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<p>We propose a novel explicit fully discrete scheme for the stochastic Allen–Cahn equation (SACE) driven by an additive <i>Q</i>-Wiener process. The scheme combines the spectral Galerkin method for spatial discretization with a tamed accelerated exponential integrator for temporal approximation. It is designed to alleviate the computational cost of implicit methods, enabling efficient long-time simulations and explicit approximation of the invariant measure. Uniform moment bounds and uniform weak convergence of the numerical solution are established, both essential for approximating the invariant measure. A refined taming strategy yields time-independent weak error estimates, and the weak convergence rate is shown to be sharp, improving upon existing results. Furthermore, by combining weak convergence analysis with exponential ergodicity, we derive an error bound for the approximation of the invariant measure. Theoretical results confirm the efficiency and accuracy of the proposed method in capturing the long-term statistical behavior of the SACE.</p>

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Approximation of the invariant measure for the stochastic Allen–Cahn equation via an explicit fully discrete scheme

  • 期刊论文

  • 发表日期：

    2026年4月20日

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  • Yibo Wang,
  • Wanrong Cao

摘要

We propose a novel explicit fully discrete scheme for the stochastic Allen–Cahn equation (SACE) driven by an additive Q-Wiener process. The scheme combines the spectral Galerkin method for spatial discretization with a tamed accelerated exponential integrator for temporal approximation. It is designed to alleviate the computational cost of implicit methods, enabling efficient long-time simulations and explicit approximation of the invariant measure. Uniform moment bounds and uniform weak convergence of the numerical solution are established, both essential for approximating the invariant measure. A refined taming strategy yields time-independent weak error estimates, and the weak convergence rate is shown to be sharp, improving upon existing results. Furthermore, by combining weak convergence analysis with exponential ergodicity, we derive an error bound for the approximation of the invariant measure. Theoretical results confirm the efficiency and accuracy of the proposed method in capturing the long-term statistical behavior of the SACE.