Milstein virtual element methods for the stochastic Allen–Cahn equation with multiplicative noise on polygonal meshes

Zhixin Liu, Minghui Song and Yuhang Zhang

Mathematics and Computers in Simulation (MATCOM), 2026, vol. 246, issue C, 356-378

Abstract: This paper proposes and analyzes two fully discrete schemes for solving the stochastic Allen–Cahn equation with multiplicative noise, employing different time-stepping strategies for the nonlinear term. The methods combine the implicit Milstein method for temporal discretization with the virtual element method for spatial discretization, offering two key advantages: higher-order temporal accuracy and compatibility with general polygonal meshes. We establish the pth (p≥2) moment stabilities and strong error estimates in the L2-norm for both schemes. The analysis relies on Hölder continuity estimates for the solution and rigorous estimates for the nonlinear terms. Finally, several numerical experiments are presented to verify the theoretical results and to investigate the influence of noise on geometric evolution.

Keywords: Stochastic Allen–Cahn equation; Virtual element method; Milstein method; Polygonal mesh (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:246:y:2026:i:c:p:356-378

DOI: 10.1016/j.matcom.2026.02.005

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