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In molecular dynamics, penalized overdamped Langevin dynamics are used tomodel the motion of a set of particles that follow constraints up to aparameter $\varepsilon$. The most used schemes for simulating these dynamicsare the Euler integrator in $\mathbb{R}^d$ and the constrained Eulerintegrator. Both have weak order one of accuracy, but work properly only inspecific regimes depending on the size of the parameter $\varepsilon$. Wepropose in this paper a new consistent method with an accuracy independent of$\varepsilon$ for solving penalized dynamics on a manifold of any dimension.Moreover, this method converges to the constrained Euler scheme when$\varepsilon$ goes to zero. The numerical experiments confirm the theoreticalfindings, in the context of weak convergence and for the invariant measure, ona torus and on the orthogonal group in high dimension and high codimension.

A uniformly accurate scheme for the numerical integration of penalized Langevin dynamics