This paper presents a unified mathematical framework for quantifying the "acceleration phenomenon" in Langevin Monte Carlo schemes. Specifically, the formalism developed in this work allows one to transfer rapid convergence results for a suitable Ito diffusion to its discretized counterpart using Runge-Kutta numerical integration. The results are of high technical quality, and the ideas presented in the paper open up a number of promising directions for further research. All three reviewers are uniformly enthusiastic about this work, which makes a strong contribution to a topic of considerable current interest.