NEARLY CONSERVATIVE MULTI-VALUE NUMERICAL METHODS FOR HAMILTONIAN PROBLEMS

This talk is devoted to the analysis of multi-value methods for the numerical integration of Hamiltonian problems. Even if the numerical flow generated by such a method cannot be symplectic, a concept of near conservation can be considered, i.e. G-symplecticity, which implies conjugate-symplecticity of the underlying one step method associated to the original multi-value scheme. It is known that multi-value methods introduce parasitic components in the numerical solution, deteriorating the overall accuracy and the ability of preserving the invariants of Hamiltonian systems; however, a remedy against the effects of parasitism is discussed, together with its longterm counterpart: indeed, a backward error analysis is presented, which permits to get sharp estimates for the parasitic solution components and for the error in the Hamiltonian. Symmetry of the numerical schemes is also analyzed as a practical tool for the construction of high order methods and an effective property for reversible mechanical systems. Numerical experiments confirming the theoretical expecations are given.