The error analysis for the Cahn-Hilliard phase field model of two-phase incompressible flows with variable density

Mingliang Liao; Danxia Wang; Chenhui Zhang; Hongen Jia; Mingliang Liao; Danxia Wang; Chenhui Zhang; Hongen Jia

doi:10.3934/math.20231595

AIMS Mathematics

2023, Volume 8, Issue 12: 31158-31185. doi: 10.3934/math.20231595

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Research article

The error analysis for the Cahn-Hilliard phase field model of two-phase incompressible flows with variable density

1.
School of Mathematics, Taiyuan University of Technology, Jinzhong 030600, China
2.
Shanxi Key Laboratory for Intelligent Optimization Computing and Blockchain Technology, Taiyuan 030000, China

Received: 20 September 2023 Revised: 10 November 2023 Accepted: 15 November 2023 Published: 23 November 2023
MSC : 35K51, 35K55, 65M12, 65M70

In this paper, we consider the numerical approximations of the Cahn-Hilliard phase field model for two-phase incompressible flows with variable density. First, a temporal semi-discrete numerical scheme is proposed by combining the fractional step method (for the momentum equation) and the convex-splitting method (for the free energy). Second, we prove that the scheme is unconditionally stable in energy. Then, the $ L^2 $ convergence rates for all variables are demonstrated through a series of rigorous error estimations. Finally, by applying the finite element method for spatial discretization, some numerical simulations were performed to demonstrate the convergence rates and energy dissipations.
- Cahn-Hilliard phase field,
- two-phase incompressible flows,
- fractional step scheme,
- energy stability,
- error estimates
Citation: Mingliang Liao, Danxia Wang, Chenhui Zhang, Hongen Jia. The error analysis for the Cahn-Hilliard phase field model of two-phase incompressible flows with variable density[J]. AIMS Mathematics, 2023, 8(12): 31158-31185. doi: 10.3934/math.20231595

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Abstract

In this paper, we consider the numerical approximations of the Cahn-Hilliard phase field model for two-phase incompressible flows with variable density. First, a temporal semi-discrete numerical scheme is proposed by combining the fractional step method (for the momentum equation) and the convex-splitting method (for the free energy). Second, we prove that the scheme is unconditionally stable in energy. Then, the $ L^2 $ convergence rates for all variables are demonstrated through a series of rigorous error estimations. Finally, by applying the finite element method for spatial discretization, some numerical simulations were performed to demonstrate the convergence rates and energy dissipations.

References

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