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A Positivity-Preserving Second-Order BDF Scheme for the Cahn-Hilliard Equation with Variable Interfacial Parameters

A Positivity-Preserving Second-Order BDF Scheme for the Cahn-Hilliard Equation with Variable Interfacial Parameters
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Year:    2020

Author:    Lixiu Dong, Cheng Wang, Hui Zhang, Zhengru Zhang

Communications in Computational Physics, Vol. 28 (2020), Iss. 3 : pp. 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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cicp.OA-2019-0037

Communications in Computational Physics, Vol. 28 (2020), Iss. 3 : pp. 967–998

Published online:    2020-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    32

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

Author Details

Lixiu Dong

Cheng Wang

Hui Zhang

Zhengru Zhang

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