### Abstract

We propose a practical estimation of a splitting parameter for a spectral method for the ternary Cahn-Hilliard system with a logarithmic free energy. We use Eyre's convex splitting scheme for the time discretization and a Fourier spectral method for the space variables. Given an absolute temperature, we find composition values that make the total free energy be minimum. Then, we find the splitting parameter value that makes the two split homogeneous free energies be convex on the neighborhood of the local minimum concentrations. For general use, we also propose a sixth-order polynomial approximation of the minimum concentration and derive a useful formula for the practical estimation of the splitting parameter in terms of the absolute temperature. The numerical tests are phase separation and total energy decrease with different temperature values. The linear stability analysis shows a good agreement between the exact and numerical solutions with an optimal value s. Various computational experiments confirm that the proposed splitting parameter estimation gives stable numerical results.

Original language | English |
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Journal | Mathematical Methods in the Applied Sciences |

DOIs | |

Publication status | Accepted/In press - 2016 |

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### Keywords

- Logarithmic free energy
- Optimal splitting parameter
- Phase separation
- Spectral method
- Ternary Cahn-Hilliard system

### ASJC Scopus subject areas

- Mathematics(all)
- Engineering(all)

### Cite this

**Practical estimation of a splitting parameter for a spectral method for the ternary Cahn-Hilliard system with a logarithmic free energy.** / Jeong, Darae; Kim, Junseok.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Practical estimation of a splitting parameter for a spectral method for the ternary Cahn-Hilliard system with a logarithmic free energy

AU - Jeong, Darae

AU - Kim, Junseok

PY - 2016

Y1 - 2016

N2 - We propose a practical estimation of a splitting parameter for a spectral method for the ternary Cahn-Hilliard system with a logarithmic free energy. We use Eyre's convex splitting scheme for the time discretization and a Fourier spectral method for the space variables. Given an absolute temperature, we find composition values that make the total free energy be minimum. Then, we find the splitting parameter value that makes the two split homogeneous free energies be convex on the neighborhood of the local minimum concentrations. For general use, we also propose a sixth-order polynomial approximation of the minimum concentration and derive a useful formula for the practical estimation of the splitting parameter in terms of the absolute temperature. The numerical tests are phase separation and total energy decrease with different temperature values. The linear stability analysis shows a good agreement between the exact and numerical solutions with an optimal value s. Various computational experiments confirm that the proposed splitting parameter estimation gives stable numerical results.

AB - We propose a practical estimation of a splitting parameter for a spectral method for the ternary Cahn-Hilliard system with a logarithmic free energy. We use Eyre's convex splitting scheme for the time discretization and a Fourier spectral method for the space variables. Given an absolute temperature, we find composition values that make the total free energy be minimum. Then, we find the splitting parameter value that makes the two split homogeneous free energies be convex on the neighborhood of the local minimum concentrations. For general use, we also propose a sixth-order polynomial approximation of the minimum concentration and derive a useful formula for the practical estimation of the splitting parameter in terms of the absolute temperature. The numerical tests are phase separation and total energy decrease with different temperature values. The linear stability analysis shows a good agreement between the exact and numerical solutions with an optimal value s. Various computational experiments confirm that the proposed splitting parameter estimation gives stable numerical results.

KW - Logarithmic free energy

KW - Optimal splitting parameter

KW - Phase separation

KW - Spectral method

KW - Ternary Cahn-Hilliard system

UR - http://www.scopus.com/inward/record.url?scp=84978653484&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84978653484&partnerID=8YFLogxK

U2 - 10.1002/mma.4093

DO - 10.1002/mma.4093

M3 - Article

AN - SCOPUS:84978653484

JO - Mathematical Methods in the Applied Sciences

JF - Mathematical Methods in the Applied Sciences

SN - 0170-4214

ER -