Long time stability and strong convergence of an efficient tamed scheme for stochastic Allen-Cahn equation driven by additive white noise

Huang and Shen [ Math. Comput. 92 (2023) 2685–2713] proposed a semi-implicit tamed scheme for the numerical approximation of stochastic Allen–Cahn equations driven by multiplicative trace-class noise. They showed that the scheme is unconditionally stable on finite time intervals and can be efficiently implemented. In this paper, we investigate the long-time stability of this tamed scheme for stochastic Allen–Cahn equations driven by additive white noise. We also address the strong convergence analysis of the associated fully discrete scheme within the Galerkin finite element framework. The main contributions of this work are as follows: (i) by constructing a suitable Lyapunov functional, we establish the unconditional long-time stability of the tamed method; (ii) we rigorously derive the strong convergence rates of the fully discrete scheme obtained by coupling the tamed approach with the finite element method. Numerical experiments are provided to validate the theoretical analysis and demonstrate the effectiveness of the proposed scheme.