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A Time Splitting Space Spectral Element Method for the Cahn-Hilliard Equation

Published online by Cambridge University Press:  28 May 2015

Lizhen Chen  and
Chuanju Xu
Lizhen Chen*
Affiliation:
Beijing Computational Science Research Center, Beijing 100084, China
Chuanju Xu*
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
*
Corresponding author. Email Address: [email protected]
Corresponding author. Email Address: [email protected]
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Abstract

We propose and analyse a class of fully discrete schemes for the Cahn-Hilliard equation with Neumann boundary conditions. The schemes combine large-time step splitting methods in time and spectral element methods in space. We are particularly interested in analysing a class of methods that split the original Cahn-Hilliard equation into lower order equations. These lower order equations are simpler and less computationally expensive to treat. For the first-order splitting scheme, the stability and convergence properties are investigated based on an energy method. It is proven that both semi-discrete and fully discrete solutions satisfy the energy dissipation and mass conservation properties hidden in the associated continuous problem. A rigorous error estimate, together with numerical confirmation, is provided. Although not yet rigorously proven, higher-order schemes are also constructed and tested by a series of numerical examples. Finally, the proposed schemes are applied to the phase field simulation in a complex domain, and some interesting simulation results are obtained.

Keywords

65M0665M1265N3065N3576A05Cahn-Hilliardtime splitting schemesspectral methodserror analysis
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 3 , Issue 4 , November 2013 , pp. 333 - 351
Copyright
Copyright © Global-Science Press 2013

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References

[1]Allen, S. and Cahn, J. W., A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall. 27, 1084–1095 (1979).Google Scholar
[2]Gottlieb, D., Costa, B., Don, W. S. and Sendersky, R., Two-dimensional multi-domain hybrid spectral-WENO methods for conservation laws, Comm. Comput. Phys. 1, 550–577 (2006).Google Scholar
[3]Cahn, J. W. and Hilliard, J. E., Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys. 28, 258–267 (1958).CrossRefGoogle Scholar
[4]Chen, L. Q., Phase-field models for microstructure evolution, Annu. Rev. Mater. Res. 32, 113–140 (2002).Google Scholar
[5]Chen, L. Q. and Shen, J., Applications of semi-implicit Fourier-spectral method to phase-field equations, Comput. Phys. Commun. 108, 147–158 (1998).CrossRefGoogle Scholar
[6]Choo, S. M. and Chung, S. K., Conservative nonlinear difference scheme for the Cahn-Hilliard equation, Comput. Math. Appl. 36, 31–39 (1998).CrossRefGoogle Scholar
[7]French, D. A., Elliott, C. M. and Milner, F. A., A second order splitting method for the Cahn-Hilliard equation, Numerische Mathematik 54, 575–590 (1989).Google Scholar
[8]Copetti, M. I. M. and Elliott, C. M., Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy, Numer. Math. 63, 39–65 (1992).Google Scholar
[9]McFadden, G. B., Anderson, D. M. and Wheeler, A. A., Diffuse-interface methods in fluid mechanics, Annu. Rev. Fluid Mech. 30, 139–165 (1998).Google Scholar
[10]Mottoni, P. de and Schatzman, M., Evolution geométrique d'interfaces, C. R. Acad. Sci. Paris, Série I, Math. 309, 453–458 (1989).Google Scholar
[11]Mottoni, P. de and Schatzman, M., Geometrical evolution of developed interfaces, Trans. Amer. Math. Soc. 347, 1533–1589 (1995).CrossRefGoogle Scholar
[12]Du, Q. and Nicolaides, R. A., Numerical analysis of a continuum model of phase transition, SIAM J. Numer. Anal. 28, 1310–1322 (1991).Google Scholar
[13]Elliott, C. M. and French, D. A., Numerical studies of the Cahn-Hilliard equation for phase separation, IMA J. Appl. Math 38, 97–128 (1987).CrossRefGoogle Scholar
[14]Elliott, C. M. and French, D. A., A nonconforming finite-element method for the two-dimensional Cahn-Hilliard equation, SIAM J. Numer. Anal. 26, 884–903 (1989).Google Scholar
[15]Elliott, C. M. and Larsson, S., Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation, Math. Comput. 58, 603–630 (1992).Google Scholar
[16]Elliott, C. M. and Songmu, Z., On the Cahn-Hilliard equation, Arch. Ration. Mech. An. 96, 339–357 (1986).Google Scholar
[17]Furihata, D., A stable and conservative finite difference scheme for the Cahn-Hilliard equation, Numerische Mathematik 87, 675–699 (2001).CrossRefGoogle Scholar
[18]He, L. P., Error estimation of a class of stable spectral approximation to the Cahn-Hilliard equation, J. Sci. Comput. 41, 461–482 (2009).Google Scholar
[19]Ilmanen, T., Convergence of the Allen-Cahn equation to Brakkeąŕs motion by mean curvature, J. Differential Geom. 38, 417–461 (1993).CrossRefGoogle Scholar
[20]Blowey, J. F., Barrett, J. W. and Garcke, H., On fully practical finite element approximations of degenerate Cahn-Hilliard systems, Math. Model. Numer. Anal. 35, 713–748 (2001).Google Scholar
[21]Kim, J., Phase-field models for multi-component fluid flows, Commun. Comput. Phys. 12, 613–661 (2012).Google Scholar
[22]Soner, H. M., Evans, L. C. and Souganidis, P. E., Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math. 45, 1097–1123 (1992).Google Scholar
[23]Liu, C. and Shen, J., A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Physica D: Nonlinear Phenomena 179, 211–228 (2003).Google Scholar
[24]Liu, C. and Walkington, N. J., An Eulerian description of fluids containing visco-elastic particles, Arch. Rational Mesh. Anal. 159, 229–252 (2001).Google Scholar
[25]Lowengrub, J. and Truskinovsky, L., Quasi-incompressible Cahn-Hilliard fluids and topological transitions, Proc. R. Soc. Lond. A 454, 2617–2654 (1998).CrossRefGoogle Scholar
[26]Liu, C., Yue, P., Feng, J. J. and Shen, J., A diffuse-interface method for simulating two-phase flows of complex fluids, J. Fluid Mech. 515, 293–317 (2004).Google Scholar
[27]Guo, B., Du, Q. and Shen, J., Fourier spectral approximation to a dissipative system modeling the flow of liquid crystals, SIAM J. Numer. Anal. 39, 735–762 (2002).Google Scholar
[28]Liu, C., Du, Q. and Wang, X., Simulating the deformation of vesicle membranes under elastic bending energyin three dimensions, J. Comput. Phys. 212, 757–777 (2005).Google Scholar
[29]Shen, J. and Yang, X. F., Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, DCDS-A 28, 1669–1691 (2010).Google Scholar
[30]Feng, X., Tang, T., and Yang, J., Stabilized Crank-Nicolson/Adams-Bashforth schemes for phase field models, East Asian J. Appl. Math. 3, 59–80 (2013).Google Scholar
[31]Liu, C., Yang, X., Feng, J. J. and Shen, J., Numerical simulations of jet pinching-off and drop formation using an energetic variational phase-field method, J. Comput. Phys. 218, 417–428 (2006).Google Scholar
[32]Ye, X. and Cheng, X., Legendre spectral approximation for the Cahn-Hilliard equation, Mathematica Numerica Sinica 2, 157–170 (2003).Google Scholar
[33]Liu, Y., He, Y. and Tang, T., On large time-stepping methods for the Cahn–Hilliard equation, Appl. Numer. Math. 57, 616–628 (2007).Google Scholar
[34]Zhuang, Q. Q. and Xu, C. J., A spectral element/Laguerre coupled method to the elliptic Helmholtz problem on the half line, Numerical Mathematics A, Journal of Chinese Universities English Series 15, 193–208 (2006).Google Scholar