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Discontinuous finite volume element method for a coupled Navier-Stokes-Cahn-Hilliard phase field model

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Abstract

In this paper, we propose a discontinuous finite volume element method to solve a phase field model for two immiscible incompressible fluids. In this finite volume element scheme, discontinuous linear finite element basis functions are used to approximate the velocity, phase function, and chemical potential while piecewise constants are used to approximate the pressure. This numerical method is efficient, optimally convergent, conserving the mass, convenient to implement, flexible for mesh refinement, and easy to handle complex geometries with different types of boundary conditions. We rigorously prove the mass conservation property and the discrete energy dissipation for the proposed fully discrete discontinuous finite volume element scheme. Using numerical tests, we verify the accuracy, confirm the mass conservation and the energy law, test the influence of surface tension and small density variations, and simulate the driven cavity, the Rayleigh-Taylor instability.

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Funding

Rui Li is financially supported by the National Natural Science Foundation of China (11901372), the Natural Science Foundation of Shaanxi Province (2019JQ-077), and the Fundamental Research Fund for the Central Universities of China (GK201903007, GK201901008); Yali Gao is financially supported by the National Natural Science Foundation of China (11901461) and the Natural Science Foundation of Shaanxi Province (2019JQ-024); Xiaoming He is financially supported by NSF grant DMS-1818642; Jie Chen is financially supported by the XJTLU Key Programme Special Fund (KSF-P-2, KSF-E-50); Zhangxin Chen is financially supported by Foundation CMG.

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Authors and Affiliations

  1. School of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710119, People’s Republic of China

    Rui Li

  2. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710129, People’s Republic of China

    Yali Gao

  3. Department of Mathematical Sciences, Xi’an Jiaotong-Liverpool University, Suzhou, 215123, People’s Republic of China

    Jie Chen

  4. School of Mathematical Science, University of Electronic Science and Technology of China, Chengdu, 611731, People’s Republic of China

    Li Zhang

  5. Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO, 65409, USA

    Xiaoming He

  6. Department of Chemical, Petroleum Engineering, Schulich School of Engineering, University of Calgary, 2500 University Drive N.W., Calgary, AB, T2N 1N4, Canada

    Zhangxin Chen

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  2. Yali Gao
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  3. Jie Chen
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  4. Li Zhang
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  5. Xiaoming He
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  6. Zhangxin Chen
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Correspondence to Yali Gao.

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Communicated by: Ilaria Perugia

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Li, R., Gao, Y., Chen, J. et al. Discontinuous finite volume element method for a coupled Navier-Stokes-Cahn-Hilliard phase field model. Adv Comput Math 46, 25 (2020). https://doi.org/10.1007/s10444-020-09764-4

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Mathematics subject classification (2010)

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