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Volume 24, Issue 3
Stabilized Predictor-Corrector Schemes for Gradient Flows with Strong Anisotropic Free Energy

Jie Shen & Jie Xu

DOI: 10.4208/cicp.OA-2017-0209

Commun. Comput. Phys., 24 (2018), pp. 635-654.

Published online: 2018-05

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

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  • Abstract

Gradient flows with strong anisotropic free energy are difficult to deal with numerically with existing approaches. We propose a stabilized predictor-corrector approach to construct schemes which are second-order accurate, easy to implement, and maintain the stability of first-order stabilized schemes. We apply the new approach to three different types of gradient flows with strong anisotropic free energy: anisotropic diffusion equation, anisotropic Cahn-Hilliard equation, and Cahn-Hilliard equation with degenerate diffusion mobility. Numerical results are presented to show that the stabilized predictor-corrector schemes are second-order accurate, unconditionally stable for the first two equations, and allow larger time step than the first-order stabilized scheme for the last equation. We also prove rigorously that, for the isotropic Cahn-Hilliard equation, the stabilized predictor-corrector scheme is of second-order.

  • Keywords

Predictor-corrector, anisotropy, Cahn-Hilliard equation, Willmore regularization, degenerate diffusion mobility.

  • AMS Subject Headings

65M12, 82C24

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-24-635, author = {Jie Shen and Jie Xu}, title = {Stabilized Predictor-Corrector Schemes for Gradient Flows with Strong Anisotropic Free Energy}, journal = {Communications in Computational Physics}, year = {2018}, volume = {24}, number = {3}, pages = {635--654}, abstract = {

Gradient flows with strong anisotropic free energy are difficult to deal with numerically with existing approaches. We propose a stabilized predictor-corrector approach to construct schemes which are second-order accurate, easy to implement, and maintain the stability of first-order stabilized schemes. We apply the new approach to three different types of gradient flows with strong anisotropic free energy: anisotropic diffusion equation, anisotropic Cahn-Hilliard equation, and Cahn-Hilliard equation with degenerate diffusion mobility. Numerical results are presented to show that the stabilized predictor-corrector schemes are second-order accurate, unconditionally stable for the first two equations, and allow larger time step than the first-order stabilized scheme for the last equation. We also prove rigorously that, for the isotropic Cahn-Hilliard equation, the stabilized predictor-corrector scheme is of second-order.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2017-0209}, url = {http://global-sci.org/intro/article_detail/cicp/12274.html} }
TY - JOUR T1 - Stabilized Predictor-Corrector Schemes for Gradient Flows with Strong Anisotropic Free Energy AU - Jie Shen & Jie Xu JO - Communications in Computational Physics VL - 3 SP - 635 EP - 654 PY - 2018 DA - 2018/05 SN - 24 DO - http://doi.org/10.4208/cicp.OA-2017-0209 UR - https://global-sci.org/intro/article_detail/cicp/12274.html KW - Predictor-corrector, anisotropy, Cahn-Hilliard equation, Willmore regularization, degenerate diffusion mobility. AB -

Gradient flows with strong anisotropic free energy are difficult to deal with numerically with existing approaches. We propose a stabilized predictor-corrector approach to construct schemes which are second-order accurate, easy to implement, and maintain the stability of first-order stabilized schemes. We apply the new approach to three different types of gradient flows with strong anisotropic free energy: anisotropic diffusion equation, anisotropic Cahn-Hilliard equation, and Cahn-Hilliard equation with degenerate diffusion mobility. Numerical results are presented to show that the stabilized predictor-corrector schemes are second-order accurate, unconditionally stable for the first two equations, and allow larger time step than the first-order stabilized scheme for the last equation. We also prove rigorously that, for the isotropic Cahn-Hilliard equation, the stabilized predictor-corrector scheme is of second-order.

Jie Shen and Jie Xu. (2018). Stabilized Predictor-Corrector Schemes for Gradient Flows with Strong Anisotropic Free Energy. Communications in Computational Physics. 24 (3). 635-654. doi:10.4208/cicp.OA-2017-0209
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