Comparison study on the different dynamics between the Allen–Cahn and the Cahn–Hilliard equations

Yibao Li; Darae Jeong; Hyundong Kim; Chaeyoung Lee; Junseok Kim

doi:10.1016/j.camwa.2018.09.034

Comparison study on the different dynamics between the Allen–Cahn and the Cahn–Hilliard equations

Yibao Li, Darae Jeong, Hyundong Kim, Chaeyoung Lee, Junseok Kim

Research output: Contribution to journal › Article › peer-review

19 Citations (Scopus)

Abstract

We perform a comparison study on the different dynamics between the Allen–Cahn (AC) and the Cahn–Hilliard (CH) equations. The AC equation describes the evolution of a non-conserved order field during anti-phase domain coarsening. The CH equation describes the process of phase separation of a conserved order field. The AC and the CH equations are second-order and fourth-order nonlinear parabolic partial differential equations, respectively. Linear stability analysis shows that growing and decaying modes for both the equations are the same. While the growth rates are monotonically decreasing with respect to the modes for the AC equation, the growth rates for the CH equation are increasing and then decreasing with respect to the modes. We perform various numerical tests using the Fourier spectral method to highlight the different evolutionary dynamics between the AC and the CH equations.

Original language	English
Pages (from-to)	311-322
Number of pages	12
Journal	Computers and Mathematics with Applications
Volume	77
Issue number	2
DOIs	https://doi.org/10.1016/j.camwa.2018.09.034
Publication status	Published - 2019 Jan 15

Bibliographical note

Funding Information:
Yibao Li is supported by National Natural Science Foundation of China (No. 11601416 , No. 11631012 ). The corresponding author (Junseok Kim) was supported by Korea University Future Research Grant. The authors thank the reviewers for the constructive and helpful comments on the revision of this article.

Publisher Copyright:
© 2018 Elsevier Ltd

Keywords

Allen–Cahn equation
Cahn–Hilliard equation
Fastest growing mode
Fourier spectral method
Linear stability analysis

ASJC Scopus subject areas

Modelling and Simulation
Computational Theory and Mathematics
Computational Mathematics

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10.1016/j.camwa.2018.09.034

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title = "Comparison study on the different dynamics between the Allen–Cahn and the Cahn–Hilliard equations",

abstract = "We perform a comparison study on the different dynamics between the Allen–Cahn (AC) and the Cahn–Hilliard (CH) equations. The AC equation describes the evolution of a non-conserved order field during anti-phase domain coarsening. The CH equation describes the process of phase separation of a conserved order field. The AC and the CH equations are second-order and fourth-order nonlinear parabolic partial differential equations, respectively. Linear stability analysis shows that growing and decaying modes for both the equations are the same. While the growth rates are monotonically decreasing with respect to the modes for the AC equation, the growth rates for the CH equation are increasing and then decreasing with respect to the modes. We perform various numerical tests using the Fourier spectral method to highlight the different evolutionary dynamics between the AC and the CH equations.",

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author = "Yibao Li and Darae Jeong and Hyundong Kim and Chaeyoung Lee and Junseok Kim",

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AU - Kim, Hyundong

AU - Lee, Chaeyoung

AU - Kim, Junseok

N1 - Funding Information: Yibao Li is supported by National Natural Science Foundation of China (No. 11601416 , No. 11631012 ). The corresponding author (Junseok Kim) was supported by Korea University Future Research Grant. The authors thank the reviewers for the constructive and helpful comments on the revision of this article. Publisher Copyright: © 2018 Elsevier Ltd

PY - 2019/1/15

Y1 - 2019/1/15

N2 - We perform a comparison study on the different dynamics between the Allen–Cahn (AC) and the Cahn–Hilliard (CH) equations. The AC equation describes the evolution of a non-conserved order field during anti-phase domain coarsening. The CH equation describes the process of phase separation of a conserved order field. The AC and the CH equations are second-order and fourth-order nonlinear parabolic partial differential equations, respectively. Linear stability analysis shows that growing and decaying modes for both the equations are the same. While the growth rates are monotonically decreasing with respect to the modes for the AC equation, the growth rates for the CH equation are increasing and then decreasing with respect to the modes. We perform various numerical tests using the Fourier spectral method to highlight the different evolutionary dynamics between the AC and the CH equations.

AB - We perform a comparison study on the different dynamics between the Allen–Cahn (AC) and the Cahn–Hilliard (CH) equations. The AC equation describes the evolution of a non-conserved order field during anti-phase domain coarsening. The CH equation describes the process of phase separation of a conserved order field. The AC and the CH equations are second-order and fourth-order nonlinear parabolic partial differential equations, respectively. Linear stability analysis shows that growing and decaying modes for both the equations are the same. While the growth rates are monotonically decreasing with respect to the modes for the AC equation, the growth rates for the CH equation are increasing and then decreasing with respect to the modes. We perform various numerical tests using the Fourier spectral method to highlight the different evolutionary dynamics between the AC and the CH equations.

KW - Allen–Cahn equation

KW - Cahn–Hilliard equation

KW - Fastest growing mode

KW - Fourier spectral method

KW - Linear stability analysis

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Comparison study on the different dynamics between the Allen–Cahn and the Cahn–Hilliard equations

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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