Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-25T17:15:59.276Z Has data issue: false hasContentIssue false
  • English
  • Français

An Efficient, Energy Stable Scheme for the Cahn-Hilliard-Brinkman System

Published online by Cambridge University Press:  03 June 2015

Craig Collins ,
Jie Shen  and
Steven M. Wise
Craig Collins*
Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, TN 37912, USA
Jie Shen*
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
Steven M. Wise*
Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, TN 37912, USA
*
Email:[email protected]
Email:[email protected]
Corresponding author.Email:[email protected]
Get access

Abstract

We present an unconditionally energy stable and uniquely solvable finite difference scheme for the Cahn-Hilliard-Brinkman (CHB) system, which is comprised of a Cahn-Hilliard-type diffusion equation and a generalized Brinkman equation mod-eling fluid flow. The CHB system is a generalization of the Cahn-Hilliard-Stokes model and describes two phase very viscous flows in porous media. The scheme is based on a convex splitting of the discrete CH energy and is semi-implicit. The equations at the implicit time level are nonlinear, but we prove that they represent the gradient of a strictly convex functional and are therefore uniquely solvable, regardless of time step size. Owing to energy stability, we show that the scheme is stable in the time and space discrete and norms. We also present an efficient, practical nonlinear multigrid method . comprised of a standard FAS method for the Cahn-Hilliard part, and a method based on the Vanka smoothing strategy for the Brinkman part . for solving these equations. In particular, we provide evidence that the solver has nearly optimal complexity in typical situations. The solver is applied to simulate spinodal decomposition of a viscous fluid in a porous medium, as well as to the more general problems of buoyancy- and boundary-driven flows.

Keywords

65M0665M1265M5576T99Cahn-Hilliard equationStokes equationsBrinkman equationfinite difference methodsnonlinear multigridconvex splittingenergy stability
Type
Research Article
Information
Communications in Computational Physics , Volume 13 , Issue 4 , April 2013 , pp. 929 - 957
Copyright
Copyright © Global Science Press Limited 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]Bertozzi, A.L., Esedoglu, S., and Gillette, A.Inpainting of binary images using the Cahn- Hilliard equation. IEEE Trans. Image Proc., 16:285–291,2007.Google Scholar
[2]Brinkman, H.C.A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res., AI:27–34,1949.CrossRefGoogle Scholar
[3]Cahn, J.W.On spinodal decomposition. Acta Metall., 9:795–801,1961.Google Scholar
[4]Cahn, J.W., Elliott, C.M., and Novick-Cohen, A.The Cahn-Hilliard equation with a concentration dependent mobility: Motion by minus the Laplacian of the mean curvature. Euro. J. Appl. Math., 7:287–301,1996.Google Scholar
[5]Cahn, J.W. and Hilliard, J.E.Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys., 28:258–267,1958.Google Scholar
[6]Elliott, C.M. and Stuart, A.M.The global dynamics of discrete semilinear parabolic equations. SIAM J. Numer. Anal, 30:1622–1663,1993.CrossRefGoogle Scholar
[7]Eyre, D.Unconditionally gradient stable time marching the Cahn-Hilliard equation. In Bullard, J. W., Kalia, R., Stoneham, M., and Chen, L.Q., editors, Computational and Mathematical Models of Microstructural Evolution, volume 53, pages 1686–1712, Warrendale, PA, USA, 1998. Materials Research Society.Google Scholar
[8]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., 44:1049–1072, 2006.Google Scholar
[9]Feng, X. and Wise, S.M.Analysis of a Darcy-Cahn-Hilliard diffuse interface model for the Hele-Shaw flow and its fully discrete finite element approximation. SIAM J. Numer. Anal., 50(3), 1320V1343,2012.Google Scholar
[10]Hu, Z., Wise, S.M., Wang, C., and Lowengrub, J.S.Stable and efficient finite-difference nonlinear-multigrid schemes for the phase-field crystal equation. J. Comput. Phys., 228:5323–5339,2009.Google Scholar
[11]Kay, D. and Welford, R.A multigrid finite element solver for the Cahn-Hilliard equation. J. Comput. Phys., 212:288–304,2006.CrossRefGoogle Scholar
[12]Kay, D. and Welford, R.Efficient numerical solution of Cahn-Hilliard-Navier Stokes fluids in 2D. SIAM J. Sci. Comput., 29:2241–2257,2007.Google Scholar
[13]Kim, J.S., Kang, K., and Lowengrub, J.S.Conservative multigrid methods for Cahn-Hilliard fluids. J. Comput. Phys., 193:511–543,2003.CrossRefGoogle Scholar
[14]Lee, H.G., Lowengrub, J.S, and Goodman, J.Modeling pinchoff and reconnection in a Hele-Shaw cell. I. The models and their calibration. Phys. Fluids, 14:492–513,2002.Google Scholar
[15]Lee, H.G., Lowengrub, J.S, and Goodman, J.Modeling pinchoff and reconnection in a Hele-Shaw cell. II. Analysis and simulation in the nonlinear regime. Phys. Fluids, 14:514–545, 2002.Google Scholar
[16]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, 179:211–228,2003.Google Scholar
[17]Lowengrub, J.S. and Truskinovsky, L.Cahn-Hilliard fluids and topological transitions. Proc. R. Soc. Lond. A, 454:2617–2654,1998.Google Scholar
[18]Ngamsaad, W., Yojina, J., and Triampo, WTheoretical studies of phase-separation kinetics in a Brinkman porous medium. J. Phys. A: Math. Theor., 43:202001,2010.Google Scholar
[19]Oosterlee, C.W. and Gaspar, F.J.Multigrid relaxation methods for systems of saddle point type. Appl. Numer. Math., 58:1933–1950,2008.Google Scholar
[20]Pham, K., Frieboes, H.B., Cristini, V., and Lowengrub, J.Predictions of tumour morphological stability and evaluation against experimental observations. J. R. Soc. Interface, 8:16–29,2011.Google Scholar
[21]Shen, J., Wang, C., Wang, X., and Wise, S.Second-order convex splitting schemes for gradient flows with Ehrlich-Schwoebel-type energy: Application to thin film epitaxy. SIAM J. Numer. Anal., 50:105–125,2012.Google Scholar
[22]Shen, J. and Yang, X.Energy stable schemes for Cahn-Hilliard phase-field model of two- phase incompressible flows. Chinese Ann. Math. Series B, 31:743–758,2010.Google Scholar
[23]Shen, J. and Yang, X.Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete Cont. Dyn. Sys. A, 28:1669–1691,2010.Google Scholar
[24]Shen, J. and Yang, X.A phase-field model and its numerical approximation for two-phase incompressible flows with different densities and viscosities. SIAM J. Sci. Comput., 32:1159–1179, 2010.Google Scholar
[25]Trottenberg, U., Oosterlee, C.W., and Schuller, A.Multigrid. Academic Press, New York, 2005.Google Scholar
[26]Vanka, S.P.Block-implicit multigrid solution of Navier-Stokes equations in primitive variables. J. Comput. Phys., 65:138–158,1986.Google Scholar
[27]Vollmayr-Lee, B.P. and Rutenberg, A.D.Fast and accurate coarsening simulation with an unconditionally stable time step. Phys. Rev. E, 68:066703, 2003.Google Scholar
[28]Wang, C., Wang, X., and Wise, S.Unconditionally stable schemes for equations of thin film epitaxy. Discrete Cont. Dyn. Sys. A, 28:405–423,2010.Google Scholar
[29]Wang, C. and Wise, S.Global smooth solution of modified phase field crystal equation. Methods Appl. Anal., 17:191–212,2010.Google Scholar
[30]Wang, C. and Wise, S.An energy stable and convergent finite-difference scheme for the modified phase field crystal equation. SIAM J. Numer. Anal., 49:945–969,2011.Google Scholar
[31]Wesseling, P.An Introduction to Multigrid Methods. R.T. Edwards, Philadelphia, 2004.Google Scholar
[32]Wise, S.M.Unconditionally stable finite difference, nonlinear multigrid simulation of the Cahn-Hilliard-Hele-Shaw system of equations. J. Sci. Comput., 44:38–68,2010.Google Scholar
[33]Wise, S.M., Lowengrub, J.S., and Cristini, V.An adaptive algorithm for simulating solid tumor growth using mixture models. Math. Comput. Model., 53:1–20,2011.Google Scholar
[34]Wise, S.M., Lowengrub, J.S., Frieboes, H.B., and Cristini, V.Three-dimensional multispecies nonlinear tumor growth -1 model and numerical method. J. Theor. Biol., 253:524–543,2008.Google Scholar
[35]Wise, S.M., Wang, C., and Lowengrub, J.S.An energy stable and convergent finite-difference scheme for the phase field crystal equation. SIAM J. Numer. Anal., 47:2269–2288,2009.Google Scholar