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A stable second-order BDF scheme for the three-dimensional Cahn–Hilliard–Hele–Shaw system

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Abstract

We propose a stable scheme to solve numerically the Cahn–Hilliard–Hele–Shaw system in three-dimensional space. In the proposed scheme, we discretize the space and time derivative terms by combining with backward differentiation formula, which turns out to be both second-order accurate in space and time. Using this method, a set of linear elliptic equations are solved instead of the complicated and high-order nonlinear equations. We prove that our proposed scheme is uniquely solvable. We use a linear multigrid solver, which is fast and convergent, to solve the resulting discrete system. The numerical tests indicate that our scheme can use a large time step. The accuracy and other capability of the proposed algorithm are demonstrated by various computational results.

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References

  1. Lee, H., Lowengrub, J., Goodman, J.: Modeling pinchoff and reconnection in a Hele–Shaw cell.I. The models and their calibration. Phys. Fluids. 14(2), 492–513 (2002)

    Article  MathSciNet  Google Scholar 

  2. Lee, H., Lowengrub, J., 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)

    Article  MathSciNet  Google Scholar 

  3. Cahn, J., Hilliard, J.: Free energy of a nonuniform system. i. Interfacial free energy. J. Chemi. Phys. 28(2), 258–267 (1958)

    Article  Google Scholar 

  4. Li, Y., Lee, H., Xia, B., Kim, J.: A compact fourth-order finite difference scheme for the three-dimensional Cahn–Hilliard equation. Comput. Phys. Commun. 200, 108–116 (2016)

    Article  MathSciNet  Google Scholar 

  5. Zhao, J., Yang, X., Jie, S., Wang, Q.: A decoupled energy stable scheme for a hydrodynamic phase-field model of mixtures of nematic liquid crystals and viscous fluids. J. Comput. Phys. 305, 539–556 (2016)

    Article  MathSciNet  Google Scholar 

  6. Li, Y., Choi, J., Kim, J.: Multi-component Cahn–Hilliard system with different boundary conditions in complex domains. J. Comput. Phys. 323, 1–16 (2016)

    Article  MathSciNet  Google Scholar 

  7. Li, Y., Jeong, D., Shin, J., Kim, J.: A conservative numerical method for the Cahn–Hilliard equation with Dirichlet boundary conditions in complex domains. Comput. Mathe. Appl. 65, 102–115 (2013)

    Article  MathSciNet  Google Scholar 

  8. Li, Q., Mei, L., Yang, X., Li, Y.: Efficient numerical schemes with unconditional energy stabilities for the modified phase field crystal equation. Adv. Comput. Math. 45, 1511–1580 (2019)

    MathSciNet  MATH  Google Scholar 

  9. Jeong, D., Choi, Y., Kim, J.: A benchmark problem for the two- and three-dimensional Cahn–Hilliard equations. Commun. Nonlinear. Sci. Numer. Simul. 61, 149–159 (2018)

    Article  MathSciNet  Google Scholar 

  10. Yang, X.F., Zhao, J.: Efficient linear schemes for the nonlocal Cahn–Hilliard equation of phase field models. Commun. Nonlinear. Sci. Numer. Simul. 235, 234–245 (2019)

    MathSciNet  Google Scholar 

  11. Garcke, H., Hinze, M., Kahle, C., Lam, K.: A phase field approach to shape optimization in navierCStokes flow with integral state constraints. Adv. Comput. Math. 44, 1345–1383 (2018)

    Article  MathSciNet  Google Scholar 

  12. Han, D.: A decoupled unconditionally stable numerical scheme for the Cahn–Hilliard–Hele–Shaw system. J. Sci. Comput. 66(3), 1102–1121 (2016)

    Article  MathSciNet  Google Scholar 

  13. Craig, C., Jie, S., Wise, S.: An efficient, energy stable scheme for the Cahn–Hilliard–Brinkman system. Commun. Comput. Phys. 13(4), 929–957 (2013)

    Article  MathSciNet  Google Scholar 

  14. Wise, S.: Unconditionally stable finite difference, nonlinear multigrid simulation of the Cahn–Hilliard–Hele–Shaw system of equations. J. Sci. Comput. 44, 33–68 (2010)

    Article  MathSciNet  Google Scholar 

  15. Wise, S., Lowengrub, J., Cristini, V.: An adaptive multigrid algorithm for simulating solid tumor growth using mixture models. Math. Comput. Model. 53(1-2), 1–20 (2011)

    Article  MathSciNet  Google Scholar 

  16. Han, D., Sun, D., Wang, X.: Two-phase flows in karstic geometry. Math. Methods. Appl. Sci. 37(18), 3048–3063 (2013)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. Guo, R., Xia, Y., Xu, Y.: An efficient fully-discrete local discountinuous Galerkin method for the Cahn–Hilliard–Hele–Shaw system. J. Comput. Phy. 264, 23–40 (2014)

    Article  Google Scholar 

  19. Dede, L., Garcke, H., Lam, K.: A Hele–Shaw–Cahn–Hilliard model for incompressible two-phase flows with different densities. J. Math. Fluid. Mech. 20, 531–567 (2018)

    Article  MathSciNet  Google Scholar 

  20. Zhao, J., Wang, Q., Yang, X.: Numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach. Int. J. Numer. Meth. Engng 110, 279–300 (2017)

    Article  MathSciNet  Google Scholar 

  21. Han, D., Wang, X.: A second order in time, decoupled, unconditionally stable numerical scheme for the Cahn–Hilliard-Darcy system. J. Sci. Comput. 77(2), 1210–1233 (2018)

    Article  MathSciNet  Google Scholar 

  22. Chen, W., Feng, W., Liu, Y., Wang, C., Wise, S.: A second order energy stable scheme for the Cahn–Hilliard–Hele–Shaw equations, 24(1),149–182 (2019)

  23. Gong, Y., Zhao, J., Wang, Q.: Linear second order in time energy stable schemes for hydrodynamic models of binary mixtures based on a spatially pseudospectral approximation. Adv. Comput. Math. 44, 1573–1600 (2018)

    Article  MathSciNet  Google Scholar 

  24. Liu, F., Anh, V., Turner, I.: Numerical solution of the space fractional Fokker–Planck equation. J. Comput. Appl. Mathe. 166(1), 209–219 (2004)

    Article  MathSciNet  Google Scholar 

  25. Feng, W., Wang, C., Wise, S., Zhang, Z.: A second-order energy stable backward differentiation formula method for the epitaxial thin film equation with slope selection. arXiv:1706.01943 (2017)

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

    Article  MathSciNet  Google Scholar 

  27. Jando, D.: Efficient goal-oriented global error estimators for BDF methods using discrete adjoints. J. Comput. Appl. Mathe. 316, 195–212 (2017)

    Article  MathSciNet  Google Scholar 

  28. Long, J., Li, Y., Luo, C., Yu, Q.: An unconditional stable compact fourth-order finite difference scheme for three dimensional Allen–Cahn equation, Comput. Mathe Appl (2018)

  29. Li, Y., Guo, S.: Triply periodic minimal surface using a modified Allen–Cahn equation. Appl. Mathe. Comput. 295, 84–94 (2017)

    Article  MathSciNet  Google Scholar 

  30. Gao, Y., Li, R., Mei, L., Lin, Y.: A second–order decoupled energy stable numerical scheme for Cahn-Hilliard-Hele-Shaw system. Appl. Numer. Math. 157, 338–355 (2020)

    Article  MathSciNet  Google Scholar 

  31. Cahn, J.: Free energy of a nonuniform system II: Thermodynamic basis. J. Chem. Phys. 30, 1121–1124 (1959)

    Article  Google Scholar 

  32. Shinozaki, A., Oono, Y.: Spinodal decomposition in a Hele–Shaw cell. Phys. Rev. A. 45(4), 2161–2164 (1992)

    Article  Google Scholar 

  33. Li, Y., Kim, J.: Phase-field simulations of crystal growth with adaptive mesh refinement. Int. J. Heat Mass Transf. 55, 7926–7932 (2012)

    Article  Google Scholar 

  34. Li, Y., Jeong, D., Kim, J.: Adaptive mesh refinement for simulation of thin film flows. Meccanica. 49, 239–252 (2014)

    Article  MathSciNet  Google Scholar 

  35. Kim, J., Kang, K., Lowengrub, J.: Conservative multigrid methods for Cahn–Hilliard fluids. J. Comput. Phys. 193, 511–543 (2004)

    Article  MathSciNet  Google Scholar 

  36. Rayleigh, W.: On the stability, or instability, of certain fluid motions. Proc. London. Math. Soc. 10, 4–13 (1878)

    Article  MathSciNet  Google Scholar 

  37. Cahn, J., Elliott, C., Novick-Cohen, A.: The Cahn–Hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the mean curvature. European J. Appl. Math. 7, 287–301 (1996)

    Article  MathSciNet  Google Scholar 

  38. Li, Y., Choi, J., Kim, J.: A phase-field fluid modeling and computation with interfacial profile correction term. Commun. Nonlinear. Sci. Numer. Simul. 30, 84–100 (2016)

    Article  MathSciNet  Google Scholar 

  39. Shen, J., Yang, X.: Numerical approximations of Allen–Cahn and Cahn–Hilliard equations. Discrete Contin. Dyn. Syst. 28(4), 1669–1691 (2010)

    Article  MathSciNet  Google Scholar 

  40. Condette, N., Melcher, C., Süli, E.: Spectral approximation of pattern–forming nonlinear evolution equations with double–well potentials of quadratic growth. Math. Comput. 80(273), 205–223 (2011)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors thank the reviewers for the constructive and helpful comments on the revision of this article.

Funding

Y.B. Li is supported by the National Natural Science Foundation of China (No. 11601416) and by the China Postdoctoral Science Foundation (No. 2018M640968). The corresponding author (J.S. Kim) was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1A2C1003053).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China

    Yibao Li, Qian Yu & Binhu Xia

  2. Department of Computational Science and Engineering, Yonsei University, Seoul, 03722, Republic of Korea

    Weiwei Fang

  3. Department of Mathematics, Korea University, Seoul, 02841, Republic of Korea

    Junseok Kim

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  1. Yibao Li
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  2. Qian Yu
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Correspondence to Junseok Kim.

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Communicated by: Long Chen

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Li, Y., Yu, Q., Fang, W. et al. A stable second-order BDF scheme for the three-dimensional Cahn–Hilliard–Hele–Shaw system. Adv Comput Math 47, 3 (2021). https://doi.org/10.1007/s10444-020-09835-6

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  • DOI: https://doi.org/10.1007/s10444-020-09835-6

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