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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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References

  1. Lee, H.-G., Lowengrub, J.S., Goodman, J.: Modeling pinchoff and reconnection in a Hele-Shaw cell. I. The models and their calibration. Phys. Fluids 14(2), 492–513 (2002). https://doi.org/10.1063/1.1425843

    Article  MathSciNet  MATH  Google Scholar 

  2. Lee, H.-G., Lowengrub, J.S., Goodman, J.: Modeling pinchoff and reconnection in a Hele–Shaw cell. II. Analysis and simulation in the nonlinear regime. Phys. Fluids 14(2), 514–545 (2002). https://doi.org/10.1063/1.1425844

    Article  MathSciNet  MATH  Google Scholar 

  3. Wise, S.M.: Unconditionally stable finite difference, nonlinear multigrid simulation of the Cahn–Hilliard–Hele–Shaw system of equations. J. Sci. Comput. 44(1), 38–68 (2010). https://doi.org/10.1007/s10915-010-9363-4

    Article  MathSciNet  MATH  Google Scholar 

  4. Bear, J.: Dynamics of Fluids in Porous Media. Courier Dover Publications, Mineola (1988)

    MATH  Google Scholar 

  5. Nield, D.A., Bejan, A.: Convection in Porous Media, 2nd edn, p. 546. Springer, New York (1999)

    Book  Google Scholar 

  6. Han, D., Wang, X.: Decoupled energy-law preserving numerical schemes for the Cahn–Hilliard–Darcy system. Numer. Methods Partial Differ. Equ. 32(3), 936–954 (2016). https://doi.org/10.1002/num.22036

    Article  MathSciNet  MATH  Google Scholar 

  7. Dedè, L., Garcke, H., Lam, K.F.: A Hele–Shaw–Cahn–Hilliard model for incompressible two-phase flows with different densities. J. Math. Fluid Mech. (2017). https://doi.org/10.1007/s00021-017-0334-5

    Article  MathSciNet  Google Scholar 

  8. Feng, X., Wise, S.: Analysis of a Darcy–Cahn–Hilliard diffuse interface model for the Hele–Shaw flow and its fully discrete finite element approximation. SIAM J. Numer. Anal. 50(3), 1320–1343 (2012). https://doi.org/10.1137/110827119

    Article  MathSciNet  MATH  Google Scholar 

  9. Wang, X., Zhang, Z.: Well-posedness of the Hele–Shaw–Cahn–Hilliard system. Ann. Inst. H. Poincaré Anal. Non Linéaire 30(3), 367–384 (2013). https://doi.org/10.1016/j.anihpc.2012.06.003

    Article  MathSciNet  MATH  Google Scholar 

  10. Wang, X., Wu, H.: Long-time behavior for the Hele–Shaw–Cahn–Hilliard system. Asymptot. Anal. 78(4), 217–245 (2012)

    MathSciNet  MATH  Google Scholar 

  11. Lowengrub, J., Titi, E., Zhao, K.: Analysis of a mixture model of tumor growth. Eur. J. Appl. Math. 24(5), 691–734 (2013). https://doi.org/10.1017/S0956792513000144

    Article  MathSciNet  MATH  Google Scholar 

  12. Jiang, J., Wu, H., Zheng, S.: Well-posedness and long-time behavior of a non-autonomous Cahn–Hilliard–Darcy system with mass source modeling tumor growth. J. Differ. Equ. 259(7), 3032–3077 (2015). https://doi.org/10.1016/j.jde.2015.04.009

    Article  MathSciNet  MATH  Google Scholar 

  13. Han, D.: A decoupled unconditionally stable numerical scheme for the Cahn–Hilliard–Hele–Shaw system. J. Sci. Comput. 1–20 (2015). https://doi.org/10.1007/s10915-015-0055-y

    Article  MathSciNet  Google Scholar 

  14. Han, D., Wang, X.: Initial-boundary layer associated with the nonlinear Darcy–Brinkman system. J. Differ. Equ. 256(2), 609–639 (2014). https://doi.org/10.1016/j.jde.2013.09.014

    Article  MathSciNet  MATH  Google Scholar 

  15. Chemetov, N., Neves, W.: The generalized Buckley–Leverett system: solvability. Arch. Ration. Mech. Anal. 208(1), 1–24 (2013). https://doi.org/10.1007/s00205-012-0591-7

    Article  MathSciNet  MATH  Google Scholar 

  16. Anderson, D.M., McFadden, G.B., Wheeler, A.A.: Diffuse-interface methods in fluid mechanics. In: Annual Review of Fluid Mechanics, Vol. 30. Annu. Rev. Fluid Mech., vol. 30, pp. 139–165. Annual Reviews, Palo Alto, CA (1998)

  17. Lowengrub, J., Truskinovsky, L.: Quasi-incompressible Cahn–Hilliard fluids and topological transitions. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454(1978), 2617–2654 (1998). https://doi.org/10.1098/rspa.1998.0273

    Article  MathSciNet  MATH  Google Scholar 

  18. Magaletti, F., Picano, F., Chinappi, M., Marino, L., Casciola, C.M.: The sharp-interface limit of the Cahn–Hilliard–Navier–Stokes model for binary fluids. J. Fluid Mech. 714, 95–126 (2013). https://doi.org/10.1017/jfm.2012.461

    Article  MathSciNet  MATH  Google Scholar 

  19. Guo, R., Xia, Y., Xu, Y.: An efficient fully-discrete local discontinuous Galerkin method for the Cahn–Hilliard–Hele–Shaw system. J. Comput. Phys. 264, 23–40 (2014). https://doi.org/10.1016/j.jcp.2014.01.037

    Article  MathSciNet  MATH  Google Scholar 

  20. Diegel, A.E., Feng, X.H., Wise, S.M.: Analysis of a mixed finite element method for a Cahn–Hilliard–Darcy–Stokes system. SIAM J. Numer. Anal. 53(1), 127–152 (2015). https://doi.org/10.1137/130950628

    Article  MathSciNet  MATH  Google Scholar 

  21. Chen, W., Liu, Y., Wang, C., Wise, S.M.: Convergence analysis of a fully discrete finite difference scheme for the Cahn–Hilliard–Hele–Shaw equation. Math. Comp. 85(301), 2231–2257 (2016). https://doi.org/10.1090/mcom3052

    Article  MathSciNet  MATH  Google Scholar 

  22. Liu, Y., Chen, W., Wang, C., Wise, S.M.: Error analysis of a mixed finite element method for a Cahn–Hilliard–Hele–Shaw system. Numer. Math. 135(3), 679–709 (2017). https://doi.org/10.1007/s00211-016-0813-2

    Article  MathSciNet  MATH  Google Scholar 

  23. Eyre, D.J.: Unconditionally gradient stable time marching the Cahn-Hilliard equation. In:Computational and Mathematical Models of Microstructural Evolution (San Francisco, CA, 1998). Mater. Res. Soc. Sympos. Proc., vol. 529, pp. 39–46. MRS, Warrendale, PA (1998)

  24. Collins, C., Shen, J., Wise, S.M.: An efficient, energy stable scheme for the Cahn–Hilliard–Brinkman system. Commun. Comput. Phys. 13(4), 929–957 (2013)

    Article  MathSciNet  Google Scholar 

  25. Feng, X.: Fully discrete finite element approximations of the Navier–Stokes–Cahn–Hilliard diffuse interface model for two-phase fluid flows. SIAM J. Numer. Anal. 44(3), 1049–1072 (2006). https://doi.org/10.1137/050638333

    Article  MathSciNet  MATH  Google Scholar 

  26. Kay, D., Styles, V., Welford, R.: Finite element approximation of a Cahn–Hilliard–Navier–Stokes system. Interf. Free Bound. 10(1), 15–43 (2008). https://doi.org/10.4171/IFB/178

    Article  MathSciNet  Google Scholar 

  27. Shen, J., Yang, X.: A phase-field model and its numerical approximation for two-phase incompressible flows with different densities and viscosities. SIAM J. Sci. Comput. 32(3), 1159–1179 (2010). https://doi.org/10.1137/09075860X

    Article  MathSciNet  MATH  Google Scholar 

  28. Shen, J.: Modeling and numerical approximation of two-phase incompressible flows by a phase-field approach. In:Multiscale Modeling and Analysis for Materials Simulation. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 22, pp. 147–195. World Sci. Publ., Hackensack, NJ, (2012)

  29. Grün, G.: On convergent schemes for diffuse interface models for two-phase flow of incompressible fluids with general mass densities. SIAM J. Numer. Anal. 51(6), 3036–3061 (2013). https://doi.org/10.1137/130908208

    Article  MathSciNet  MATH  Google Scholar 

  30. Guo, Z., Lin, P., Lowengrub, J.S.: A numerical method for the quasi-incompressible Cahn–Hilliard–Navier–Stokes equations for variable density flows with a discrete energy law. J. Comput. Phys. 276, 486–507 (2014). https://doi.org/10.1016/j.jcp.2014.07.038

    Article  MathSciNet  MATH  Google Scholar 

  31. Minjeaud, S.: An unconditionally stable uncoupled scheme for a triphasic Cahn–Hilliard/Navier–Stokes model. Numer. Methods Partial Differ. Equ. 29(2), 584–618 (2013). https://doi.org/10.1002/num.21721

    Article  MathSciNet  MATH  Google Scholar 

  32. Shen, J., Yang, X.: Decoupled, energy stable schemes for phase-field models of two-phase incompressible flows. SIAM J. Numer. Anal. 53(1), 279–296 (2015). https://doi.org/10.1137/140971154

    Article  MathSciNet  MATH  Google Scholar 

  33. Shen, J., Yang, X., Yu, H.: Efficient energy stable numerical schemes for a phase field moving contact line model. J. Comput. Phys. 284, 617–630 (2015). https://doi.org/10.1016/j.jcp.2014.12.046

    Article  MathSciNet  MATH  Google Scholar 

  34. Guermond, J.-L., Salgado, A.: A splitting method for incompressible flows with variable density based on a pressure Poisson equation. J. Comput. Phys. 228(8), 2834–2846 (2009). https://doi.org/10.1016/j.jcp.2008.12.036

    Article  MathSciNet  MATH  Google Scholar 

  35. Hu, Z., Wise, S.M., Wang, C., Lowengrub, J.S.: Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation. J. Comput. Phys. 228(15), 5323–5339 (2009). https://doi.org/10.1016/j.jcp.2009.04.020

    Article  MathSciNet  MATH  Google Scholar 

  36. Baskaran, A., Lowengrub, J.S., Wang, C., Wise, S.M.: Convergence analysis of a second order convex splitting scheme for the modified phase field crystal equation. SIAM J. Numer. Anal. 51(5), 2851–2873 (2013). https://doi.org/10.1137/120880677

    Article  MathSciNet  MATH  Google Scholar 

  37. Shen, J., Wang, C., Wang, X., Wise, S.M.: Second-order convex splitting schemes for gradient flows with Ehrlich–Schwoebel type energy: application to thin film epitaxy. SIAM J. Numer. Anal. 50(1), 105–125 (2012). https://doi.org/10.1137/110822839

    Article  MathSciNet  MATH  Google Scholar 

  38. Han, D., Wang, X.: A second order in time, uniquely solvable, unconditionally stable numerical scheme for Cahn–Hilliard–Navier–Stokes equation. J. Comput. Phys. 290, 139–156 (2015). https://doi.org/10.1016/j.jcp.2015.02.046

    Article  MathSciNet  MATH  Google Scholar 

  39. van Kan, J.: A second-order accurate pressure-correction scheme for viscous incompressible flow. SIAM J. Sci. Statist. Comput. 7(3), 870–891 (1986). https://doi.org/10.1137/0907059

    Article  MathSciNet  MATH  Google Scholar 

  40. Kim, J., Kang, K., Lowengrub, J.: Conservative multigrid methods for Cahn–Hilliard fluids. J. Comput. Phys. 193(2), 511–543 (2004). https://doi.org/10.1016/j.jcp.2003.07.035

    Article  MathSciNet  MATH  Google Scholar 

  41. Diegel, A.E., Wang, C., Wang, X., Wise, S.M.: Convergence analysis and error estimates for a second order accurate finite element method for the Cahn–Hilliard–Navier–Stokes system. Numerische Mathematik, 1–40 (2017). https://doi.org/10.1007/s00211-017-0887-5

    Article  MathSciNet  Google Scholar 

  42. Dong, S., Shen, J.: A time-stepping scheme involving constant coefficient matrices for phase-field simulations of two-phase incompressible flows with large density ratios. J. Comput. Phys. 231(17), 5788–5804 (2012). https://doi.org/10.1016/j.jcp.2012.04.041

    Article  MathSciNet  MATH  Google Scholar 

  43. Aland, S.: Time integration for diffuse interface models for two-phase flow. J. Comput. Phys. 262, 58–71 (2014). https://doi.org/10.1016/j.jcp.2013.12.055

    Article  MathSciNet  MATH  Google Scholar 

  44. Guillén-González, F., Tierra, G.: On linear schemes for a Cahn–Hilliard diffuse interface model. J. Comput. Phys. 234, 140–171 (2013). https://doi.org/10.1016/j.jcp.2012.09.020

    Article  MathSciNet  MATH  Google Scholar 

  45. Yang, X., Zhao, J., Wang, Q., Shen, J.: Numerical approximations for a three-components cahnhilliard phase-field model based on the invariant energy quadratization method. Math. Models Methods Appl. Sci. 1–38 (2017). https://doi.org/10.1142/S0218202517500373. http://www.worldscientific.com/doi/pdf/10.1142/S0218202517500373

    Article  MathSciNet  Google Scholar 

  46. Cheng, Q., Yang, X., Shen, J.: Efficient and accurate numerical schemes for a hydro-dynamically coupled phase field diblock copolymer model. J. Comput. Phys. 341, 44–60 (2017). https://doi.org/10.1016/j.jcp.2017.04.010

    Article  MathSciNet  MATH  Google Scholar 

  47. Yang, X., Ju, L.: Linear and unconditionally energy stable schemes for the binary fluid-surfactant phase field model. Comput. Methods Appl. Mech. Eng. 318, 1005–1029 (2017). https://doi.org/10.1016/j.cma.2017.02.011

    Article  MathSciNet  Google Scholar 

  48. Zhao, J., Yang, X., Gong, Y., Wang, Q.: A novel linear second order unconditionally energy stable scheme for a hydrodynamic Q-tensor model of liquid crystals. Comput. Methods Appl. Mech. Eng. 318, 803–825 (2017). https://doi.org/10.1016/j.cma.2017.01.031

    Article  MathSciNet  Google Scholar 

  49. Yang, X., Zhao, J., Wang, Q.: Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method. J. Comput. Phys. 333, 104–127 (2017). https://doi.org/10.1016/j.jcp.2016.12.025

    Article  MathSciNet  MATH  Google Scholar 

  50. Yang, X., Han, D.: Linearly first- and second-order, unconditionally energy stable schemes for the phase field crystal equation. J. Comput. Phys. 330, 1116–1134 (2017)

    Article  MathSciNet  Google Scholar 

  51. Guermond, J.L., Minev, P., Shen, J.: An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Engrg. 195(44–47), 6011–6045 (2006). https://doi.org/10.1016/j.cma.2005.10.010

    Article  MathSciNet  MATH  Google Scholar 

  52. Guermond, J.-L., Quartapelle, L.: On the approximation of the unsteady Navier-Stokes equations by finite element projection methods. Numer. Math. 80(2), 207–238 (1998). https://doi.org/10.1007/s002110050366

    Article  MathSciNet  MATH  Google Scholar 

  53. Han, D., Wang, X., Wu, H.: Existence and uniqueness of global weak solutions to a Cahn–Hilliard–Stokes–Darcy system for two phase incompressible flows in karstic geometry. J. Differ. Equ. 257(10), 3887–3933 (2014). https://doi.org/10.1016/j.jde.2014.07.013

    Article  MathSciNet  MATH  Google Scholar 

  54. Diegel, A.E., Wang, C., Wise, S.M.: Stability and convergence of a second-order mixed finite element method for the Cahn–Hilliard equation. IMA J. Numer. Anal. 36(4), 1867–1897 (2016). https://doi.org/10.1093/imanum/drv065

    Article  MathSciNet  MATH  Google Scholar 

  55. Hecht, F.: New development in freefem++. J. Numer. Math. 20(3–4), 251–265 (2012)

    MathSciNet  MATH  Google Scholar 

  56. Christlieb, A., Jones, J., Promislow, K., Wetton, B., Willoughby, M.: High accuracy solutions to energy gradient flows from material science models. J. Comput. Phys. 257(part A), 193–215 (2014). https://doi.org/10.1016/j.jcp.2013.09.049

    Article  MathSciNet  Google Scholar 

  57. Glasner, K., Orizaga, S.: Improving the accuracy of convexity splitting methods for gradient flow equations. J. Comput. Phys. 315, 52–64 (2016). https://doi.org/10.1016/j.jcp.2016.03.042

    Article  MathSciNet  MATH  Google Scholar 

  58. Le Bars, M., Worster, M.G.: Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification. J. Fluid Mech. 550, 149–173 (2006). https://doi.org/10.1017/S0022112005007998

    Article  MathSciNet  MATH  Google Scholar 

  59. Saffman, P.G., Taylor, G.: The penetration of a fluid into a porous medium or Hele–Shaw cell containing a more viscous liquid. Proc. Roy. Soc. Lond. Ser. A 245, 312–3292 (1958)

    Article  MathSciNet  Google Scholar 

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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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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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