Skip to main content
Springer Nature Link
Log in

Unconditionally Maximum Bound Principle Preserving Linear Schemes for the Conservative Allen–Cahn Equation with Nonlocal Constraint

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In comparison with the Cahn–Hilliard equation, the classic Allen-Cahn equation satisfies the maximum bound principle (MBP) but fails to conserve the mass along the time. In this paper, we consider the MBP and corresponding numerical schemes for the modified Allen–Cahn equation, which is formed by introducing a nonlocal Lagrange multiplier term to enforce the mass conservation. We first study sufficient conditions on the nonlinear potentials under which the MBP holds and provide some concrete examples of nonlinear functions. Then we propose first and second order stabilized exponential time differencing schemes for time integration, which are linear schemes and unconditionally preserve the MBP in the time discrete level. Convergence of these schemes is analyzed as well as their energy stability. Various two and three dimensional numerical experiments are also carried out to validate the theoretical results and demonstrate the performance of the proposed schemes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

Similar content being viewed by others

References

  1. Allen, S.M., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metallurgica 27(6), 1085–1095 (1979)

    Article  Google Scholar 

  2. Beylkin, G., Keiser, J.M., Vozovoi, L.: A new class of time discretization schemes for the solution of nonlinear PDEs. J. Comput. Phys. 147(2), 362–387 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  4. 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 

  5. Chen, X., Hilhorst, D., Logak, E.: Mass conserving Allen–Cahn equation and volume preserving mean curvature flow. Interf. Free Boundaries 12, 527–549 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Copetti, M.I.M., Elliott, C.M.: Numerical analysis of the Cahn–Hilliard equation with a logarithmic free energy. Numerische Mathematik 63(1), 39–65 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cox, S.M., Matthews, P.C.: Exponential time differencing for stiff systems. J. Comput. Phys. 176(2), 430–455 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Debussche, A., Dettori, L.: On the Cahn–Hilliard equation with a logarithmic free energy. Nonlinear Anal. Theory Methods Appl. 24(10), 1491–1514 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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(4), 921–939 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  10. Du Q.: Nonlocal modeling, analysis, and computation. CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM (2019)

  11. Du, Q., Ju, L., Li, X., Qiao, Z.: Maximum principle preserving exponential time differencing schemes for the nonlocal Allen–Cahn equation. SIAM J. Numer. Anal. 57(2), 875–898 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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)

  13. Feng, X., Prohl, A.: Numerical analysis of the Allen–Cahn equation and approximation for mean curvature flows. Numerische Mathematik 94(1), 33–65 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fernandez-Real, X., Ros-Oton, X.: Boundary regularity for the fractional heat equation. Revista de la Real Academia de Ciencias Exactas, Fisicasy Naturales. Serie A Matematicas 110(1), 49–64 (2016)

    MathSciNet  Google Scholar 

  15. Huang, J., Ju, L., Wu, B.: A fast compact time integrator method for a family of general order semilinear evolution equations. J. Comput.Phys. 393, 313–336 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hochbruck, M., Ostermann, A.: Explicit exponential Runge–Kutta methods for semilinear parabolic problems. SIAM J. Numer. Anal. 43(3), 1069–1090 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lee, D., Kim, J.: Comparison study of the conservative Allen–Cahn and the Cahn–Hilliard equations. Math. Comput. Simul. 119, 35–56 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lee, H.G.: High-order and mass conservative methods for the conservative Allen–Cahn equation. Comput. Math. Appl. 72(3), 620–631 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Li, J., Li, X., Ju, L., Feng, X.: Stabilized integrating factor Runge–Kutta method and unconditional preservation of maximum bound principle. SIAM J. Sci. Comput. (2021). https://doi.org/10.1137/20M1340678

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

    Article  MathSciNet  Google Scholar 

  21. Liao, H., Tang, T., Zhou, T.: A second-order and nonuniform time-stepping maximum-principle preserving scheme for time-fractional Allen-Cahn equations. J. Comput. Phys. 414, 109473 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ju, L., Li, X., Qiao, Z., Zhang, H.: Energy stability and error estimates of exponential time differencing schemes for the epitaxial growth model without slope selection. Math. Comput. 87(312), 1859–1885 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ju, L., Li, X., Qiao, Z., Yang, J.: Maximum bound principle preserving integrating factor Runge-Kutta methods for semilinear parabolic equations. J. Comput. Phys. (2021). https://doi.org/10.1016/j.jcp.2021.110405

  24. Ju, L., Zhang, J., Du, Q.: Fast and accurate algorithms for simulating coarsening dynamics of Cahn–Hilliard equations. Comput. Mater. Sci. 108, 272–282 (2015)

    Article  Google Scholar 

  25. Ju, L., Zhang, J., Zhu, L.: Fast explicit integration factor methods for semilinear parabolic equations. J. Sci. Comput. 62(2), 431–455 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

  27. Peng, G., Gao, Z., Yan, W., Feng, X.: A positivity-preserving nonlinear finite volume scheme for radionuclide transport calculations in geological radioactive waste repository. Int. J. Numer. Methods Heat Fluid Flow 30(2), 516–534 (2019)

    Article  Google Scholar 

  28. Peng, G., Gao, Z., Feng, X.: A stabilized extremum-preserving scheme for nonlinear parabolic equation on polygonal meshes. Int. J. Numer. Methods Fluids 90(7), 340–356 (2019)

    Article  MathSciNet  Google Scholar 

  29. Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Springer, New York (1984)

    Book  MATH  Google Scholar 

  30. Qian, Y., Wang, C., Zhou, S.: A positive and energy stable numerical scheme for the Poisson-Nernst-Planck-Cahn-Hilliard equations with steric interactions. J. Comput. Phys. 426, 109908 (2021)

    Article  MathSciNet  Google Scholar 

  31. Rubinstein, J., Sternberg, P.: Nonlocal reaction-diffusion equations and nucleation. IMA J. Appl. Math. 48(3), 249–264 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  32. Shen, J., Tang, T., Yang, J.: On the maximum principle preserving schemes for the generalized Allen–Cahn equation. Commun. Math. Sci. 14(6), 1517–1534 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  33. Shen, J., Xu, J.: Unconditionally bound preserving and energy dissipative schemes for a class of Keller–Segel equations. SIAM J. Numer. Anal. 58(3), 1674–1695 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  34. Shen, J., Xu, J., Yang, J.: A new class of efficient and robust energy stable schemes for gradient flows. SIAM Rev. 61(3), 474–506 (2019)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  36. Shin, J., Park, S.K., Kim, J.: A hybrid FEM for solving the Allen–Cahn equation. Appl. Math. Comput. 244, 606–612 (2014)

    MathSciNet  MATH  Google Scholar 

  37. Tang, T., Yang, J.: Implicit-explicit scheme for the Allen–Cahn equation preserves the maximum principle. J. Comput. Math. 34(5), 471–481 (2016)

    MathSciNet  MATH  Google Scholar 

  38. Xiao, X., Feng, X., Yuan, J.: The stabilized semi-implicit finite element method for the surface Allen-Cahn equation. Discrete Contin. Dyn. Syst. B 22(7), 2857 (2017)

    MathSciNet  MATH  Google Scholar 

  39. Xiao, X., Feng, X., He, Y.: Numerical simulations for the chemotaxis models on surfaces via a novel characteristic finite element method. Comput. Math. Appl. 78(1), 20–34 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  40. Xiao, X., He, R., Feng, X.: Unconditionally maximum principle preserving finite element schemes for the surface Allen–Cahn type equations. Numer. Methods Partial Differ. Equ. 36(2), 418–438 (2020)

    Article  MathSciNet  Google Scholar 

  41. Xiao, X., Dai, Z., Feng, X.: A positivity preserving characteristic finite element method for solving the transport and convection-diffusion-reaction equations on general surfaces. Comput. Phys. Commun. 247, 106941 (2020)

    Article  MathSciNet  Google Scholar 

  42. Xu, C., Tang, T.: Stability analysis of large time-stepping methods for epitaxial growth models. SIAM J. Numer. Anal. 44(4), 1759–1779 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  43. Yang, J., Du, Q., Zhang, W.: Uniform \(L^p\)-bound of the Allen-Cahn equation and its numerical discretization. Int. J. Numer. Anal. Model. 15, 213–227 (2018)

    MathSciNet  MATH  Google Scholar 

  44. Yang X, Zhang G. Numerical approximations of the Cahn–Hilliard and Allen–Cahn equations with general nonlinear potential using the Invariant Energy Quadratization approach. arXiv preprint arXiv:1712.02760 (2017)

  45. Zhai, S., Weng, Z., Feng, X.: Investigations on several numerical methods for the nonlocal Allen-Cahn equation. Int. J. Heat Mass Transf. 87, 111–118 (2015)

    Article  Google Scholar 

  46. Zhai, S., Weng, Z., Feng, X.: Fast explicit operator splitting method and time-step adaptivity for fractional nonlocal Allen–Cahn model. Appl. Math. Model. 40(2), 1315–1324 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  47. Zhang, J., Yang, X.: Numerical approximations for a new \(L^2\)-gradient flow based phase field crystal model with precise nonlocal mass conservation. Comput. Phys. Commun. 243, 51–67 (2019)

  48. Zhang, J., Yang, X.: 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 (2020)

    Article  MATH  Google Scholar 

  49. Zhang, J., Chen, C., Yang, X., Chu, Y., Xia, Y.: Efficient, non-iterative, and second-order accurate numerical algorithms for the anisotropic Allen–Cahn Equation with precise nonlocal mass conservation. J. Comput. Appl. Math. 363, 444–463 (2020)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratory of Mathematics and Complex Systems and School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    Jingwei Li & Yongyong Cai

  2. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    Lili Ju

  3. College of Mathematics and System Science, Xinjiang University, Urumqi, 830046, China

    Xinlong Feng

Authors
  1. Jingwei Li
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Lili Ju
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Yongyong Cai
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Xinlong Feng
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Lili Ju.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Jingwei Li’ work was partially supported by National Natural Science Foundation of China grant 61962056. Lili Ju’s work was partially supported by US National Science Foundation Grant DMS-1818438 and US Department of Energy grant DE-SC0020270. Yongyong Cai’s work was partially supported by National Natural Science Foundation of China Grants 11771036 and 91630204. Xinlong Feng’s work was partially supported by Research Fund from Key Laboratory of Xinjiang Province Grant 2020D04002 and National Natural Science Foundation of China Grants U19A2079 and 12071406.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, J., Ju, L., Cai, Y. et al. Unconditionally Maximum Bound Principle Preserving Linear Schemes for the Conservative Allen–Cahn Equation with Nonlocal Constraint. J Sci Comput 87, 98 (2021). https://doi.org/10.1007/s10915-021-01512-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-021-01512-0

Keywords

Mathematics Subject Classification