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

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  • Liu, Zhixin
  • Song, Minghui
  • Zhang, Yuhang

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.

Suggested Citation

  • Liu, Zhixin & Song, Minghui & Zhang, Yuhang, 2026. "Milstein virtual element methods for the stochastic Allen–Cahn equation with multiplicative noise on polygonal meshes," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 246(C), pages 356-378.
    • Handle: RePEc:eee:matcom:v:246:y:2026:i:c:p:356-378
    DOI: 10.1016/j.matcom.2026.02.005
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    References listed on IDEAS

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    1. Becker, Sebastian & Jentzen, Arnulf, 2019. "Strong convergence rates for nonlinearity-truncated Euler-type approximations of stochastic Ginzburg–Landau equations," Stochastic Processes and their Applications, Elsevier, vol. 129(1), pages 28-69.
    2. Adak, D. & Natarajan, S., 2020. "Virtual element method for semilinear sine–Gordon equation over polygonal mesh using product approximation technique," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 172(C), pages 224-243.
    3. Cui, Jianbo & Hong, Jialin & Sun, Liying, 2021. "Weak convergence and invariant measure of a full discretization for parabolic SPDEs with non-globally Lipschitz coefficients," Stochastic Processes and their Applications, Elsevier, vol. 134(C), pages 55-93.
    4. Wang, Xiaojie, 2020. "An efficient explicit full-discrete scheme for strong approximation of stochastic Allen–Cahn equation," Stochastic Processes and their Applications, Elsevier, vol. 130(10), pages 6271-6299.
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