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Consider the approximation of stochastic Allen-Cahn-type equations (i.e.$1+1$-dimensional space-time white noise-driven stochastic PDEs with polynomialnonlinearities $F$ such that $F(\pm \infty)=\mp \infty$) by a fully discretespace-time explicit finite difference scheme. The consensus in literature,supported by rigorous lower bounds, is that strong convergence rate $1/2$ withrespect to the parabolic grid meshsize is expected to be optimal. We show thatone can reach almost sure convergence rate $1$ (and no better) when measuringthe error in appropriate negative Besov norms, by temporarily `pretending' thatthe SPDE is singular.

Strong convergence of parabolic rate $1$ of discretisations of stochastic Allen-Cahn-type equations