THERMAL SCIENCE

International Scientific Journal

Authors of this Paper

External Links

Jun Zhou

,

Ke-Long Cheng

ENERGY-STABLE BACKWARD DIFFERENTIATION FORMULA TYPE FOURIER COLLOCATION SPECTRAL SCHEMES FOR THE CAHN-HILLIARD EQUATION

ABSTRACT
We present a variant of second order accurate in time backward differentiation formula schemes for the Cahn-Hilliard equation, with a Fourier collocation spectral approximation in space. A three-point stencil is applied in the temporal discretization, and the concave term diffusion term is treated explicitly. An addition-al Douglas-Dupont regularization term is introduced, which ensures the energy stability with a mild requirement. Various numerical simulations including the verification of accuracy, coarsening process and energy decay rate are presented to demonstrate the efficiency and the robustness of proposed schemes.
KEYWORDS
Cahn-Hilliard equation, backward differentiation formula schemes, energy stability, Fourier collocation spectral method
PAPER SUBMITTED: 2021-06-01
PAPER REVISED: 2021-07-20
PAPER ACCEPTED: 2021-07-29
PUBLISHED ONLINE: 2022-04-09
DOI REFERENCE: https://doi.org/10.2298/TSCI2202095Z
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2022, VOLUME 26, ISSUE Issue 2, PAGES [1095 - 1104]
REFERENCES
  1. Cahn, J. W., Hilliard, J. E., Free Energy of a Nonuniform System, I. Interfacial Free Energy, Journal of Chemical Physics, 28 (1958), 2, pp. 258-267
  2. Chen, F., Shen, J., Efficient Energy Stable Schemes with Spectral Discretization in Space for Aniso-tropic Cahn-Hilliard systems, Communications in Computational Physics, 13 (2013), 5, pp. 1189-1208
  3. Cheng, K., et al., A Second-Order, Weakly Energy-Stable Pseudo-Spectral Scheme for the Cahn-Hilliard Equation and Its Solution by the Homogeneous Linear Iteration Method, Journal of Scientific Compu-ting, 69 (2016), 3, pp. 1083-1114
  4. Cheng, K., et al., An Energy Stable Fourth Order Finite Difference Scheme for the Cahn-Hilliard Equa-tion, Journal of Computational and Applied and Mathematics, 362 (2019), Dec., pp. 574-595
  5. Cheng, K., et al., A Third Order Exponential Time Differencing Numerical Scheme for No-Slope-Selection Epitaxial Thin Film Model with Energy Stability, Journal of Scientific Computing, 81 (2019), 1, pp. 154-185
  6. Furihata, D., A Stable and Conservative Finite Difference Scheme for the Cahn-Hilliard Equation, Numerische Mathematik, 87 (2001), 4, pp. 675-699
  7. Diegel, A., et al., Stability and Convergence of a Second Order Mixed Finite Element Method for the Cahn-Hilliard Equation, IMA Journal of Numerical Analysis, 36 (2016), 4, pp. 1867-1897
  8. Yan, Y., et al., A Second-Order Energy Stable BDF Numerical Scheme for the Cahn-Hilliard Equation, Communications in Computational Physics, 23 (2018), 2, pp. 572-602
  9. Li, D., Qiao, Z., On Second Order Semi-Implicit Fourier Spectral Methods for 2-D Cahn-Hilliard Equa-tions, Journal of Scientific Computing, 70 (2017), 1, pp. 301-341
PDF VERSION [DOWNLOAD]
Energy-stable backward differentiation formula type fourier collocation spectral schemes for the Cahn-Hilliard equation

© 2024 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence