Advertisement

A conservative finite difference scheme for the N-component Cahn–Hilliard system on curved surfaces in 3D

  • Junxiang Yang
  • Yibao Li
  • Chaeyoung Lee
  • Darae Jeong
  • Junseok KimEmail author
Article
  • 8 Downloads

Abstract

This paper presents a conservative finite difference scheme for solving the N-component Cahn–Hilliard (CH) system on curved surfaces in three-dimensional (3D) space. Inspired by the closest point method (Macdonald and Ruuth, SIAM J Sci Comput 31(6):4330–4350, 2019), we use the standard seven-point finite difference discretization for the Laplacian operator instead of the Laplacian–Beltrami operator. We only need to independently solve (\(N-1\)) CH equations in a narrow band domain around the surface because the solution for the Nth component can be obtained directly. The N-component CH system is discretized using an unconditionally stable nonlinear splitting numerical scheme, and it is solved by using a Jacobi-type iteration. Several numerical tests are performed to demonstrate the capability of the proposed numerical scheme. The proposed multicomponent model can be simply modified to simulate phase separation in a complex domain on 3D surfaces.

Keywords

Closest point method Conservative scheme N-component Cahn–Hilliard equation Narrow band domain 

Notes

Acknowledgements

The author (D. Jeong) was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIP) (NRF-2017R1E1A1A03070953). Y.B. Li is supported by National Natural Science Foundation of China (Nos. 11601416, 11631012). The corresponding author (J.S. Kim) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B03933243). The authors appreciate the reviewers for their constructive comments, which have improved the quality of this paper.

References

  1. 1.
    Cahn JW (1965) Phase separation by spinodal decomposition in isotropic systems. J Chem Phys 42(1):93–99CrossRefGoogle Scholar
  2. 2.
    Wise SM, Lowengrub JS, Frieboes HB, Cristini V (2008) Three-dimensional multispecies nonlinear tumor growth - 1. Model and numerical method. J Theor Biol 253:524–543MathSciNetCrossRefGoogle Scholar
  3. 3.
    Dehghan M, Mohammadi V (2017) Comparison between two meshless methods based on collocation technique for the numerical solution of four-species tumor growth model. Commun Nonlinear Sci Numer Simul 44:204–219MathSciNetCrossRefGoogle Scholar
  4. 4.
    Deng Y, Liu Z, Wu Y (2017) Topology optimization of capillary, two-phase flow problems. Commun Comput Phys 22:1413–1438MathSciNetCrossRefGoogle Scholar
  5. 5.
    Zhang Y, Ye W (2017) A flux-corrected phase-field method for surface diffusion. Commun Comput Phys 22:422–440MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ju L, Zhang J, Du Q (2015) Fast and accurate algorithms for simulating coarsening dynamics of Cahn–Hilliard equations. Comput Mater Sci 108:272–282CrossRefGoogle Scholar
  7. 7.
    Kang D, Chugunova M, Nadim A, Waring AJ, Walther FJ (2018) Modeling coating flow and surfactant dynamics inside the alveolar compartment. J Eng Math 113(1):23–43MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lee D, Huh JY, Jeong D, Shin J, Yun A, Kim JS (2014) Physical, mathematical, and numerical derivations of the Cahn–Hilliard euqation. Comput Mater Sci 81:216–225CrossRefGoogle Scholar
  9. 9.
    Kim J, Lee S, Choi Y, lee SM, Jeong D (2016) Basic principles and practical applications of the Cahn–Hilliard equation. Math Probl Eng 2016:9532608MathSciNetzbMATHGoogle Scholar
  10. 10.
    Lee HG, Kim J (2013) Buoyancy-driven mixing of multi-component fluids in two-dimensional titled channels. Eur J Mech B 42:37–46CrossRefGoogle Scholar
  11. 11.
    Park JM, Anderson PD (2012) A ternary model for double-emulsion formation in a capillary microfluidic device. Lab Chip 12:2672–2677CrossRefGoogle Scholar
  12. 12.
    Lee HG, Kim J (2015) Two-dimesional Kelvin–Helmholtz instabilities of multi-component fluids. Eur J Mech B 49:77–88CrossRefGoogle Scholar
  13. 13.
    Bhattacharyya S, Abinanadanan TA (2003) A study of phase separation in ternary alloys. Bull Mater Sci 26:193CrossRefGoogle Scholar
  14. 14.
    Lee HG, Kim J (2008) A second-order accurate non-linear difference scheme for the \(N\)-component Cahn–Hilliard system. Physica A 387:4787–4799MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lee HG, Choi JW, Kim J (2012) A practically unconditionally gradient stable scheme for the \(N\)-component Cahn–Hilliard system. Physica A 391:1009–1019CrossRefGoogle Scholar
  16. 16.
    Li Y, Choi JI, Kim J (2016) Multi-component Cahn–Hilliard system with different boundary conditions in complex domains. J Comput Phys 323:1–16MathSciNetCrossRefGoogle Scholar
  17. 17.
    Jeong D, Yang J, Kim J (2019) A practical and efficient numerical method for the Cahn–Hilliard equation in complex domains. Commun Nonlinear Sci Numer Simul 73:217–228MathSciNetCrossRefGoogle Scholar
  18. 18.
    Du Q, Ju L, Tian L (2011) Finite element approximation of the Cahn–Hilliard equation on surfaces. Comput Methods Appl Mech Eng 200(29–32):2458–2470MathSciNetCrossRefGoogle Scholar
  19. 19.
    Li Y, Kim J, Wang N (2017) An unconditionally energy-stable second-order time-accurte scheme for the Cahn-Hilliard equation on surfaces. Commun Nonlinear Sci Numer Simul 53:213–227MathSciNetCrossRefGoogle Scholar
  20. 20.
    Li Y, Qi X, Kim J (2018) Direct discretization method for the Cahn–Hilliard equation on an evolving surface. J Sci Comput 77:1147–1163MathSciNetCrossRefGoogle Scholar
  21. 21.
    Li Y, Luo C, Xia B, Kim J (2019) An efficient linear second order unconditionally stable direct discretization method for the phase-field crystal equation on surfaces. Appl Math Model 67:477–490MathSciNetCrossRefGoogle Scholar
  22. 22.
    Dziuk G, Elliott CM (2007) Surface finite elements for parabolic equations. J Comput Math 25(4):385–407MathSciNetGoogle Scholar
  23. 23.
    Green JB, Bertozzi AL, Sapiro G (2006) Fourth order paratial differential equations on general geometries. J Comput Phys 216(1):216–246MathSciNetCrossRefGoogle Scholar
  24. 24.
    Jeong D, Li Y, Lee C, Yang J, Kim J (2019) A conservative numerical method for the Cahn–Hilliard euqation with generalized mobilities on curved surfaces in three-dimensional space. Commun Comput Phys (in press)Google Scholar
  25. 25.
    Eyre DJ (1998) Unconditionally gradient stable time marching the Cahn–Hilliard equation. In: Computational and mathematical models of microstructural evolution. MRS proceedings, vol 529, pp 39–46Google Scholar
  26. 26.
    Kim J (2009) A generalized continuous surface force formulation for phase-field models for multi-component immiscible fluid flows. Comput Methods Appl Mech Eng 198:3105–3112MathSciNetCrossRefGoogle Scholar
  27. 27.
    Macdonald CB, Brandman J, Ruuth SJ (2011) Solving eigenvalue problems on curved surfaces using the closet point method. J Comput Phys 230:7944–7956MathSciNetzbMATHGoogle Scholar
  28. 28.
    Ruuth SJ, Merriman B (2008) A simple embedding method for solving partial differential equations on surfaces. J Comput Phys 227(3):1943–1961MathSciNetCrossRefGoogle Scholar
  29. 29.
    Greer JB (2006) An improvment of a recent Eulerian method for solving PDEs on general geometries. J Sci Comput 29:321–352MathSciNetCrossRefGoogle Scholar
  30. 30.
    Macdonald CB, Ruuth SJ (2009) The implicit closest point method for the numerical solution of partial differential equations on surfaces. SIAM J Sci Comput 31(6):4330–4350MathSciNetCrossRefGoogle Scholar
  31. 31.
    Choi JW, Lee HG, Jeong D, Kim J (2009) An unconditionally gradient stable numerical method for solving the Allen–Cahn equation. Physica A 338(9):1791–1803MathSciNetCrossRefGoogle Scholar
  32. 32.
    Jeong D, Li Y, Choi Y, Yoo M, Kang D, Park J, Choi J, Kim J (2017) Numerical simulation of the zebra pattern formation on a three-dimensional model. Physica A 475:106–116MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsKorea UniversitySeoulRepublic of Korea
  2. 2.School of Mathematics and StatisticsXi’an Jiaotong UniversityXi’anChina
  3. 3.Department of MathematicsKangwon National UniversityGangwon-doRepublic of Korea

Personalised recommendations