Volume 28, Issue 3
A Positivity-Preserving Second-Order BDF Scheme for the Cahn-Hilliard Equation with Variable Interfacial Parameters

Lixiu Dong, Cheng Wang, Hui Zhang & Zhengru Zhang

Commun. Comput. Phys., 28 (2020), pp. 967-998.

Published online: 2020-07

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  • Abstract

We present and analyze a new second-order finite difference scheme for the Macromolecular Microsphere Composite hydrogel, Time-Dependent Ginzburg-Landau (MMC-TDGL) equation, a Cahn-Hilliard equation with Flory-Huggins-deGennes energy potential. This numerical scheme with unconditional energy stability is based on the Backward Differentiation Formula (BDF) method in time derivation combining with Douglas-Dupont regularization term. In addition, we present a pointwise bound of the numerical solution for the proposed scheme in the theoretical level. For the convergent analysis, we treat three nonlinear logarithmic terms as a whole and deal with all logarithmic terms directly by using the property that the nonlinear error inner product is always non-negative. Moreover, we present the detailed convergent analysis in $ℓ^∞$(0,$T$;$H_h^{-1}$)∩$ℓ^2$(0,$T$;$H_h^1$)  norm. At last, we use the local Newton approximation and multigrid method to solve the nonlinear numerical scheme, and various numerical results are presented, including the numerical convergence test, positivity-preserving property test, spinodal decomposition, energy dissipation and mass conservation properties.

  • Keywords

Cahn-Hilliard equation, Flory-Huggins energy, deGennes diffusive coefficient, energy stability, positivity preserving, convergence analysis.

  • AMS Subject Headings

35K35, 65M06, 65M12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-28-967, author = {Lixiu Dong , and Cheng Wang , and Hui Zhang , and Zhengru Zhang , }, title = {A Positivity-Preserving Second-Order BDF Scheme for the Cahn-Hilliard Equation with Variable Interfacial Parameters}, journal = {Communications in Computational Physics}, year = {2020}, volume = {28}, number = {3}, pages = {967--998}, abstract = {

We present and analyze a new second-order finite difference scheme for the Macromolecular Microsphere Composite hydrogel, Time-Dependent Ginzburg-Landau (MMC-TDGL) equation, a Cahn-Hilliard equation with Flory-Huggins-deGennes energy potential. This numerical scheme with unconditional energy stability is based on the Backward Differentiation Formula (BDF) method in time derivation combining with Douglas-Dupont regularization term. In addition, we present a pointwise bound of the numerical solution for the proposed scheme in the theoretical level. For the convergent analysis, we treat three nonlinear logarithmic terms as a whole and deal with all logarithmic terms directly by using the property that the nonlinear error inner product is always non-negative. Moreover, we present the detailed convergent analysis in $ℓ^∞$(0,$T$;$H_h^{-1}$)∩$ℓ^2$(0,$T$;$H_h^1$)  norm. At last, we use the local Newton approximation and multigrid method to solve the nonlinear numerical scheme, and various numerical results are presented, including the numerical convergence test, positivity-preserving property test, spinodal decomposition, energy dissipation and mass conservation properties.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2019-0037}, url = {http://global-sci.org/intro/article_detail/cicp/17667.html} }
TY - JOUR T1 - A Positivity-Preserving Second-Order BDF Scheme for the Cahn-Hilliard Equation with Variable Interfacial Parameters AU - Lixiu Dong , AU - Cheng Wang , AU - Hui Zhang , AU - Zhengru Zhang , JO - Communications in Computational Physics VL - 3 SP - 967 EP - 998 PY - 2020 DA - 2020/07 SN - 28 DO - http://doi.org/10.4208/cicp.OA-2019-0037 UR - https://global-sci.org/intro/article_detail/cicp/17667.html KW - Cahn-Hilliard equation, Flory-Huggins energy, deGennes diffusive coefficient, energy stability, positivity preserving, convergence analysis. AB -

We present and analyze a new second-order finite difference scheme for the Macromolecular Microsphere Composite hydrogel, Time-Dependent Ginzburg-Landau (MMC-TDGL) equation, a Cahn-Hilliard equation with Flory-Huggins-deGennes energy potential. This numerical scheme with unconditional energy stability is based on the Backward Differentiation Formula (BDF) method in time derivation combining with Douglas-Dupont regularization term. In addition, we present a pointwise bound of the numerical solution for the proposed scheme in the theoretical level. For the convergent analysis, we treat three nonlinear logarithmic terms as a whole and deal with all logarithmic terms directly by using the property that the nonlinear error inner product is always non-negative. Moreover, we present the detailed convergent analysis in $ℓ^∞$(0,$T$;$H_h^{-1}$)∩$ℓ^2$(0,$T$;$H_h^1$)  norm. At last, we use the local Newton approximation and multigrid method to solve the nonlinear numerical scheme, and various numerical results are presented, including the numerical convergence test, positivity-preserving property test, spinodal decomposition, energy dissipation and mass conservation properties.

Lixiu Dong, Cheng Wang, Hui Zhang & Zhengru Zhang. (2020). A Positivity-Preserving Second-Order BDF Scheme for the Cahn-Hilliard Equation with Variable Interfacial Parameters. Communications in Computational Physics. 28 (3). 967-998. doi:10.4208/cicp.OA-2019-0037
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