Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2019; 56(1): 265-276

Online first article January 11, 2019      Printed January 31, 2019

https://doi.org/10.4134/BKMS.b180238

Copyright © The Korean Mathematical Society.

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

Junseok Kim, Hyun Geun Lee

Korea University; Kwangwoon University

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.

Keywords: nonlinear convex splitting scheme, Fourier spectral method, Cahn--Hilliard equation, logarithmic free energy, phase separation

MSC numbers: 35Q99, 65M70

Supported by: 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).

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