In long-time numerical integration of Hamiltonian systems,
and especially in molecular dynamics simulation,
it is important that the energy is well conserved. For symplectic
integrators applied with sufficiently small step size, this
is guaranteed by the existence of a modified
Hamiltonian that is exactly conserved up to exponentially small
terms. This article is concerned with the simplified
Takahashi-Imada method, which is a modification
of the Störmer-Verlet method that is as easy to implement but
has improved accuracy. This integrator is symmetric and
volume-preserving, but no longer symplectic. We study its
long-time energy conservation and give theoretical
arguments, supported by numerical experiments, which
show the possibility of a drift in the energy (linear or like a random walk).
With respect to energy conservation, this article provides empirical
and theoretical data concerning the importance of using a symplectic
integrator.