Unconditionally stable finite difference, nonlinear multigrid simulation of the cahn-hilliard-hele-shaw system of equations

Journal of Scientific Computing

Volumn 44, Issue 1, 2010, Pages 38-68

  Wise, S M ^a  

^a UNIVERSITY OF TENNESSEE   (United States)

Author keywords

Cahn Hilliard equation; Convex splitting; Darcy's law; Energy stability; Finite difference methods; Hele Shaw flow; Nonlinear partial differential equations

Indexed keywords

CAHN-HILLIARD EQUATION; DARCY'S LAW; ENERGY STABILITY; HELE-SHAW FLOW; NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS;

COMPUTATIONAL FLUID DYNAMICS; DIFFERENCE EQUATIONS; ENERGY CONSERVATION; FINITE DIFFERENCE METHOD; FLUID DYNAMICS; GROUNDWATER FLOW; PARTIAL DIFFERENTIAL EQUATIONS; POROUS MATERIALS; SEEPAGE; SPINODAL DECOMPOSITION; SYSTEM STABILITY; TWO PHASE FLOW;

NONLINEAR EQUATIONS;

EID: 77953019931     PISSN: 08857474     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10915-010-9363-4     Document Type: Article

References (27)

1. A. Bertozzi S. Esedoglu A. Gillette 2007 Inpainting of binary images using the Cahn-Hilliard equation IEEE Trans. Image Process. 16 285 291 10.1109/TIP.2006.887728 2460167 (Pubitemid 46442121)
2. J. Bramble 1966 A second order finite difference analog of the first biharmonic boundary value problem Numer. Math. 9 236 249 0154.41105 10.1007/BF02162087 205478
3. Brinkman, H.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. AI, 27-34 (1949)
4. J. Cahn 1961 On spinodal decomposition Acta Metall. 9 795 10.1016/0001-6160(61)90182-1
5. J. Cahn J. Hilliard 1958 Free energy of a nonuniform system. i. interfacial free energy J. Chem. Phys. 28 258 10.1063/1.1744102
6. C. Elliot A. Stuart 1993 The global dynamics of discrete semilinear parabolic equations SIAM J. Numer. Anal. 30 1622 1663 10.1137/0730084 1249036
7. Eyre, D.: Unconditionally gradient stable time marching the Cahn-Hilliard equation. In: Bullard, J.W., Kalia, R., Stoneham, M., Chen, L. (eds.) Computational and Mathematical Models of Microstructural Evolution, vol. 53, pp. 1686-1712. Materials Research Society, Warrendale (1998)
8. X. Feng 2006 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 05167765 10.1137/050638333 2231855 (Pubitemid 47135914)
9. Feng, X., Wise, S.: Analysis of a Fully Discrete Finite Element Approximation of a Darcy-Cahn-Hilliard Diffuse Interface Model for the Hele-Shaw Flow (in preparation)
10. D. Furihata 2001 A stable and conservative finite difference scheme for the Cahn-Hilliard equation Numer. Math. 87 675 699 0974.65086 10.1007/PL00005429 1815731 (Pubitemid 33589610)
11. Z. Hu S. Wise C. Wang J. Lowengrub 2009 Stable and efficient finite-difference nonlinear-multigrid schemes for the phase-field crystal equation J. Comput. Phys. 228 5323 5339 1171.82015 10.1016/j.jcp.2009.04.020 2541456
12. D. Kay R. Welford 2006 A multigrid finite element solver for the Cahn-Hilliard equation J. Comput. Phys. 212 288 304 1081.65091 10.1016/j.jcp.2005.07.004 2183612 (Pubitemid 41799815)
13. D. Kay R. Welford 2007 Efficient numerical solution of Cahn-Hilliard-Navier Stokes fluids in 2d SIAM J. Sci. Comput. 29 2241 2257 1154.76033 10.1137/050648110 2357613
14. J. Kim K. Kang J. Lowengrub 2003 Conservative multigrid methods for Cahn-Hilliard fluids J. Comput. Phys. 193 511 543 10.1016/j.jcp.2003.07.035 2030475
15. H. Lee J. Lowengrub J. Goodman 2002 Modeling pinchoff and reconnection in a Hele-Shaw cell. I. The models and their calibration Phys. Fluids 14 492 513 10.1063/1.1425843 1878351 (Pubitemid 34143755)
16. H. Lee J. Lowengrub J. Goodman 2002 Modeling pinchoff and reconnection in a Hele-Shaw cell. II. Analysis and simulation in the nonlinear regime Phys. Fluids 14 514 545 10.1063/1.1425844 1878907 (Pubitemid 34143756)
17. C. Liu J. Shen 2003 A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method Physica D 179 211 228 1092.76069 10.1016/S0167-2789(03)00030-7 1984386
18. J. Lowengrub L. Truskinovsky 1998 Cahn-Hilliard fluids and topological transitions Proc. R. Soc. Lond. A 454 2617 2654 0927.76007 10.1098/rspa.1998. 0273 1650795
19. A. Shinozaki Y. Oono 1992 Spinodal decomposition in a Hele-Shaw cell Phys. Rev. A 45 R2161 R2164 10.1103/PhysRevA.45.R2161
20. Trottenberg, U., Oosterlee, C., Schüller, A.: Multigrid. Academic Press, New York (2005)
21. B. Vollmayr-Lee A. Rutenberg 2003 Fast and accurate coarsening simulation with an unconditionally stable time step Phys. Rev. E 68 066,703 10.1103/PhysRevE.68.066703
22. C. Wang X. Wang S. Wise 2010 Unconditionally stable schemes for equations of thin film epitaxy Discrete Cont. Dyn. Sys. A 28 405 423 10.3934/dcds.2010. 28.405
23. Wang, C., Wise, S.: An energy stable and convergent finite-difference scheme for the modified phase field crystal equation. SIAM J. Numer. Anal. (in review)
24. Wise, S., Lowengrub, J., Cristini, V.: An adaptive algorithm for simulating solid tumor growth using mixture models. Math. Comput. Model. (in review)
25. S. Wise J. Lowengrub H. Frieboes V. Cristini 2008 Three-dimensional multispecies nonlinear tumor growth-I model and numerical method J. Theor. Biol. 253 524 543 10.1016/j.jtbi.2008.03.027
26. S. Wise C. Wang J. Lowengrub 2009 An energy stable and convergent finite-difference scheme for the phase field crystal equation SIAM J. Numer. Anal. 47 2269 2288 10.1137/080738143 2519603
27. X. Zheng S. Wise V. Cristini 2005 Nonlinear simulation of tumor necrosis, neo-vascularization and tissue invasion via an adaptive finite-element/level- set method Bull. Math. Biol. 67 211 259 10.1016/j.bulm.2004.08.001 2197392 (Pubitemid 40234527)