A nonlinear convex splitting Fourier spectral scheme for the Cahn--Hilliard equation with a logarithmic free energy
Bull. Korean Math. Soc. 2019 Vol. 56, No. 1, 265-276
https://doi.org/10.4134/BKMS.b180238
Published online 2019 Jan 31
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
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