TY - JOUR
T1 - A Modified Crank-Nicolson Numerical Scheme for the Flory-Huggins Cahn-Hilliard Model
AU - Chen , Wenbin
AU - Jianyu Jing , 
AU - Cheng Wang , 
AU - Wang , Xiaoming
AU - Wise , Steven M.
JO - Communications in Computational Physics
VL - 1
SP - 60
EP - 93
PY - 2021
DA - 2021/12
SN - 31
DO - http://doi.org/10.4208/cicp.OA-2021-0074
UR - https://global-sci.org/intro/article_detail/cicp/20018.html
KW - Cahn-Hilliard equation, Flory Huggins energy potential, positivity preserving, energy stability, second order accuracy, optimal rate convergence estimate.
AB - <p style="text-align: justify;">In this paper we propose and analyze a second order accurate numerical
scheme 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 Adams-Bashforth 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 phase
variable 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 form
of the logarithmic approximation term. In addition, an optimal rate convergence estimate is provided for the proposed numerical scheme, with the help of linearized
stability analysis. A few numerical results, including both the constant-mobility and
solution-dependent mobility flows, are presented to validate the robustness of the proposed numerical scheme.</p>