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Numerical analysis of second order, fully discrete energy stable schemes for phase field models of two-phase incompressible flows. (English) Zbl 1397.76070
Summary: In this paper, we propose several second order in time, fully discrete, linear and nonlinear numerical schemes for solving the phase field model of two-phase incompressible flows, in the framework of finite element method. The schemes are based on the second order Crank-Nicolson method for time discretization, projection method for Navier-Stokes equations, as well as several implicit-explicit treatments for phase field equations. The energy stability and unique solvability of the proposed schemes are proved. Ample numerical experiments are performed to validate the accuracy and efficiency of the proposed schemes.

76M10 Finite element methods applied to problems in fluid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76T99 Multiphase and multicomponent flows
Full Text: DOI
[1] Anderson, D.M., McFadden, G.B., Wheeler, A.A.: Diffuse-interface methods in fluid mechanics. In: Annual Review of Fluid Mechanics, Annu. Rev. Fluid Mech., vol. 30, pp. 139-165. Annual Reviews, Palo Alto, CA (1998) · Zbl 1327.65178
[2] Arnold, D.N., Brezzi, F., Fortin, M.: A stable finite element for the Stokes equations. Calcolo 21(4), 337-344 (1985) (1984). doi:10.1007/BF02576171 · Zbl 0593.76039
[3] Baskaran, A; Lowengrub, JS; Wang, C; Wise, SM, Convergence analysis of a second order convex splitting scheme for the modified phase field crystal equation, SIAM J. Numer. Anal., 51, 2851-2873, (2013) · Zbl 1401.82046
[4] Bray, A, Theory of phase-ordering kinetics, Adv. Phys., 43, 509-523, (1994)
[5] Brown, DL; Cortez, R; Minion, ML, Accurate projection methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 168, 464-499, (2001) · Zbl 1153.76339
[6] Caginalp, G; Chen, X, Convergence of the phase field model to its sharp interface limits, Eur. J. Appl. Math., 9, 417-445, (1998) · Zbl 0930.35024
[7] Cahn, JW, Free energy of a nonuniform system. II. thermodynamic basis, J. Chem. Phys., 30, 1121-1124, (1959)
[8] Cahn, JW; Hilliard, JE, Free energy of a nonuniform system. I. interfatial free energy, J. Chem. Phys., 28, 258-267, (1958)
[9] Chaikin, P.M., Lubensky, T.C.: Principles of Condensed Matter Physics. Cambridge Univ Press, Cambridge (1995)
[10] Chen, L; Shen, J, Applications of semi-implicit Fourier-spectral method to phase field equations, Comput. Phys. Commun., 108, 147-158, (1998) · Zbl 1017.65533
[11] Chen, R; Ji, G; Yang, X; Zhang, H, Decoupled energy stable schemes for phase-field vesicle membrane model, J. Comput. Phys., 302, 509-523, (2015) · Zbl 1349.76842
[12] Ciarlet, P.G.: The finite element method for elliptic problems, Classics in Applied Mathematics, vol. 40. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2002) · Zbl 0999.65129
[13] Collins, C; Shen, J; Wise, SM, An efficient, energy stable scheme for the Cahn-Hilliard-Brinkman system, Commun. Comput. Phys., 13, 929-957, (2013) · Zbl 1373.76161
[14] Dong, S; Shen, J, A time-stepping scheme involving constant coefficient matrices for phase-field simulations of two-phase incompressible flows with large density ratios, J. Comput. Phys., 231, 5788-5804, (2012) · Zbl 1277.76118
[15] Du, Q; Liu, C; Wang, X, A phase field approach in the numerical study of the elastic bending energy for vesicle membranes, J. Comput. Phys., 198, 450-468, (2004) · Zbl 1116.74384
[16] Et, W; Liu, JG, Projection method. I. convergence and numerical boundary layers, SIAM J. Numer. Anal, 32, 1017-1057, (1995) · Zbl 0842.76052
[17] Elder, KR; Grant, M; Provatas, N; Kosterlitz, JM, Sharp interface limits of phase-field models, Phys. Rev. E., 64, 021604, (2001)
[18] Eyre, D.J.: 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., vol. 529, pp. 39-46. MRS, Warrendale, PA (1998) · Zbl 1247.65088
[19] Fick, A.E.: Über diffusion. Poggend. Ann. d. Physik u. Chem. 94, 59-86 (1855) · Zbl 0927.76007
[20] Guermond, J.L., Quartapelle, L.: On stability and convergence of projection methods based on pressure Poisson equation. Int. J. Numer. Methods Fluids 26(9), 1039-1053 (1998). doi:10.1002/(SICI)1097-0363(19980515)26:9<1039:AID-FLD675>3.0.CO;2-U · Zbl 0912.76054
[21] Guermond, JL; Quartapelle, L, On the approximation of the unsteady Navier-Stokes equations by finite element projection methods, Numer. Math., 80, 207-238, (1998) · Zbl 0914.76051
[22] Guermond, JL; Shen, J, Velocity-correction projection methods for incompressible flows, SIAM J. Numer. Anal., 41, 112-134, (2003) · Zbl 1130.76395
[23] Guillén-González, F; Tierra, G, On linear schemes for a Cahn-Hilliard diffuse interface model, J. Comput. Phys., 234, 140-171, (2013) · Zbl 1284.35025
[24] Gurtin, ME; Polignone, D; Viñals, J, Two-phase binary fluids and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci., 6, 815-831, (1996) · Zbl 0857.76008
[25] Han, D; Wang, X, A second order in time, uniquely solvable, unconditionally stable numerical scheme for Cahn-Hilliard-Navier-Stokes equation, J. Comput. Phys., 290, 139-156, (2015) · Zbl 1349.76213
[26] Hecht, F, New development in freefem\(++\), J. Numer. Math., 20, 251-265, (2012) · Zbl 1266.68090
[27] Hohenberg, P; Halperin, B, Theory of dynamic critical phenomena, Rev. Mod. Phys., 49, 435-479, (1977)
[28] Hu, Z; Wise, SM; Wang, C; Lowengrub, JS, Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation, J. Comput. Phys., 228, 5323-5339, (2009) · Zbl 1171.82015
[29] Hua, J; Lin, P; Liu, C; Wang, Q, Energy law preserving c0 finite element schemes for phase field models in two-phase flow computations, J. Comput. Phys., 230, 7115-7131, (2011) · Zbl 1408.76550
[30] Ingram, R, A new linearly extrapolated Crank-Nicolson time-stepping scheme for the Navier-Stokes equations, Math. Comput., 82, 1953-1973, (2013) · Zbl 06195242
[31] Kay, D; Styles, V; Welford, R, Finite element approximation of a Cahn-Hilliard-Navier-Stokes system, Interfaces Free Bound., 10, 15-43, (2008) · Zbl 1144.35043
[32] Kim, J; Kang, K; Lowengrub, J, Conservative multigrid methods for Cahn-Hilliard fluids, J. Comput. Phys., 193, 511-543, (2004) · Zbl 1109.76348
[33] Lin, P; Liu, C, Simulation of singularity dynamics in liquid crystal flows: a C0 finite element approach, J. Comput. Phys., 215, 348-362, (2006) · Zbl 1101.82039
[34] Liu, C; Shen, J, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Phys. D Nonlinear Phenom., 179, 211-228, (2003) · Zbl 1092.76069
[35] Liu, F; Shen, J, Stabilized semi-implicit spectral deferred correction methods for Allen-Cahn and Cahn-Hilliard equations, Math. Methods Appl. Sci., 38, 4564-4575, (2015) · Zbl 1335.65078
[36] Lowengrub, J; Truskinovsky, L, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454, 2617-2654, (1998) · Zbl 0927.76007
[37] Minjeaud, S, An unconditionally stable uncoupled scheme for a triphasic Cahn-Hilliard/Navier-Stokes model, Numer. Methods Partial Differ. Equ., 29, 584-618, (2013) · Zbl 1364.76091
[38] Shen, J, On error estimates of the projection methods for the Navier-Stokes equations: second-order schemes, Math. Comput., 65, 1039-1065, (1996) · Zbl 0855.76049
[39] Shen, J.: Modeling and numerical approximation of two-phase incompressible flows by a phase-field approach. In: Multiscale Modeling and Analysis for Materials Simulation, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 22, pp. 147-195. World Sci. Publ., Hackensack, NJ (2012) · Zbl 1130.76437
[40] Shen, J; Wang, C; Wang, X; Wise, SM, 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) · Zbl 1247.65088
[41] Shen, J; Yang, X, An efficient moving mesh spectral method for the phase-field model of two-phase flows, J. Comput. Phys., 228, 2978-2992, (2009) · Zbl 1159.76032
[42] Shen, J; Yang, X, Numerical approximation of Allen-Cahn and Cahn-Hilliard equations, Discrete Contin. Dyn. Syst. Ser. B, 28, 1669-1691, (2010) · Zbl 1201.65184
[43] Shen, J; 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) · Zbl 1410.76464
[44] Shen, J; Yang, X, Decoupled energy stable schemes for phase field models of two phase incompressible flows, SIAM J. Numer. Anal., 53, 279-296, (2015) · Zbl 1327.65178
[45] Témam, R, Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires II, Arch. Ration. Mech. Anal., 33, 377-385, (1969) · Zbl 0207.16904
[46] Temam, R.: Navier-Stokes Equations. Theory and Numerical Analysis. Studies in Mathematics and Its Applications, vol. 2. North-Holland Publishing Co., Amsterdam (1977) · Zbl 0383.35057
[47] Kan, J, A second-order accurate pressure-correction scheme for viscous incompressible flow, SIAM J. Sci. Stat. Comput., 7, 870-891, (1986) · Zbl 0594.76023
[48] Wang, C; Wang, X; Wise, SM, Unconditionally stable schemes for equations of thin film epitaxy, Discrete Contin. Dyn. Syst., 28, 405-423, (2010) · Zbl 1201.65166
[49] Wang, C; Wise, SM, An energy stable and convergent finite-difference scheme for the modified phase field crystal equation, SIAM J. Numer. Anal., 49, 945-969, (2011) · Zbl 1230.82005
[50] Wise, SM, Unconditionally stable finite difference, nonlinear multigrid simulation of the Cahn-Hilliard-Hele-Shaw system of equations, J. Sci. Comput., 44, 38-68, (2010) · Zbl 1203.76153
[51] Wu, X; Zwieten, GJ; Zee, KG, Stabilized second-order convex splitting schemes for Cahn-Hilliard models with application to diffuse-interface tumor-growth models, Int. J. Numer. Methods Biomed. Eng., 30, 180-203, (2014)
[52] Yang, X; Feng, JJ; Liu, C; 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) · Zbl 1158.76319
[53] Yue, P; Feng, JJ; Liu, C; Shen, J, A diffuse interface method for simulating two phase flows of complex fluids, J. Fluid Mech., 515, 293-317, (2004) · Zbl 1130.76437
[54] Zhang, J; Das, S; Du, Q, A phase field model for vesicle-substrate adhesion, J. Comput. Phys., 228, 7837-7849, (2009) · Zbl 1173.74028
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