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An efficient time-dependent auxiliary variable approach for the three-phase conservative Allen–Cahn fluids

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  • Tan, Zhijun
  • Yang, Junxiang
  • Chen, Jianjun
  • Kim, Junseok

Abstract

Based on a time-dependent auxiliary variable approach, we propose linear, totally decoupled, and energy dissipative methods for the three-phase conservative Allen–Cahn (CAC) fluid system. The three-phase CAC equation has been extensively applied in the simulation of multi-component fluid flows because of the following advantages: (i) Total mass is conserved, (ii) Topological change of the interface can be implicitly captured. Compared with the ternary Cahn–Hilliard (CH) model, the CAC-type model is simple to solve. When we solve the CAC model by using the classical scalar auxiliary variable (SAV) approach, extra computational time is needed because we must decouple the local and non-local variables. The variant of SAV approach considered in the present study not only leads to linear and energy stable schemes, but also achieves highly efficient computation. Linear and decoupled equations need to be updated at each time step. We adopt the linear multigrid algorithm to speed up the convergence. Extensive numerical experiments with and without fluid flows are conducted to validate the temporal accuracy, mass conservation, and energy law.

Suggested Citation

  • Tan, Zhijun & Yang, Junxiang & Chen, Jianjun & Kim, Junseok, 2023. "An efficient time-dependent auxiliary variable approach for the three-phase conservative Allen–Cahn fluids," Applied Mathematics and Computation, Elsevier, vol. 438(C).
    • Handle: RePEc:eee:apmaco:v:438:y:2023:i:c:s0096300322006725
    DOI: 10.1016/j.amc.2022.127599
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    References listed on IDEAS

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    1. Junseok Kim & Seunggyu Lee & Yongho Choi & Seok-Min Lee & Darae Jeong, 2016. "Basic Principles and Practical Applications of the Cahn–Hilliard Equation," Mathematical Problems in Engineering, Hindawi, vol. 2016, pages 1-11, October.
    2. Chaeyoung Lee & Darae Jeong & Junxiang Yang & Junseok Kim, 2020. "Nonlinear Multigrid Implementation for the Two-Dimensional Cahn–Hilliard Equation," Mathematics, MDPI, vol. 8(1), pages 1-23, January.
    3. Yu, Qian & Wang, Kunyang & Xia, Binhu & Li, Yibao, 2021. "First and second order unconditionally energy stable schemes for topology optimization based on phase field method," Applied Mathematics and Computation, Elsevier, vol. 405(C).
    4. Dehghan, Mehdi & Gharibi, Zeinab, 2021. "Numerical analysis of fully discrete energy stable weak Galerkin finite element Scheme for a coupled Cahn-Hilliard-Navier-Stokes phase-field model," Applied Mathematics and Computation, Elsevier, vol. 410(C).
    5. Park, Jang Min, 2018. "Mathematical modeling and computational simulation of phase separation in ternary mixtures," Applied Mathematics and Computation, Elsevier, vol. 330(C), pages 11-22.
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    Cited by:

    1. Wang, Jian & Han, Ziwei & Jiang, Wenjing & Kim, Junseok, 2023. "A fast, efficient, and explicit phase-field model for 3D mesh denoising," Applied Mathematics and Computation, Elsevier, vol. 458(C).

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