Numerical Approximations of Flow Coupled Binary Phase Field Crystal System: Fully Discrete Finite Element Scheme with Second-Order Temporal Accuracy and Decoupling Structure
In this article, we first establish a new flow-coupled binary phase-field crystal model and prove its energy law. Then by using some newly introduced variables, we reformulate this three-phase model into an equivalent form, which makes it possible to construct a fully discrete linearized decoupling scheme with unconditional energy stability and second-order time accuracy to solve this model for the first time. The energy law of the reformulated model is also proved. Then we incorporate the explicit-IEQ (invariant energy quadratization) method for the nonlinear potentials, the projection method for the Navier-Stokes equations, the Crank-Nicolson method for time marching, and the finite element method for spatial discretization together to develop the fully discrete scheme for the reformulated and equivalent system. By using the nonlocal splitting technique, at each time step, only a few decoupled constant-coefficient elliptic equations are required to be solved, even though the original and reformulated models are much more complicated in the form. The developed algorithm is further proved to be unconditionally energy stable, and a detailed implementation process is also provided. Various numerical experiments in 2D and 3D are carried out to verify the effectiveness of the developed scheme, including the binary crystal growth under the action of shear flow and the sedimentation process of many binary particles.
X. Yang and X. He, "Numerical Approximations of Flow Coupled Binary Phase Field Crystal System: Fully Discrete Finite Element Scheme with Second-Order Temporal Accuracy and Decoupling Structure," Journal of Computational Physics, vol. 467, article no. 111448, Elsevier, Oct 2022.
The definitive version is available at https://doi.org/10.1016/j.jcp.2022.111448
Mathematics and Statistics
Keywords and Phrases
Binary crystal model; Fully-decoupled; Phase-field; Second-order accuracy; Three-phase; Unconditional energy stability
International Standard Serial Number (ISSN)
Article - Journal
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15 Oct 2022
National Science Foundation, Grant DMS-1818642