On the Conservation of Fractional Nonlinear Schrödinger Equation’s Invariants by the Local Discontinuous Galerkin Method

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Abstract

Using the primal formulation of the Local Discontinuous Galerkin (LDG) method, discrete analogues of the energy and the Hamiltonian of a general class of fractional nonlinear Schrödinger equation are shown to be conserved for two stabilized version of the method. Accuracy of these invariants is numerically studied with respect to the stabilization parameter and two different projection operators applied to the initial conditions. The fully discrete problem is analyzed for two implicit time step schemes: the midpoint and the modified Crank–Nicolson; and the explicit circularly exact Leapfrog scheme. Stability conditions for the Leapfrog scheme and a stabilized version of the LDG method applied to the fractional linear Schrödinger equation are derived using a von Neumann stability analysis. A series of numerical experiments with different nonlinear potentials are presented.

Keywords

Fractional nonlinear Schrödinger equation (FNLS) Local discontinuous Galerkin (LDG) Energy and Hamiltonian conservation CFL 

Mathematics Subject Classification

65M12 65M20 65M60 

Notes

Acknowledgements

We kindly thank the anonymous reviewers for their valuable suggestions.

References

  1. 1.
    Aboelenen, T.: A high-order nodal discontinuous Galerkin method for nonlinear fractional Schrödinger type equations. Commun. Nonlinear Sci. Numer. Simul. 54, 428–452 (2018)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ardila, A.: Existence and stability of standing waves for nonlinear fractional Schrödinger equation with logarithmic nonlinearity. Nonlinear Anal. 155, 52–64 (2017)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742–760 (1982)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2002)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bhrawy, A.H., Zaky, M.A.: Highly accurate numerical schemes for multi-dimensional space variable-order fractional Schrödinger equations. Computers & Mathematics with Applications 73(6), 1100–1117 (2017). (Advances in Fractional Differential Equations (IV): Time-fractional PDEs) MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bhrawy, A.H., Zaky, M.A.: An improved collocation method for multi-dimensional spacetime variable-order fractional Schrödinger equations. Appl. Numer. Math. 111, 197–218 (2017)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Castillo, P.: An optimal error estimate for the local discontinuous Galerkin method. In: Cockburn, B., Karniadakis, G.E., Shu, C.-W. (eds.) Discontinuous Galerkin Methods: Theory, Computation and Applications, Volume 11 of Lecture Notes in Computational Science and Engineering, pp. 285–290. Springer, Berlin (2000)CrossRefGoogle Scholar
  8. 8.
    Castillo, P., Cockburn, B., Perugia, I., Schötzau, D.: An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38(5), 1676–1706 (2000)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Castillo, P., Cockburn, B., Schötzau, D., Schwab, Ch.: An optimal a priori error estimate for the \(hp\)-version of the local discontinuous Galerkin method for convection-diffusion problems. Math. Comput. 71(238), 455–478 (2001)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Cheng, Y., Shu, C.W.: A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives. Math. Comput. 77(262), 699–730 (2008)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Cockburn, B., Shu, C.W.: The local discontinuous Galerkin method for time-dependent convection–diffusion systems. SIAM J. Numer. Anal. 35(6), 2440–2463 (1998)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    D’Avenia, P., Squassina, M., Zenari, M.: Fractional logarithmic Schrödinger equations. Math. Methods Appl. Sci. 38(18), 5207–5216 (2015)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Delfour, M., Fortin, M., Payré, G.: Finite-difference solutions of a non-linear Schrödinger equation. J. Comput. Phys. 44(2), 277–288 (1981)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Deng, W.H.: Finite element method for the space and time fractional Fokker–Planck equation. SIAM J. Numer. Anal. 47(1), 204–226 (2008)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Deng, W.H., Hesthaven, J.S.: Local Discontinuous Galerkin methods for fractional diffusion equations. ESAIM: M2AN 47(6), 1845–1864 (2013)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Griffiths, D.F., Mitchell, A.R., Morris, JLi: A numerical study of the nonlinear Schrödinger equation. Comput. Methods Appl. Mech. Eng. 45(1), 177–215 (1984)CrossRefMATHGoogle Scholar
  17. 17.
    Guo, B., Han, Y., Xin, J.: Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation. Appl. Math. Comput. 204(1), 468–477 (2008)MathSciNetMATHGoogle Scholar
  18. 18.
    Guo, X., Xu, M.: Some physical applications of fractional Schrödinger equation. J. Math. Phys. 47(8), 082104 (2006)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Herbst, B.M., Morris, JLi, Mitchell, A.R.: Numerical experience with the nonlinear Schrödinger equation. J. Comput. Phys. 60, 282–305 (1985)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Klein, C., Sparber, C., Markowich, P.: Numerical study of fractional nonlinear Schrödinger equations. Proc. R. Soc. A 470, 20140364 (2014)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Laskin, N.: Fractional quantum mechanics. Phys. Rev. E 62(3), 3135 (2000)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Li, M., Gu, X.M., Huang, C., Fei, M., Zhang, G.: A fast linearized conservative finite element method for the strongly coupled nonlinear fractional equations. J. Comput. Phys. 358(1), 256–282 (2018)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Li, M., Huang, C., Wang, P.: Galerkin finite element method for nonlinear fractional Schrödinger equations. Numer. Algorithms 74(2), 499–525 (2017)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198. Academic press, Cambridge (1998)MATHGoogle Scholar
  25. 25.
    Ran, M., Zhang, C.: A conservative difference scheme for solving the strongly coupled nonlinear fractional Schrödinger equations. Commun. Nonlinear Sci. Numer. Simul. 41, 64–83 (2016)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Sanz-Serna, J.M.: Methods for the numerical solution of the nonlinear Schrödinger equation. Math. Comput. 43(167), 21–27 (1984)CrossRefMATHGoogle Scholar
  27. 27.
    Sanz-Serna, J.M., Manoranjan, V.S.: A method for the integration in time of certain partial differential equations. J. Comput. Phys. 52(2), 273–289 (1983)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Sanz-Serna, J.M., Verwer, J.G.: Conservative and nonconservative schemes for the solution of the nonlinear Schrödinger equation. IMA J. Numer. Anal. 6, 25–42 (1986)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Shabat, A., Zakharov, V.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP 34(1), 62 (1972)MathSciNetGoogle Scholar
  30. 30.
    Strauss, W., Vazquez, L.: Numerical solution of a nonlinear Klein–Gordon equation. J. Comput. Phys. 28(2), 271–278 (1978)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Sulem, C., Sulem, P.L.: The Nonlinear Schrödinger Equation, Volume 139 of Applied Mathematical Sciences. Springer, Berlin (1999)MATHGoogle Scholar
  32. 32.
    Verwer, J.G., Dekker, K.: Step by step stability in the numerical solution of partial differential equations. Technical Report 161-83, Centre for Mathematics and Computer Science, Amsterdam (1983)Google Scholar
  33. 33.
    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(1), 670–681 (2013)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Wang, P., Huang, C.: An energy conservative difference scheme for the nonlinear fractional Schrödinger equations. J. Comput. Phys. 293, 238–251 (2015)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Wang, P., Huang, C.: Split-step alternating direction implicit difference scheme for the fractional Schrödinger equation in two dimensions. Comput. Math. Appl. 71(5), 1114–1128 (2016)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Wei, L., Zhang, X., Kumar, S., Yildirim, A.: A numerical study based on an implicit fully discrete local discontinuous galerkin method for the time-fractional coupled Schrödinger system. Comput. Math. Appl. 64(8), 2603–2615 (2012)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Weideman, J.A.C., Herbst, B.M.: Split-step alternating direction implicit difference scheme for the fractional schrödinger equation in two dimensions. SIAM J. Numer. Anal. 23(3), 485–507 (1986)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Xu, Q., Hesthaven, J.S.: Discontinuous Galerkin method for fractional convection-diffusion equations. SIAM J. Numer. Anal. 52(1), 405–423 (2014)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Xu, Y., Shu, C.W.: Local discontinuous Galerkin methods for nonlinear Schrödinger equations. J. Comput. Phys. 205(1), 72–97 (2005)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Xu, Y., Shu, C.W.: Optimal error estimates of the semidiscrete local discontinuous Galerkin methods for high order wave equations. SIAM J. Numer. Anal. 50(1), 79–104 (2012)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Zhang, H., Hu, Q.: Existence of the global solution for fractional logarithmic Schrödinger equation. Comput. Math. Appl. 75(1), 161–169 (2018)Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of Puerto RicoMayagüezPuerto Rico

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