric methods are proposed which are, in the particular case of the linear oscillator, symplectic, energy-preserving and explicit. On the non-linear oscillator, the energy preservation and the simplecticity are not obtained exactly but retained at some extent. The proposed methods are semi-explicit in the sense that they require, as additional computational effort, the search for a zero of a scalar function with respect to a scalar variable.