Efficient energy-stable schemes for the hydrodynamics coupled phase-field model

Guangpu Zhu, Huangxin Chen, Jun Yao*, Shuyu Sun

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

79 Scopus citations


In this article, several efficient and energy-stable semi–implicit schemes are presented for the Cahn–Hilliard phase-field model of two-phase incompressible flows. A scalar auxiliary variable (SAV) approach is implemented to solve the Cahn–Hilliard equation, while a splitting method based on pressure stabilization is used to solve the Navier–Stokes equation. At each time step, the schemes involve solving only a sequence of linear elliptic equations, and computations of the phase-field variable, velocity, and pressure are totally decoupled. A finite-difference method on staggered grids is adopted to spatially discretize the proposed time-marching schemes. We rigorously prove the unconditional energy stability for the semi-implicit schemes and the fully discrete scheme. Numerical results in both two and three dimensions are obtained, which demonstrate the accuracy and effectiveness of the proposed schemes. Using our numerical schemes, we compare the SAV, invariant energy quadratization (IEQ), and stabilization approaches. Bubble rising dynamics and coarsening dynamics are also investigated in detail. The results demonstrate that the SAV approach is more accurate than the IEQ approach and that the stabilization approach is the least accurate among the three approaches. The energy stability of the SAV approach appears to be better than that of the other approaches at large time steps.

Original languageEnglish (US)
Pages (from-to)82-108
Number of pages27
JournalApplied Mathematical Modelling
StatePublished - Jun 2019

Bibliographical note

Funding Information:
The work of Jun Yao and Guangpu Zhu is supported by the National Science and Technology Major Project (2016ZX05011-001), the National Natural Science Foundation of China (grants 51490654 , 51504276 , and 51304232 ), and the Innovative Project of the China University of Petroleum (YCX2017021). The work of Huangxin Chen is supported by the National Natural Science Foundation of China (grants 11771363 , 91630204 , and 51661135011 ), the Fundamental Research Funds for Central Universities (Grant No. 20720180003), and the Program for Prominent Young Talents in Fujian Province University. The work of Shuyu Sun is supported by King Abdullah University of Science and Technology research funding awarded to the Computational Transport Phenomena Laboratory through grant BAS/1/1351-01-01 .

Publisher Copyright:
© 2018 Elsevier Inc.


  • Energy stability
  • Navier–Stokes equation
  • Phase-field modeling
  • Scalar auxiliary variable
  • Two-phase flows

ASJC Scopus subject areas

  • Applied Mathematics
  • Modeling and Simulation


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