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A Structure-Preserving, Upwind-SAV Scheme for the Degenerate Cahn–Hilliard Equation with Applications to Simulating Surface Diffusion

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Abstract

This paper establishes a structure-preserving numerical scheme for the Cahn–Hilliard equation with degenerate mobility. First, by applying a finite volume method with upwind numerical fluxes to the degenerate Cahn–Hilliard equation rewritten by the scalar auxiliary variable (SAV) approach, we creatively obtain an unconditionally bound-preserving, energy-stable and fully-discrete scheme, which, for the first time, addresses the boundedness of the classical SAV approach under \(H^{-1}\)-gradient flow. Then, a dimensional-splitting technique is introduced in high-dimensional cases, which greatly reduces the computational complexity while preserves original structural properties. Numerical experiments are presented to verify the bound-preserving and energy-stable properties of the proposed scheme. Finally, by applying the proposed structure-preserving scheme, we numerically demonstrate that surface diffusion is approximated by the Cahn–Hilliard equation with degenerate mobility and Flory–Huggins potential, when the absolute temperature is sufficiently low, which agrees well with the theoretical result by using formal asymptotic analysis.

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The code for the current study is available from the corresponding author on reasonable request.

References

  1. Acosta-Soba, D., Guillén-González, F., Rodríguez-Galván, J.R.: An upwind DG scheme preserving the maximum principle for the convective Cahn–Hilliard model. Numer. Algorithms 92, 1589–1619 (2023)

    MathSciNet  MATH  Google Scholar 

  2. Alikakos, N.D., Bates, P.W., Chen, X.: Convergence of the Cahn–Hilliard equation to the Hele–Shaw model. Arch. Ration. Mech. Anal. 128, 165–205 (1994)

    MathSciNet  MATH  Google Scholar 

  3. Backofen, R., Wise, S.M., Salvalaglio, M., Voigt, A.: Convexity splitting in a phase field model for surface diffusion. Int. J. Numer. Anal. Mod. 16, 192–209 (2019)

    MathSciNet  MATH  Google Scholar 

  4. Bailo, R., Carrillo, J.A., Kalliadasis, S., Perez, S.P.: Unconditional bound-preserving and energy-dissipating finite-volume schemes for the Cahn–Hilliard equation. Commun. Comput. Phys. 34, 713–748 (2023)

  5. Barrett, J.W., Blowey, J.F., Garcke, H.: Finite element approximation of the Cahn–Hilliard equation with degenerate mobility. SIAM J. Numer. Anal. 37, 286–318 (1999)

    MathSciNet  MATH  Google Scholar 

  6. Bertozzi, A.L., Esedoglu, S., Gillette, A.: Inpainting of binary images using the Cahn–Hilliard equation. IEEE Trans. Image Process. 16, 285–291 (2007)

    MathSciNet  MATH  Google Scholar 

  7. Bessemoulin-Chatard, M., Filbet, F.: A finite volume scheme for nonlinear degenerate parabolic equations. SIAM J. Sci. Comput. 34, B559–B583 (2012)

    MathSciNet  MATH  Google Scholar 

  8. Bretin, E., Masnou, S., Sengers, A., Terii, G.: Approximation of surface diffusion flow: a second order variational Cahn–Hilliard model with degenerate mobilities. Math. Mod. Methods Appl. Sci. 32, 793–829 (2022)

    MathSciNet  MATH  Google Scholar 

  9. Brown, G., Chakrabarti, A.: Surface-directed spinodal decomposition in a two-dimensional model. Phys. Rev. A 46, 4829–4835 (1992)

    Google Scholar 

  10. Burger, M., He, L., Schönlieb, C.-B.: Cahn–Hilliard inpainting and a generalization for gray value images. SIAM J. Imaging Sci. 2, 1129–1167 (2009)

    MathSciNet  MATH  Google Scholar 

  11. Cahn, J.W., Elliott, C.M., Novickcohen, A.: The Cahn–Hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the mean curvature. Eur. J. Appl. Math. 7, 287–301 (1996)

    MathSciNet  MATH  Google Scholar 

  12. Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)

    MATH  Google Scholar 

  13. Cahn, J.W., Taylor, J.E.: Surface motion by surface diffusion. Acta Metall. Mater. 42, 1045–1063 (1994)

    Google Scholar 

  14. Chen, W., Wang, C., Wang, X., Wise, S.M.: Positivity-preserving, energy stable numerical schemes for the Cahn–Hilliard equation with logarithmic potential. J. Comput. Phys. X 3, 100031 (2019)

    MathSciNet  Google Scholar 

  15. Chen, C., Yang, X.: Fast, provably unconditionally energy stable, and second-order accurate algorithms for the anisotropic Cahn–Hilliard model. Comput. Methods Appl. Mech. Eng. 351, 35–59 (2019)

    MathSciNet  MATH  Google Scholar 

  16. Cheng, Q., Liu, C., Shen, J.: A new Lagrange multiplier approach for gradient flows. Comput. Methods Appl. Mech. Eng. 367, 113070 (2020)

    MathSciNet  MATH  Google Scholar 

  17. Cheng, Q., Shen, J.: A new Lagrange multiplier approach for constructing structure preserving schemes, I. Positivity preserving. Comput. Methods Appl. Mech. Eng. 391, 114585 (2022)

    MathSciNet  MATH  Google Scholar 

  18. Cheng, Q., Shen, J.: A new Lagrange multiplier approach for constructing structure preserving schemes, II. Bound preserving. SIAM J. Numer. Anal. 60, 970–998 (2022)

    MathSciNet  MATH  Google Scholar 

  19. Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. J. Symb. Comput. 9, 251–280 (1990)

    MathSciNet  MATH  Google Scholar 

  20. Cox, S.M., Matthews, P.C.: Exponential time differencing for stiff systems. J. Comput. Phys. 176, 430–455 (2002)

    MathSciNet  MATH  Google Scholar 

  21. Dai, S., Du, Q.: Weak solutions for the Cahn–Hilliard equation with degenerate mobility. Arch. Ration. Mech. Anal. 219, 1161–1184 (2016)

    MathSciNet  MATH  Google Scholar 

  22. Dong, L., Wang, C., Zhang, H., Zhang, Z.: A positivity-preserving, energy stable and convergent numerical scheme for the Cahn–Hilliard equation with a Flory–Huggins–deGennes energy. Commun. Math. Sci. 17, 921–939 (2019)

    MathSciNet  MATH  Google Scholar 

  23. Du, Q., Ju, L., Li, X., Qiao, Z.: Maximum bound principles for a class of semilinear parabolic equations and exponential time-differencing schemes. SIAM Rev. 63, 317–359 (2021)

    MathSciNet  MATH  Google Scholar 

  24. Du, Q., Nicolaides, R.A.: Numerical analysis of a continuum model of phase transition. SIAM J. Numer. Anal. 28, 1310–1322 (1991)

    MathSciNet  MATH  Google Scholar 

  25. Elliott, C.M., Garcke, H.: On the Cahn–Hilliard equation with degenerate mobility. SIAM J. Math. Anal. 27, 404–423 (1996)

    MathSciNet  MATH  Google Scholar 

  26. Eyre, D.J.: Unconditionally gradient stable time marching the Cahn–Hilliard equation. Mater. Res. Soc. Symp. Proc. 529, 39–46 (1998)

    MathSciNet  Google Scholar 

  27. Fu, Z., Yang, J.: Energy-decreasing exponential time differencing Runge–Kutta methods for phase-field models. J. Comput. Phys. 454, 110943 (2022)

    MathSciNet  MATH  Google Scholar 

  28. Garcke, H., Lam, K.F., Nürnberg, R., Sitka, E.: A multiphase Cahn–Hilliard–Darcy model for tumour growth with necrosis. Math. Methods Appl. Sci. 28, 525–577 (2018)

    MathSciNet  MATH  Google Scholar 

  29. Gong, Y., Hong, Q., Wang, Q.: Supplementary variable method for thermodynamically consistent partial differential equations. Comput. Methods Appl. Mech. Eng. 381, 113746 (2021)

    MathSciNet  MATH  Google Scholar 

  30. Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43(1), 89–112 (2001)

    MathSciNet  MATH  Google Scholar 

  31. Hong, Q., Li, J., Wang, Q.: Supplementary variable method for structure-preserving approximations to partial differential equations with deduced equations. Appl. Math. Lett. 110, 106576 (2020)

    MathSciNet  MATH  Google Scholar 

  32. Huang, Q.-A., Jiang, W., Yang, J.Z.: An unconditionally energy stable scheme for simulating wrinkling phenomena of elastic thin films on a compliant substrate. J. Comput. Phys. 388, 123–143 (2019)

    MathSciNet  MATH  Google Scholar 

  33. Huang, Q.-A., Jiang, W., Yang, J.Z.: An efficient and unconditionally energy stable scheme for simulating solid-state dewetting of thin films with isotropic surface energy. Commun. Comput. Phys. 26, 1444–1470 (2019)

    MathSciNet  MATH  Google Scholar 

  34. Huang, Q.-A., Jiang, W., Yang, J.Z., Zhang, G.: Linear multi-step methods and their numerical stability for solving gradient flow equations. Adv. Comput. Math. 49, 39 (2023)

    MathSciNet  MATH  Google Scholar 

  35. Huang, F., Shen, J., Wu, K.: Bound/positivity preserving and unconditionally stable schemes for a class of fourth order nonlinear equations. J. Comput. Phys. 460, 111177 (2022)

    MathSciNet  MATH  Google Scholar 

  36. Huang, F., Shen, J., Yang, Z.: A highly efficient and accurate new scalar auxiliary variable approach for gradient flows. SIAM J. Sci. Comput. 42, A2514–A2536 (2020)

    MathSciNet  MATH  Google Scholar 

  37. Huang, J., Yang, C., Wei, Y.: Parallel energy-stable solver for a coupled Allen–Cahn and Cahn–Hilliard system. SIAM J. Sci. Comput. 42, C294–C312 (2020)

    MathSciNet  MATH  Google Scholar 

  38. Ipocoana, E.: On a non-isothermal Cahn–Hilliard model for tumor growth. J. Math. Anal. Appl. 506, 125665 (2022)

    MathSciNet  MATH  Google Scholar 

  39. Jiang, W., Bao, W., Thompson, C.V., Srolovitz, D.J.: Phase field approach for simulating solid-state dewetting problems. Acta Mater. 60, 5578–5592 (2012)

    Google Scholar 

  40. Jiang, M., Zhang, Z., Zhao, J.: Improving the accuracy and consistency of the scalar auxiliary variable (SAV) method with relaxation. J. Comput. Phys. 456, 110954 (2022)

    MathSciNet  MATH  Google Scholar 

  41. Jiang, W., Zhao, Q.: Sharp-interface approach for simulating solid-state dewetting in two dimensions: a Cahn–Hoffman \({\varvec {\xi }}\)-vector formulation. Physica D 390, 69–83 (2019)

    MathSciNet  MATH  Google Scholar 

  42. Ju, L., Li, X., Qiao, Z.: Stabilized exponential-SAV schemes preserving energy dissipation law and maximum bound principle for the Allen–Cahn type equations. J. Sci. Comput. 92, 66 (2022)

    MathSciNet  MATH  Google Scholar 

  43. Ju, L., Li, X., Qiao, Z.: Generalized SAV-exponential integrator schemes for Allen–Cahn type gradient flows. SIAM J. Numer. Anal. 60, 1905–1931 (2022)

    MathSciNet  MATH  Google Scholar 

  44. Jüngel, A.: A positivity-preserving numerical scheme for a nonlinear fourth order parabolic system. SIAM J. Numer. Anal. 39, 385–406 (2001)

    MathSciNet  MATH  Google Scholar 

  45. Lee, A.A., Münch, A., Süli, E.: Sharp-interface limits of the Cahn–Hilliard equation with degenerate mobility. SIAM J. Appl. Math. 76, 433–456 (2016)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  47. Li, B., Yang, J., Zhou, Z.: Arbitrarily high-order exponential cut-off methods for preserving maximum principle of parabolic equations. SIAM J. Sci. Comput. 42, A3957–A3978 (2020)

    MathSciNet  MATH  Google Scholar 

  48. Liao, H.-L., Tang, T., Zhou, T.: On energy stable, maximum-principle preserving, second-order BDF scheme with variable steps for the Allen–Cahn equation. SIAM J. Numer. Anal. 58, 2294–2314 (2020)

    MathSciNet  MATH  Google Scholar 

  49. Liu, Z., Li, X.: The exponential scalar auxiliary variable (E-SAV) approach for phase field models and its explicit computing. SIAM J. Sci. Comput. 42, B630–B655 (2020)

    MathSciNet  MATH  Google Scholar 

  50. Lu, C., Huang, W., Vleck, E.S.V.: The cutoff method for the numerical computation of nonnegative solutions of parabolic PDEs with application to anisotropic diffusion and Lubrication-type equations. J. Comput. Phys. 242, 24–36 (2013)

    MathSciNet  MATH  Google Scholar 

  51. Müller, G., Schwahn, D., Eckerlebe, H., Rieger, J., Springer, T.: Deviation of early stage of spinodal decomposition from the Cahn–Hilliard–Cook theory observed in an isotopic polymer blend. Physica B 234–236, 245–246 (1997)

    Google Scholar 

  52. Mullins, W.W., Sekerka, R.F.: Morphological stability of a particle growing by diffusion or heat flow. J. Appl. Phys. 34, 323–329 (1963)

    Google Scholar 

  53. Park, J.-H., Salgado, A.J., Wise, S.M.: Benchmark computations of the phase field crystal and functionalized Cahn–Hilliard equations via fully implicit. Nesterov accelerated schemes. Commun. Comput. Phys. 33, 367–398 (2023)

    MathSciNet  MATH  Google Scholar 

  54. Pego, R.L.: Front migration in the nonlinear Cahn–Hilliard equation. Proc. R. Soc. Lond. A 422, 261–278 (1989)

    MathSciNet  MATH  Google Scholar 

  55. Pesce, C., Muench, A.: How do degenerate mobilities determine singularity formation in Cahn–Hilliard equations? Multiscale Model. Simul. 19, 1143–1166 (2021)

    MathSciNet  MATH  Google Scholar 

  56. Shen, J., Xu, J., Yang, J.: The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys. 353, 407–416 (2018)

    MathSciNet  MATH  Google Scholar 

  57. Shen, J., Xu, J., Yang, J.: A new class of efficient and robust energy stable scheme for gradient flows. SIAM Rev. 61, 474–506 (2019)

    MathSciNet  MATH  Google Scholar 

  58. Strassen, V.: Gaussian elimination is not optimal. Numer. Math. 13, 354–356 (1969)

    MathSciNet  MATH  Google Scholar 

  59. Tang, T., Yang, J.: Implicit-explicit scheme for the Allen–Cahn equation preserves the maximum principle. J. Comput. Math. 34, 451–461 (2016)

    MathSciNet  MATH  Google Scholar 

  60. Tang, T., Yu, H., Zhou, T.: On energy dissipation theory and numerical stability for time-fractional phase-field equations. SIAM J. Sci. Comput. 41, A3757–A3778 (2019)

    MathSciNet  MATH  Google Scholar 

  61. Yan, J., Shu, C.-W.: A local discontinuous Galerkin method for KdV type equations. SIAM J. Numer. Anal. 40(2), 769–791 (2002)

    MathSciNet  MATH  Google Scholar 

  62. Yang, J., Yuan, Z., Zhou, Z.: Arbitrarily high-order maximum bound preserving schemes with cut-off postprocessing for Allen–Cahn equations. J. Sci. Comput. 90, 76 (2022)

    MathSciNet  MATH  Google Scholar 

  63. Yang, X., Zhao, J.: On linear and unconditionally energy stable algorithms for variable mobility Cahn–Hilliard type equation with logarithmic Flory–Huggins potential. Commun. Comput. Phys. 25, 703–728 (2019)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The numerical calculations in this paper have been done on the supercomputing system in the Supercomputing Center of Wuhan University.

Funding

This work was partially supported by the National Natural Science Foundation of China (Nos. 12001210, 12131010, 12271414, 11871384, 12125103, 12071362, 12301558).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Henan University, Kaifeng, 475004, China

    Qiong-Ao Huang

  2. Center for Applied Mathematics of Henan Province, Henan University, Zhengzhou, 450046, China

    Qiong-Ao Huang

  3. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Wei Jiang, Jerry Zhijian Yang & Cheng Yuan

  4. Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan, 430072, China

    Wei Jiang & Jerry Zhijian Yang

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  1. Qiong-Ao Huang
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Correspondence to Cheng Yuan.

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Huang, QA., Jiang, W., Yang, J.Z. et al. A Structure-Preserving, Upwind-SAV Scheme for the Degenerate Cahn–Hilliard Equation with Applications to Simulating Surface Diffusion. J Sci Comput 97, 64 (2023). https://doi.org/10.1007/s10915-023-02380-6

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  • DOI: https://doi.org/10.1007/s10915-023-02380-6

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