Japan Journal of Industrial and Applied Mathematics

, Volume 35, Issue 3, pp 1245–1281 | Cite as

A stable and structure-preserving scheme for a non-local Allen–Cahn equation

  • Makoto OkumuraEmail author
Original Paper Area 2


We propose a stable and structure-preserving finite difference scheme for a non-local Allen–Cahn equation which describes a process of phase separation in a binary mixture. The proposed scheme inherits characteristic properties, the conservation of mass and the decrease of the global energy from the equation. We show the stability and unique existence of the solution of the scheme. We also prove the error estimate for the scheme. Numerical experiments demonstrate the effectiveness of the proposed scheme.


Non-local Allen–Cahn equation Discrete variational derivative method 

Mathematics Subject Classification




I thank the reviewer for helpful and attentive comments that helped me to improve this manuscript. I also thank Prof. D. Furihata of Osaka University, Prof. N. Yamazaki of Kanagawa University, Prof. T. Fukao of Kyoto University of Education and Assoc. Prof. K. Takasao of Kyoto University for helpful advice and discussions.


  1. 1.
    Allen, S.M., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1085–1095 (1979)CrossRefGoogle Scholar
  2. 2.
    Bao, W.: Approximation and comparison for motion by mean curvature with intersection points. Comput. Math. Appl. 46, 1211–1228 (2003)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Beneš, M.: Diffuse-interface treatment of the anisotropic mean-curvature flow. Appl. Math. 48, 437–453 (2003)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Beneš, M., Chalupecky, V., Mikula, K.: Geometrical image segmentation by the Allen–Cahn equation. Appl. Numer. Math. 51, 187–205 (2004)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Beneš, M., Yazaki, S., Kimura, M.: Computational studies of non-local anisotropic Allen–Cahn equation. Math. Bohem. 136, 429–437 (2011)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Brassel, M., Bretin, E.: A modified phase field approximation for mean curvature flow with conservation of the volume. Math. Methods Appl. Sci. 34, 1157–1180 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bronsard, L., Stoth, B.: Volume-preserving mean curvature flow as a limit of a nonlocal Ginzburg–Landau equation. SIAM J. Math. Anal. 28, 769–807 (1997)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chen, L.Q.: Phase-field models for microstructure evolution. Annu. Rev. Mater. Res. 32, 113–140 (2002)CrossRefGoogle Scholar
  9. 9.
    Chen, X., Hilhorst, D., Logak, E.: Mass conserving Allen–Cahn equation and volume preserving mean curvature flow. Interface Free Bound. 12, 527–549 (2010)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Conti, M., Meerson, B., Peleg, A., Sasorov, P.V.: Phase ordering with a global conservation law: Ostwald ripening and coalescence. Phys. Rev. E 65, 046117 (2002)CrossRefGoogle Scholar
  11. 11.
    Dobrosotskaya, J.A., Bertozzi, A.L.: A wavelet-Laplace variational technique for image deconvolution and inpainting. IEEE Trans. Image Process. 17, 657–663 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Evans, L.C., Soner, H.M., Souganidis, P.E.: Phase transitions and generalized motion by mean curvature. Commun. Pure Appl. Math. 45, 1097–1123 (1992)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Feng, X., Prohl, A.: Numerical analysis of the Allen–Cahn equation and approximation for mean curvature flows. Numer. Math. 94, 33–65 (2003)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Golovaty, D.: The volume-preserving motion by mean curvature as an asymptotic limit of reaction-diffusion equations. Q. Appl. Math. 55, 243–298 (1997)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Furihata, D.: A stable and conservative finite difference scheme for the Cahn–Hilliard equation. Numer. Math. 87, 675–699 (2001)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Furihata, D., Matsuo, T.: Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations. CRC Press, Boca Raton (2010)CrossRefGoogle Scholar
  17. 17.
    Ilmanen, T.: Convergence of the Allen–Cahn equation to Brakke’s motion by mean curvature. J. Differ. Geom. 38, 417–461 (1993)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Katsoulakis, M., Kossioris, G.T., Reitich, F.: Generalized motion by mean curvature with Neumann conditions and the Allen–Cahn model for phase transitions. J. Geom. Anal. 5, 255–279 (1995)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kim, J., Lee, S., Choi, Y.: A conservative Allen–Cahn equation with a space-time dependent Lagrange multiplier. Int. J. Eng. Sci. 84, 11–17 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Lee, H.G.: High-order and mass conservative methods for the conservative Allen–Cahn equation. Comput. Math. Appl. 72, 620–631 (2016)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Li, Y., Kim, J.: Multiphase image segmentation using a phase-field model. Comput. Math. Appl. 62, 737–745 (2011)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ohtsuka, T.: Motion of interfaces by an Allen–Cahn type equation with multiple-well potentials. Asymptot. Anal. 56, 87–123 (2008)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Rubinstein, J., Sternberg, P.: Nonlocal reaction-diffusion equations and nucleation. IMA J. Appl. Math. 48, 249–264 (1992)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Stafford, D., Ward, M.J., Wetton, B.: The dynamics of drops and attached interfaces for the constrained Allen–Cahn equation. Eur. J. Appl. Math. 12, 1–24 (2001)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Takasao, K.: Existence of weak solution for volume preserving mean curvature flow via phase field method, pp. 1–16 (2015). arXiv:1511.01687 [math.AP]
  26. 26.
    Ward, M.J.: Metastable bubble solutions for the Allen–Cahn equation with mass conservation. SIAM J. Appl. Math. 56, 1247–1279 (1996)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Zhai, S., Weng, Z., Feng, X.: Investigations on several numerical methods for the non-local Allen–Cahn equation. Int. J. Heat Mass Transfer 87, 111–118 (2015)CrossRefGoogle Scholar

Copyright information

© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Pure and Applied Mathematics, Graduate School of Information Science and TechnologyOsaka UniversitySuitaJapan

Personalised recommendations