The BDF2-Maruyama method for the stochastic Allen-Cahn equation with multiplicative noise

We investigate the numerical approximation of the stochastic Allen-Cahn equation on a bounded domain D under Dirichlet boundary conditions and with multiplicative noise. The considered numerical method combines the two-step backward differentiation formula (BDF2) for the temporal discretization in conjunction with an abstract Galerkin scheme for the spatial approximation. In dependence on the regularity of the exact solution we derive a rate of convergence for the BDF2-Maruyama method with respect to the root-mean-square error in discrete analogues of the spaces L-infinity([0, T]; L-2(D)) and L-2([0, T]; H-0(1)(D)). Our error analysis is based on the variational approach for stochastic evolution equations. Finally, several numerical experiments illustrate our theoretical results, where a finite element method is used as an example for a Galerkin scheme. (C) 2022 The Author(s). Published by Elsevier B.V.