Analysis of some splitting schemes for the stochastic Allen-Cahn equation

We introduce and analyze an explicit time discretization scheme for the one-dimensional stochastic Allen-Cahn, driven by space-time white noise. The scheme is based on a splitting strategy, and uses the exact solution for the nonlinear term contribution. We first prove boundedness of moments of the numerical solution. We then prove strong convergence results: first, <tex-math id="M1">{document}L²(Ω){document}</tex-math>-convergence of order almost <tex-math id="M2">{document}$1/4${document}</tex-math>, localized on an event of arbitrarily large probability, then convergence in probability of order almost <tex-math id="M3">{document}$1/4${document}</tex-math>. The theoretical analysis is supported by numerical experiments, concerning strong and weak orders of convergence.