We consider the application of a symmetric Boundary Value Method to solve the linear Hamiltonian Boundary Value Problem

on a uniform mesh with stepsize h (S is symmetric and positive definite). Concerning the approximation of the Hamiltonian function, symmetric schemes of even order (ETRs, ETR₂s, TOMs) exhibit a superconvergence property. Furthermore, if we neglect the effect of the additional methods (that act as perturbations on the behavior of the symmetric formula), we can state the following correspondences between the continuous and the discrete problems:

• the solution of (1), projected on the given mesh, may be regarded as a solution generated by a symmetric BVM of even order p applied to a Hamiltonian problem obtained perturbing (1) by an Hamiltonian term O(h^p);
• the solution of a symmetric BVM of even order p applied to (1) may be regarded as the projection on the mesh of the solution of a Hamiltonian problem obtained perturbing (1) by a Hamiltonian term O(h^p).

The additional methods work as a perturbation over this favourable situation, but their effect disappear very quickly as long as we move towards the middle of the time integration interval.

Footnotes:

¹ Joint work with Pierluigi AMODIO and Donato TRIGIANTE.