Abstract
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 language | English (US) |
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Pages (from-to) | 82-108 |
Number of pages | 27 |
Journal | Applied Mathematical Modelling |
Volume | 70 |
DOIs | |
State | Published - Jun 2019 |
Keywords
- Energy stability
- Navier–Stokes equation
- Phase-field modeling
- Scalar auxiliary variable
- Two-phase flows
ASJC Scopus subject areas
- Modeling and Simulation
- Applied Mathematics