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

Abstract 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 $$d\in \{1,2,3\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>{</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> . 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 $$d\in \{1,2\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>{</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> . 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.