Movatterモバイル変換

[0]ホーム
URL:

Skip to main content
Springer Nature Link
Log in

Analysis of the Energy Stability for Stabilized Semi-implicit Schemes of the Functionalized Cahn-Hilliard Mass-conserving Gradient Flow Equation

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

A stabilized semi-implicit scheme was designed in [3] to solve the Functionalized Cahn-Hilliard (CCH) equation, but there is a lack of theoretical analysis of the energy stability. In this paper, we generalize this scheme to solve the general FCH mass-conserving gradient flow (FCH-MCGF) equation and show the theoretical analysis results about the unique solvability and energy stability. We successfully prove that this scheme is uniquely solvable and energy stable in theory by rewriting the double-well potential function to satisfy the Lipschitz-type condition. The range of stabilization parameters is theoretically given as well. In addition, another similar energy stable scheme is proposed, which slightly widens the range of stabilization parameters in theory and has almost the same precision as the previous one. Both the detailed numerical procedure and the selection of stabilization parameters are presented. Finally, several numerical experiments are performed for the FCH-MCGF equation based on these schemes. Specially, the adaptive time step size is considered in the scheme for the simulations of the phase separation in 2D and 3D, since any time step size can be used according to our theoretical results. Numerical results show that these schemes are energy stable and the large time step size indeed can be used in computations. Moreover, by comprehensive comparisons of stability and accuracy among the stabilized semi-implicit scheme, the convex splitting scheme, and the fully implicit scheme, we conclude that the performance of the stabilized semi-implicit scheme is the best, and the convex splitting scheme performs better than the fully implicit scheme.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Badalassi, V., Ceniceros, H., Banerjee, S.: Computation of multiphase systems with phase field models. J. Comput. Phys.190(2), 371–397 (2003)

    Article MathSciNet MATH  Google Scholar 

  2. Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys.28(2), 258–267 (1958)

    Article MATH  Google Scholar 

  3. Chen, F., Shen, J.: Efficient spectral-Galerkin methods for systems of coupled second-order equations and their applications. J. Comput. Phys.231(15), 5016–5028 (2012)

    Article MathSciNet MATH  Google Scholar 

  4. Chen, L.Q., Shen, J.: Applications of semi-implicit Fourier-spectral method to phase field equations. Comput. Phys. Commun.108(2), 147–158 (1998)

    Article MATH  Google Scholar 

  5. 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)

    Article MathSciNet MATH  Google Scholar 

  6. Chen, Y., Lowengrub, J., Shen, J., Wang, C., Wise, S.: Efficient energy stable schemes for isotropic and strongly anisotropic Cahn-Hilliard systems with the willmore regularization. J. Comput. Phys.365, 56–73 (2018)

    Article MathSciNet MATH  Google Scholar 

  7. Chen, Y., Shen, J.: Efficient, adaptive energy stable schemes for the incompressible Cahn-Hilliard Navier-Stokes phase-field models. J. Comput. Phys.308, 40–56 (2016)

    Article MathSciNet MATH  Google Scholar 

  8. Cheng, K., Wang, C., Wise, S.M., Yuan, Z.: Global-in-time Gevrey regularity solutions for the functionalized Cahn-Hilliard equation. Discret. Contin. Dyn. Syst. S13(8), 2211–2229 (2020)

    Article MathSciNet MATH  Google Scholar 

  9. Cheng, K., Wang, C., Wise, S.M., Yue, X.: A second-order, weakly energy-stable pseudo-spectral scheme for the Cahn-Hilliard equation and its solution by the homogeneous linear iteration method. J. Sci. Comput.69, 1083–1114 (2016)

    Article MathSciNet MATH  Google Scholar 

  10. Cheng, X., Promislow, K., Wetton, B.: Asymptotic behaviour of time stepping methods for phase field models. arXiv preprintarXiv:1905.02299 (2020)

  11. 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, 193–215 (2014)

    Article MathSciNet MATH  Google Scholar 

  12. Dai, S., Promislow, K.: Geometric evolution of bilayers under the functionalized Cahn-Hilliard equation. Proc. R. Soc. A469(2153), 20120505 (2013)

    Article MathSciNet MATH  Google Scholar 

  13. Doelman, A., Hayrapetyan, G., Promislow, K., Wetton, B.: Meander and pearling of single-curvature bilayer interfaces in the functionalized Cahn-Hilliard equation. SIAM J. Math. Anal.46(6), 3640–3677 (2014)

    Article MathSciNet MATH  Google Scholar 

  14. Du, Q., Ju, L., Li, X., Qiao, Z.: Stabilized linear semi-implicit schemes for the nonlocal Cahn-Hilliard equation. J. Comput. Phys.363, 39–54 (2018)

    Article MathSciNet MATH  Google Scholar 

  15. Eyre, D.J.: Unconditionally gradient stable time marching the Cahn-Hilliard equation. In: Bullard, J.W., Kalia, R., Stoneham, M., Chen, L. (eds.) Computational and Mathematical Models of Microstructural Evolution, vol. 53, pp. 1686–1712. Materials Research Society, Warrendale (1998)

    Google Scholar 

  16. Feng, W., Guan, Z., Lowengrub, J., Wang, C., Wise, S.M., Chen, Y.: A uniquely solvable, energy stable numerical scheme for the functionalized Cahn-Hilliard equation and its convergence analysis. J. Sci. Comput.76(3), 1938–1967 (2018)

    Article MathSciNet MATH  Google Scholar 

  17. Feng, W., Salgado, A.J., Wang, C., Wise, S.M.: Preconditioned steepest descent methods for some nonlinear elliptic equations involving p-Laplacian terms. J. Comput. Phys.334, 45–67 (2017)

    Article MathSciNet MATH  Google Scholar 

  18. Feng, X., Prohl, A.: Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows. Numer. Math.94(1), 33–65 (2003)

    Article MathSciNet MATH  Google Scholar 

  19. Gavish, N., Hayrapetyan, G., Promislow, K., Yang, L.: Curvature driven flow of bi-layer interfaces. Phys. D Nonlinear Phenom.240(7), 675–693 (2011)

    Article MATH  Google Scholar 

  20. Gavish, N., Jones, J., Xu, Z., Christlieb, A., Promislow, K.: Variational models of network formation and ion transport: applications to perfluorosulfonate ionomer membranes. Polymers4(1), 630–655 (2012)

    Article  Google Scholar 

  21. Guan, Z., Lowengrub, J.S., Wang, C., Wise, S.M.: Second order convex splitting schemes for periodic nonlocal Cahn-Hilliard and Allen-Cahn equations. J. Comput. Phys.277, 48–71 (2014)

    Article MathSciNet MATH  Google Scholar 

  22. Guo, J., Wang, C., Wise, S.M., Yue, X.: An\(H^2\) convergence of a second-order convex-splitting, finite difference scheme for the three-dimensional Cahn-Hilliard equation. Commun. Math. Sci.14(2), 489–515 (2016)

    Article MathSciNet MATH  Google Scholar 

  23. Guo, R., Xu, Y., Xu, Z.: Local discontinuous Galerkin methods for the Functionalized Cahn-Hilliard equation. J. Sci. Comput.63(3), 913–937 (2015)

    Article MathSciNet MATH  Google Scholar 

  24. Jain, S., Bates, F.S.: On the origins of morphological complexity in block copolymer surfactants. Science300(5618), 460–464 (2003)

    Article  Google Scholar 

  25. Jain, S., Bates, F.S.: Consequences of nonergodicity in aqueous binary PEO-PB micellar dispersions. Macromolecules37(4), 1511–1523 (2004)

    Article  Google Scholar 

  26. Ju, L., Zhang, J., Zhu, L., Du, Q.: Fast explicit integration factor methods for semilinear parabolic equations. J. Sci. Comput.62(2), 431–455 (2015)

    Article MathSciNet MATH  Google Scholar 

  27. Kraitzman, N., Promislow, K.: An overview of network bifurcations in the functionalized Cahn-Hilliard free energy. In: Mathematics of Energy and Climate Change, pp. 191–214. Springer (2015)

  28. Kraitzman, N., Promislow, K.: Pearling bifurcations in the strong Functionalized Cahn-Hilliard free energy. SIAM J. Math. Anal.50(3), 3395–3426 (2018)

    Article MathSciNet MATH  Google Scholar 

  29. Li, D., Qiao, Z.: On second order semi-implicit Fourier spectral methods for 2D Cahn-Hilliard equations. J. Sci. Comput.70(1), 301–341 (2017)

    Article MathSciNet MATH  Google Scholar 

  30. Li, D., Qiao, Z., Tang, T.: Characterizing the stabilization size for semi-implicit Fourier-spectral method to phase field equations. SIAM J. Numer. Anal.54(3), 1653–1681 (2016)

    Article MathSciNet MATH  Google Scholar 

  31. Li, X., Qiao, Z., Zhang, H.: A second-order convex splitting scheme for a Cahn-Hilliard equation with variable interfacial parameters. J. Comput. Math.35(6) (2017)

  32. Promislow, K., Wetton, B.: Pem fuel cells: a mathematical overview. SIAM J. Appl. Math.70(2), 369–409 (2009)

    Article MathSciNet MATH  Google Scholar 

  33. Promislow, K., Wu, Q.: Existence of pearled patterns in the planar functionalized Cahn-Hilliard equation. J. Differ. Equ.259(7), 3298–3343 (2015)

    Article MathSciNet MATH  Google Scholar 

  34. 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)

    Article MathSciNet MATH  Google Scholar 

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

    Article MathSciNet MATH  Google Scholar 

  36. 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)

    Article MathSciNet MATH  Google Scholar 

  37. Shen, J., Yang, X.: Energy stable schemes for Cahn-Hilliard phase-field model of two-phase incompressible flows. Chin. Ann. Math. Ser. B31(5), 743–758 (2010)

    Article MathSciNet MATH  Google Scholar 

  38. Shen, J., Yang, X.: Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discret. Contin. Dyn. Syst. A28(4), 1669–1691 (2010)

    Article MathSciNet MATH  Google Scholar 

  39. Stuart, A.M., Humphries, A.R.: Model problems in numerical stability theory for initial value problems. SIAM Rev.36(2), 226–257 (1994)

    Article MathSciNet MATH  Google Scholar 

  40. Torabi, S., Lowengrub, J., Voigt, A., Wise, S.: A new phase-field model for strongly anisotropic systems. Proceedings of the Royal Society of A465(2105), 1337–1359 (2009)

    Article MathSciNet MATH  Google Scholar 

  41. Wang, L., Yu, H.: On efficient second order stabilized semi-implicit schemes for the Cahn-Hilliard phase-field equation. J. Sci. Comput.77(2), 1185–1209 (2018)

    Article MathSciNet MATH  Google Scholar 

  42. Wise, S., Kim, J., Lowengrub, J.: Solving the regularized, strongly anisotropic Cahn-Hilliard equation by an adaptive nonlinear multigrid method. J. Comput. Phys.226(1), 414–446 (2007)

    Article MathSciNet MATH  Google Scholar 

  43. 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)

    Article MathSciNet MATH  Google Scholar 

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

    Article MathSciNet MATH  Google Scholar 

  45. Yang, X., Zhang, G.: Convergence analysis for the Invariant Energy Quadratization (IEQ) schemes for solving the Cahn-Hilliard and Allen-Cahn equations with general nonlinear potential. J. Sci. Comput.82(3), 1–28 (2020)

    Article MathSciNet MATH  Google Scholar 

  46. Zhang, Z., Qiao, Z.: An adaptive time-stepping strategy for the Cahn-Hilliard equation. Comm. Comput. Phys.11(4), 1261–1278 (2012)

    Article MathSciNet MATH  Google Scholar 

  47. Zhu, J., Chen, L.Q., Shen, J., Tikare, V.: Coarsening kinetics from a variable-mobility cahn-hilliard equation: Application of a semi-implicit fourier spectral method. Phys. Rev. E60(4), 3564 (1999)

    Article  Google Scholar 

  48. Zhu, T.F., Adamala, K., Zhang, N., Szostak, J.W.: Photochemically driven redox chemistry induces protocell membrane pearling and division. Proc. Natl. Acad. Sci. U.S.A.109(25), 9828–9832 (2012)

    Article  Google Scholar 

Download references

Acknowledgements

C. Zhang, J. Ouyang, X. Wang, Y. Chai and M. Ma are grateful to the support from the National Natural Science Foundation of China (Grant No. 11671321, 11971387, 11901051, 91434201). C. Zhang also acknowledges partial support from the Natural Science Foundation of Shaanxi Province (Grant No.2020JQ-338).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710129, China

    Chenhui Zhang, Jie Ouyang, Xiaodong Wang, Yong Chai & Mengxia Ma

Authors
  1. Chenhui Zhang

    You can also search for this author inPubMed Google Scholar

  2. Jie Ouyang

    You can also search for this author inPubMed Google Scholar

  3. Xiaodong Wang

    You can also search for this author inPubMed Google Scholar

  4. Yong Chai

    You can also search for this author inPubMed Google Scholar

  5. Mengxia Ma

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence toXiaodong Wang.

Additional information

Publisher's Note

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

Rights and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, C., Ouyang, J., Wang, X.et al. Analysis of the Energy Stability for Stabilized Semi-implicit Schemes of the Functionalized Cahn-Hilliard Mass-conserving Gradient Flow Equation.J Sci Comput87, 34 (2021). https://doi.org/10.1007/s10915-021-01430-1

Download citation

Keywords

Mathematics Subject Classification

Access this article

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Advertisement

[8]ページ先頭
©2009-2025 Movatter.jp