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We discretize the stochastic Allen-Cahn equation with additive noise by meansof a spectral Galerkin method in space and a tamed version of the exponentialEuler method in time. The resulting error bounds are analyzed for thespatio-temporal full discretization in both strong and weak senses. Differentfrom existing works, we develop a new and direct approach for the weak erroranalysis, which does not rely on the use of the associated Kolmogorov equationor It\^{o}'s formula and is therefore non-Markovian in nature. Such an approachthus has a potential to be applied to non-Markovian equations such asstochastic Volterra equations or other types of fractional SPDEs, which sufferfrom the lack of Kolmogorov equations. It turns out that the obtained weakconvergence rates are, in both spatial and temporal direction, essentiallytwice as high as the strong convergence rates. Also, it is revealed how theweak convergence rates depend on the regularity of the noise. Numericalexperiments are finally reported to confirm the theoretical conclusion.

Weak convergence rates for an explicit full-discretization of stochastic Allen-Cahn equation with additive noise