An efficient approximation to the stochastic Allen-Cahn equation with random diffusion coefficient field and multiplicative noise

This paper studies the stochastic Allen-Cahn equation involving random diffusion coefficient field and multiplicative force noise. A new time-stepping method based on auxiliary variable approach is proposed and analyzed. The proposed method is efficient thanks to its low computational complexity. Furthermore, it is unconditionally stable in the sense that a discrete energy is dissipative when the multiplicative noise is absent. Our numerical experiments show that the new scheme is much more robust than the classical semi-implicit Euler-Maruyama scheme, particularly when the interface width parameter is small. Several numerical examples are provided to demonstrate the performance of the proposed method.