### 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 |
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Journal | Computers and Mathematics with Applications |

DOIs | |

Publication status | Accepted/In press - 2018 Jan 1 |

### 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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## Cite this

*Computers and Mathematics with Applications*. https://doi.org/10.1016/j.camwa.2018.09.034