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

  • Craig Collins (a1), Jie Shen (a2) and Steven M. Wise (a1)


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.


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[1]Bertozzi, A.L., Esedoglu, S., and Gillette, A.Inpainting of binary images using the Cahn- Hilliard equation. IEEE Trans. Image Proc., 16:285291,2007.
[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:2734,1949.
[3]Cahn, J.W.On spinodal decomposition. Acta Metall., 9:795801,1961.
[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., 7:287301,1996.
[5]Cahn, J.W. and Hilliard, J.E.Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys., 28:258267,1958.
[6]Elliott, C.M. and Stuart, A.M.The global dynamics of discrete semilinear parabolic equations. SIAM J. Numer. Anal, 30:16221663,1993.
[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 16861712, Warrendale, PA, USA, 1998. Materials Research Society.
[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., 44:10491072, 2006.
[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.
[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., 228:53235339,2009.
[11]Kay, D. and Welford, R.A multigrid finite element solver for the Cahn-Hilliard equation. J. Comput. Phys., 212:288304,2006.
[12]Kay, D. and Welford, R.Efficient numerical solution of Cahn-Hilliard-Navier Stokes fluids in 2D. SIAM J. Sci. Comput., 29:22412257,2007.
[13]Kim, J.S., Kang, K., and Lowengrub, J.S.Conservative multigrid methods for Cahn-Hilliard fluids. J. Comput. Phys., 193:511543,2003.
[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, 14:492513,2002.
[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, 14:514545, 2002.
[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, 179:211228,2003.
[17]Lowengrub, J.S. and Truskinovsky, L.Cahn-Hilliard fluids and topological transitions. Proc. R. Soc. Lond. A, 454:26172654,1998.
[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.
[19]Oosterlee, C.W. and Gaspar, F.J.Multigrid relaxation methods for systems of saddle point type. Appl. Numer. Math., 58:19331950,2008.
[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, 8:1629,2011.
[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., 50:105125,2012.
[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, 31:743758,2010.
[23]Shen, J. and Yang, X.Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete Cont. Dyn. Sys. A, 28:16691691,2010.
[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., 32:11591179, 2010.
[25]Trottenberg, U., Oosterlee, C.W., and Schuller, A.Multigrid. Academic Press, New York, 2005.
[26]Vanka, S.P.Block-implicit multigrid solution of Navier-Stokes equations in primitive variables. J. Comput. Phys., 65:138158,1986.
[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.
[28]Wang, C., Wang, X., and Wise, S.Unconditionally stable schemes for equations of thin film epitaxy. Discrete Cont. Dyn. Sys. A, 28:405423,2010.
[29]Wang, C. and Wise, S.Global smooth solution of modified phase field crystal equation. Methods Appl. Anal., 17:191212,2010.
[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., 49:945969,2011.
[31]Wesseling, P.An Introduction to Multigrid Methods. R.T. Edwards, Philadelphia, 2004.
[32]Wise, S.M.Unconditionally stable finite difference, nonlinear multigrid simulation of the Cahn-Hilliard-Hele-Shaw system of equations. J. Sci. Comput., 44:3868,2010.
[33]Wise, S.M., Lowengrub, J.S., and Cristini, V.An adaptive algorithm for simulating solid tumor growth using mixture models. Math. Comput. Model., 53:120,2011.
[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., 253:524543,2008.
[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., 47:22692288,2009.
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Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
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