  In this article, we study the density function of the numerical solution of
the splitting averaged vector field (AVF) scheme for the stochastic Langevin
equation. To deal with the non-globally monotone coefficient in the considered
equation, we first present the exponential integrability properties of the
exact and numerical solutions. Then we show the existence and smoothness of the
density function of the numerical solution by proving its uniform
non-degeneracy in Malliavin sense. In order to analyze the approximate error
between the density function of the exact solution and that of the numerical
solution, we derive the optimal strong convergence rate in every
Malliavin--Sobolev norm of the numerical scheme via Malliavin calculus.
Combining the approximation result of Donsker's delta function and the
smoothness of the density functions, we prove that the convergence rate in
density coincides with the optimal strong convergence rate of the numerical
scheme.
