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Highly efficient variant of SAV approach for two-phase incompressible conservative Allen–Cahn fluids

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Abstract

Herein, we construct efficient linear, totally decoupled, and energy dissipative schemes for two-phase incompressible conservative Allen–Cahn (CAC) fluid model. The binary CAC model has good potential in simulating fluid flow with interface because of the following merits: (i) mass conservation is satisfied, (ii) interfacial position can be easily captured. Comparing with the well-known fourth-order Cahn–Hilliard (CH) equation, the CAC equation is easy to solve since its second-order property. The scalar auxiliary variable (SAV)-type methods provide practical approach to develop linearly energy-stable schemes for phase-field problems. A variant of SAV approach considered in this work not only leads to accurate schemes for CAC fluid system, but also achieves highly efficient calculation. In each time step, only several linear and decoupled equations need to be computed. The linear multigrid algorithm is adopted to accelerate convergence. The unique solvability, modified energy dissipation law, and mass conservation in time-discretized version are analytically proved. Extensive numerical experiments are performed to validate the superior performance of the proposed methods.

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References

  1. Ren H, Zhuang X, Oterkus E, Zhu H, Rabczuk T (2021) Nonlocal strong forms of thin plate, gradient elasticity, magneto-electro-elasticity and phase-field fracture by nonlocal operator method. Eng Comput. https://doi.org/10.1007/s00366-021-01502-8

  2. Abbaszadeh M, Dehghan M (2021) The fourth-order time-discrete scheme and split-step direct meshless finite volume method for solving cubic-quintic complex Ginzburg-Landau equations on complicated geometries. Eng Comput. https://doi.org/10.1007/s00366-020-01089-6

  3. Zong Y, Zhang C, Liang H, Wang L, Xu J (2020) Modeling surfactant-laden droplet dynamics by lattice Boltzmann method. Phys Fluids 32:122105

    Google Scholar 

  4. Qiao Y, Qian L, Feng X (2021) Fast numerical approximation for the space-fractional semilinear parabolic equations on surfaces. Eng Comput. https://doi.org/10.1007/s00366-021-01357-z

  5. Cahn JW, Hilliard JE (1958) Free energy of a nonuniform system. I. Interfacial free energy. J Chem Phys 28(2):258–267

    MATH  Google Scholar 

  6. Xia Q, Yu Q, Li Y (2021) A second-order accurate, unconditionally energy stable numerical scheme for binary fluid flows on arbitrarily curved surfaces. Comput Methods Appl Mech Eng 384:113987

    MathSciNet  MATH  Google Scholar 

  7. Gong Y, Zhao J, Wang Q (2017) An energy stable algorithm for a quasi-incompressible hydrodynamic phase-field model of viscous fluid mixtures with variable densities and viscosities. Commun Comput Phys 219:20–34

    MathSciNet  MATH  Google Scholar 

  8. Liang H, Xu J, Chen J, Chai Z, Shi B (2019) Lattice Boltzmann modeling of wall-bounded ternary fluid flows. Appl Math Model 73:487–513

    MathSciNet  MATH  Google Scholar 

  9. Chiu P-H (2019) A coupled phase field framework for solving incompressible two-phase flows. J Comput Phys 392:115–140

    MathSciNet  MATH  Google Scholar 

  10. Han D, Brylev A, Yang X, Tan Z (2017) Numerical analysis of second-order, fully discrete energy stable schemes for phase field models of two-phase incompressible flows. J Sci Comput 70:965–989

    MathSciNet  MATH  Google Scholar 

  11. Li H-L, Liu H-R, Ding H (2020) A fully 3D simulation of fluid-structure interaction with dynamic wetting and contact angle hysteresis. J Comput Phys 420:109709

    MathSciNet  MATH  Google Scholar 

  12. Bai F, Han D, He X, Yang X (2020) Deformation and coalescence of ferrodroplets in Rosensweig model using the phase field and modified level set approaches under uniform magnetic fields. Commun Nonlinear Sci Numer Simul 85:105213

    MathSciNet  MATH  Google Scholar 

  13. Yan Y, Chen W, Wang C, Wise SM (2018) A second-order energy stable BDF numerical scheme for the Cahn–Hilliard equation. Commun Comput Phys 23(2):572–602

    MathSciNet  MATH  Google Scholar 

  14. Cheng K, Feng W, Wang C, Wise SM (2019) An energy stable fourth order finite difference scheme for the Cahn–Hilliard equation. J Comput Appl Math 362:574–595

    MathSciNet  MATH  Google Scholar 

  15. Chen W, Feng W, Liu Y, Wang C, Wise SM (2019) A second order energy stable scheme for the Cahn–Hilliard–Hele–Shaw equations. Discrete Contin Dyn Syst Ser B 24(1):149–182

    MathSciNet  MATH  Google Scholar 

  16. Chen W, Wang C, Wang S, Wang X, Wise SM (2020) Energy stable numerical schemes for ternary Cahn–Hilliard system. J Sci Comput 84:27

    MathSciNet  MATH  Google Scholar 

  17. Guo J, Wang C, Wise SM, Yue X (2016) An \(H^2\) convergence of a second-order convex-splitting, finite difference scheme for the three-dimensional Cahn–Hilliard equation. Commun Math Sci 14(2):489–515

    MathSciNet  MATH  Google Scholar 

  18. Diegel AE, Wang C, Wise SM (2016) Stability and convergence of a second-order mixed finite element method for the Cahn–Hilliard equation. IMA J Numer Anal 36(4):1867–1897

    MathSciNet  MATH  Google Scholar 

  19. Cheng K, Wang C, Wise SM, Yue X (2016) A second-order, weakly energy-stable pseudo-spectral scheme for the Cahn–Hilliard equation and its solution by the homogeneous linear iteration method. J Sci Comput 69:1083–1114

    MathSciNet  MATH  Google Scholar 

  20. Guo J, Wang C, Wise SM, Yue X (2021) An improved error analysis for a second-order numerical scheme for the Cahn–Hilliard equation. J Comput Appl Math 388:113300

    MathSciNet  MATH  Google Scholar 

  21. Zhao S, Xiao X, Feng X (2020) A efficient time adaptivity based on chemical potential for surface Cahn–Hilliard equation using finite element approximation. Appl Math Comput 369:124901

    MathSciNet  MATH  Google Scholar 

  22. Yang J, Kim J (2020) An unconditionally stable second-order accurate method for systems of Cahn–Hilliard equations. Commun Nonlinear Sci Numer Simulat 87:105276

    MathSciNet  MATH  Google Scholar 

  23. Li X, Ju L, Meng X (2019) Convergence analysis of exponential time differencing schemes for the Cahn–Hilliard equation. Commun Comput Phys 26(5):1510–1529

    MathSciNet  MATH  Google Scholar 

  24. Gong Y, Zhao J, Wang Q (2020) Arbitrarily high-order linear energy stable schemes for gradient flow models. J Comput Phys 419:109610

    MathSciNet  MATH  Google Scholar 

  25. Liu Z, Li X (2019) Efficient modified techniques of invariant energy quadratization approach for gradient flows. Appl Math Lett 98:206–214

    MathSciNet  MATH  Google Scholar 

  26. Zhu G, Chen H, Yao J, Sun S (2019) Efficient energy-stable schemes for the hydrodynamics coupled phase-field model. Appl Math Model 70:82–108

    MathSciNet  MATH  Google Scholar 

  27. Liu Z, Li X (2020) The fast scalar auxiliary variable approach with unconditional energy stability for nonlocal Cahn–Hilliard equation. Methods Partial Differ Equ Numer. https://doi.org/10.1002/num.22527

  28. Sun M, Feng X, Wang K (2020) Numerical simulation of binary fluid-surfactant phase field model coupled with geometric curvature on the curved surface. Comput Methods Appl Mech Eng 367:113123

    MathSciNet  MATH  Google Scholar 

  29. Zhang C, Ouyang J, Wang C, Wise SM (2020) Numerical comparison of modified-energy stable SAV-type schemes and classical BDF methods on benchmark problems for the functionalized Cahn–Hilliard equation. J Comput Phys 423:109772

    MathSciNet  MATH  Google Scholar 

  30. Han D, Jiang N (2020) A second order, linear, unconditionally stable, Crank–Nicolson–Leapfrog scheme for phase field models of two-phase incompressible flows. Appl Math Lett 108:106521

    MathSciNet  MATH  Google Scholar 

  31. Chen L, Zhao J (2020) A novel second-order linear scheme for the Cahn–Hilliard–Navier–Stokes equations. J Comput Phys 423:109782

    MathSciNet  MATH  Google Scholar 

  32. Wang X, Kou J, Gao H (2021) Linear energy stable and miximum pribciple preserving semi-implicit scheme for Allen–Cahn equation with double well potential. Commun Nonlinear Sci Numer Simulat 98:105766

    MATH  Google Scholar 

  33. Kim J, Lee S, Choi Y (2014) A conservative Allen–Cahn equation with a space-time dependent Lagrange multiplier. Int J Eng Sci 84:11–17

    MathSciNet  MATH  Google Scholar 

  34. Jeong D, Kim J (2017) Conservative Allen–Cahn–Navier–Stokes systems for incompressible two-phase fluid flows. Comput Fluid 156:239–246

    MathSciNet  MATH  Google Scholar 

  35. Lee HG (2016) High-order and mass conservative methods for the conservative Allen–Cahn equation. Comput Math Appl 72:620–631

    MathSciNet  MATH  Google Scholar 

  36. Yang J, Jeong D, Kim J (2021) A fast and practical adaptive finite difference method for the conservative Allen–Cahn model in two-phase flow system. Int J Multiphase Flow 137:103561

    MathSciNet  Google Scholar 

  37. Joshi V, Jaiman RK (2018) An adaptive variational procedure for the conservative and positivity preserving Allen–Cahn phase-field model. J Comput Phys 366:478–504

    MathSciNet  MATH  Google Scholar 

  38. Huang Z, Lin G, Ardenaki AM (2020) Consistent and conservative scheme for incompressible two-phase flows using the conservative Allen–Cahn model. J Comput Phys 420:109718

    MathSciNet  MATH  Google Scholar 

  39. Aihara S, Takaki T, Takada N (2019) Multi-phase-field modeling using a conservative Allen–Cahn equation for multiphase flow. Comput Fluid 178:141–151

    MathSciNet  MATH  Google Scholar 

  40. Li J, Ju L, Cai Y, Feng X (2021) Unconditionally maximum bound principle preserving linear schemes for the conservative Allen–Cahn equation with nonlocal constraint. J Sci Comput 87:98

    MathSciNet  MATH  Google Scholar 

  41. Jiang K, Ju L, Li J, Li X (2021) Unconditionally stable exponential time differencing schemes for the mass-conserving Allen-Cahn equation with nonlocal and local effects. Numer Method Partial Differential Equ. https://doi.org/10.1002/num.22827

  42. Zhang J, Yang X (2020) Unconditionally energy stable large time stepping method for the \(L^2\)-gradient flow based ternary phase-field model with precise nonlocal volume conservation. Comput Methods Appl Mech Eng 361:112743

    MATH  Google Scholar 

  43. Yang X (2021) Efficient, second-order in time, and energy stable scheme for a new hydrodynamically coupled three components volume-conserved Allen–Cahn phase-field model. Math Model Method Appl Sci 31(4):753–787

    MathSciNet  MATH  Google Scholar 

  44. Liu Z, Li X (2021) A highly efficient and accurate exponential semi-implicit scalar auxiliary variable (ESI-SAV) approach for dissipative system. J Comput Phys 447:110703

    MathSciNet  MATH  Google Scholar 

  45. Deville MO, Fischer PF, Mund EH (2002) High-order methods for incompressible fluid flow. Cambridge University Press, Cambridge, p 9

    MATH  Google Scholar 

  46. Chen W, Liu Y, Wang C, Wise SM (2016) Convergence analysis of a fully discrete finite difference scheme for the Cahn–Hilliard–Hele–Shaw equation. Math Comput 85:2231–2257

    MathSciNet  MATH  Google Scholar 

  47. Liu Y, Chen W, Wang C, Wise SM (2017) Error analysis of a mixed finite element method for a Cahn–Hilliard–Hele–Shaw system. Numer Math 135:679–709

    MathSciNet  MATH  Google Scholar 

  48. Diegel AE, Wang C, Wang X, Wise SM (2017) Convergence analysis and error estimates for a second order accurate finite element method for the Cahn–Hilliard–Navier–Stokes system. Numer Math 137:495–534

    MathSciNet  MATH  Google Scholar 

  49. Shen J, Xu J (2018) Convergence and error analysis for the scalar auxiliary variable (SAV) schemes to gradient flows. SIAM J Numer Anal 56(5):2895–2912

    MathSciNet  MATH  Google Scholar 

  50. Li X, Shen J, Rui H (2019) Energy stability and convergence of SAV block-centered finite difference method for gradient flows. Math Comput 88:2047–2068

    MathSciNet  MATH  Google Scholar 

  51. Wang M, Huang Q, Wang C (2021) A second order accurate scalar auxiliary variable (SAV) numerical method for the square phase field crystal equation. J Sci Comput 88:33

    MathSciNet  MATH  Google Scholar 

  52. Huang F, Shen J (2021) Stability and error analysis of a class of high-order IMEX schemes for Navier–Stokes equations with periodic boundary conditions. SIAM J Numer Anal 59(6):2926–2954

    MathSciNet  MATH  Google Scholar 

  53. Li X, Shen J (2020) On a SAV-MAC scheme for the Cahn–Hilliard–Navier-Stokes phase-field model and its error analysis for the corresponding Cahn–Hilliard–Stokes case. Math Model Meth Appl Sci 30(12):2263–2297

    MathSciNet  MATH  Google Scholar 

  54. Trottenberg U, Oosterlee C, Schüller A (2001) Multigrid. Academic press, New York

    MATH  Google Scholar 

  55. Shu CW, Osher S (1989) Efficient implementation of essentially non-oscillatory shock capturing schemes II. J Comput Phys 83:32–78

    MathSciNet  MATH  Google Scholar 

  56. Bronsard L, Stoth B (1997) Volume-preserving mean curvature flow as a limit of a nonlocal Ginzburg–Landau equation. SIAM J Math Anal 28:769–807

    MathSciNet  MATH  Google Scholar 

  57. Lee HG, Kim J (2012) A comparison study of the boussinesq and the variable density models on buoyancy-driven flows. J Eng Math 75:15–27

    MathSciNet  MATH  Google Scholar 

  58. Zhu G, Kou J, Yao B, Wu YS, Yao J, Sun S (2019) Thermodynamically consistent modelling of two-phase flows with moving contact line and soluble surfactants. J Fluid Mech 879:327–359

    MathSciNet  MATH  Google Scholar 

  59. Yang J, Kim J (2021) A variant of stabilized-scalar auxiliary variable (S-SAV) approach for a modified phase-field surfactant model. Comput Phys Commun 261:107825

    MathSciNet  Google Scholar 

  60. Zhu G, Kou J, Yao J, Li A, Sun S (2020) A phase-field moving contact line model with soluble surfactants. J Comput Phys 405:109170

    MathSciNet  MATH  Google Scholar 

  61. Qin Y, Xu Z, Zhang H, Zhang Z (2020) Fully decoupled, linear and unconditionally energy stable schemes for the binary fluid-surfactant model. Commun Comput Phys 28:1389–1414

    MathSciNet  MATH  Google Scholar 

  62. Yang J, Kim J (2021) An efficient stabilized multiple auxiliary variables method for the Cahn–Hilliard–Darcy two-phase flow system. Comput Fluid 223:104948

  63. Zheng L, Zheng S, Zhai Q (2020) Multiphase flows of \(N\) immiscible incompressible fluids: Conservative Allen–Cahn equation and lattice Boltzmann equation method. Phys Rev E 101:013305

    Google Scholar 

  64. Yang J, Kim J (2021) Numerical study of the ternary Cahn–Hilliard fluids by using an efficient modified scalar auxiliary variable approach. Commun Nonlinear Sci Numer Simulat 102:105923

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The work of Z. Tan is supported by the National Nature Science Foundation of China (11971502), Guangdong Province Key Laboratory of Computational Science at the Sun Yat-sen University (2020B1212060032), and Key-Area Research and Development Program of Guangdong Province (2021B0101190003). The authors wish to thank the reviewers for the constructive comments on the revision of this article.

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  1. School of Computer Science and Engineering, Sun Yat-sen University, Guangzhou, 510275, China

    Junxiang Yang, Jianjun Chen & Zhijun Tan

  2. Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou, 510275, China

    Zhijun Tan

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  1. Junxiang Yang
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Yang, J., Chen, J. & Tan, Z. Highly efficient variant of SAV approach for two-phase incompressible conservative Allen–Cahn fluids. Engineering with Computers 38, 5339–5357 (2022). https://doi.org/10.1007/s00366-022-01618-5

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