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Uniquely solvable and energy stable decoupled numerical schemes for the Cahn–Hilliard–Stokes–Darcy system for two-phase flows in karstic geometry

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Abstract

We propose and analyze two novel decoupled numerical schemes for solving the Cahn–Hilliard–Stokes–Darcy (CHSD) model for two-phase flows in karstic geometry. In the first numerical scheme, we explore a fractional step method (operator splitting) to decouple the phase-field (Cahn–Hilliard equation) from the velocity field (Stokes–Darcy fluid equations). To further decouple the Stokes–Darcy system, we introduce a first order pressure stabilization term in the Darcy solver in the second numerical scheme so that the Stokes system is decoupled from the Darcy system and hence the CHSD system can be solved in a fully decoupled manner. We show that both decoupled numerical schemes are uniquely solvable, energy stable, and mass conservative. Ample numerical results are presented to demonstrate the accuracy and efficiency of our schemes.

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Notes

  1. Invertibility is equivalent to a trivial kernel for a square matrix.

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Authors and Affiliations

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

    Wenbin Chen

  2. Department of Mathematics, Indiana University, Bloomington, IN, 47405, USA

    Daozhi Han

  3. Department of Mathematics, Florida State University, Tallahassee, FL, 32306-4510, USA

    Xiaoming Wang

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  1. Wenbin Chen
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Correspondence to Xiaoming Wang.

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Supported in part by an NSF Grant DMS-1312701. Wenbin Chen is supported by the NSFC (11671098,11331004, 91630309), and a 111 Project (B08018).

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Chen, W., Han, D. & Wang, X. Uniquely solvable and energy stable decoupled numerical schemes for the Cahn–Hilliard–Stokes–Darcy system for two-phase flows in karstic geometry. Numer. Math. 137, 229–255 (2017). https://doi.org/10.1007/s00211-017-0870-1

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