  We consider the long-time behavior of an explicit tamed exponential Euler
scheme applied to a class of parabolic semilinear stochastic partial
differential equations driven by additive noise, under a one-sided Lipschitz
continuity condition. The setting encompasses 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 exponential
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 SPDEs with
non-globally Lipschitz coefficients using an explicit tamed scheme.
