  We propose a modification of the standard linear implicit Euler integrator
for the weak approximation of parabolic semilinear stochastic PDEs driven by
additive space-time white noise. The new method can easily be combined with a
finite difference method for the spatial discretization. The proposed method is
shown to have improved qualitative properties compared with the standard
method. First, for any time-step size, the spatial regularity of the solution
is preserved, at all times. Second, the proposed method preserves the Gaussian
invariant distribution of the infinite dimensional Ornstein--Uhlenbeck process
obtained when the nonlinearity is absent, for any time-step size. The weak
order of convergence of the proposed method is shown to be equal to $1/2$ in a
general setting, like for the standard Euler scheme. A stronger weak
approximation result is obtained when considering the approximation of a Gibbs
invariant distribution, when the nonlinearity is a gradient: one obtains an
approximation in total variation distance of order $1/2$, which does not hold
for the standard method. This is the first result of this type in the
literature. A key point in the analysis is the interpretation of the proposed
modified Euler scheme as the accelerated exponential Euler scheme applied to a
modified stochastic evolution equation. Finally, it is shown that the proposed
method can be applied to design an asymptotic preserving scheme for a class of
slow-fast multiscale systems, and to construct a Markov Chain Monte Carlo
method which is well-defined in infinite dimension. We also revisit the
analysis of the standard and the accelerated exponential Euler scheme, and we
prove new results with approximation in the total variation distance, which
serve to illustrate the behavior of the proposed modified Euler scheme.
