Hostname: page-component-588bc86c8c-pzvk2 Total loading time: 0 Render date: 2023-12-01T05:33:58.995Z Has data issue: false Feature Flags: { "corePageComponentGetUserInfoFromSharedSession": true, "coreDisableEcommerce": false, "useRatesEcommerce": true } hasContentIssue false
  • English
  • Français

Numerical schemes for a threecomponent Cahn-Hilliard model

Published online by Cambridge University Press:  10 December 2010

Franck Boyer  and
Sebastian Minjeaud
Franck Boyer
Affiliation:
Université Paul Cézanne, FST Saint-Jérôme, Case cour A, LATP, Avenue Escadrille Normandie-Niemen, 13397 Marseille Cedex 20, France. fboyer@latp.univ-mrs.fr
Sebastian Minjeaud
Affiliation:
Institut de Radioprotection et de Sûreté Nucléaire, Bât. 702, BP3, 13115 Saint Paul lez Durance, France. sebastian.minjeaud@irsn.fr
Get access

Abstract

In this article, we investigate numerical schemes for solving a three component Cahn-Hilliard model. The space discretization is performed by using a Galerkin formulation and the finite element method. Concerning the time discretization, the main difficulty is to write a scheme ensuring, at the discrete level, the decrease of the free energy and thus the stability of the method. We study three different schemes and prove existence and convergence theorems. Theoretical results are illustrated by various numerical examples showing that the new semi-implicit discretization that we propose seems to be a good compromise between robustness and accuracy.

Keywords

Finite elementCahn-Hilliard modelnumerical schemeenergy estimate
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 4 , July 2011 , pp. 697 - 738
Copyright
© EDP Sciences, SMAI, 2010

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

Barrett, J.W. and Blowey, J.F., An improved error bound for a finite element approximation of a model for phase separation of a multi-component alloy. IMA J. Numer. Anal. 19 (1999) 147–168. CrossRef
Barrett, J.W. and Blowey, J.F., Finite element approximation of an Allen-Cahn/Cahn-Hilliard system. IMA J. Numer. Anal. 22 (2002) 11–71. CrossRef
Barrett, J.W., Blowey, J.F. and Garcke, H., Finite element approximation of the Cahn-Hilliard equation with degenerate mobility. SIAM J. Numer. Anal. 37 (1999) 286–318. CrossRef
Barrett, J.W., Blowey, J.F. and Garcke, H., On fully practical finite element approximations of degenerate Cahn-Hilliard systems. ESAIM: M2AN 35 (2001) 713–748. CrossRef
Blowey, J.F. and Elliott, C.M., The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy. II. Numerical analysis. Eur. J. Appl. Math. 3 (1992) 147–179. CrossRef
Blowey, J.F., Copetti, M.I.M. and Elliott, C.M., Numerical analysis of a model for phase separation of a multi-component alloy. IMA J. Numer. Anal. 16 (1996) 111–139. CrossRef
Boyer, F., A theoretical and numerical model for the study of incompressible mixture flows. Comput. Fluids 31 (2002) 41–68. CrossRef
Boyer, F. and Lapuerta, C., Study of a three component Cahn-Hilliard flow model. ESAIM: M2AN 40 (2006) 653–687. CrossRef
F. Boyer, C. Lapuerta, S. Minjeaud and B. Piar, A local adaptive refinement method with multigrid preconditioning illustrated by multiphase flows simulations, in CANUM 2008, ESAIM Proc. 27, EDP Sciences, Les Ulis (2009) 15–53.
Boyer, F., Lapuerta, C., Minjeaud, S., Piar, B. and Quintard, M., Cahn-Hilliard/Navier-Stokes model for the simulation of three-phase flows. Transp. Porous Media 82 (2010) 463–483. CrossRef
K. Deimling, Nonlinear functional analysis. Springer-Verlag (1985).
Du, Q. and Nicolaides, R.A., Numerical analysis of a continuum model of phase transition. SIAM J. Numer. Anal. 28 (1991) 1310–1322. CrossRef
C.M. Elliott, The Cahn-Hilliard model for the kinetics of phase separation, in Mathematical models for phase change problems, Óbidos, 1988, Internat. Ser. Numer. Math. 88, Birkhäuser, Basel (1989) 35–73.
Elliott, C.M. and Garcke, H., Diffusional phase transitions in multicomponent systems with a concentration dependent mobility matrix. Physica D 109 (1997) 242–256. CrossRef
C.M. Elliott and S. Luckhaus, A generalised diffusion equation for phase separation of a multi-component mixture with interfacial free energy. IMA Preprint Series # 887 (1991).
Elliott, C.M. and Stuart, A.M., The global dynamics of discrete semilinear parabolic equations. SIAM J. Numer. Anal. 30 (1993) 1622–1663. CrossRef
A. Ern and J.-L. Guermond, Theory and Pratice of Finite Elements, Applied Mathematical Sciences 159. Springer (2004).
D.J. Eyre, Unconditionally gradient stable time marching the Cahn-Hilliard equation, in Computational and mathematical models of microstructural evolution, San Francisco, CA, 1998, Mater. Res. Soc. Sympos. Proc. 529, MRS, Warrendale, PA (1998) 39–46.
Feng, X., Fully discrete finite element approximations of the Navier-Stokes-Cahn-Hilliard diffuse interface model for two-phase fluid flows. SIAM J. Numer. Anal. 44 (2006) 1049–1072. CrossRef
Feng, X., He, Y. and Liu, C., Analysis of finite element approximations of a phase field model for two-phase fluids. Math. Comp. 76 (2007) 539–571. CrossRef
Garcke, H. and Stinner, B., Second order phase field asymptotics for multi-component systems. Interface Free Boundaries 8 (2006) 131–157. CrossRef
Garcke, H., Nestler, B. and Stoth, B., A multiphase field concept: numerical simulations of moving phase boundaries and multiple junctions. SIAM J. Appl. Math. 60 (2000) 295–315. CrossRef
Kim, J. and Lowengrub, J., Phase field modeling and simulation of three-phase flows. Interfaces Free Boundaries 7 (2005) 435–466. CrossRef
Kim, J., Kang, K. and Lowengrub, J., Conservative multigrid methods for Cahn-Hilliard fluids. J. Comput. Phys. 193 (2004) 511–543. CrossRef
Kim, J., Kang, K. and Lowengrub, J., Conservative multigrid methods for ternary Cahn-Hilliard systems. Commun. Math. Sci. 2 (2004) 53–77.
C. Lapuerta, Échanges de masse et de chaleur entre deux phases liquides stratifiées dans un écoulement à bulles. Mathématiques appliquées, Université de Provence, France (2006).
Lee, H.G. and Kim, J., A second-order accurate non-linear difference scheme for the n-component Cahn-Hilliard system. Physica A 387 (2008) 4787–4799. CrossRef
Nestler, B., Garcke, H. and Stinner, B., Multicomponent alloy solidification: Phase-field modeling and simulations. Phys. Rev. E 71 (2005) 041609.
PELICANS, Collaborative Development environment, https://gforge.irsn.fr/gf/project/pelicans/.
J.S. Rowlinson and B. Widom, Molecular theory of capillarity. Clarendon Press (1982).
Seiler, J.M. and Froment, K., Material effects on multiphase phenomena in late phases of severe accidents of nuclear reactors. Multiph. Sci. Technol. 12 (2000) 117–257. CrossRef
J. Simon, Compact sets in the space L p(0, T; B). Ann. Mat. Pura Appl. 146 (1987) 65–96.