论文标题
随机cahn--hilliard方程的完全离散有限差异方法的密度收敛
Density convergence of a fully discrete finite difference method for stochastic Cahn--Hilliard equation
论文作者
论文摘要
本文侧重于研究完全离散有限差异方法的密度收敛性，用于求解由乘法时空白音驱动的随机cahn--hilliard方程。主要困难在于控制既不是全球Lipschitz也不是单方面Lipschitz的漂移系数。为了解决这一难度，我们提出了一个新的本地化参数，并得出了数值解的强收敛速率，以估计精确和数值解之间的总变化距离。这与数值解的密度的存在最终产生了数值解的$ l^1（\ Mathbb {r}）$中的密度的收敛性。我们的结果部分地回答了[J. Cui和J. Hong，J。微分方程（2020）]在数值上计算精确溶液的密度。
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.
