This article investigates the weak approximation towards the invariant
measure of semi-linear stochastic differential equations (SDEs) under
non-globally Lipschitz coefficients. For this purpose, we propose a
linear-theta-projected Euler (LTPE) scheme, which also admits an invariant
measure, to handle the potential influence of the linear stiffness. Under
certain assumptions, both the SDE and the corresponding LTPE method are shown
to converge exponentially to the underlying invariant measures, respectively.
Moreover, with time-independent regularity estimates for the corresponding
Kolmogorov equation, the weak error between the numerical invariant measure and
the original one can be guaranteed with an order one. Numerical experiments are
provided to verify our theoretical findings.Comment: 45 pages, 7 figure