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A Second Order in Time, Decoupled, Unconditionally Stable Numerical Scheme for the Cahn–Hilliard–Darcy System

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Abstract

We propose a novel second order in time, fully decoupled and unconditionally stable numerical scheme for solving the Cahn–Hilliard–Darcy system which models two-phase flow in porous medium or in a Hele–Shaw cell. The scheme is based on the ideas of second order convex-splitting for the Cahn–Hilliard equation and pressure-correction for the Darcy equation. The computation of order parameter, pressure and velocity is completely decoupled in our scheme. We show that the scheme is uniquely solvable, unconditionally energy stable and mass-conservative. Ample numerical results are presented to gauge the efficiency and robustness of our scheme.

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Acknowledgements

The work of X. Wang is supported in part by DMS 1715504 and grants from Fudan University. The work of D. Han is supported by a seed fund from the Material Research Center at Missouri University of Science and Technology. The authors wish to thank Wenbin Chen, Mike Jolly and Jie Shen for helpful discussions.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO, 65409, USA

    Daozhi Han

  2. Shanghai Center for Mathematical Sciences and School for Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Xiaoming Wang

  3. Department of Mathematics, Florida State University, Tallahassee, FL, 32306, USA

    Xiaoming Wang

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  1. Daozhi Han
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Han, D., Wang, X. A Second Order in Time, Decoupled, Unconditionally Stable Numerical Scheme for the Cahn–Hilliard–Darcy System. J Sci Comput 77, 1210–1233 (2018). https://doi.org/10.1007/s10915-018-0748-0

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