Skip to main content
SpringerLink
Log in

Non-iterative, unconditionally energy stable and large time-stepping method for the Cahn-Hilliard phase-field model with Flory-Huggins-de Gennes free energy

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we consider numerical approximations for solving the Cahn-Hilliard phase-field model with the Flory-Huggins-de Gennes free energy for homopolymer blends. We develop an efficient, second-order accurate, and unconditionally energy stable scheme that combines the SAV approach with the stabilization technique, in which the H1 norm is split from the total free energy and two extra linear stabilization terms are added to enhance the stability and keeping the required accuracy while using large time steps. The scheme is very easy to implement and non-iterative where one only needs to solve two decoupled fourth-order biharmonic equations with constant coefficients at each time step. We further prove the unconditional energy stability of the scheme rigorously. Through the comparisons with some other prevalent schemes like the non-stabilized-SAV and MSAV schemes for some benchmark numerical examples in 2D and 3D, we demonstrate the stability and the accuracy of the developed scheme numerically.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Binder, K.: Collective diffusion, nucleation, and spinodal decomposition in polymer mixtures. J. Chem. Phys. 79, 6387 (1983)

    Article  Google Scholar 

  2. Chen, C., Yang, X.: Efficient numerical scheme for a dendritic solidification phase field model with melt convection. J Comput. Phys. 388, 41–62 (2019)

    Article  MathSciNet  Google Scholar 

  3. Chen, C., Yang, X.: Fast, provably unconditionally energy stable, and second-order accurate algorithms for the anisotropic Cahn-Hilliard model. Comput. Meth. Appl. Mech. Engrg 351, 35–59 (2019)

    Article  MathSciNet  Google Scholar 

  4. Chen, F., Shen, J.: Efficient energy stable schemes with spectral discretization in space for anisotropic Cahn-Hilliard systems. Commun Comput. Phys. 05, 1189–1208 (2013)

    Article  MathSciNet  Google Scholar 

  5. Chen, L., Zhao, J., Yang, X.: Regularized linear schemes for the molecular beam epitaxy model with slope selection. Appl. Num Math. 128, 139–156 (2018)

    Article  MathSciNet  Google Scholar 

  6. Cheng, Q., Shen, J.: Multiple scalar auxiliary variable (MSAV) approach and its application to the phase-field vesicle membrane model. SIAM J. Sci. Comput. 40, A3982–A4006 (2018)

    Article  MathSciNet  Google Scholar 

  7. Cheng, Q., Yang, X., Shen, J.: Efficient and accurate numerical schemes for a hydro-dynamically coupled phase field diblock copolymer model. J. Comp. Phys. 341, 44–60 (2017)

    Article  MathSciNet  Google Scholar 

  8. Cook, H. E.: Brownian motion in spinodal decomposition mouvement brownien dans la decomposition spinodale brownsche bewegung bei der spinodalen entmischung. Acta Metall 18, 297 (1970)

    Article  Google Scholar 

  9. Copetti, M. I. M., Elliott, C. M.: Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy. Numer Math. 63(4), 39–65 (1992)

    Article  MathSciNet  Google Scholar 

  10. de Gennes, P. G.: Scaling Concepts in Polymer Physics. Cornell University Press, Ithaca (1979)

    Google Scholar 

  11. de Gennes, P. G.: Dynamics of fluctuations and spinodal decomposition in polymer blends. J. Chem. Phys. 7, 4756 (1980)

    Article  MathSciNet  Google Scholar 

  12. Debussche, A., Dettori, L.: On the Cahn-Hilliard equation with a logarithmic free energy. Nonlinear analysis: Theory. Methods & Applications 24(10), 1491–1514 (1995)

    MathSciNet  MATH  Google Scholar 

  13. Elliott, C. M., Garcke, H.: On the Cahn-Hilliard equation with degenerate mobility. SIAMJ Math. Anal. 27, 404–423 (1996)

    Article  MathSciNet  Google Scholar 

  14. Eyre, D. J.: Unconditionally Gradient Stable Time Marching the Cahn-Hilliard Equation. In: Computational and Mathematical Models of Microstructural Evolution (San Francisco, CA, 1998), volume 529 of Mater. Res. Soc. Sympos. Proc., pp 39–46. MRS, Warrendale (1998)

  15. Feng, X., Prol, A.: Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows. Numer. Math. 94, 33–65 (2003)

    Article  MathSciNet  Google Scholar 

  16. Fialkowski, M., Holyst, R.: Dynamics of phase separation in polymer blends revisited: morphology, spinodal, noise, and nucleation. Macromol. Theory Simul. 17, 263 (2008)

    Article  Google Scholar 

  17. Forster, S., Khandpur, A. K., Zhao, J., Bates, F. S.: Complex phase behavior of polyisoprene-polystyrene diblock copolymers near the order-disorder transition. Macromolecules 27, 6922–6935 (1994)

    Article  Google Scholar 

  18. Gao, Y., He, X., Mei, L., Yang, X.: Decoupled, linear, and energy stable finite element method for Cahn-Hilliard-Navier-Stokes-Darcy phase field model. SIAM J. Sci. Comput. 40, B110–B137 (2018)

    Article  MathSciNet  Google Scholar 

  19. Gomez, H., Calo, V. M., Bazilevs, Y., Hughes, T.J.R.: Isogeometric analystis of the Cahn-Hilliard phase-field model. Comput. Methods Appl. Mech. Eng. 197, 4333–4352 (2008)

    Article  Google Scholar 

  20. Gomez, H., Van der, Z., Kristoffer, G.: Computational phase-field modeling. In: Encyclopedia of Computational Mechanics, Second Edition. John Wiley & Sons, Ltd, ISBN 978-1-119-00379-3 (2017)

  21. Gomez, H., Hughes, T.J.R.: Provably unconditionally stable, second-order time-accurate, mixed variational methods for phase-field models. J. Comput. Phys. 230, 5310–5327 (2011)

    Article  MathSciNet  Google Scholar 

  22. 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 (2017)

    Article  MathSciNet  Google Scholar 

  23. 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)

    Article  MathSciNet  Google Scholar 

  24. Huang, Q., Yang, X., He, X.: Numerical approximations for a smectic–a liquid crystal flow model first-order, linear, decoupled and energy stable schemes. Disc. Conti. Dyn. Sys.-B 23, 2177–2192 (2018)

    MathSciNet  MATH  Google Scholar 

  25. Kim, J., Lowengrub, J.: Phase field modeling and simulation of three-phase flows. Interfaces and Free Boundaries 7, 435–466 (2005)

    Article  MathSciNet  Google Scholar 

  26. 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)

    Article  MathSciNet  Google Scholar 

  27. Romero, I.: Thermodynamically consistent time stepping algorithms for nonlinear thermomechanical systems. Int. J. Numer. Meth. Engng. 79, 706–732 (2009)

    Article  Google Scholar 

  28. Schimperna, G., Pawlow, I.: On a class of Cahn-Hilliard models with nonlinear diffusion. SIAM J. Math. Anal. 45(4), 31–63 (2013)

    Article  MathSciNet  Google Scholar 

  29. Shen, J., Wang, C., Wang, S., Wang, X.: Second-order convex splitting schemes for gradient flows with Ehrlich-Schwoebel type energy: application to thin film epitaxy. in press, SIAM J. Numer. Anal (2011)

  30. Shen, J., Xu, J.: Convergence and error analysis for the scalar auxiliary variable (SAV) schemes to gradient flows. SIAM J. Numer Anal. 56, 2895–2912 (2019)

    Article  MathSciNet  Google Scholar 

  31. Shen, J., Xue, J., Yang, J.: The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys. 353, 407–416 (2018)

    Article  MathSciNet  Google Scholar 

  32. Shen, J., Yang, X.: Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Disc. Conti. Dyn. Sys.-A 28, 1669–1691 (2010)

    Article  MathSciNet  Google Scholar 

  33. Wodo, O., Ganapathysubramanian, B.: Computationally efficient solution to the Cahn-Hilliard equation adaptive implicit time schemes, mesh sensitivity analysis and the 3D isoperimetric problem. J. Comp. Phys. 230(15), 6037–6060 (2011)

    Article  MathSciNet  Google Scholar 

  34. Xu, Z., Yang, X., Zhang, H., Xie, Z.: Efficient and linear schemes for anisotropic Cahn-Hilliard model using the stabilized-invariant energy quadratization (s-IEQ) approach. Comput. Phys. Commun. 238, 36–49 (2019)

    Article  MathSciNet  Google Scholar 

  35. Yang, X.: Linear, first and second order and unconditionally energy stable numerical schemes for the phase field model of homopolymer blends. J. Comput. Phys. 327, 294–316 (2016)

    Article  MathSciNet  Google Scholar 

  36. Yang, X.: Numerical approximations for the Cahn-Hilliard phase field model of the binary fluid-surfactant system. J. Sci. Comput 74, 1533–1553 (2017)

    Article  MathSciNet  Google Scholar 

  37. Yang, X.: Efficient Linear, stabilized, second order time marching schemes for an anisotropic phase field dendritic crystal growth model. Comput. Meth. Appl. Mech. Engrg. 347, 316–339 (2019)

    Article  MathSciNet  Google Scholar 

  38. Yang, X., Zhang, G. -D.: Numerical approximations of the Cahn-Hilliard and Allen-Cahn equations with general nonlinear potential using the invariant energy quadratization approach submitted (2018)

  39. Yang, X., Zhao, J., He, X.: Linear, second order and unconditionally energy stable schemes for the viscous Cahn-Hilliard equation with hyperbolic relaxation using the invariant energy quadratization method. J. Comput. Appl. Math. 343, 80–97 (2018)

    Article  MathSciNet  Google Scholar 

  40. Yang, X., Zhao, J., Wang, Q., Shen, J.: Numerical approximations for a three components Cahn-Hilliard phase-field model based on the invariant energy quadratization method. M3AS: Mathematical Models and Methods in Applied Sciences 27, 1993–2030 (2017)

    MathSciNet  MATH  Google Scholar 

  41. Yuan, C., Zhang, H.: Self-consistent mean field model of hydrogel and its numerical simulation. J. Theor. Comput. Chem. 12, 1350048 (2013)

    Article  Google Scholar 

  42. 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. Meth. Appl. Mech Engrg. 318, 803–825 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Funding

J. Zhang was supported by the Science and Technology Program of Guizhou Province (No.[2020]1Y013), and Guizhou Key Laboratory of Big Data Statistics Analysis (No. BDSA20190107) .X. Yang was partially supported by NSF-DMS-1720212 and DMS-1818783.

Author information

Authors and Affiliations

  1. Guizhou Key Laboratory of Big Data Statistics Analysis, Guizhou University of Finance and Economics, Guizhou, 550025, Guiyang, China

    Jun Zhang

  2. Computational Mathematics Research Center, Guizhou University of Finance and Economics, Guizhou, 550025, Guiyang, China

    Jun Zhang & Xiaofeng Yang

Authors
  1. Jun Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaofeng Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaofeng Yang.

Additional information

Communicated by: Long Chen

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, J., Yang, X. Non-iterative, unconditionally energy stable and large time-stepping method for the Cahn-Hilliard phase-field model with Flory-Huggins-de Gennes free energy. Adv Comput Math 46, 47 (2020). https://doi.org/10.1007/s10444-020-09793-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10444-020-09793-z

Keywords

Mathematics Subject Classification (2010)

Search

Navigation