Numerical approximations of allen-cahn and cahn-hilliard equations

Discrete and Continuous Dynamical Systems

Volumn 28, Issue 4, 2010, Pages 1669-1691

  Shen, Jie ^a,b   Yang, Xiaofeng ^c  

^a XIAMEN UNIVERSITY   (China)

^b PURDUE UNIVERSITY   (United States)

^c UNIVERSITY OF SOUTH CAROLINA   (United States)

Author keywords

Allen Cahn; Cahn Hilliard; Error analysis; Spectral method; Stability

Indexed keywords

EID: 77957918727     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.2010.28.1669     Document Type: Article

References (28)

1. S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall. Mater., 27 (1979), 1085-1095.
2. D. M. Anderson, G. B. McFadden and A. A. Wheeler, Diffuse-interface methods in fluid mechanics, Annual Review Of Fluid Mechanics, 30, 139-165, Annu. Rev. Fluid Mech., 30, Annual Reviews, Palo Alto, CA, (1998).
3. Luis A. Caffarelli and Nora E. Muler, An L∞; bound for solutions of the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 133 (1995), 129-144.
4. J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.
5. C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, "Spectral Methods," Fundamentals in single domains. Scientific Computation, Springer-Verlag, Berlin, 2006.
6. L. Q. Chen, Phase-field models for microstructure evolution, Annual Review of Material Research, 32 (2002), 113.
7. L. Q. Chen and J. Shen, Applications of semi-implicit Fourier-spectral method to phase-field equations, Comput. Phys. Comm., 108 (1998), 147-158.
8. N. Condette, C. Melcher and E. Süli, Spectral approximation of pattern-forming nonlinear evolution equations with double-well potentials of quadratic growth, to appear in Math. Comp.
9. Q. Du, C. Liu and X. Wang, Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions, J. Comput. Phys., 212 (2006), 757-777.
10. Qiang Du and R. A. Nicolaides, Numerical analysis of a continuum model of phase transition, SIAM J. Numer. Anal., 28 (1991), 1310-1322.
11. David J. Eyre, Unconditionally gradient stable time marching the Cahn-Hilliard equation, In "Computational and Mathematical Models of Microstructural Evolution" (San Francisco, CA, 1998), 39-46, Mater. Res. Soc. Sympos. Proc. 529, MRS, Warrendale, PA, 1998.
12. Xiaobing Feng and Andreas Prohl, Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows, Numer. Math., 94 (2003), 33-65.
13. Xiaobing Feng and Andreas Prohl, Error analysis of a mixed finite element method for the Cahn-Hilliard equation, Numer. Math., 99 (2004), 47-84.
14. Z. Hu, S. M. Wise, C. Wang and J. S. Lowengrub, Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation, J. Comput. Phys., 228 (2009), 5323-5339.
15. Daniel Kessler, Ricardo H. Nochetto and Alfred Schmidt, A posteriori error control for the Allen-Cahn problem: circumventing Gronwall's inequality, M2AN Math. Model. Numer. Anal., 38 (2004), 129-142.
16. D. Khismatullin, Y. Renardy and M. Renardy, Development and implementation of vof-prost for 3d viscoelastic liquid-liquid simulations, J. Non-Newt. Fluid Mech., 140 (1992), 120-131.
17. C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Physica D, 179 (2003), 211-228.
18. C. Liu, J. Shen and X. Yang, Dynamics of deject motion in nematic liquid crystal flow: Modeling and numerical simulation, Comm. Comput. Phys, 2 (2007), 1184-1198.
19. J. Lowengrub and L. Truskinovsky, Quasi-incompressible cahn-hilliard fluids and topological transitions, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454 (1998), 2617-2654.
20. J. Shen, Efficient spectral-Galerkin method I. direct solvers for second- and fourth-order equations by using Legendre polynomials, SIAM J. Sci. Comput., 15 (1994), 1489-1505.
21. Jie Shen, On error estimates of the projection methods for the Navier-Stokes equations: First-order schemes, SIAM J. Numer. Anal., 29 (1992), 57-77.
22. 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 (2009), 2269-2288.
23. Chuanju Xu and Tao Tang, Stability analysis of large time-stepping methods for epitaxial growth models, SIAM J. Numer. Anal., 44 (2006), 1759-1779. (Pubitemid 47264169)
24. X. Yang, Error analysis of stabilized semi-implicit method of Allen-Cahn equation, Disc. Conti. Dyn. Sys.-B, 11 (2009), 1057-1070.
25. X. Yang, J. J. Feng, C. Liu and J. Shen, Numerical simulations of jet pinching-off and drop formation using an energetic variational phase-field method, J. Comput. Phys., 218 (2006), 417-428.
26. Xingde Ye, The Legendre collocation method for the Cahn-Hilliard equation, J. Comput. Appl. Math., 150 (2003), 87-108.
27. P. Yue, J. J. Feng, C. Liu and J. Shen, A diffuse-interface method for simulating two-phase flows of complex fluids, J. Fluid Mech., 515 (2004), 293-317.
28. J. Zhang and Q. Du, Numerical studies of discrete approximations to the Allen-Cahn equation in the sharp interface limit, to appear in SIAM J. Scient. Comput.