A Modified Crank-Nicolson Numerical Scheme for the Flory-Huggins Cahn-Hilliard Model  

作　　者：Wenbin Chen Jianyu Jing Cheng Wang Xiaoming Wang Steven M.Wise 

机构地区：[1]Shanghai Key Laboratory for Contemporary Applied Mathematics,Fudan University,Shanghai 200433,P.R.China [2]School of Mathematical Sciences,Fudan University,Shanghai 200433,P.R.China [3]Mathematics Department,University of Massachusetts,North Dartmouth,MA 02747,USA [4]International Center for Mathematics and Department of Mathematics,and Guangdong Provincial Key Laboratory of Computational Science and Material Design,and National Center for Applied Mathematics Shenzhen,Southern University of Science and Technology,Shenzhen 518055,P.R.China [5]Mathematics Department,University of Tennessee,Knoxville,TN 37996,USA

出　　处：《Communications in Computational Physics》2022年第1期60-93,共34页计算物理通讯（英文）

基　　金：This work is supported in part by the grants NSFC 12071090(W.Chen);NSF DMS-2012669(C.Wang);NSFC 11871159;Guangdong Provincial Key Laboratory for Computational Science and Material Design 2019B030301001(X.Wang);NSF DMS-1719854,DMS-2012634(S.Wise).C.Wang also thanks the Key Laboratory of Mathematics for Nonlinear Sciences,Fudan University,for the support.

摘　　要：In this paper we propose and analyze a second order accurate numericalscheme for the Cahn-Hilliard equation with logarithmic Flory Huggins energy potential. A modified Crank-Nicolson approximation is applied to the logarithmic nonlinear term, while the expansive term is updated by an explicit second order AdamsBashforth extrapolation, and an alternate temporal stencil is used for the surface diffusion term. A nonlinear artificial regularization term is added in the numerical scheme,which ensures the positivity-preserving property, i.e., the numerical value of the phasevariable is always between -1 and 1 at a point-wise level. Furthermore, an unconditional energy stability of the numerical scheme is derived, leveraging the special formof the logarithmic approximation term. In addition, an optimal rate convergence estimate is provided for the proposed numerical scheme, with the help of linearizedstability analysis. A few numerical results, including both the constant-mobility andsolution-dependent mobility flows, are presented to validate the robustness of the proposed numerical scheme.

关 键 词：Cahn-Hilliard equation Flory Huggins energy potential positivity preserving energy stability second order accuracy optimal rate convergence estimate 

分 类 号：O17[理学—数学]