On an asymptotic method for computing the modified energy for symplectic methods

We revisit an algorithm by Skeel et al. [5,16] for computing the modified, or shadow, energy associated with symplectic discretizations of Hamiltonian systems. We amend the algorithm to use Richardson extrapolation in order to obtain arbitrarily high order of accuracy. Error estimates show that the new method captures the exponentially small drift associated with such discretizations. Several numerical examples illustrate the theory