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

N1 - Funding Information:
Received March 16, 2018; Revised July 10, 2018; Accepted October 11, 2018. 2010 Mathematics Subject Classification. 35Q99, 65M70. Key words and phrases. nonlinear convex splitting scheme, Fourier spectral method, Cahn–Hilliard equation, logarithmic free energy, phase separation. J. S. Kim was supported by Korea University Future Research Grant. H. G. Lee was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A1B03034619).
Publisher Copyright:
© 2019 Korean Mathematical Society.
Copyright:
Copyright 2019 Elsevier B.V., All rights reserved.

PY - 2019

Y1 - 2019

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

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U2 - 10.4134/BKMS.b180238

DO - 10.4134/BKMS.b180238

M3 - Article

AN - SCOPUS:85062345252

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 -