A posteriori error estimates for leap-frog and cosine methods for second order evolution problems
Provides rigorous error control for popular time-stepping schemes in wave equation simulations, though the contribution is incremental as it extends existing a posteriori techniques to a specific class of methods.
The paper derives optimal order a posteriori error estimates for leap-frog (Verlet) and cosine-type two-step time discretization methods for wave-type equations, with numerical experiments confirming the estimators' convergence rates match the theoretical error rate.
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.