A numerical method for the Cahn-Hilliard equation with a variable mobility. (English) Zbl 1118.35049
Summary: We consider a conservative nonlinear multigrid method for the Cahn-Hilliard equation with a variable mobility of a model for phase separation in a binary mixture. The method uses the standard finite difference approximation in spatial discretization and the Crank-Nicholson semi-implicit scheme in temporal discretization. And the resulting discretized equations are solved by an efficient nonlinear multigrid method. The continuous problem satisfies conservation of mass and decrease of the total energy. It is proved that these properties hold for the discrete problem. Also, we show the proposed scheme has a second-order convergence in space and time numerically. For numerical experiments, we investigate the effects of a variable mobility.
MSC:
| 35Q72 | Other PDE from mechanics (MSC2000) |
| 37L65 | Special approximation methods (nonlinear Galerkin, etc.) for infinite-dimensional dissipative dynamical systems |
| 65M55 | Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs |
References:
| [1] | Bai, F. S.; Spence, A.; Stuart, A. M., Numerical computations of coarsening in the one-dimensional Cahn-Hilliard model of phase separation, Physica D, 78, 155-165 (1994) · Zbl 0824.35127 |
| [2] | Cahn, J. W., On spinodal decomposition, Acta Metall, 9, 795-801 (1961) |
| [3] | Choo, S. M.; Chung, S. K., Conservative nonlinear difference scheme for the Cahn-Hilliard equation, Comput Math Appl, 36, 31-39 (1998) · Zbl 0933.65098 |
| [4] | Choo, S. M.; Chung, S. K.; Kim, K. I., Conservative nonlinear difference scheme for the Cahn-Hilliard equation. II, Comput Math Appl, 39, 229-243 (2000) · Zbl 0973.65067 |
| [5] | Copetti, M.; Elliott, C. M., Kinetics of phase decomposition processes: numerical solutions to the Cahn-Hilliard equation, Materials Science and Technology, 6, 273-283 (1990) |
| [6] | Cahn, J. W.; Hilliard, J. E., Free energy of a non-uniform system. I. Interfacial free energy, J Chem Phys, 28, 258-267 (1958) · Zbl 1431.35066 |
| [7] | Cahn, J. W.; Hilliard, J. E., Spinodal decomposition: A reprise, Acta Metall, 19, 151-161 (1971) |
| [8] | Cahn, J. W.; Taylor, J. E., Surface motion by surface diffusion, Acta Metall, 42, 1045-1063 (1994) |
| [9] | Elliott, C. M.; French, D. A., Numerical studies of the Cahn-Hilliard equation for phase separation, IMA J Appl Math, 38, 97-128 (1987) · Zbl 0632.65113 |
| [10] | Eggleston, J. J.; McFadden, G. B.; Voorhees, P. W., A phase-field model for highly anisotropic interfacial energy, Physica D, 150, 91-103 (2001) · Zbl 0979.35140 |
| [11] | Furihata, D., A stable and conservative finite difference scheme for the Cahn-Hilliard Equation, Numer Math, 87, 675-699 (2001) · Zbl 0974.65086 |
| [12] | Barrett, J. W.; Blowey, J. F.; Garcke, H., Finite element approximation of the Cahn-Hilliard equation with degenerate mobility, SIAM J Numer Anal, 37, 286-318 (1999) · Zbl 0947.65109 |
| [13] | Trottenberg, U.; Oosterlee, C.; Schüller, A., Multigrid (2001), Academic press · Zbl 0976.65106 |
| [14] | Ye, X., The Legendre collocation method for the Cahn-Hilliard equation, J Comput Appl Math, 150, 87-108 (2003) · Zbl 1043.65116 |
| [15] | Zhu, J.; Chen, L. Q.; Shen, J.; Tikare, V., Coarsening kinetics from a variable mobility Cahn-Hilliard equation – application of semi-implicit Fourier spectral method, Phys Rev E, 60, 3564-3572 (1999) |
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