Skip to main content
SpringerLink
Log in

Time-Fractional Allen–Cahn Equations: Analysis and Numerical Methods

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

Abstract

In this work, we consider a time-fractional Allen–Cahn equation, where the conventional first order time derivative is replaced by a Caputo fractional derivative with order \(\alpha \in (0,1)\). First, the well-posedness and (limited) smoothing property are studied, by using the maximal \(L^p\) regularity of fractional evolution equations and the fractional Grönwall’s inequality. We also show the maximum principle like their conventional local-in-time counterpart, that is, the time-fractional equation preserves the property that the solution only takes value between the wells of the double-well potential when the initial data does the same. Second, after discretizing the fractional derivative by backward Euler convolution quadrature, we develop several unconditionally solvable and stable time stepping schemes, such as a convex splitting scheme, a weighted convex splitting scheme and a linear weighted stabilized scheme. Meanwhile, we study the discrete energy dissipation property (in a weighted average sense), which is important for gradient flow type models, for the two weighted schemes. In addition, we prove the fractional energy dissipation law for the gradient flow associated with a convex free energy. Finally, using a discrete version of fractional Grönwall’s inequality and maximal \(\ell ^p\) regularity, we prove that the convergence rates of those time-stepping schemes are \(O(\tau ^\alpha )\) without any extra regularity assumption on the solution. We also present extensive numerical results to support our theoretical findings and to offer new insight on the time-fractional Allen–Cahn dynamics.

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

Access this article

Log in via an institution

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

Similar content being viewed by others

References

  1. Ainthworth, M., Mao, Z.: Analysis and approximation of a fractional Cahn–Hilliard equation. SIAM J. Numer. Anal. 55(4), 1689–1718 (2017)

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  3. Akagi, Goro, Schimperna, Giulio, Segatti, Antonio: Fractional Cahn–Hilliard, Allen–Cahn and porous medium equations. J. Diff. Equ. 261(6), 2935–2985 (2016)

    MathSciNet  MATH  Google Scholar 

  4. Anderson, D.M., McFadden, G.B., Wheeler, A.A.: Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30(1), 139–165 (1998)

    MathSciNet  MATH  Google Scholar 

  5. Antil, Harbir, Bartels, Sören: Spectral approximation of fractional PDEs in image processing and phase field modeling. Comput. Methods Appl. Math. 17(4), 661–678 (2017)

    MathSciNet  MATH  Google Scholar 

  6. Bates, P.W.: On some nonlocal evolution equations arising in materials science. Nonlinear Dyn. Evol. Equ. 48, 13–52 (2006)

    MathSciNet  MATH  Google Scholar 

  7. Bates, W.P., Sarah, B., Jianlong, H.: Numerical analysis for a nonlocal Allen–Cahn equation. Int. J. Numer. Anal. Model 6(1), 33–49 (2009)

    MathSciNet  MATH  Google Scholar 

  8. Caffarelli, L., Roquejoffre, J.-M., Savin, O.: Nonlocal minimal surfaces. Commun. Pure Appl. Math. 63(9), 1111–1144 (2010)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  10. Clarke, S., Vvedensky, D.D.: Origin of reflection high-energy electron–diffraction intensity oscillations during molecular-beam epitaxy: A computational modeling approach. Phys. Rev. Lett. 58(21), 2235 (1987)

    Google Scholar 

  11. Chechkin, A.V., Gorenflo, R., Sokolov, I.M.: Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Phys. Rev. E 66, 046129 (2002)

    Google Scholar 

  12. Chen, L., Zhao, J., Cao, W., Wang, H., Zhang, J.: An accurate and efficient algorithm for the time-fractional molecular beam epitaxy model with slope selection. Comput. Phys. Commun. 245, 106842 (2019)

    MathSciNet  Google Scholar 

  13. Du, Q.: Nonlocal Modeling, Analysis and Computation. In: CBMS-NSF regional conference series, vol. 94. SIAM, (2019)

  14. Du, Q., Feng, X.-B.: The phase field method for geometric moving interfaces and their numerical approximations. arXiv preprint arXiv:1902.04924 (2019)

  15. Du, Q., Ju, L., Li, X., Qiao, Z.: Stabilized linear semi-implicit schemes for the nonlocal Cahn–Hilliard equation. J. Comput. Phys. 363, 39–54 (2018)

    MathSciNet  MATH  Google Scholar 

  16. 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, 875–898 (2019)

    MathSciNet  MATH  Google Scholar 

  17. Du, Q., Ju, L., Li, X., Qiao, Z.: Maximum bound principles for a class of semilinear parabolic equations and exponential time differencing schemes. SIAM Review (2020). arXiv preprint arXiv:2005.11465

  18. Du, Q., Liu, C., Wang, X.: A phase field approach in the numerical study of the elastic bending energy for vesicle membranes. J. Comput. Phys. 198(2), 450–468 (2004)

    MathSciNet  MATH  Google Scholar 

  19. Du, Q., Yang, J., Zhou, Z.: Analysis of a nonlocal-in-time parabolic equation. Discrete Contin. Dyn. Syst. Ser. B 22(2), 339–368 (2017)

    MathSciNet  MATH  Google Scholar 

  20. Du, Q., Yang, J.: Asymptotically compatible Fourier spectral approximations of nonlocal Allen–Cahn equations. SIAM J. Numer. Anal. 54(3), 1899–1919 (2016)

    MathSciNet  MATH  Google Scholar 

  21. Golding, I., Cox, E.C.: Physical nature of bacterial cytoplasm. Phys. Rev. Lett. 96(9), 098102 (2006)

    Google Scholar 

  22. Guan, Z., Lowengrub, J., Wang, C., Wise, S.: Second order convex splitting schemes for periodic nonlocal Cahn–Hilliard and Allen–Cahn equations. J. Comput. Phys. 277, 48–71 (2014)

    MathSciNet  MATH  Google Scholar 

  23. Gui, C., Zhao, M.: Traveling wave solutions of Allen–Cahn equation with a fractional laplacian. Annales de l’Institut Henri Poincare C Non Linear Anal. 32(4), 785–812 (2015)

    MathSciNet  MATH  Google Scholar 

  24. Lev Davidovich Landau and Evgenii Mikhailovich Lifshitz: Course of Theoretical Physics. Elsevier, Amsterdam (2013)

    Google Scholar 

  25. Jin, B., Lazarov, R., Zhou, Z.: Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data. SIAM J. Sci. Comput. 38(1), A146–A170 (2016)

    MathSciNet  MATH  Google Scholar 

  26. Jin, B., Li, B., Zhou, Z.: Discrete maximal regularity of time-stepping schemes for fractional evolution equations. Numer. Math. 138(1), 101–131 (2018)

    MathSciNet  MATH  Google Scholar 

  27. Jin, B., Li, B., Zhou, Z.: Numerical analysis of nonlinear subdiffusion equations. SIAM J. Numer. Anal. 56(1), 1–23 (2018)

    MathSciNet  MATH  Google Scholar 

  28. Jin, B., Li, B., Zhou, Z.: Subdiffusion with a time-dependent coefficient: analysis and numerical solution. Math. Comput. 88(319), 2157–2186 (2019)

    MathSciNet  MATH  Google Scholar 

  29. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  30. Kirchner, J.W., Feng, X., Neal, C.: Fractal stream chemistry and its implications for contaminant transport in catchments. Nature 403(6769), 524 (2000)

    Google Scholar 

  31. Liu, H., Cheng, A., Wang, H., Zhao, J.: Time-fractional Allen–Cahn and Cahn–Hilliard phase-field models and their numerical investigation. Comput. Math. Appl., 76(8):1876–1892

  32. Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225(2), 1533–1552 (2007)

    MathSciNet  MATH  Google Scholar 

  33. Liu, C., Shen, J.: A phase field model for the mixture of two incompressible fluids and its approximation by a fourier-spectral method. Phys. D 179(3–4), 211–228 (2003)

    MathSciNet  MATH  Google Scholar 

  34. Lubich, C.: Convolution quadrature and discretized operational calculus I. Numer. Math. 52(2), 129–145 (1988)

    MathSciNet  MATH  Google Scholar 

  35. Mustapha, K., Abdallah, B., Furati, K.M.: A discontinuous Petrov–Galerkin method for time-fractional diffusion equations. SIAM J. Numer. Anal. 52(5), 2512–2529 (2014)

    MathSciNet  MATH  Google Scholar 

  36. Nigmatullin, R.: The realization of the generalized transfer equation in a medium with fractal geometry. Phys. Status Solidi (b) 133(1), 425–430 (1986)

    Google Scholar 

  37. Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)

    MATH  Google Scholar 

  38. Paragios, N. Mellina-Gottardo, O., Ramesh, V.: Gradient vector flow fast geodesic active contours. In: Proceedings Eighth IEEE International Conference on Computer Vision. ICCV 2001, volume 1, pages 67–73. IEEE (2001)

  39. Pismen, L.M.: Nonlocal diffuse interface theory of thin films and the moving contact line. Phys. Rev. E 64(2), 021603 (2001)

    Google Scholar 

  40. Qian, T., Wang, X.-P., Sheng, P.: Molecular scale contact line hydrodynamics of immiscible flows. Phys. Rev. E 68(1), 016306 (2003)

    Google Scholar 

  41. Sabzikar, F., Meerschaert, M.M., Chen, J.: Tempered fractional calculus. J. Comput. Phys. 293, 14–28 (2015)

    MathSciNet  MATH  Google Scholar 

  42. Shao, D., Rappel, W.J., Levine, H.: Computational model for cell morphodynamics. Phys. Rev. Lett. 105(10), 108104 (2010)

    Google Scholar 

  43. 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)

    MathSciNet  MATH  Google Scholar 

  44. 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)

    MathSciNet  MATH  Google Scholar 

  45. Song, F., Xu, C., Karniadakis, G.: A fractional phase-field model for two-phase flows with tunable sharpness: algorithms and simulations. Comput. Methods Appl. Mech. Eng. 305, 376–404 (2016)

    MathSciNet  MATH  Google Scholar 

  46. Sun, Z.-Z., Wu, X.: A fully discrete scheme for a diffusion wave system. Appl. Numer. Math. 56(2), 193–209 (2006)

    MathSciNet  MATH  Google Scholar 

  47. 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 

  48. Tang, T., Yu, H., Zhou, T.: On energy dissipation theory and numerical stability for time-fractional phase field equations. SIAM J. Sci. Comput. 41(6), A3757–A3778 (2019)

    MathSciNet  MATH  Google Scholar 

  49. Valdinoci, Enrico: A fractional framework for perimeters and phase transitions. Milan J. Math. 81(1), 1–23 (2013)

    MathSciNet  MATH  Google Scholar 

  50. Van der Waals, J.D.: Thermodynamic theory of capillarity assuming steady density change. J. Phys. Chem. 13(1), 657–725 (1894)

    Google Scholar 

  51. Wise, S.M., Lowengrub, J.S., Frieboes, H.B., Cristini, V.: Three-dimensional multispecies nonlinear tumor growth I: model and numerical method. J. Theor. Biol. 253(3), 524–543 (2008)

    MathSciNet  MATH  Google Scholar 

  52. Xu, C., Prince, J.L.: Snakes, shapes, and gradient vector flow. IEEE Trans. Image Process. 7(3), 359–369 (1998)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY, 10027, USA

    Qiang Du

  2. Department of Mathematics, SUSTech International Center for Mathematics, Southern University of Science and Technology, Shenzhen, China

    Jiang Yang

  3. Guangdong Provincial Key Laboratory of Computational Science and Material Design, Southern University of Science and Technology, Shenzhen, China

    Jiang Yang

  4. Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong

    Zhi Zhou

Authors
  1. Qiang Du
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jiang Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhi Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiang Yang.

Additional information

Publisher's Note

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

The work of Q. Du is supported in part by NSF DMS-2012562 and ARO MURI Grant W911NF-15-1-0562. The work of J. Yang is supported by National Natural Science Foundation of China (NSFC) Grant No. 11871264, the Guangdong Basic and Applied Basic Research Foundation (2018A0303130123) and and the fund of the Guangdong Provincial Key Laboratory of Computational Science and Material Design (No. 2019B030301001). The work of Z. Zhou is supported by the start-up Grant from the Hong Kong Polytechnic University and Hong Kong RGC Grant No. 25300818.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Du, Q., Yang, J. & Zhou, Z. Time-Fractional Allen–Cahn Equations: Analysis and Numerical Methods. J Sci Comput 85, 42 (2020). https://doi.org/10.1007/s10915-020-01351-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-020-01351-5

Keywords

Mathematics Subject Classification

Search

Navigation