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COMPARISON OF DIFFERENT NUMERICAL SCHEMES FOR THE CAHN-HILLIARD EQUATION

  • Received : 2013.06.08
  • Accepted : 2013.07.01
  • Published : 2013.09.25

Abstract

The Cahn-Hilliard equation was proposed as a phenomenological model for describing the process of phase separation of a binary alloy. The equation has been applied to many physical applications such as amorphological instability caused by elastic non-equilibrium, image inpainting, two- and three-phase fluid flow, phase separation, flow visualization and the formation of the quantum dots. To solve the Cahn-Hillard equation, many numerical methods have been proposed such as the explicit Euler's, the implicit Euler's, the Crank-Nicolson, the semi-implicit Euler's, the linearly stabilized splitting and the non-linearly stabilized splitting schemes. In this paper, we investigate each scheme in finite-difference schemes by comparing their performances, especially stability and efficiency. Except the explicit Euler's method, we use the fast solver which is called a multigrid method. Our numerical investigation shows that the linearly stabilized stabilized splitting scheme is not unconditionally gradient stable in time unlike the known result. And the Crank-Nicolson scheme is accurate but unstable in time, whereas the non-linearly stabilized splitting scheme has advantage over other schemes on the time step restriction.

Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

References

  1. J.W. Cahn, On spinodal decomposition, Acta. Metall., 9 (1961), 795-801. https://doi.org/10.1016/0001-6160(61)90182-1
  2. J.W. Cahn and J.E. Hilliard, Free energy of a non-uniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. https://doi.org/10.1063/1.1744102
  3. J.Y. Kim, J.K. Yoon and P.R. Cha, Phase-field model of a morphological instability caused by elastic nonequilibrium, J. Korean Phys. Soc., 49 (2006), 1501-1509.
  4. A. Bertozzi, S. Esedoglu and A. Gillette, Inpainting of Binary Images Using the Cahn-Hilliard equation, IEEE Trans. Image Process., 16 (2007), 285-291. https://doi.org/10.1109/TIP.2006.887728
  5. J.S. Kim, A continuous surface tension force formulation for diffuse-interface models, J. Comput. Phys., 204 (2005), 784-804. https://doi.org/10.1016/j.jcp.2004.10.032
  6. J.S. Kim, A diffuse-interface model for axisymmetric immiscible two-phase flow. Appl. Math. Comput. 160, 589-606 (2005). https://doi.org/10.1016/j.amc.2003.11.020
  7. J.S. Kim, K.K. Kang and J.S. Lowengrub, Conservative multigrid methods for Cahn-Hilliard fluids, J. Comput. Phys., 193 (2004), 511-543. https://doi.org/10.1016/j.jcp.2003.07.035
  8. J.S. Kim and J.S. Lowengrub, Phase field modeling and simulation of three-phase flows, Interfaces Free Bound., 7 (2005), 435-466.
  9. C.M. Elliott and D.A. French, Numerical studies of the Cahn-Hilliard equation for phase separation, IMA J. Appl. Math., 38 (1987), 97-128. https://doi.org/10.1093/imamat/38.2.97
  10. H. Garcke, T. Preusser, M. Rumpf, A. Telea, U. Weikard and J. van Wijk, A phase field model for continuous clustering on vectot fields, IEEE Trans. Visual. Comput. Graph., 7 (2001), 230-241. https://doi.org/10.1109/2945.942691
  11. S.M.Wise, J.S. Lowengrub, J.S. Kim, K. Thornton, P.W. Voorhees andW.C. Johnson, Quantum dot formation on a strain-patterned epitaxial thin film, Appl. Phys. Lett., 87 (2005), 133102. https://doi.org/10.1063/1.2061852
  12. J.S. Kim, A numerical method for the Cahn-Hilliard equation with a variable mobility, Comm. Nonlinear Sci. Numer. Simulat., 12 (2007), 1560-1571. https://doi.org/10.1016/j.cnsns.2006.02.010
  13. J.J. Eggleston, G.B. McFadden and P.W. Voorhees, A phase-field model for highly anisotropic interfacial energy, Phys. D, 150 (2001), 91-103. https://doi.org/10.1016/S0167-2789(00)00222-0
  14. T. Zhang and Q.Wang, Cahn-Hilliard vs singular Cahn-Hilliard equations in phase field modeling, Commun. Comput. Phys., 7 (2010), 362-382.
  15. Q. Du and M. Li, On the stochastic immersed boundary method with an implicit interface formulation, DCDSB., 15 (2011), 373-389.
  16. J.W. Kim, D.J. Kim and H.C. Choi, An immersed-boundary finite-volume method for simulations of flow in complex geometries, J. Comput. Phys., 171 (2001), 132-150. https://doi.org/10.1006/jcph.2001.6778
  17. C.R. Hirt and B.D. Nichols, Volume of fluid (VOF) method for the dynamics of free boundaries, J. Comput. Phys, 39 (1981), 201-225. https://doi.org/10.1016/0021-9991(81)90145-5
  18. S.Welch, J.Wilson, A volume of fluid based method for fluid flows with phase change, J. Comput. Phys., 160 (2000), 662-682. https://doi.org/10.1006/jcph.2000.6481
  19. J. Du, B. Fix, J. Glimm, X.C. Jia, X.L. Li, Y.H. Li, Y.H. and L.L. Wu, A simple package for front tracking, J. Comput. Phys., 213 (2006), 613-628. https://doi.org/10.1016/j.jcp.2005.08.034
  20. S.O. Unverdi and G. Tryggvason, A front-tracking method for viscous, incompressible, multi-fluid flows. J. Comput. Phys., 100 (1992), 25. https://doi.org/10.1016/0021-9991(92)90307-K
  21. M. Dehghan and D. Mirzaei, A numerical method based on the boundary integral equation and dual reciprocity methods for one-dimensional Cahn-Hilliard equation, Eng. Anal. Bound. Elem., 33 (2009), 522-528. https://doi.org/10.1016/j.enganabound.2008.08.008
  22. T.Y. Hou, J.S. Lowengrub and M.J. Shelley, Boundary integral methods for multicomponent fluids and multiphase materials, J. Comput. Phys., 169 (2001), 302-362. https://doi.org/10.1006/jcph.2000.6626
  23. R.J. Leveque, Z. Li, The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal., 31 (1994), 1001-1025.
  24. J. Sethian and Y. Shan, Solving partial differential equations on irregular domains with moving interfaces, with applications to superconformal electrodeposition in semiconductor manufacturing, J. Comput. Phys., 227 (2008), 6411-6447. https://doi.org/10.1016/j.jcp.2008.03.001
  25. S.J. Osher and R.P. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer Verlag, 2003.
  26. J.A. Sethian, Level Set Methods and Fast Marching Methods Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge University Press, 1999.
  27. F. Liu and H. Metiu, Dynamics of phase-separation of crystal-surfaces, Phys. Rev. B, 48 (1993), 5808-5817. https://doi.org/10.1103/PhysRevB.48.5808
  28. D.J. Eyre, Systems for Cahn-Hilliard equations, SIAM J. Appl. Math., 53 (1993), 1686-1712. https://doi.org/10.1137/0153078
  29. N. Khiari, T. Achouri, M.L. Ben Mohamed and K. Omrani, Finite difference approximate solutions for the Cahn-Hilliard equation, Numer. Methods Partial Differ. Equ., 23 (2007), 437-455. https://doi.org/10.1002/num.20189
  30. H.D. Ceniceros and A.M. Roma, A nonstiff, adaptive mesh refinement-based method for the Cahn-Hilliard equation, J. Comput. Phys., 225 (2007), 1849-1862. https://doi.org/10.1016/j.jcp.2007.02.019
  31. M. Copetti and C.M. Elliott, Kinetics of phase decomposition processes: numerical solutions to the Cahn-Hilliard equation, Mater. Sci. Technol., 6 (1990), 273-283. https://doi.org/10.1179/mst.1990.6.3.273
  32. Q. Du and R. Nicolaides, Numerical studies of a continuummodel of phase transition, SIAM J. Numer. Anal., 28 (1991), 1310-1322. https://doi.org/10.1137/0728069
  33. L. Zhang, Long time behavior of difference approximations for the two-dimensional complex Ginzburg-Landau equation, Numer. Funct. Anal. Optim., 31 (2010), 1190-1211. https://doi.org/10.1080/01630563.2010.510974
  34. R. Acar, Simulation of interface dynamics: a diffuse-interface model, Vis. Comput., 25 (2009), 101-115. https://doi.org/10.1007/s00371-008-0208-1
  35. D.J. Eyre, An unconditionally stable one-step scheme for gradient systems, http://www.math.utah.edu/∼eyre/research/methods/stable.ps, 1998.
  36. D.J. Eyre, Unconditionally gradient stable time marching the Cahn-Hilliard equation, In: J.W. Bullard, R. Kalia, M. Stoneham and L. Chen (eds.) Comput. Math. Model. Microstructural Evolut., 1686-1712, Mater. Res. Soc., Pennsylvania, 1998.
  37. J.S. Kim and H.O. Bae, An unconditionally stable adaptive mesh refinement for Cahn-Hilliard equation, J. Korean Phys. Soc., 53 (2008), 672-679. https://doi.org/10.3938/jkps.53.672
  38. J.S. Kim, Phase-field models for multi-component fluid flows, Communications in Computational Physics, 12 (2012), 613-661. https://doi.org/10.4208/cicp.301110.040811a
  39. S.M. Wise, C. Wang and J.S. Lowengrub, An energy-stable and convergent finite-difference scheme for the phase field crystal equation, SIAM J. Numer. Anal., 47(3) (2009), 2269-2288. https://doi.org/10.1137/080738143