  In this paper, we consider the stochastic Langevin equation with additive
noises, which possesses both conformal symplectic geometric structure and
ergodicity. We propose a methodology of constructing high weak order conformal
symplectic schemes by converting the equation into an equivalent autonomous
stochastic Hamiltonian system and modifying the associated generating function.
To illustrate this approach, we construct a specific second order numerical
scheme, and prove that its symplectic form dissipates exponentially. Moreover,
for the linear case, the proposed scheme is also shown to inherit the
ergodicity of the original system, and the temporal average of the numerical
solution is a proper approximation of the ergodic limit over long time.
Numerical experiments are given to verify these theoretical results.
