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A Positivity Preserving, Energy Stable Finite Difference Scheme for the Flory-Huggins-Cahn-Hilliard-Navier-Stokes System

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Abstract

In this paper, we propose and analyze a finite difference numerical scheme for the Cahn-Hilliard-Navier-Stokes system, with logarithmic Flory-Huggins energy potential. In the numerical approximation to the singular chemical potential, the logarithmic term and the surface diffusion term are implicitly updated, while an explicit computation is applied to the concave expansive term. Moreover, the convective term in the phase field evolutionary equation is approximated in a semi-implicit manner. Similarly, the fluid momentum equation is computed by a semi-implicit algorithm: implicit treatment for the kinematic diffusion term, explicit update for the pressure gradient, combined with semi-implicit approximations to the fluid convection and the phase field coupled term, respectively. Such a semi-implicit method gives an intermediate velocity field. Subsequently, a Helmholtz projection into the divergence-free vector field yields the velocity vector and the pressure variable at the next time step. This approach decouples the Stokes solver, which in turn drastically improves the numerical efficiency. The positivity-preserving property and the unique solvability of the proposed numerical scheme is theoretically justified, i.e., the phase variable is always between -1 and 1, following the singular nature of the logarithmic term as the phase variable approaches the singular limit values. In addition, an iteration construction technique is applied in the positivity-preserving and unique solvability analysis, motivated by the non-symmetric nature of the fluid convection term. The energy stability of the proposed numerical scheme could be derived by a careful estimate. A few numerical results are presented to validate the robustness of the proposed numerical scheme.

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Acknowledgements

This work is supported in part by the grants NSFC 12071090 (W. Chen), NSF DMS-2012269 (C. Wang), NSFC 11871159, Guangdong Provincial Key Laboratory for Computational Science and Material Design 2019B030301001 (X. Wang). C. Wang also thanks the Key Laboratory of Mathematics for Nonlinear Sciences, Fudan University, for the support.

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Authors and Affiliations

  1. Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, 200433, Shanghai, China

    Wenbin Chen

  2. School of Mathematical Sciences, Fudan University, 200433, Shanghai, China

    Jianyu Jing

  3. Mathematics Department, University of Massachusetts, North Dartmouth, MA, 02747, USA

    Cheng Wang

  4. International Center for Mathematics and Department of Mathematics, and Guangdong Provincial Key Laboratory of Computational Science and Material Design, and National Center for Applied Mathematics Shenzhen, Southern University of Science and Technology, 518055, Shenzhen, China

    Xiaoming Wang

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  1. Wenbin Chen
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Correspondence to Cheng Wang.

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Chen, W., Jing, J., Wang, C. et al. A Positivity Preserving, Energy Stable Finite Difference Scheme for the Flory-Huggins-Cahn-Hilliard-Navier-Stokes System. J Sci Comput 92, 31 (2022). https://doi.org/10.1007/s10915-022-01872-1

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