A posteriori error estimates for leap-frog and cosine methods for second order evolution problems

Abstract

We consider second order explicit and implicit two-step time-discrete schemes for wave-type equations. We derive optimal order a posteriori estimates controlling the time discretization error. Our analysis has been motivated by the need to provide a posteriori estimates for the popular leap-frog method (also known as Verlet's method in the 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 the convergence rates of the proposed estimators and the theoretical convergence rate of the true error.