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Numerical Approximation of the Two-Component PFC Models for Binary Colloidal Crystals: Efficient, Decoupled, and Second-Order Unconditionally Energy Stable Schemes

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Abstract

In this paper, we consider numerical approximations for the two-component PFC models for binary colloidal crystals. In addition to the Cahn–Hilliard type two-component PFC model that is commonly used for considering mass conservation, we also derived a new Allen–Cahn type two-component PFC model by using the \(L^2\)-gradient flow and add two nonlocal Lagrange multipliers to the system to conserve the mass for each component. For these two types of two-component PFC models, the stabilized scalar auxiliary variable (SAV) approach is adopted to develop efficient, decoupled, second-order accurate, and linear numerical schemes, where a new SAV is introduced to reformulate the models, and two extra linear stabilization terms are added to improve the stability and keep the required accuracy thus allowing large time steps. These schemes are unconditionally energy stable, mass conservative and require solving only four linear equations with constant coefficients at each time step. Numerical examples are performed to demonstrate the accuracy and energy stability of the proposed schemes, and numerous 2D and 3D simulations are also presented to show a variety of complex binary ordered patterns of phase transformation.

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Acknowledgements

The authors would like to thank the editor and reviewers for careful reading, helpful comments and suggestions. The first author would like to thank Xiaofeng Yang at the University of South Carolina for the valuable discussions. Q. Li’s work is supported by the Fundamental Research Funds for the Central Universities, CHD (No. 300102121302). L. Mei’s work is supported by the Science Challenge Project (No. TZ2016002).

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  1. School of Science, Chang’an University, Xi’an, 710064, Shaanxi, People’s Republic of China

    Qi Li

  2. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, Shaanxi, People’s Republic of China

    Liquan Mei

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  1. Qi Li
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Li, Q., Mei, L. Numerical Approximation of the Two-Component PFC Models for Binary Colloidal Crystals: Efficient, Decoupled, and Second-Order Unconditionally Energy Stable Schemes. J Sci Comput 88, 60 (2021). https://doi.org/10.1007/s10915-021-01564-2

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