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Energy Stable Semi-implicit Schemes for Allen–Cahn–Ohta–Kawasaki Model in Binary System

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Abstract

In this paper, we propose a first order energy stable linear semi-implicit method for solving the Allen–Cahn–Ohta–Kawasaki equation. By introducing a new nonlinear term in the Ohta–Kawasaki free energy functional, all the system forces in the dynamics are localized near the interfaces which results in the desired hyperbolic tangent profile. In our numerical method, the time discretization is done by some stabilization technique in which some extra nonlocal but linear term is introduced and treated explicitly together with other linear terms, while other nonlinear and nonlocal terms are treated implicitly. The spatial discretization is performed by the Fourier collocation method with FFT-based fast implementations. The energy stabilities are proved for this method in both semi-discretization and full discretization levels. Numerical experiments indicate the force localization and desire hyperbolic tangent profile due to the new nonlinear term. We test the first order temporal convergence rate of the proposed scheme. We also present hexagonal bubble assembly as one type of equilibrium for the Ohta–Kawasaki model. Additionally, the two-third law between the number of bubbles and the strength of long-range interaction is verified which agrees with the theoretical studies.

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Acknowledgements

Y. Zhao’s work is supported by Columbian College Facilitating Funds (CCFF 2018) of George Washington University and a grant from the Simons Foundation through Grant No. 357963.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA, 23529, USA

    Xiang Xu

  2. Department of Mathematics, George Washington University, Washington, DC, 20052, USA

    Yanxiang Zhao

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  1. Xiang Xu
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  2. Yanxiang Zhao
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Correspondence to Yanxiang Zhao.

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Xu, X., Zhao, Y. Energy Stable Semi-implicit Schemes for Allen–Cahn–Ohta–Kawasaki Model in Binary System. J Sci Comput 80, 1656–1680 (2019). https://doi.org/10.1007/s10915-019-00993-4

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