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

Zhijun Tan, Junxiang Yang, Jianjun Chen, Junseok Kim

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


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.

Original languageEnglish
Article number127599
JournalApplied Mathematics and Computation
Publication statusPublished - 2023 Feb 1

Bibliographical note

Funding Information:
The work of Z. Tan is supported by the National Nature Science Foundation of China (11971502), Guangdong Natural Science Foundation (2022A1515010426), Guangdong Province Key Laboratory of Computational Science at the Sun Yat-sen University (2020B1212060032), and Key-Area Research and Development Program of Guangdong Province (2021B0101190003). J. Yang is supported by the National Natural Science Foundation of China (No. 12201657 ), the China Postdoctoral Science Foundation (No. 2022M713639 ), and the 2022 International Postdoctoral Exchange Fellowship Program (Talent-Introduction Program) (No. YJ20220221). The corresponding author (J.S. Kim) supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. 2022R1A2C1003844). The authors thank the reviewers for the constructive comments on the revision of this article.

Publisher Copyright:
© 2022 Elsevier Inc.


  • Dissipation law
  • Finite difference method
  • Multigrid algorithm
  • Three-phase conservative allen–Cahn fluids

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics


Dive into the research topics of 'An efficient time-dependent auxiliary variable approach for the three-phase conservative Allen–Cahn fluids'. Together they form a unique fingerprint.

Cite this