Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-10T16:34:03.876Z Has data issue: false hasContentIssue false
  • English
  • Français

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: lzchen@csrc.ac.cn
Corresponding author. Email Address: cjxu@xmu.edu.cn
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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