  We consider second order explicit and implicit two-step time-discrete schemes
for wave-type equations. We derive optimal order aposteriori estimates
controlling the time discretization error. Our analysis, has been motivated by
the need to provide aposteriori estimates for the popular leap-frog method
(also known as Verlet's method in molecular dynamics literature); it is
extended, however, to general cosine-type second order methods. The estimators
are based on a novel reconstruction of the time-dependent component of the
approximation. Numerical experiments confirm similarity of convergence rates of
the proposed estimators and of the theoretical convergence rate of the true
error.
