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Numerical Analysis of Second Order, Fully Discrete Energy Stable Schemes for Phase Field Models of Two-Phase Incompressible Flows

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Abstract

In this paper, we propose several second order in time, fully discrete, linear and nonlinear numerical schemes for solving the phase field model of two-phase incompressible flows, in the framework of finite element method. The schemes are based on the second order Crank–Nicolson method for time discretization, projection method for Navier–Stokes equations, as well as several implicit–explicit treatments for phase field equations. The energy stability and unique solvability of the proposed schemes are proved. Ample numerical experiments are performed to validate the accuracy and efficiency of the proposed schemes.

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Acknowledgments

D. Han is partially supported by NSF DMS-1312701. Alex. Brylev is partially supported by NSF DMS-1418898. X. Yang is partially supported by NSF-DMS-1200487, NSF-DMS-1418898, AFOSR-FA9550-12-1-0178, NSFC-11471372, and NSFC-11571385. Z. Tan is partially supported by the NSFC-11571385, the special project “High performance computing” of National Key Research and Development Program (No. 2016YFB0200604), the Fundamental Research Funds for the Central Universities (15lgjc17), and Guangdong Province Key Laboratory of Computational Science at the Sun Yat-sen University.

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Authors and Affiliations

  1. Department of Mathematics, Indiana University Bloomington, Bloomington, IN, 47405, USA

    Daozhi Han

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

    Alex Brylev & Xiaofeng Yang

  3. Guangdong Province Key Laboratory of Computational Science and School of Data and Computer Science, Sun Yat-sen University, Guangzhou, 510275, China

    Zhijun Tan

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  1. Daozhi Han
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Han, D., Brylev, A., Yang, X. et al. Numerical Analysis of Second Order, Fully Discrete Energy Stable Schemes for Phase Field Models of Two-Phase Incompressible Flows. J Sci Comput 70, 965–989 (2017). https://doi.org/10.1007/s10915-016-0279-5

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