  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.
