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An efficient fourth-order in space difference scheme for the nonlinear fractional Ginzburg–Landau equation

Article

Abstract

This paper proposes and analyzes a high-order implicit-explicit difference scheme for the nonlinear complex fractional Ginzburg–Landau equation involving the Riesz fractional derivative. For the time discretization, the second-order backward differentiation formula combined with the explicit second-order Gear’s extrapolation is adopted. While for the space discretization, a fourth-order fractional quasi-compact method is used to approximate the Riesz fractional derivative. The scheme is efficient in the sense that, at each time step, only a linear system with a coefficient matrix independent of the time level needs to be solved. Despite of the explicit treatment of the nonlinear term, the scheme is shown to be unconditionally convergent in the \(l^2_h\) norm, semi-\(H^{\alpha /2}_h\) norm and \(l^\infty _h\) norm at the order of \(O(\tau ^2+h^4)\) with \(\tau \) time step and h mesh size. Numerical tests are provided to confirm the accuracy and efficiency of the scheme.

Keywords

Fractional Ginzburg–Landau equation Riesz fractional derivative Implicit-explicit method Fractional compact scheme Convergence 

Mathematics Subject Classification

65M06 65M12 65M15 

Notes

Acknowledgements

This work was supported by National Natural Science Foundation of China (No. 11771163).

References

  1. 1.
    Aranson, I.S., Kramer, L.: The world of the complex Ginzburg–Landau equation. Rev. Mod. Phys. 74(1), 99–143 (2002)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ascher, U.M., Ruuth, S.J., Wetton, B.T.R.: Implicit-explicit methods for time-dependent partial differential equations. SIAM J. Numer. Anal. 32, 797–823 (1995)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bueno-Orovio, A., Kay, D., Burrage, K.: Fourier spectral methods for fractional-in-space reaction–diffusion equations. BIT 54, 937–954 (2014)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Çelik, C., Duman, M.: Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative. J. Comput. Phys. 231(4), 1743–1750 (2012)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Chen, M., Deng, W.: Fourth order accurate scheme for the space fractional diffusion equations. SIAM J. Numer. Anal. 52, 1418–1438 (2014)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Defterli, O., D’Elia, M., Du, Q., Gunzburger, M., Lehoucq, R., Meerschaert, M.M.: Fractional diffusion on bounded domains. Fract. Calc. Appl. Anal. 18, 342–360 (2015)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Ding, H., Li, C., Chen, Y.: High-order algorithms for Riesz derivative and their applications (II). J. Comput. Phys. 293, 218–237 (2015)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Ding, H., Li, C., Chen, Y.: High-order algorithms for Riesz derivative and their applications (III). Fract. Calc. Appl. Anal. 19, 19–55 (2016)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Du, Q., Gunzburger, M., Lehoucq, R.B., Zhou, K.: Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev. 54(4), 667–696 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ervin, V.J., Heuer, N., Roop, J.P.: Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation. SIAM J. Numer. Anal. 45, 572–591 (2007)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Guo, B., Huo, Z.: Well-posedness for the nonlinear fractional Schrödinger equation and inviscid limit behavior of solution for the fractional Ginzburg-Landau equation. Fract. Calc. Appl. Anal. 16, 226–242 (2013)MathSciNetMATHGoogle Scholar
  12. 12.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer Series in Computational Mathematics, vol. 14, 2nd edn. Springer, Berlin (1996)CrossRefMATHGoogle Scholar
  13. 13.
    Hao, Z.P., Sun, Z.Z.: A linearized high-order difference scheme for the fractional Ginzburg–Landau equation. Numer. Methods Partial Differ. Equ. 33, 105–124 (2017)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Hao, Z.P., Sun, Z.Z., Cao, W.R.: A fourth-order approximation of fractional derivatives with its applications. J. Comput. Phys. 281, 787–805 (2015)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Hao, Z.P., Sun, Z.Z., Cao, W.R.: A three-level linearized compact difference scheme for the Ginzburg–Landau equation. Numer. Methods Partial Differ. Equ. 31, 876–899 (2015)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kirkpatrick, K., Lenzmann, E., Staffilani, G.: On the continuum limit for discrete NLS with long-range lattice interactions. Commun. Math. Phys. 317, 563–591 (2013)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Lord, G.J.: Attractors and inertial manifolds for finite-difference approximations of the complex Ginzburg–Landau equation. SIAM J. Numer. Anal. 34(4), 1483–1512 (1997)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Lu, H., Bates, P.W., Lü, S., Zhang, M.: Dynamics of the 3-D fractional complex Ginzburg–Landau equation. J. Differ. Equ. 259(10), 5276–5301 (2015)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection–dispersion flow equations. J. Comput. Appl. Math. 172, 65–77 (2004)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Milovanov, A., Rasmussen, J.: Fractional generalization of the Ginzburg–Landau equation: an unconventional approach to critical phenomena in complex media. Phys. Lett. A 337, 75–80 (2005)CrossRefMATHGoogle Scholar
  21. 21.
    Mvogo, A., Ben-Bolie, G.H., Kofané, T.C.: Coupled fractional nonlinear differential equations and exact Jacobian elliptic solutions for excitoncphonon dynamics. Phys. Lett. A 378, 2509–2517 (2014)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Mvogo, A., Tambue, A., Ben-Bolie, G.H., Kofané, T.C.: Localized numerical impulse solutions in diffuse neural networks modeled by the complex fractional Ginzburg–Landau equation. Commun. Nonlinear Sci. Numer. Simul. 39, 396–410 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)MATHGoogle Scholar
  24. 24.
    Pu, X., Guo, B.: Well-posedness and dynamics for the fractional Ginzburg–Landau equation. Appl. Anal. 92(2), 318–334 (2013)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer Series in Computational Mathematics, vol. 23. Springer, Berlin (1994)MATHGoogle Scholar
  26. 26.
    Sun, Z.Z., Zhu, Q.: On Tsertsvadze’s difference scheme for the Kuramoto–Tsuzuki equation. J. Comput. Appl. Math. 98(2), 289–304 (1998)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Tadjeran, C., Meerschaert, M.M., Scheffler, H.P.: A second-order accurate numerical approximation for the fractional diffusion equation. J. Comput. Phys. 213, 205–213 (2006)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Tarasov, V., Zaslavsky, G.: Fractional Ginzburg–Landau equation for fractal media. Phys. A 354, 249–261 (2005)CrossRefGoogle Scholar
  29. 29.
    Tarasov, V., Zaslavsky, G.: Fractional dynamics of coupled oscillators with long-range interaction. Chaos 16(2), 023110 (2006)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Tian, W., Zhou, H., Deng, W.: A class of second order difference approximation for solving space fractional diffusion equations. Math. Comput. 84, 1703–1727 (2015)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Wang, D., Xiao, A., Yang, W.: Crank-Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative. J. Comput. Phys. 242, 670–681 (2013)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Wang, D., Xiao, A., Yang, W.: A linearly implicit conservative difference scheme for the space fractional coupled nonlinear Schrödinger equations. J. Comput. Phys. 272, 644–655 (2014)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Wang, D., Xiao, A., Yang, W.: Maximum-norm error analysis of a difference scheme for the space fractional CNLS. Appl. Math. Comput. 257, 241–251 (2015)MathSciNetMATHGoogle Scholar
  34. 34.
    Wang, H., Basu, T.S.: A fast finite difference method for two-dimensional space-fractional diffusion equations. SIAM J. Sci. Comput. 34, A2444–A2458 (2012)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Wang, P., Huang, C.: A conservative linearized difference scheme for the nonlinear fractional Schrödinger equation. Numer. Algorithm 69, 625–641 (2015)CrossRefMATHGoogle Scholar
  36. 36.
    Wang, P., Huang, C.: An energy conservative difference scheme for the nonlinear fractional Schrödinger equations. J. Comput. Phys. 293, 238–251 (2015)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Wang, P., Huang, C.: An implicit midpoint difference scheme for the fractional Ginzburg–Landau equation. J. Comput. Phys. 312, 31–49 (2016)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Wang, P., Huang, C., Zhao, L.: Point-wise error estimate of a conservative difference scheme for the fractional Schrödinger equation. J. Comput. Appl. Math. 306, 231–247 (2016)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Wang, T., Guo, B.: Analysis of some finite difference schemes for two-dimensional Ginzburg–Landau equation. Numer. Methods Partial Differ. Equ. 27, 1340–1363 (2011)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Xu, Q., Hesthaven, J.S.: Discontinuous Galerkin method for fractional convection–diffusion equations. SIAM J. Numer. Anal. 52, 405–423 (2014)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Yang, Q., Liu, F., Turner, I.: Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. Appl. Math. Model. 34(1), 200–218 (2010)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Zeng, F., Liu, F., Li, C., 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)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Zhao, X., Sun, Z.Z., Hao, Z.P.: A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional Schrödinger equation. SIAM J. Sci. Comput. 36(6), A2865–A2886 (2014)CrossRefMATHGoogle Scholar
  44. 44.
    Zhou, H., Tian, W., Deng, W.: Quasi-compact finite difference schemes for space fractional diffusion equations. J. Sci. Comput. 56, 45–66 (2013)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Zhuang, P., Liu, F., Anh, V., Turner, I.: Numerical methods for the variable-order fractional advection–diffusion equation with a nonlinear source term. SIAM J. Numer. Anal. 47(3), 1760–1781 (2009)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceHenan University of Economics and LawZhengzhouChina
  2. 2.School of Mathematics and StatisticsHuazhong University of Science and TechnologyWuhanChina
  3. 3.Hubei Key Laboratory of Engineering Modeling and Scientific ComputingHuazhong University of Science and TechnologyWuhanChina

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