In this paper, we propose and analyze an explicit time-stepping scheme for a spatial discretization of stochastic Cahn--Hilliard equation with additive noise. The fully discrete approximation combines a spectral Galerkin method in space with a tamed exponential Euler method in time. In contrast to implicit schemes in the literature, the explicit scheme here is easily implementable and produces significant improvement in the computational efficiency. It is shown that the fully discrete approximation converges strongly to the exact solution, with strong convergence rates identified. Different from the tamed time-stepping schemes for stochastic Allen--Cahn equations, essential difficulties arise in the analysis due to the presence of the unbounded linear operator in front of the nonlinearity. To overcome them, new and non-trivial arguments are developed in the present work. To the best of our knowledge, it is the first result concerning an explicit scheme for the stochastic Cahn--Hilliard equation. Numerical experiments are finally performed to confirm the theoretical results.

翻译：本文提出并分析了一种用于空间离散化的带加性噪声随机Cahn-Hilliard方程的显式时间步进格式。全离散近似结合了空间上的谱Galerkin方法与时间上的驯化指数Euler方法。与文献中隐式格式相比，本文的显式格式易于实现，并显著提升了计算效率。研究表明，该全离散近似强收敛于精确解，并确定了强收敛率。与随机Allen-Cahn方程的驯化时间步进格式不同，由于非线性项前存在无界线性算子，分析中出现了本质性困难。为克服这些困难，本文发展了新颖且非平凡的论证方法。据我们所知，这是关于随机Cahn-Hilliard方程显式格式的首个研究成果。最后通过数值实验验证了理论结果。