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

We show expansion $\grave{\mathit{\text{a}}}\mathit{\text{la}}\mathit{\text{Talay-Tubaro}}$ 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.