Conservative Block-Boundary Value Methods for the solution of Polynomial Hamiltonian Systems

We devise a technique to derive high order one-step methods suitable for the precise conservation of Hamiltonian functions of polynomial type. In many cases of interest Hamiltonian functions are polynomials of the variables p (conjugate momenta) and q (generalized coordinates), or they may be well approximated by polynomials. In deriving this class of methods, the key idea is to exploit the relation between the method itself and what we called discrete line integral, the discrete counterpart of the the line integral in conservative vector fields. This approach naturally suggests a formulation of such methods in terms of block Boundary Value Methods.