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Highly efficient and unconditionally energy stable semi-discrete time-marching numerical scheme for the two-phase incompressible flow phase-field system with variable-density and viscosity

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Abstract

For the variable-density/viscosity Cahn-Hilliard phase-field model of the binary-phase incompressible fluid flow system, the development of easy-to-implement numerical schemes has long been known as a challenging problem. We develop a novel fully-decoupled numerical technique in this article which can achieve unconditional energy stability while explicitly discretizing nonlinear coupling items. The idea is invented on the basis of combining the Strang operator splitting method and the novel decoupling method by using the zero-energy-contribution property. The scheme only needs to solve a series of completely independent linear elliptic equations at each time step, in which the Cahn-Hilliard equation and the pressure Poisson equation are the constant coefficient. To demonstrate the effectiveness of the scheme, we provide the rigorous proof of the energy stability/solvability, and also perform ample accuracy and stability tests and 2D/3D numerical simulations, including the Rayleigh-Taylor instability and bubble rising dynamics.

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Acknowledgements

The first author was supported by National Natural Science Foundation of China (Grant No. 11771375). The second author was supported by National Science Foundation of USA (Grant No. DMS-2012490).

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Authors and Affiliations

  1. School of Mathematics and Information Sciences, Yantai University, Yantai, 264005, China

    Chuanjun Chen

  2. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    Xiaofeng Yang

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  1. Chuanjun Chen
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  2. Xiaofeng Yang
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Correspondence to Xiaofeng Yang.

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Chen, C., Yang, X. Highly efficient and unconditionally energy stable semi-discrete time-marching numerical scheme for the two-phase incompressible flow phase-field system with variable-density and viscosity. Sci. China Math. 65, 2631–2656 (2022). https://doi.org/10.1007/s11425-021-1932-x

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  • DOI: https://doi.org/10.1007/s11425-021-1932-x

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