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A second order, linear, unconditionally stable, Crank-Nicolson-Leapfrog scheme for phase field models of two-phase incompressible flows. (English) Zbl 07258071
Summary: In this article we propose a second order, linear, unconditionally stable, implicit-explicit scheme based on the Crank-Nicolson-Leapfrog discretization and the artificial compression method for solving phase field models of two-phase incompressible flows. We show that the scheme is unconditionally long-time stable. Numerical examples are provided to demonstrate the accuracy and long-time stability.

76 Fluid mechanics
65 Numerical analysis
Full Text: DOI
[1] Hohenberg, P. C.; Halperin, B. I., Theory of dynamic critical phenomena, Rev. Modern Phys., 49, 435-479 (1977)
[2] Gurtin, Morton E.; Polignone, Debra; Viñals, Jorge, Two-phase binary fluids and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci., 6, 6, 815-831 (1996) · Zbl 0857.76008
[3] Liu, Chun; Shen, Jie, A phase field model for the mixture of two incompressible fluids and its approximation by a fourier-spectral method, Physica D, 179, 34, 211-228 (2003) · Zbl 1092.76069
[4] Lee, Hyeong-Gi; Lowengrub, J. S.; Goodman, J., Modeling pinchoff and reconnection in a Hele-Shaw cell. I. The models and their calibration, Phys. Fluids, 14, 2, 492-513 (2002) · Zbl 1184.76316
[5] Lee, Hyeong-Gi; Lowengrub, J. S.; Goodman, J., Modeling pinchoff and reconnection in a Hele-Shaw cell. II. Analysis and simulation in the nonlinear regime, Phys. Fluids, 14, 2, 514-545 (2002) · Zbl 1184.76317
[6] Han, Daozhi; Sun, Dong; Wang, Xiaoming, Two-phase flows in karstic geometry, Math. Methods Appl. Sci., 37, 18, 3048-3063 (2014) · Zbl 1309.76204
[7] Lowengrub, J.; Truskinovsky, L., Quasi-incompressible Cahn-Hilliard fluids and topological transitions, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454, 1978, 2617-2654 (1998) · Zbl 0927.76007
[8] Abels, Helmut; Garcke, Harald; Grün, Günther, Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities, Math. Models Methods Appl. Sci., 22, 3, 1150013 (2012), 40 · Zbl 1242.76342
[9] Anderson, D. M.; McFadden, G. B.; Wheeler, A. A., Diffuse-interface methods in fluid mechanics, (Annual Review of Fluid Mechanics, Vol. 30. Annual Review of Fluid Mechanics, Vol. 30, Annu. Rev. Fluid Mech., vol. 30 (1998), Annual Reviews: Annual Reviews Palo Alto, CA), 139-165 · Zbl 1398.76051
[10] Lamorgese, Andrea G.; Molin, Dafne; Mauri, Roberto, Phase field approach to multiphase flow modeling, Milan J. Math., 79, 2, 597-642 (2011) · Zbl 1237.82031
[11] Shen, Jie; Wang, Cheng; Wang, Xiaoming; Wise, Steven M., Second-order convex splitting schemes for gradient flows with Ehrlich-Schwoebel type energy: application to thin film epitaxy, SIAM J. Numer. Anal., 50, 1, 105-125 (2012) · Zbl 1247.65088
[12] Han, Daozhi; Wang, Xiaoming, 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
[13] Yan, Yue; Chen, Wenbin; Wang, Cheng; Wise, Steven M., A second-order energy stable BDF numerical scheme for the Cahn-Hilliard equation, Commun. Comput. Phys., 23, 2, 572-602 (2018)
[14] Wu, X.; van Zwieten, G. J.; van der Zee, K. G., 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, 2, 180-203 (2014)
[15] 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
[16] Yang, Xiaofeng; Zhao, Jia; Wang, Qi; Shen, Jie, Numerical approximations for a three-components Cahn-Hilliard phase-field model based on the invariant energy quadratization method, Math. Models Methods Appl. Sci., 1-38 (2017) · Zbl 1393.80003
[17] Gong, Yuezheng; Zhao, Jia; Yang, Xiaogang; Wang, Qi, Fully discrete second-order linear schemes for hydrodynamic phase field models of binary viscous fluid flows with variable densities, SIAM J. Sci. Comput., 40, 1, B138-B167 (2018) · Zbl 06835747
[18] Shen, Jie; Xu, Jie; Yang, Jiang, The scalar auxiliary variable (SAV) approach for gradient flows, J. Comput. Phys., 353, 407-416 (2018) · Zbl 1380.65181
[19] Layton, W.; Trenchea, C., Stability of two IMEX methods, CNLF and BDF2-AB2, for uncoupling systems of evolution equations, Appl. Numer. Math., 62, 2, 112-120 (2012) · Zbl 1237.65101
[20] Hurl, Nicholas; Layton, William; Li, Yong; Trenchea, Catalin, Stability analysis of the Crank-Nicolson-Leapfrog method with the Robert-Asselin-Williams time filter, BIT, 54, 4, 1009-1021 (2014) · Zbl 1325.65113
[21] Jiang, Nan; Kubacki, Michaela; Layton, William; Moraiti, Marina; Tran, Hoang, A Crank-Nicolson Leapfrog stabilization: unconditional stability and two applications, J. Comput. Appl. Math., 281, 263-276 (2015) · Zbl 1308.65144
[22] Jiang, Nan; Tran, Hoang, Analysis of a stabilized CNLF method with fast slow wave splittings for flow problems, Comput. Methods Appl. Math., 15, 3, 307-330 (2015) · Zbl 1317.65185
[23] DeCaria, Victor; Layton, William; McLaughlin, Michael, A conservative, second order, unconditionally stable artificial compression method, Comput. Methods Appl. Mech. Engrg., 325, 733-747 (2017) · Zbl 1439.76059
[24] Temam, Roger, Une méthode d’approximation de la solution des équations de Navier-Stokes, Bull. Soc. Math. France, 96, 115-152 (1968) · Zbl 0181.18903
[25] 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
[26] Chorin, Alexandre Joel, Numerical solution of the Navier-Stokes equations, Math. Comp., 22, 745-762 (1968) · Zbl 0198.50103
[27] Yang, Xiaofeng; Han, Daozhi, Linearly first- and second-order, unconditionally energy stable schemes for the phase field crystal equation, J. Comput. Phys., 330, 1116-1134 (2017) · Zbl 1380.65209
[28] Cheng, Qing; Yang, Xiaofeng; Shen, Jie, Efficient and accurate numerical schemes for a hydro-dynamically coupled phase field diblock copolymer model, J. Comput. Phys., 341, 44-60 (2017) · Zbl 1380.65203
[29] Yang, Xiaofeng; Ju, Lili, Linear and unconditionally energy stable schemes for the binary fluid-surfactant phase field model, Comput. Methods Appl. Mech. Engrg., 318, 1005-1029 (2017) · Zbl 1439.76029
[30] Zhao, Jia; Yang, Xiaofeng; Gong, Yuezheng; Wang, Qi, A novel linear second order unconditionally energy stable scheme for a hydrodynamic \(\mathbf{Q} \)-tensor model of liquid crystals, Comput. Methods Appl. Mech. Engrg., 318, 803-825 (2017) · Zbl 1439.76124
[31] Yang, Xiaofeng; Zhao, Jia; Wang, Qi, Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method, J. Comput. Phys., 333, 104-127 (2017) · Zbl 1375.82121
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