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}. 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}. 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.

Citation

Qi, X., Dewhirst, G., & Yan, Y. (2026). An efficient tamed Milstein Scheme for the Stochastic Allen-Cahn equation with multiplicative noise. Journal of Scientific Computing, 107(1), 10. https://doi.org/10.1007/s10915-026-03218-7

Publisher

Springer Nature

Journal

Journal of Scientific Computing

URI

https://chesterrep.openrepository.com/handle/10034/629897

DOI

10.1007/s10915-026-03218-7

Type

Article

Language

en

Description

©The Author(s) 2026.
The version of record of this article, first published in [Journal of Scientific Computing], is available online at Publisher’s website: http://dx.doi.org/10.1007/s10915-026-03218-7

ISSN

0885-7474

EISSN

1573-7691

Additional Links

https://link.springer.com/article/10.1007/s10915-026-03218-7