COMPARISON OF DIFFERENT NUMERICAL SCHEMES FOR THE CAHN-HILLIARD EQUATION |
Lee, Seunggyu
(DEPARTMENT OF MATHEMATICS, KOREA UNIVERSITY)
Lee, Chaeyoung (DEPARTMENT OF MATHEMATICS, KOREA UNIVERSITY) Lee, Hyun Geun (INSTITUTE OF MATHEMATICAL SCIENCES, EWHA W. UNIVERSITY) Kim, Junseok (DEPARTMENT OF MATHEMATICS, KOREA UNIVERSITY) |
1 | J.W. Cahn, On spinodal decomposition, Acta. Metall., 9 (1961), 795-801. DOI ScienceOn |
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. DOI |
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. DOI ScienceOn |
5 | J.S. Kim, A continuous surface tension force formulation for diffuse-interface models, J. Comput. Phys., 204 (2005), 784-804. DOI ScienceOn |
6 | J.S. Kim, A diffuse-interface model for axisymmetric immiscible two-phase flow. Appl. Math. Comput. 160, 589-606 (2005). DOI ScienceOn |
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. DOI ScienceOn |
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. DOI |
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. DOI ScienceOn |
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. DOI ScienceOn |
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. DOI ScienceOn |
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. DOI ScienceOn |
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. DOI ScienceOn |
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. DOI ScienceOn |
18 | S.Welch, J.Wilson, A volume of fluid based method for fluid flows with phase change, J. Comput. Phys., 160 (2000), 662-682. DOI ScienceOn |
19 | S.O. Unverdi and G. Tryggvason, A front-tracking method for viscous, incompressible, multi-fluid flows. J. Comput. Phys., 100 (1992), 25. DOI ScienceOn |
20 | 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. DOI ScienceOn |
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. DOI ScienceOn |
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. DOI ScienceOn |
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. DOI ScienceOn |
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. DOI ScienceOn |
28 | D.J. Eyre, Systems for Cahn-Hilliard equations, SIAM J. Appl. Math., 53 (1993), 1686-1712. DOI ScienceOn |
29 | 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. DOI ScienceOn |
30 | 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. DOI ScienceOn |
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. DOI |
32 | Q. Du and R. Nicolaides, Numerical studies of a continuummodel of phase transition, SIAM J. Numer. Anal., 28 (1991), 1310-1322. DOI ScienceOn |
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. DOI ScienceOn |
34 | R. Acar, Simulation of interface dynamics: a diffuse-interface model, Vis. Comput., 25 (2009), 101-115. DOI |
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. 과학기술학회마을 DOI ScienceOn |
38 | J.S. Kim, Phase-field models for multi-component fluid flows, Communications in Computational Physics, 12 (2012), 613-661. DOI |
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. DOI ScienceOn |