Uniquely Solvable and Energy Stable Decoupled Numerical Schemes for the Cahn–Hilliard–Navier–Stokes–Darcy–Boussinesq System

Abstract

In this article we propose the Cahn–Hilliard–Navier–Stokes–Darcy–Boussinesq system that models thermal convection of two-phase flows in a fluid layer overlying a porous medium. Based on operator splitting and pressure stabilization we propose a family of fully decoupled numerical schemes such that the Navier–Stokes equations, the Darcy equations, the heat equation and the Cahn–Hilliard equation are solved independently at each time step, thus significantly reducing the computational cost. We show that the schemes preserve the underlying energy law and hence are unconditionally long-time stable. Numerical results are presented to demonstrate the accuracy and stability of the algorithms.

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Correspondence to Daozhi Han.

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W. Chen and Y. Zhang are supported by the National Science Foundation of China (11671098, 91630309, 12071090). D. Han acknowledges support from NSF-DMS-1912715. X. Wang thanks support from NSFC11871159 and Guangdong Provincial Key Laboratory via 2019B030301001.

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Chen, W., Han, D., Wang, X. et al. Uniquely Solvable and Energy Stable Decoupled Numerical Schemes for the Cahn–Hilliard–Navier–Stokes–Darcy–Boussinesq System. J Sci Comput 85, 45 (2020). https://doi.org/10.1007/s10915-020-01341-7

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Keywords

  • Phase field model
  • Two-phase flow
  • Convection
  • Unconditional stability

Mathematics Subject Classification

  • 35K61
  • 76T99
  • 76S05
  • 76D07