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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.
}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v56n4.23.05}, url = {http://global-sci.org/intro/article_detail/jms/22255.html} }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.