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It is known that when the diffuse interface thickness $\epsilon$ vanishes,the sharp interface limit of the stochastic reaction-diffusion equation isformally a stochastic geometric flow. To capture and simulate such geometricflow, it is crucial to develop numerical approximations whose error boundsdepends on $\frac 1\epsilon$ polynomially. However, due to loss of spectralestimate of the linearized stochastic reaction-diffusion equation, how to getsuch error bound of numerical approximation has been an open problem.  In this paper, we solve this weak error bound problem for stochasticreaction-diffusion equations near sharp interface limit. We first introduce aregularized problem which enjoys the exponential ergodicity. Then we presentthe regularity analysis of the regularized Kolmogorov and Poisson equationswhich only depends on $\frac 1{\epsilon}$ polynomially. Furthermore, weestablish such weak error bound. This phenomenon could be viewed as a kind ofthe regularization effect of noise on the numerical approximation of stochasticpartial differential equation (SPDE). As a by-product, a central limit theoremof the weak approximation is shown near sharp interface limit. Our method ofproof could be extended to a number of other spatial and temporal numericalapproximations for semilinear SPDEs.

Weak approximation for stochastic Reaction-diffusion equation near sharp interface limit