Numerische Mathematik

, Volume 137, Issue 1, pp 229–255 | Cite as

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

  • Wenbin Chen
  • Daozhi Han
  • Xiaoming Wang


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.

Mathematics Subject Classification

35K61 76T99 76S05 76D07 


  1. 1.
    Beavers, G.S., Joseph, D.D.: Boundary conditions at a naturally permeable wall. J. Fluid Mech. 3, 197–207 (1967). doi: 10.1017/S0022112067001375 CrossRefGoogle Scholar
  2. 2.
    Cao, Y., Gunzburger, M., Hua, F., Wang, X.: Coupled Stokes–Darcy model with Beavers–Joseph interface boundary condition. Commun. Math. Sci. 8(1), 1–25 (2010)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Cao, Y., Gunzburger, M., Hua, F., Wang, X.: Analysis and finite element approximation of a coupled, continuum pipe-flow/Darcy model for flow in porous media with embedded conduits. Numer. Methods Partial Differ. Equ. 27(5), 1242–1252 (2011). doi: 10.1002/num.20579 MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cesmelioglu, A., Girault, V., Rivière, B.: Time-dependent coupling of Navier–Stokes and Darcy flows. ESAIM Math. Model. Numer. Anal. 47(2), 539–554 (2013). doi: 10.1051/m2an/2012034 MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Çeşmelioğlu, A., Rivière, B.: Analysis of time-dependent Navier–Stokes flow coupled with Darcy flow. J. Numer. Math. 16(4), 249–280 (2008). doi: 10.1515/JNUM.2008.012 MathSciNetMATHGoogle Scholar
  6. 6.
    Çeşmelioğlu, A., Rivière, B.: Existence of a weak solution for the fully coupled Navie-r-Stokes/Darcy-transport problem. J. Differ. Equ. 252(7), 4138–4175 (2012). doi: 10.1016/j.jde.2011.12.001 CrossRefMATHGoogle Scholar
  7. 7.
    Chen, J., Sun, S., Wang, X.P.: A numerical method for a model of two-phase flow in a coupled free flow and porous media system. J. Comput. Phys. 268, 1–16 (2014). doi: 10.1016/ MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chen, N., Gunzburger, M., Wang, X.: Asymptotic analysis of the differences between the Stokes–Darcy system with different interface conditions and the Stokes–Brinkman system. J. Math. Anal. Appl. 368(2), 658–676 (2010). doi: 10.1016/j.jmaa.2010.02.022 MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Chen, W., Gunzburger, M., Hua, F., Wang, X.: A parallel Robin–Robin domain decomposition method for the Stokes–Darcy system. SIAM J. Numer. Anal. 49(3), 1064–1084 (2011). doi: 10.1137/080740556 MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Chen, W., Gunzburger, M., Sun, D., Wang, X.: Efficient and long-time accurate second-order methods for the Stokes–Darcy system. SIAM J. Numer. Anal. 51(5), 2563–2584 (2013). doi: 10.1137/120897705 MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Chen, W., Gunzburger, M., Sun, D., Wang, X.: An efficient and long-time accurate third-order algorithm for the Stokes-Darcy system. Numer. Math. (2015). doi: 10.1007/s00211-015-0789-3 MATHGoogle Scholar
  12. 12.
    Chidyagwai, P., Rivière, B.: On the solution of the coupled Navier–Stokes and Darcy equations. Comput. Methods Appl. Mech. Eng. 198(47–48), 3806–3820 (2009). doi: 10.1016/j.cma.2009.08.012 MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Chorin, A.J.: The numerical solution of the Navier–Stokes equations for an incompressible fluid. Bull. Am. Math. Soc. 73, 928–931 (1967)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    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). doi: 10.4208/cicp.171211.130412a MathSciNetCrossRefGoogle Scholar
  15. 15.
    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). doi: 10.1137/130950628 MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Discacciati, M., Miglio, E., Quarteroni, A.: Mathematical and numerical models for coupling surface and groundwater flows. Appl. Numer. Math. 43(1–2), 57–74 (2002). doi: 10.1016/S0168-9274(02)00125-3 MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Discacciati, M., Quarteroni, A.: Analysis of a domain decomposition method for the coupling of the Stokes and Darcy equations. In: Brezzi, F., Buffa, A., Corsaro, S., Murli, A. (eds.) Numerical Mathematics and Advanced Applications, vol. 320, pp. 3–20. Springer, Milan (2003)Google Scholar
  18. 18.
    Discacciati, M., Quarteroni, A.: Navier–Stokes/Darcy coupling: modeling, analysis, and numerical approximation. Rev. Mat. Complut. 22(2), 315–426 (2009). doi: 10.5209/rev_REMA.2009.v22.n2.16263 MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    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 (1998). doi: 10.1557/PROC-529-39
  20. 20.
    Girault, V., Raviart, P.A.: Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms. Springer Series in Computational Mathematics, vol. 5. Springer, Berlin (1986). doi: 10.1007/978-3-642-61623-5 CrossRefMATHGoogle Scholar
  21. 21.
    Grn, 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). doi: 10.1137/130908208 MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Guermond, J.L., Shen, J.: Velocity-correction projection methods for incompressible flows. SIAM J. Numer. Anal. 41(1), 112–134 (2003). doi: 10.1137/S0036142901395400. (electronic)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Guillén-González, F., Tierra, G.: On linear schemes for a Cahn–Hilliard diffuse interface model. J. Comput. Phys. 234, 140–171 (2013). doi: 10.1016/ MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    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). doi: 10.1016/ MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Han, D., Sun, D., Wang, X.: Two-phase flows in karstic geometry. Math. Methods Appl. Sci. 37(18), 3048–3063 (2014). doi: 10.1002/mma.3043 MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    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). doi: 10.1016/ MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    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). doi: 10.1002/num.22036 MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    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). doi: 10.1016/j.jde.2014.07.013 MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Hecht, F.: New development in freefem++. J. Numer. Math. 20(3–4), 251–265 (2012)MathSciNetMATHGoogle Scholar
  30. 30.
    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). doi: 10.1016/ MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Jäger, W., Mikelić, A.: On the interface boundary condition of Beavers, Joseph, and Saffman. SIAM J. Appl. Math. 60(4), 1111–1127 (2000). doi: 10.1137/S003613999833678X. (electronic)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Jones, I.P.: Low Reynolds number flow past a porous spherical shell. Math. Proc. Camb. Philos. Soc. 73, 231–238 (1973). doi: 10.1017/S0305004100047642 CrossRefMATHGoogle Scholar
  33. 33.
    Kay, D., Styles, V., Welford, R.: Finite element approximation of a Cahn–Hilliard–Navier–Stokes system. Interfaces Free Bound 10(1), 15–43 (2008). doi: 10.4171/IFB/178 MathSciNetCrossRefGoogle Scholar
  34. 34.
    Kim, J., Kang, K., Lowengrub, J.: Conservative multigrid methods for Cahn–Hilliard fluids. J. Comput. Phys. 193(2), 511–543 (2004). doi: 10.1016/ MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Kim, J., Moin, P.: Application of a fractional-step method to incompressible Navier–Stokes equations. J. Comput. Phys. 59(2), 308–323 (1985). doi: 10.1016/0021-9991(85)90148-2 MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Layton, W.J., Schieweck, F., Yotov, I.: Coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 40(6), 2195–2218 (2002). doi: 10.1137/S0036142901392766 MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    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). doi: 10.1017/jfm.2012.461 MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Marchuk, G.: 3-The splitting-up method. In: Marchuk, G. (ed.) Numerical Methods in Weather Prediction, pp. 84–115. Academic Press, Cambridge (1974). doi: 10.1016/B978-0-12-470650-7.50008-6 Google Scholar
  39. 39.
    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). doi: 10.1002/num.21721 MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Mu, M., Zhu, X.: Decoupled schemes for a non-stationary mixed Stokes–Darcy model. Math. Comput. 79(270), 707–731 (2010). doi: 10.1090/S0025-5718-09-02302-3 MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Saffman, P.G.: On the boundary condition at the interface of a porous medium. Stud. Appl. Math. 1, 93–101 (1971)CrossRefMATHGoogle Scholar
  42. 42.
    Shen, J.: On error estimates of the projection methods for the Navier–Stokes equations: second-order schemes. Math. Comput. 65(215), 1039–1065 (1996). doi: 10.1090/S0025-5718-96-00750-8 MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    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(2012). doi: 10.1142/9789814360906_0003
  44. 44.
    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). doi: 10.1137/110822839 MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Shen, J., Yang, X.: Numerical approximations of Allen–Cahn and Cahn–Hilliard equations. Discrete Contin. Dyn. Syst. 28(4), 1669–1691 (2010). doi: 10.3934/dcds.2010.28.1669 MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    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). doi: 10.1137/09075860X MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    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). doi: 10.1137/140971154 MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Strang, G.: On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5, 506–517 (1968)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Temam, R.: Une méthode d’approximation de la solution des équations de Navier–Stokes. Bull. Soc. Math. France 96, 115–152 (1968)MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Témam, R.: Sur l’approximation de la solution des équations de Navier–Stokes par la méthode des pas fractionnaires II. Arch. Ration. Mech. Anal. 33, 377–385 (1969)CrossRefMATHGoogle Scholar
  51. 51.
    van Kan, J.: A second-order accurate pressure-correction scheme for viscous incompressible flow. SIAM J. Sci. Stat. Comput. 7(3), 870–891 (1986). doi: 10.1137/0907059 MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Wang, X.: Numerical algorithms for stationary statistical properties of dissipative dynamical systems. Discrete Contin. Dyn. Syst. 36(8), 4599–4618 (2016). doi: 10.3934/dcds.2016.36.4599 MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Wise, S.M.: Unconditionally stable finite difference, nonlinearmultigrid simulation of the Cahn–Hilliard–Hele–Shaw systemof equations. J. Sci. Comput. 44(1), 38–68 (2010). doi: 10.1007/s10915-010-9363-4 MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Yanenko, N.N.: The method of fractional steps. The solution of problems of mathematical physics in several variables. Springer, New York (1971). (Translated from the Russian by T. Cheron. English translation edited by M. Holt)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.School of Mathematical SciencesFudan UniversityShanghaiChina
  2. 2.Department of MathematicsIndiana UniversityBloomingtonUSA
  3. 3.Department of MathematicsFlorida State UniversityTallahasseeUSA

Personalised recommendations