  We compare the performance of several discretizations of the simple pendulum
equation in a series of numerical experiments. The stress is put on the
long-time behaviour. We choose for the comparison numerical schemes which
preserve the qualitative features of solutions (like periodicity). All these
schemes are either symplectic maps or integrable (preserving the energy
integral) maps, or both. We describe and explain systematic errors (produced by
any method) in numerical computations of the period and the amplitude of
oscillations. We propose a new numerical scheme which is a modification of the
discrete gradient method. This discretization preserves (almost exactly) the
period of small oscillations for any time step.
