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We propose a modification of the standard linear implicit Euler integratorfor the weak approximation of parabolic semilinear stochastic PDEs driven byadditive space-time white noise. The new method can easily be combined with afinite difference method for the spatial discretization. The proposed method isshown to have improved qualitative properties compared with the standardmethod. First, for any time-step size, the spatial regularity of the solutionis preserved, at all times. Second, the proposed method preserves the Gaussianinvariant distribution of the infinite dimensional Ornstein--Uhlenbeck processobtained when the nonlinearity is absent, for any time-step size. The weakorder of convergence of the proposed method is shown to be equal to $1/2$ in ageneral setting, like for the standard Euler scheme. A stronger weakapproximation result is obtained when considering the approximation of a Gibbsinvariant distribution, when the nonlinearity is a gradient: one obtains anapproximation in total variation distance of order $1/2$, which does not holdfor the standard method. This is the first result of this type in theliterature. A key point in the analysis is the interpretation of the proposedmodified Euler scheme as the accelerated exponential Euler scheme applied to amodified stochastic evolution equation. Finally, it is shown that the proposedmethod can be applied to design an asymptotic preserving scheme for a class ofslow-fast multiscale systems, and to construct a Markov Chain Monte Carlomethod which is well-defined in infinite dimension. We also revisit theanalysis of the standard and the accelerated exponential Euler scheme, and weprove new results with approximation in the total variation distance, whichserve to illustrate the behavior of the proposed modified Euler scheme.

Analysis of a modified Euler scheme for parabolic semilinear stochastic PDEs