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

2021 ◽  
Vol 88 (3) ◽  
Author(s):  
Qi Li ◽  
Liquan Mei
Keyword(s):  
Numerical Approximation ◽  
Colloidal Crystals ◽  
Second Order ◽  
Two Component ◽  
Energy Stable Schemes ◽  
Energy Stable

scholarly journals Numerical Approximation of the Space Fractional Cahn-Hilliard Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Zhifeng Weng ◽  
Langyang Huang ◽  
Rong Wu
Keyword(s):  
Numerical Approximation ◽  
Second Order ◽  
Differentiation Formula ◽  
Numerical Examples ◽  
Backward Differentiation Formula ◽  
Hilliard Equation ◽  
Order Case ◽  
Energy Stable ◽  
Cahn Hilliard Equation ◽  
Fourier Spectral Methods

In this paper, a second-order accurate (in time) energy stable Fourier spectral scheme for the fractional-in-space Cahn-Hilliard (CH) equation is considered. The time is discretized by the implicit backward differentiation formula (BDF), along with a linear stabilized term which represents a second-order Douglas-Dupont-type regularization. The semidiscrete schemes are shown to be energy stable and to be mass conservative. Then we further use Fourier-spectral methods to discretize the space. Some numerical examples are included to testify the effectiveness of our proposed method. In addition, it shows that the fractional order controls the thickness and the lifetime of the interface, which is typically diffusive in integer order case.

scholarly journals Second-order stabilized semi-implicit energy stable schemes for bubble assemblies in binary and ternary systems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Hyunjung Choi ◽  
Yanxiang Zhao
Keyword(s):  
Ternary Systems ◽  
Second Order ◽  
Implicit Methods ◽  
Coarsening Dynamics ◽  
Energy Stable Schemes ◽  
Energy Stable ◽  
The Difference ◽  
Nonlocal Terms ◽  
Fourier Collocation ◽  
Binary And Ternary Systems

<p style='text-indent:20px;'>In this paper, we propose some second-order stabilized semi-implicit methods for solving the Allen-Cahn-Ohta-Kawasaki and the Allen-Cahn-Ohta-Nakazawa equations. In the numerical methods, some nonlocal linear stabilizing terms are introduced and treated implicitly with other linear terms, while other nonlinear and nonlocal terms are treated explicitly. We consider two different forms of such stabilizers and compare the difference regarding the energy stability. The spatial discretization is performed by the Fourier collocation method with FFT-based fast implementations. Numerically, we verify the second order temporal convergence rate of the proposed schemes. In both binary and ternary systems, the coarsening dynamics is visualized as bubble assemblies in hexagonal or square patterns.</p>

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