Energy and quadratic invariants preserving integrators of Gaussian type

Recently, a whole class of evergy-preserving integrators has been derived for the numerical solution of Hamiltonian problems [3, 2, 4]. In the mainstream of this research [6], we have defined a now family of symplectic integrators depending On a real parameter alpha [7]. For alpha = 0, the corresponding method in the family becomes the classical Gauss collocation formula of order 2s, where s denotes the number of the internal stages. For any given non-null alpha, the corresponding method remains symplectic and has order 2s - 2: hence it may be interpreted as a O((2s-2)) (symplectic) perturbation of the Gauss method. Under suitable assumptions, it can be shown that the parameter alpha may be properly tuned, at each step of the integration procedure, so as to guarantee energy conservation in the numerical solution. The resulting method shares the same order 2s as the generating Gauss formula, and is able to preserve both energy and quadratic invariants.