Weak error expansion of a stopped numerical scheme for singular Langevin process

TL;DR

This paper develops a weak error expansion for a stopped numerical scheme applied to a singular Langevin process, including Lennard-Jones potentials, by analyzing the associated semi-group to improve understanding of numerical accuracy in molecular dynamics simulations.

Contribution

It introduces a weak error expansion for a numerical scheme in the context of singular potentials, extending previous methods to more complex, physically relevant cases.

Findings

Derived error estimates for the scheme with Lennard-Jones potentials

Provided semi-group estimates for singular Langevin processes

Validated the expansion approach for molecular dynamics applications

Abstract

We show expansion \textit{\`a la Talay-Tubaro} of a stopped numerical scheme for the Langevin process in the case of a singular potential. In order to achieve this, we provide estimates on the associated semi-group of the process. The class of admissible potentials includes the Lennard-Jones interaction with confinement, which is an important potential in molecular dynamics and served as the primary motivation for this study.