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Volume 31, Issue 1
A Modified Crank-Nicolson Numerical Scheme for the Flory-Huggins Cahn-Hilliard Model

Wenbin Chen, Jianyu Jing, Cheng Wang, Xiaoming Wang & Steven M. Wise

DOI: 10.4208/cicp.OA-2021-0074

Commun. Comput. Phys., 31 (2022), pp. 60-93.

Published online: 2021-12

[An open-access article; the PDF is free to any online user.]

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  • Abstract

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.

  • Keywords

Cahn-Hilliard equation, Flory Huggins energy potential, positivity preserving, energy stability, second order accuracy, optimal rate convergence estimate.

  • AMS Subject Headings

35K35, 35K55, 49J40, 65K10, 65M06, 65M12

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

wbchen@fudan.edu.cn (Wenbin Chen)

cwang1@umassd.edu ( Cheng Wang)

wangxm@sustech.edu.cn (Xiaoming Wang)

  • BibTex
  • RIS
  • TXT
@Article{CiCP-31-60, author = {Chen , WenbinJianyu Jing , Cheng Wang , Wang , Xiaoming and Wise , Steven M.}, title = {A Modified Crank-Nicolson Numerical Scheme for the Flory-Huggins Cahn-Hilliard Model}, journal = {Communications in Computational Physics}, year = {2021}, volume = {31}, number = {1}, pages = {60--93}, abstract = {

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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2021-0074}, url = {http://global-sci.org/intro/article_detail/cicp/20018.html} }
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 -

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.

Chen , WenbinJianyu Jing , Cheng Wang , Wang , Xiaoming and Wise , Steven M.. (2021). A Modified Crank-Nicolson Numerical Scheme for the Flory-Huggins Cahn-Hilliard Model. Communications in Computational Physics. 31 (1). 60-93. doi:10.4208/cicp.OA-2021-0074
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