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Journal of Scientific Computing

, Volume 75, Issue 2, pp 993–1015 | Cite as

Unconditionally Energy Stable Linear Schemes for the Diffuse Interface Model with Peng–Robinson Equation of State

  • Hongwei Li
  • Lili Ju
  • Chenfei Zhang
  • Qiujin Peng
Article

Abstract

In this paper, we investigate numerical solution of the diffuse interface model with Peng–Robinson equation of state, that describes real states of hydrocarbon fluids in the petroleum industry. Due to the strong nonlinearity of the source terms in this model, how to design appropriate time discretizations to preserve the energy dissipation law of the system at the discrete level is a major challenge. Based on the “Invariant Energy Quadratization” approach and the penalty formulation, we develop efficient first and second order time stepping schemes for solving the single-component two-phase fluid problem. In both schemes the resulted temporal semi-discretizations lead to linear systems with symmetric positive definite spatial operators at each time step. We rigorously prove their unconditional energy stabilities in the time discrete sense. Various numerical simulations in 2D and 3D spaces are also presented to validate accuracy and stability of the proposed linear schemes and to investigate physical reliability of the target model by comparisons with laboratory data.

Keywords

Diffuse interface Linear scheme Peng–Robinson equation of state Invariant energy quadratization Energy stability Penalty formulation 

Mathematics Subject Classification

65N30 65N50 49S05 

References

  1. 1.
    Chen, R., Ji, G., Yang, X., Zhang, H.: Decoupled energy stable schemes for phase-field vesicle membrane model. J. Comput. Phys. 302, 509–523 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Copetti, M., Elliott, C.: Numerical analysis of the Cahn–Hilliard equation with a logarithmic free energy. Numer. Math. 63, 39–65 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Davis, H.T.: Statistical Mechanics of Phases, Interfaces, and Thin Films. VCH, New York (1996)zbMATHGoogle Scholar
  4. 4.
    Du, Q., Li, M., Liu, C.: Analysis of a phase field Navier–Stokes vesicle-fluid interaction model. Dis. Contin. Dyn. Syst. B 8, 539–556 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Du, Q., Liu, C., Wang, X.: A phase field approach in the numerical study of the elastic bending energy for vesicle membranes. J. Comput. Phys. 198, 450–468 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Du, Q., Liu, C., Wang, X.: Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions. J. Comput. Phys. 212, 757–777 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Elliott, C.M., Garcke, H.: On the Cahn–Hilliard equation with degenerate mobility. SIAM J. Math. Anal. 27, 404–423 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Eyre, D.J.: Unconditionally gradient stable time marching the Cahn-Hilliard equation. In: Computational and Mathematical Models of Microstructural Evolution (San Francisco, CA, 1998), Materials Research Society Symposia Proceedings, vol. 529, pp. 39–46. MRS, Warrendale, PA (1998)Google Scholar
  9. 9.
    Firoozabadi, A.: Thermodynamics of Hydrocarbon Reservoirs. McGraw-Hill, New York (1999)Google Scholar
  10. 10.
    Frenkel, D., Smit, B.: Understanding Molecular Simulation: From Algorithms to Applications. Academic Press, San Diego, CA (2001)zbMATHGoogle Scholar
  11. 11.
    Guillén-González, F., Tierra, G.: On linear schemes for a Cahn–Hilliard diffuse interface model. J. Comput. Phys. 234, 140–171 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Han, D., Brylev, A., Yang, X., Tan, Z.: Numerical analysis of second order, fully discrete energy stable schemes for phase field models of two phase incompressible flows. J. Sci. Comput. 70, 965–989 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Haugen, K.B., Firoozabadi, A.: Composition at the interface between multicomponent nonequilibrium fluid phases. J. Chem. Phys. 130, 064707 (2009)CrossRefGoogle Scholar
  14. 14.
    Kim, J.: Phase-field models for multi-component fluid flows. Commun. Comput. Phys. 12, 613–661 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kou, J., Sun, S.: An adaptive finite element method for simulating surface tension with the gradient theory of fluid interfaces. J. Comput. Appl. Math. 255, 593–604 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kou, J., Sun, S.: Numerical methods for a multicomponent two-phase interface model with geometric mean influence parameters. SIAM J. Sci. Comput. 37, B543–B569 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kou, J., Sun, S.: Unconditionally stable methods for simulating multi-component two-phase interface models with Peng–Robinson equation of state and various boundary conditions. J. Comput. Appl. Math. 291, 158–182 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kou, J., Sun, S., Wang, X.: Efficient numerical methods for simulating surface tension of multi-component mixtures with the gradient theory of fluid interfaces. Comput. Methods Appl. Mech. Eng. 292, 92–106 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lin, H., Duan, Y.Y.: Surface tension measurements of propane (r-290) and isobutane (r-600a) from (253 to 333)K. J. Chem. Eng. Data 48, 1360–1363 (2003)CrossRefGoogle Scholar
  20. 20.
    Miqueu, C., Mendiboure, B., Graciaa, A., Lachaise, J.: Modelling of the surface tension of multicomponent mixtures with the gradient theory of fluid interfaces. Ind. Eng. Chem. Res. 44, 3321–3329 (2005)CrossRefGoogle Scholar
  21. 21.
    Peng, D., Robinson, D.: A new two-constant equation of state. Ind. Eng. Chem. Fundam. 15, 59–64 (1976)CrossRefGoogle Scholar
  22. 22.
    Qiao, Z., Sun, S.: Two-phase fluid simulation using a diffuse interface model with Peng–Robinson equation of state. SIAM J. Sci. Comput. 36, B708–B728 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Rongy, L., Haugen, K.B., Firoozabadi, A.: Mixing from Fickian diffusion and natural convection in binary non-equilibrium fluid phases. AIChE J. 58, 1336–1345 (2012)CrossRefGoogle Scholar
  24. 24.
    Shen, J., Yang, X.: Numerical approximations of Allen–Cahn and Cahn–Hilliard equations. Dis. Conti. Dyn. Syst. A 28, 1669–1691 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Sun, S., Wheeler, M.F.: Symmetric and nonsymmetric discontinuous Galerkin methods for reactive transport in porous media. SIAM J. Numer. Anal. 43, 195–219 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Sun, S., Wheeler, M.F.: Local problem-based a posteriori error estimators for discontinuous Galerkin approximations of reactive transport. Comput. Geosci. 11, 87–101 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    van der Waals, J.: The thermodynamic theory of capillarity under the hypothesis of a continuous density variation. J. Stat. Phys. 20, 197–244 (1893)Google Scholar
  28. 28.
    Wang, X., Ju, L., Du, Q.: Efficient and stable exponential time differencing Runge–Kutta methods for phase field elastic bending energy models. J. Comput. Phys. 316, 21–38 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Wang, C., Wise, S.M.: An energy stable and convergent finite-difference scheme for the modified phase field crystal equation. SIAM J. Numer. Anal. 49, 945–969 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Wheeler, M.F., Wick, T., Wollner, W.: An augment-Lagrangian method for the phase-field approach for pressurized fractures. Comput. Methods Appl. Mech. Eng. 271, 69–85 (2014)CrossRefzbMATHGoogle Scholar
  31. 31.
    Yang, X.: Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends. J. Comput. Phys. 327, 294–316 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Yang, X.: Numerical approximations for the Cahn-Hilliard phase field model of the binary fluid-surfactant system. J. Sci. Comput. (2017). doi: 10.1007/s10915-017-0508-6 Google Scholar
  33. 33.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Yang, X., Ju, L.: Efficient linear schemes with unconditionally energy stability for the phase field elastic bending energy model. Comput. Methods Appl. Mech. Eng. 315, 691–712 (2017)CrossRefGoogle Scholar
  35. 35.
    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)MathSciNetCrossRefGoogle Scholar
  36. 36.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Zhao, J., Wang, Q., Yang, X.: Numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach. Inter. J. Numer. Methods Eng. 110, 279–300 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsShandong Normal UniversityJinanChina
  2. 2.Department of MathematicsUniversity of South CarolinaColumbiaUSA
  3. 3.Institute for Mathematical SciencesRenmin University of ChinaBeijingChina

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