An Efficient Tamed Milstein Scheme for the Stochastic Allen-Cahn Equation with Multiplicative Noise

This paper investigates the strong convergence of a fully discrete scheme for the stochastic Allen–Cahn equation with multiplicative noise, combining a tamed Milstein method for the temporal discretization with the finite element method in space. The proposed method is shown to be unconditionally stable in spatial dimensions d∈ \1,2,3\ d ∈ { 1 , 2 , 3 } . Beyond the inherent challenges caused by, see, e.g., [1], the cubic non-globally Lipschitz drift term and multiplicative driving noise in the convergence analysis, the Milstein scheme further complicates the error estimation of the noise term compared to the Euler-Maruyama discretization. By introducing a novel auxiliary process, we rigorously establish strong convergence rates in both space and time under mild assumptions for d∈ \1,2\ d ∈ { 1 , 2 } . Our analysis shows that the temporal convergence order is doubled compared to that of tamed Euler-Maruyama scheme. Numerical experiments are provided to confirm the theoretical results and to demonstrate that the proposed scheme exhibits improved robustness over the pure semi-implicit Milstein method.