This paper focuses on investigating the density convergence of a fully discrete finite difference method when applied to numerically solve the stochastic Cahn--Hilliard equation driven by multiplicative space-time white noises. The main difficulty lies in the control of the drift coefficient that is neither globally Lipschitz nor one-sided Lipschitz. To handle this difficulty, we propose a novel localization argument and derive the strong convergence rate of the numerical solution to estimate the total variation distance between the exact and numerical solutions. This along with the existence of the density of the numerical solution finally yields the convergence of density in $L^1(\mathbb{R})$ of the numerical solution. Our results partially answer positively to the open problem emerged in [J. Cui and J. Hong, J. Differential Equations (2020)] on computing the density of the exact solution numerically.

翻译：本文聚焦于研究数值求解由乘性时空白噪声驱动的随机Cahn-Hilliard方程时，全离散有限差分方法的密度收敛性。主要难点在于漂移系数既非全局Lipschitz连续也非单边Lipschitz连续。为克服这一困难，我们提出了一种新颖的局部化论证方法，并推导出数值解的强收敛速度以估计精确解与数值解之间的总变差距离。结合数值解密度的存在性，最终得到数值解在$L^1(\mathbb{R})$空间中的密度收敛性。我们的结果部分正面回应了[J. Cui and J. Hong, J. Differential Equations (2020)]中关于数值计算精确解密度的开放性问题。