### Abstract

For a simple implementation, a linear convex splitting scheme was coupled with the Fourier spectral method for the Cahn–Hilliard equation with a logarithmic free energy. However, an inappropriate value of the splitting parameter of the linear scheme may lead to incorrect morphologies in the phase separation process. In order to overcome this problem, we present a nonlinear convex splitting Fourier spectral scheme for the Cahn–Hilliard equation with a logarithmic free energy, which is an appropriate extension of Eyre’s idea of convex-concave decomposition of the energy functional. Using the nonlinear scheme, we derive a useful formula for the relation between the gradient energy coefficient and the thickness of the interfacial layer. And we present numerical simulations showing the different evolution of the solution using the linear and nonlinear schemes. The numerical results demonstrate that the nonlinear scheme is more accurate than the linear one.

Original language | English |
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Pages (from-to) | 265-276 |

Number of pages | 12 |

Journal | Bulletin of the Korean Mathematical Society |

Volume | 56 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2019 Jan 1 |

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### Keywords

- Cahn
- Fourier spectral method
- Hilliard equation
- Logarithmic free energy
- Nonlinear convex splitting scheme
- Phase separation

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**A nonlinear convex splitting fourier spectral scheme for the Cahn–Hilliard equation with a logarithmic free energy.** / Kim, Junseok; Lee, Hyun Geun.

Research output: Contribution to journal › Article

*Bulletin of the Korean Mathematical Society*, vol. 56, no. 1, pp. 265-276. https://doi.org/10.4134/BKMS.b180238

}

TY - JOUR

T1 - A nonlinear convex splitting fourier spectral scheme for the Cahn–Hilliard equation with a logarithmic free energy

AU - Kim, Junseok

AU - Lee, Hyun Geun

PY - 2019/1/1

Y1 - 2019/1/1

N2 - For a simple implementation, a linear convex splitting scheme was coupled with the Fourier spectral method for the Cahn–Hilliard equation with a logarithmic free energy. However, an inappropriate value of the splitting parameter of the linear scheme may lead to incorrect morphologies in the phase separation process. In order to overcome this problem, we present a nonlinear convex splitting Fourier spectral scheme for the Cahn–Hilliard equation with a logarithmic free energy, which is an appropriate extension of Eyre’s idea of convex-concave decomposition of the energy functional. Using the nonlinear scheme, we derive a useful formula for the relation between the gradient energy coefficient and the thickness of the interfacial layer. And we present numerical simulations showing the different evolution of the solution using the linear and nonlinear schemes. The numerical results demonstrate that the nonlinear scheme is more accurate than the linear one.

AB - For a simple implementation, a linear convex splitting scheme was coupled with the Fourier spectral method for the Cahn–Hilliard equation with a logarithmic free energy. However, an inappropriate value of the splitting parameter of the linear scheme may lead to incorrect morphologies in the phase separation process. In order to overcome this problem, we present a nonlinear convex splitting Fourier spectral scheme for the Cahn–Hilliard equation with a logarithmic free energy, which is an appropriate extension of Eyre’s idea of convex-concave decomposition of the energy functional. Using the nonlinear scheme, we derive a useful formula for the relation between the gradient energy coefficient and the thickness of the interfacial layer. And we present numerical simulations showing the different evolution of the solution using the linear and nonlinear schemes. The numerical results demonstrate that the nonlinear scheme is more accurate than the linear one.

KW - Cahn

KW - Fourier spectral method

KW - Hilliard equation

KW - Logarithmic free energy

KW - Nonlinear convex splitting scheme

KW - Phase separation

UR - http://www.scopus.com/inward/record.url?scp=85062345252&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85062345252&partnerID=8YFLogxK

U2 - 10.4134/BKMS.b180238

DO - 10.4134/BKMS.b180238

M3 - Article

VL - 56

SP - 265

EP - 276

JO - Bulletin of the Korean Mathematical Society

JF - Bulletin of the Korean Mathematical Society

SN - 1015-8634

IS - 1

ER -