Skip to main content
SpringerLink
Log in

Local Discontinuous Galerkin Scheme for Space Fractional Allen–Cahn Equation

  • Original Paper
  • Published:
Communications on Applied Mathematics and Computation Aims and scope Submit manuscript

Abstract

This paper is concerned with the efficient numerical solution for a space fractional Allen–Cahn (AC) equation. Based on the features of the fractional derivative, we design and analyze a semi-discrete local discontinuous Galerkin (LDG) scheme for the initial-boundary problem of the space fractional AC equation. We prove the optimal convergence rates of the semi-discrete LDG approximation for smooth solutions. Finally, we test the accuracy and efficiency of the designed numerical scheme on a uniform grid by three examples. Numerical simulations show that the space fractional AC equation displays abundant dynamical behaviors.

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
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)

    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, 1085–1095 (1979)

    Article  Google Scholar 

  3. Achleitner, F., Kuehn, Ch.: Analysis and numerics of traveling waves for asymmetric fractional reaction–diffusion equations. arXiv:1405.5779v1 [math.NA] 22 (May 2014)

  4. Baeumer, B., Kovacsa, M., Meerschaert, M.M.: Numerical solutions for fractional reaction–diffusion equations. Comput. Math. Appl. 55, 2212–2226 (2008)

    Article  MathSciNet  Google Scholar 

  5. Bueno-Orovio, A., Kay, D., Burrage, K.: Fourier spectral methods for fractional-in-space reaction–diffusion equations. BIT Numer. Math. 54, 937–954 (2014)

    Article  MathSciNet  Google Scholar 

  6. Cabré, X., Roquejoffre, J.-M.: The influence of fractional diffusion in Fisher-KPP equations. Commun. Math. Phys. 320, 679–722 (2013)

    Article  MathSciNet  Google Scholar 

  7. Castillo, P., Gómez, S.: On the conservation of fractional nonlinear Schrödinger equation’s invariants by the local discontinuous Galerkin method. J Sci Comput. 5, 1–24 (2018)

    MATH  Google Scholar 

  8. Cifani, S., Jakobsen, E.R., Karlsen, K.H.: The discontinuous Galerkin method for fractal conservation laws. IMA J. Numer. Anal. 31, 1090–1122 (2011)

    Article  MathSciNet  Google Scholar 

  9. Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection–diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)

    Article  MathSciNet  Google Scholar 

  10. Castillo, P., Cockburn, B., Schötzau, D., Schwab, C.: Optimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for convection–diffusion problems. Math. Comp. 71, 455–478 (2003)

    Article  MathSciNet  Google Scholar 

  11. Deng, W.H., Hesthaven, J.S.: Local discontinuous Galerkin methods for fractional diffusion equations. ESAIM Math. Model. Numer. Anal. 47, 1845–1864 (2013)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. Ervin, V.J., Roop, J.P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Diff. Equ. 22, 558–576 (2006)

    Article  MathSciNet  Google Scholar 

  14. Feng, X.B., Prohl, A.: Numerical analysis of the Allen–Cahn equation and approximation for mean curvature flows. Numer. Math. 94, 33–65 (2003)

    Article  MathSciNet  Google Scholar 

  15. Feng, X.L., Song, H., Tang, T., Yang, J.: Nonlinear stability of the implicit–explicit methods for the Allen–Cahn equation. Inverse Probl. Imaging 7, 679–695 (2013)

    Article  MathSciNet  Google Scholar 

  16. Guo, R.H., Ji, L.Y., Xu, Y.: High order local discontinuous Galerkin methods for the Allen–Cahn equation: analysis and simulation. J. Comput. Math. 34, 135–158 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. Hou, T., Tang, T., Yang, J.: Numerical analysis of fully discretized Crank–Nicolson scheme for fractional-in-space Allen–Cahn equations. J. Sci. Comput. 72, 1214–1231 (2017)

    Article  MathSciNet  Google Scholar 

  19. Ji, X., Tang, H.Z.: High-order accurate Runge–Kutta (local) discontinuous Galerkin methods for one-and two-dimensional fractional diffusion equations. Numer. Math. Theor. Meth. Appl. 5, 333–358 (2012)

    Article  MathSciNet  Google Scholar 

  20. Li, Z., Wang, H., Yang, D.P.: A space–time fractional phase-field model with tunable sharpness and decay behavior and its efficient numerical simulation. J. Comput. Phys. 347, 20–38 (2017)

    Article  MathSciNet  Google Scholar 

  21. Mao, Z.P., Kamiadakis, G.E.: Fractional Burgers equation with nonlinear non-locality: spectral vanishing viscosity and local discontinuous Galerkin methods. J. Comput. Phys. 336, 143–163 (2017)

    Article  MathSciNet  Google Scholar 

  22. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. Shen, J.: Modeling and numerical approximation of two-phase incompressible flows by a phase-field approach. In: Bao, W., Du, Q. (eds.) Multiscale Modeling and Analysis for Materials Simulation. Lecture Note Series, vol. 9, pp. 147–196. National University of Singapore, Singapore (2011)

    Chapter  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  26. Shu, C.-W.: High order WENO and DG methods for time-dependent convection-dominated PDEs: a brief survey of several recent developments. J. Comput. Phys. 316, 598–613 (2016)

    Article  MathSciNet  Google Scholar 

  27. Volpert, V.A., Nec, Y., Nepomnyashchy, A.A.: Fronts in anomalous diffusion–reaction systems. Phil. Trans. R. Soc. A 371, 20120179 (2013)

    Article  MathSciNet  Google Scholar 

  28. Xu, Q.W., Hesthaven, J.S.: Discontinuous Galerkin method for fractional convection–diffusion equations. SIAM J. Numer. Anal. 52, 405–423 (2014)

    Article  MathSciNet  Google Scholar 

  29. Yang, Q.Q., Liu, F.W., Turner, I.: Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. Appl. Math. Model. 34, 200–218 (2010)

    Article  MathSciNet  Google Scholar 

  30. Zeng, F.H., Li, C.P., Liu, F.W., Burrage, K., Turner, I., Anh, V.: A Crank Nicolson ADI spectral method for a two dimensional Riesz space fractional nonlinear reaction diffusion equation. SIAM J. Numer. Anal. 52, 2599–2622 (2014)

    Article  MathSciNet  Google Scholar 

  31. Zhuang, P.H., Liu, F.W., Anh, V., Turner, I.: Numerical methods for the variable-order fractional advection–diffusion equation with a nonlinear source term. SIAM J. Numer. Anal. 47, 1760–1781 (2009)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referees for their valuable comments and suggestions that have vastly improved the original manuscript of this paper. The research is supported by the National Natural Science Foundations of China (Grant number 11426174) and  the Natural Science Basic Research Plan in Shaanxi Province of China (Grant number 2018JM1016).

Author information

Authors and Affiliations

  1. College of Sciences, Xi’an University of Technology, Xi’an, 710048, China

    Can Li & Shuming Liu

Authors
  1. Can Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shuming Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Can Li.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, C., Liu, S. Local Discontinuous Galerkin Scheme for Space Fractional Allen–Cahn Equation. Commun. Appl. Math. Comput. 2, 73–91 (2020). https://doi.org/10.1007/s42967-019-00034-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s42967-019-00034-9

Keywords

Mathematics Subject Classification

Search

Navigation