Numerical approximation of the stochastic Cahn–Hilliard equation with space–time white noise near the sharp-interface limit

We consider the stochastic Cahn–Hilliard equation with additive space–time white noise ε γẆ in dimension $d=2,3$, where ε>0 is an interfacial width parameter. We study a numerical approximation of the equation that combines a structure-preserving implicit time-discretization scheme with a discrete approximation of the space–time white noise. We derive a strong error estimate for the considered numerical approximation that is robust with respect to the inverse of the interfacial width parameter ε. Furthermore, by a splitting approach we show that for sufficiently large scaling parameter γ the numerical approximation of the stochastic Cahn–Hilliard equation converges uniformly to the deterministic Hele-Shaw/Mullins–Sekerka problem in the sharp-interface limit ε → 0.