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.
Similar content being viewed by others
References
Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)
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)
Achleitner, F., Kuehn, Ch.: Analysis and numerics of traveling waves for asymmetric fractional reaction–diffusion equations. arXiv:1405.5779v1 [math.NA] 22 (May 2014)
Baeumer, B., Kovacsa, M., Meerschaert, M.M.: Numerical solutions for fractional reaction–diffusion equations. Comput. Math. Appl. 55, 2212–2226 (2008)
Bueno-Orovio, A., Kay, D., Burrage, K.: Fourier spectral methods for fractional-in-space reaction–diffusion equations. BIT Numer. Math. 54, 937–954 (2014)
Cabré, X., Roquejoffre, J.-M.: The influence of fractional diffusion in Fisher-KPP equations. Commun. Math. Phys. 320, 679–722 (2013)
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)
Cifani, S., Jakobsen, E.R., Karlsen, K.H.: The discontinuous Galerkin method for fractal conservation laws. IMA J. Numer. Anal. 31, 1090–1122 (2011)
Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection–diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)
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)
Deng, W.H., Hesthaven, J.S.: Local discontinuous Galerkin methods for fractional diffusion equations. ESAIM Math. Model. Numer. Anal. 47, 1845–1864 (2013)
Du, Q., Yang, J.: Asymptotic compatible Fourier spectral approximations of nonlocal Allen–Cahn equations. SIAM J. Numer. Anal. 54, 1899–1919 (2016)
Ervin, V.J., Roop, J.P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Diff. Equ. 22, 558–576 (2006)
Feng, X.B., Prohl, A.: Numerical analysis of the Allen–Cahn equation and approximation for mean curvature flows. Numer. Math. 94, 33–65 (2003)
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)
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)
Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001)
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)
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)
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)
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)
Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)
Shen, J., Yang, X.F.: Numerical approximations of Allen–Cahn and Cahn–Hilliard equations. Discret. Contin. Dyn. Syst. 28, 1669–1691 (2010)
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)
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)
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)
Volpert, V.A., Nec, Y., Nepomnyashchy, A.A.: Fronts in anomalous diffusion–reaction systems. Phil. Trans. R. Soc. A 371, 20120179 (2013)
Xu, Q.W., Hesthaven, J.S.: Discontinuous Galerkin method for fractional convection–diffusion equations. SIAM J. Numer. Anal. 52, 405–423 (2014)
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)
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)
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)
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
Corresponding author
Rights and permissions
About this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s42967-019-00034-9