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Linear second order in time energy stable schemes for hydrodynamic models of binary mixtures based on a spatially pseudospectral approximation

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Abstract

We develop two linear, second order energy stable schemes for solving the governing system of partial differential equations of a hydrodynamic phase field model of binary fluid mixtures. We first apply the Fourier pseudo-spectral approximation to the partial differential equations in space to obtain a semi-discrete, time-dependent, ordinary differential and algebraic equation (DAE) system, which preserves the energy dissipation law at the semi-discrete level. Then, we discretize the DAE system by the Crank-Nicolson (CN) and the second-order backward differentiation/extrapolation (BDF/EP) method in time, respectively, to obtain two fully discrete systems. We show that the CN method preserves the energy dissipation law while the BDF/EP method does not preserve it exactly but respects the energy dissipation property of the hydrodynamic model. The two new fully discrete schemes are linear, unconditional stable, second order accurate in time and high order in space, and uniquely solvable as linear systems. Numerical examples are presented to show the convergence property as well as the efficiency and accuracy of the new schemes in simulating mixing dynamics of binary polymeric solutions.

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Acknowledgements

Yuezheng Gong’s work was partially supported by the China Postdoctoral Science Foundation through grant 2016M591054 and the Foundation of Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems (201703). Jia Zhao’s research is partially supported by a Research Catalyst Grant from Office of Research and Graduate Studies at Utah State University. Qi Wang’s work was partially supported by grants NSF- DMS-1517347 and NSFC awards 11571032, 91630207, and NSAF-U1530401.

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Authors and Affiliations

  1. College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Yuezheng Gong

  2. Department of Mathematics and Statistics, Utah State University, Logan, UT, 84322, USA

    Jia Zhao

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

    Qi Wang

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

    Qi Wang

  5. School of Materials and Engineering, Nankai University, Tianjian, 300350, China

    Qi Wang

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  1. Yuezheng Gong
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  2. Jia Zhao
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Correspondence to Qi Wang.

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Communicated by: John Lowengrub

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Gong, Y., Zhao, J. & Wang, Q. Linear second order in time energy stable schemes for hydrodynamic models of binary mixtures based on a spatially pseudospectral approximation. Adv Comput Math 44, 1573–1600 (2018). https://doi.org/10.1007/s10444-018-9597-5

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  • DOI: https://doi.org/10.1007/s10444-018-9597-5

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