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

TL;DR

This paper develops optimal a posteriori error estimates for leap-frog and cosine methods applied to second order wave equations, enabling better error control and adaptive time-stepping.

Contribution

It introduces a novel reconstruction-based a posteriori error estimator for leap-frog and cosine methods, extending existing analysis to general second order schemes.

Findings

Estimators accurately predict the true error with similar convergence rates.

Numerical experiments validate the effectiveness of the proposed error estimators.

The approach applies to both explicit and implicit second order schemes.

Abstract

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.