TY - JOUR T1 - Extended Milstein Approximation to the Stochastic Allen-Cahn Equation with Random Diffusion Coefficient Field and Multiplicative Noise AU - Qi , Xiao JO - Journal of Mathematical Study VL - 4 SP - 366 EP - 391 PY - 2023 DA - 2023/12 SN - 56 DO - http://doi.org/10.4208/jms.v56n4.23.05 UR - https://global-sci.org/intro/article_detail/jms/22255.html KW - Stochastic Allen-Cahn equation, multiplicative noise, strong convergence, extended Milstein scheme, stability. AB -

This paper studies the stochastic Allen-Cahn equation driven by a random diffusion coefficient field and multiplicative force noise. A new time-stepping scheme based on a stabilized approach and Milstein scheme is proposed and analyzed. The proposed method is unconditionally stable in the sense that a discrete energy is dissipative when the multiplicative noise is absent. The strong convergence rate of a spatio-temporal fully discrete scheme is derived. Numerical experiments are finally reported to confirm the theoretical result and show that the new scheme is much more robust than the classical semi-implicit Euler-Maruyama scheme, especially when the interface width parameter is small.