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Fully Discretized Energy Stable Schemes for Hydrodynamic Equations Governing Two-Phase Viscous Fluid Flows

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Abstract

We develop systematically a numerical approximation strategy to discretize a hydrodynamic phase field model for a binary fluid mixture of two immiscible viscous fluids, derived using the generalized Onsager principle that warrants not only the variational structure but also the energy dissipation property. We first discretize the governing equations in space to arrive at a semi-discretized, time-dependent ordinary differential and algebraic equation (DAE) system in which a corresponding discrete energy dissipation law is preserved. Then, we discretize the DAE system in time to obtain a fully discretized system using a structure preserving finite difference method like the Crank–Nicolson method, which satisfies a fully discretized energy dissipation law. Alternatively, we solve the first order DAE system using the integration factor method after the algebraic equation is solved firstly. The integration factor method, which treats the linear, spatial derivative terms explicitly. Finally, two numerical examples are presented to compare the efficiency and accuracy of the two proposed methods.

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Author information

Authors and Affiliations

  1. Beijing Computational Science Research Center, Beijing, 100193, China

    Yuezheng Gong & Qi Wang

  2. Department of Mathematics, Interdisciplinary Mathematics Institute and NanoCenter at USC, University of South Carolina, Columbia, SC, 29208, USA

    Xinfeng Liu & Qi Wang

  3. School of Materials Science and Engineering, Nankai University, Tianjin, 300071, China

    Qi Wang

Authors
  1. Yuezheng Gong
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  2. Xinfeng Liu
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  3. Qi Wang
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Corresponding author

Correspondence to Qi Wang.

Additional information

Yuezheng Gong’s work is partially supported by China Postdoctoral Science Foundation through Grants 2016M591054. Xinfeng Liu’s work is partially supported by US National Science Foundation through Grant DMS1308948. Qi Wang’s work is partially supported by US National Science Foundation through Grants DMS-1200487 and DMS-1517347, AFOSR Grant FA9550-12-1-0178, and SC EPSCOR GEAR awards.

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Gong, Y., Liu, X. & Wang, Q. Fully Discretized Energy Stable Schemes for Hydrodynamic Equations Governing Two-Phase Viscous Fluid Flows. J Sci Comput 69, 921–945 (2016). https://doi.org/10.1007/s10915-016-0224-7

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  • DOI: https://doi.org/10.1007/s10915-016-0224-7

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