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 - <p style="text-align: justify;">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.</p>