Multi-value Numerical Methods for Hamiltonian Systems

We discuss the effectiveness of multi-value numerical methods in the numerical treatment of Hamiltonian problems. Multi-value (or general linear) methods extend the well-known families of Runge-Kutta and linear multistep methods and can be considered as a general framework for the numerical solution of ordinary differential equations. There are some features that needs to be achieved by reliable geometric numerical integrators based on multi-value methods: G-symplecticity, symmetry and boundedness of the parasitic components. In particular, we analyze the effects of the mentioned features for the long term conservation of the energy and provide the numerical evidence confirming the theoretical expectations.