A posteriori error estimates for leap-frog and cosine methods for second   order evolution problems
  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. 
Emmanuil H. Georgoulis and Omar Lakkis and Charalambos Makridakis and   Juha M. Virtanen
math.NA