THERMAL SCIENCE

International Scientific Journal

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Xiao-Rong Kang

,

Yan-Mei Wu

,

Ke-Long Cheng

CONVERGENCE ANALYSIS OF THE ENERGY-STABLE NUMERICAL SCHEMES FOR THE CAHN-HILLIARD EQUATION

ABSTRACT
In this paper we present a second order numerical scheme for the Cahn-Hilliard equation, with a Fourier pseudo-spectral approximation in space. An additional Douglas-Dupont regularization term is introduced, which ensures the energy stability. The bound of numerical solution in H2h and l∞ norms are obtained at a theoretical level. Moreover, for the global nature of the pseudo-spectral method, we propose a linear iteration algorithm to solve the non-linear system, due to the implicit treatment for the non-linear term. Some numerical simulations verify the efficiency of iteration algorithm.
KEYWORDS
Cahn-Hilliard equation, energy stability scheme, convergence, linear iteration
PAPER SUBMITTED: 2021-06-15
PAPER REVISED: 2021-07-12
PAPER ACCEPTED: 2021-07-21
PUBLISHED ONLINE: 2022-04-09
DOI REFERENCE: https://doi.org/10.2298/TSCI2202037K
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2022, VOLUME 26, ISSUE Issue 2, PAGES [1037 - 1046]
REFERENCES
  1. Cahn, J. W., Hilliard, J. E., Free Energy of a Nonuniform System. I. Interfacial Free Energy, Journal of Chemical Physics, 28 (1958), 3, pp. 258-267
  2. Wu, X., et al., Stabilized Second-Order Convex Splitting Schemes for Cahn-Hilliard Models with Application to Diffuse-Interface Tumor-Growth Models, International Journal for Numerical Methods in Biomedical Engineering, 2 (2014), 4, pp. 180-203
  3. Guillen-Gonzalez, F., Tierra, G., Second Order Schemes and Time-Step Adaptivity for Allen-Cahn and Cahn-Hilliard Models, Computers & Mathematics with Applications, 8 (2014), 4, pp. 821-846
  4. Guo, J., et al., An H2 Convergence of a Second-Order Convex Splitting, Finite Difference Scheme for the Three-Dimensional Cahn-Hilliard Equation, Communications in Mathematical Sciences, 2 (2016), 3, pp. 489-515
  5. He, Y., et al., On Large Time-Stepping Methods for the Cahn-Hilliard Equation, Applied Numerical Mathematics, 5 (2017), 3, pp. 616-628
  6. Ju, L., et al., Fast and Accurate Algorithms for Simulating Coarsening Dynamics of Cahn-Hilliard Equations, Computational Materials Science, 2015 (2015), 4, pp. 272-282
  7. Khiari, N., et al., Finite Difference Approximate Solutions for the Cahn-Hilliard Equation, Numerical Methods for Partial Differential Equations, 2 (2007), 3, pp. 437-455
  8. 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 Computing, 69 (2016), 3, pp. 1083-1114
  9. Gottlieb, S., Wang, C., Stability and Convergence Analysis of Fully Discrete Fourier Collocation Spectral Method for 3-D Viscous Burgers' Equation, Journal of Scientific Computing, 1 (2012), 5, pp. 102-128
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Convergence analysis of the energy-stable numerical 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