Efficient and structure-preserving time-dependent auxiliary variable method for a conservative Allen–Cahn type surfactant system

Junxiang Yang, Junseok Kim

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)


In this study, we establish a phase-field two-phase surfactant system using two conservative Allen–Cahn type equations. Two nonlocal Lagrange multipliers are used to achieve the mass conservations. Comparing with the Cahn–Hilliard-type binary surfactant models which consist of two fourth-order nonlinear partial differential equations, the present model is easier because we solve two second-order nonlinear equations. In phase-field surfactant models, the existences of nonlinear terms lead to high challenges in energy estimation and numerical computation. To deal with these problems, we present first- and second-order time-accurate methods using a new time-dependent auxiliary variable approach. Due to the introduction of a new auxiliary variable, all nonlinear terms are explicitly solved. The energy dissipation law can be proved. To achieve linear and totally decoupled computation, we describe an efficient splitting algorithm. Various two- and three-dimensional computational tests are presented to show that our proposed schemes have desired accuracy, energy dissipation property, and work well for surfactant-laden coarsening.

Original languageEnglish
Pages (from-to)5231-5250
Number of pages20
JournalEngineering with Computers
Issue number6
Publication statusPublished - 2022 Dec

Bibliographical note

Funding Information:
The corresponding author (J.S. Kim) is supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(NRF-2019R1A2C1003053). The authors wish to thank the reviewers for the constructive and helpful comments on the revision of this manuscript.

Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag London Ltd., part of Springer Nature.


  • Conservative Allen–Cahn model
  • Energy dissipation law
  • New Lagrange multiplier approach
  • Two-phase surfactant system

ASJC Scopus subject areas

  • Software
  • Modelling and Simulation
  • General Engineering
  • Computer Science Applications


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