Symmetric schemes, time reversal symmetry and conservative methods for Hamiltonian systems

It is important, when integrating numerically Hamiltonian problems, that the numerical methods retain some properties of the continuous problem such as the constants of motion and the time reversal symmetry. This may be a difficult task for multistep numerical methods. In the present paper we discuss the problem in the case of linear autonomous Hamiltonian systems and we show the equivalence among the symmetry of the numerical methods and the above-mentioned requirements. In particular, the analysis is carried out for the class of methods known as boundary value methods (BVMs) (Brugnano, Trigiante. Solving Differential Problems by Multistep Initial and Boundary Value Methods. Gordon and Breach Science Publishers, Amsterdam, 1998).