论文标题
随机cahn--hilliard方程的有限差异方法的收敛分析
Convergence analysis of a finite difference method for stochastic Cahn--Hilliard equation
论文作者
论文摘要
本文介绍了随机cahn- hilliard方程的空间有限差法（FDM）的收敛分析，具有Lipschitz非线性和乘法噪声。基于对离散绿色函数的良好估计，我们证明了空间半差异溶液及其Malliavin衍生产品都具有强收敛顺序$ 1 $。此外，通过显示精确解决方案的负矩估计，我们可以获得空间半差异溶液的密度将$ l^1（\ mathbb r）$收敛到确切的元素。最后，我们应用指数的Euler方法来分散空间半差异数值解决方案，并表明暂时的强收敛顺序几乎是$ \ frac38 $，我们克服的困难就是得出空间半脱污方法的最佳Hölder连续性。
This paper presents the convergence analysis of the spatial finite difference method (FDM) for the stochastic Cahn--Hilliard equation with Lipschitz nonlinearity and multiplicative noise. Based on fine estimates of the discrete Green function, we prove that both the spatial semi-discrete numerical solution and its Malliavin derivative have strong convergence order $1$. Further, by showing the negative moment estimates of the exact solution, we obtain that the density of the spatial semi-discrete numerical solution converges in $L^1(\mathbb R)$ to the exact one. Finally, we apply an exponential Euler method to discretize the spatial semi-discrete numerical solution in time and show that the temporal strong convergence order is nearly $\frac38$, where a difficulty we overcome is to derive the optimal Hölder continuity of the spatial semi-discrete numerical solution.
