  We consider the long-time behavior of an explicit tamed Euler scheme applied
to a class of stochastic differential equations driven by additive noise, under
a one-sided Lipschitz continuity condition. The setting encompasses drift
nonlinearities with polynomial growth. First, we prove that moment bounds for
the numerical scheme hold, with at most polynomial dependence with respect to
the time horizon. Second, we apply this result to obtain error estimates, in
the weak sense, in terms of the time-step size and of the time horizon, to
quantify the error to approximate averages with respect to the invariant
distribution of the continuous-time process. We justify the efficiency of using
the explicit tamed Euler scheme to approximate the invariant distribution,
since the computational cost does not suffer from the at most polynomial growth
of the moment bounds. To the best of our knowledge, this is the first result in
the literature concerning the approximation of the invariant distribution for
SDEs with non-globally Lipschitz coefficients using an explicit tamed scheme.
