Energy- and Quadratic Invariants-Preserving Integrators Based upon Gauss Collocation Formulae

We introduce a new family of symplectic integrators for canonical Hamiltonian systems. Each method in the family depends on a real parameter α. When α = 0 we obtain the classical Gauss collocation formula of order 2s, where s denotes the number of the internal stages. For any given non-null α, the corresponding method remains symplectic and has order 2s−2; hence it may be interpreted as an O(h2s−2) (symplectic) perturbation of the Gauss method. Under suitable assumptions, we show that the parameter α may be properly tuned, at each step of the integration procedure, so as to guarantee energy conservation in the numerical solution, as well as to maintain the original order 2s as the generating Gauss formula.