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A direct discontinuous Galerkin method for fractional convection-diffusion and Schrödinger-type equations

  • Tarek Aboelenen
Regular Article
  • 33 Downloads

Abstract.

A direct discontinuous Galerkin (DDG) finite element method is developed for solving fractional convection-diffusion and Schrödinger-type equations with a fractional Laplacian operator of order \(\alpha\) \((1 < \alpha < 2)\). The fractional operator of order \(\alpha\) is expressed as a composite of second-order derivative and a fractional integral of order \(2-\alpha\). These problems have been expressed as a system of parabolic equations and low-order integral equation. This allows us to apply the DDG method, which is based on the direct weak formulation for solutions of fractional convection-diffusion and Schrödinger-type equations in each computational cell, letting cells communicate via the numerical flux \((\partial_{x}u)^{\ast}\) only. Moreover, we prove stability and optimal order of convergence \( O(h^{N+1})\) for the general fractional convection-diffusion and Schrödinger problems, where h, N are the space step size and polynomial degree. The DDG method has the advantage of easier formulation and implementation as well as the high-order accuracy. Finally, numerical experiments confirm the theoretical results of the method.

References

  1. 1.
    B.B. Mandelbrot, R. Pignoni, The Fractal Geometry of Nature (WH Freeman, New York, 1983)Google Scholar
  2. 2.
    V.E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media (Springer Science & Business Media, 2011)Google Scholar
  3. 3.
    R. Herrmann, Fractional Calculus: An Introduction for Physicists (World Scientific, 2014)Google Scholar
  4. 4.
    V.V. Uchaikin, Fractional Derivatives for Physicists and Engineers (Springer, 2013)Google Scholar
  5. 5.
    H.K. Moffatt, G. Zaslavsky, P. Comte, M. Tabor, Topological Aspects of the Dynamics of Fluids and Plasmas, Vol. 218 (Springer Science & Business Media, 2013)Google Scholar
  6. 6.
    G.M. Zaslavsky, Phys. Rep. 371, 461 (2002)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    A.I. Saichev, G.M. Zaslavsky, Chaos 7, 753 (1997)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    G. Zaslavsky, M. Edelman, Chaos 11, 295 (2001)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    R. Metzler, J. Klafter, J. Phys. A 37, R161 (2004)ADSCrossRefGoogle Scholar
  10. 10.
    K.B. Oldham, Adv. Eng. Softw. 41, 9 (2010)CrossRefGoogle Scholar
  11. 11.
    E. Cuesta, M. Kirane, S.A. Malik, Signal Process. 92, 553 (2012)CrossRefGoogle Scholar
  12. 12.
    J. Cai, Appl. Math. Comput. 216, 2417 (2010)MathSciNetGoogle Scholar
  13. 13.
    A. El-Sayed, M. Gaber, Electron. J. Theor. Phys. 3, 81 (2006)Google Scholar
  14. 14.
    S.I. Muslih, O.P. Agrawal, Int. J. Theor. Phys. 49, 270 (2010)CrossRefGoogle Scholar
  15. 15.
    Q. Yang, F. Liu, I. Turner, Appl. Math. Modell. 34, 200 (2010)CrossRefGoogle Scholar
  16. 16.
    A.C. Fowler, Rev. R. Acad. Cien, Ser. A. Mat. 96, 377 (2002)Google Scholar
  17. 17.
    M. Alfaro, J. Droniou, Appl. Math. Res. eXpress 2012, 127 (2012)Google Scholar
  18. 18.
    P. Azerad, A. Bouharguane, J.-F. Crouzet, Commun. Nonlinear Sci. Numer. Simul. 17, 867 (2012)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    P. Clavin, Instabilities and nonlinear patterns of overdriven detonations in gases, in Nonlinear PDEs in Condensed Matter and Reactive Flows (Springer, 2002) pp. 49--97Google Scholar
  20. 20.
    M.F. Shlesinger, G.M. Zaslavsky, U. Frisch, Lévy flights and related topics in physics, in Levy Flights and Related Topics in Physics, Vol. 450, (1995)Google Scholar
  21. 21.
    M. Cui, J. Comput. Appl. Math. 255, 404 (2014)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Y. Lin, C. Xu, J. Comput. Phys. 225, 1533 (2007)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Z. Wang, S. Vong, Comput. Math. Appl. 68, 185 (2014)MathSciNetCrossRefGoogle Scholar
  24. 24.
    S. Zhai, X. Feng, Y. He, J. Comput. Phys. 269, 138 (2014)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    F. Liu, P. Zhuang, V. Anh, I. Turner, K. Burrage, Appl. Math. Comput. 191, 12 (2007)MathSciNetGoogle Scholar
  26. 26.
    M. Chen, W. Deng, Appl. Math. Modell. 38, 3244 (2014)CrossRefGoogle Scholar
  27. 27.
    W. Deng, SIAM J. Numer. Anal. 47, 204 (2008)MathSciNetCrossRefGoogle Scholar
  28. 28.
    H.-f. Ding, Y.-x. Zhang, Comput. Math. Appl. 63, 1135 (2012)MathSciNetCrossRefGoogle Scholar
  29. 29.
    A.-M. Matache, C. Schwab, T.P. Wihler, SIAM J. Sci. Comput. 27, 369 (2005)MathSciNetCrossRefGoogle Scholar
  30. 30.
    R. Cont, E. Voltchkova, SIAM J. Numer. Anal. 43, 1596 (2005)MathSciNetCrossRefGoogle Scholar
  31. 31.
    D. Benney, A. Newell, J. Math. Phys. 46, 133 (1967)MathSciNetCrossRefGoogle Scholar
  32. 32.
    R. Bullough, P. Jack, P. Kitchenside, R. Saunders, Phys. Scr. 20, 364 (1979)ADSCrossRefGoogle Scholar
  33. 33.
    J. Yang, Physica D 108, 92 (1997)ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    M. Ran, C. Zhang, Commun. Nonlinear Sci. Numer. Simul. 41, 64 (2016)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    P. Wang, C. Huang, L. Zhao, J. Comput. Appl. Math. 306, 231 (2016)MathSciNetCrossRefGoogle Scholar
  36. 36.
    G. Zhang, C. Huang, M. Li, Eur. Phys. J. Plus 133, 155 (2018)CrossRefGoogle Scholar
  37. 37.
    P. Wang, C. Huang, Appl. Numer. Math. 129, 137 (2018)MathSciNetCrossRefGoogle Scholar
  38. 38.
    F. Bassi, S. Rebay, J. Comput. Phys. 131, 267 (1997)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    B. Cockburn, C.-W. Shu, Math. Comput. 52, 411 (1989)Google Scholar
  40. 40.
    B. Cockburn, C. Dawson, Comput. Geosci. 6, 505 (2002)MathSciNetCrossRefGoogle Scholar
  41. 41.
    B. Cockburn, G. Kanschat, D. Schötzau, Math. Comput. 74, 1067 (2005)ADSCrossRefGoogle Scholar
  42. 42.
    Q. Xu, J.S. Hesthaven, SIAM J. Numer. Anal. 52, 405 (2014)MathSciNetCrossRefGoogle Scholar
  43. 43.
    T. Aboelenen, arXiv:1708.04546 (2017)Google Scholar
  44. 44.
    K. Mustapha, W. McLean, Numer. Algorithms 56, 159 (2011)MathSciNetCrossRefGoogle Scholar
  45. 45.
    K. Mustapha, W. McLean, IMA J. Numer. Anal. 32, 906 (2012)MathSciNetCrossRefGoogle Scholar
  46. 46.
    K. Mustapha, W. McLean, SIAM J. Numer. Anal. 51, 491 (2013)MathSciNetCrossRefGoogle Scholar
  47. 47.
    W. Deng, J.S. Hesthaven, ESAIM: Math. Modell. Numer. Anal. 47, 1845 (2013)CrossRefGoogle Scholar
  48. 48.
    T. Aboelenen, Commun. Nonlinear Sci. Numer. Simul. 54, 428 (2018)ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    T. Aboelenen, H. El-Hawary, Comput. Math. Appl. 73, 1197 (2017)MathSciNetCrossRefGoogle Scholar
  50. 50.
    T. Aboelenen, Nonlinear Dyn. 92, 395 (2018)CrossRefGoogle Scholar
  51. 51.
    L. Wei, X. Zhang, S. Kumar, A. Yildirim, Comput. Math. Appl. 64, 2603 (2012)MathSciNetCrossRefGoogle Scholar
  52. 52.
    H. Liu, J. Yan, SIAM J. Numer. Anal. 47, 675 (2009)CrossRefGoogle Scholar
  53. 53.
    H. Liu, J. Yan, Commun. Comput. Phys. 8, 541 (2010)MathSciNetGoogle Scholar
  54. 54.
    H. Liu, Math. Comput. 84, 2263 (2015)CrossRefGoogle Scholar
  55. 55.
    K. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (Wiley, 1993)Google Scholar
  56. 56.
    V.J. Ervin, J.P. Roop, Numer. Methods Partial Differ. Equ. 22, 558 (2006)CrossRefGoogle Scholar
  57. 57.
    P.G. Ciarlet, Finite Element Method for Elliptic Problems (SIAM, Philadelphia, 2002)Google Scholar
  58. 58.
    Q. Zhang, C.-W. Shu, SIAM J. Numer. Anal. 42, 641 (2004)MathSciNetCrossRefGoogle Scholar
  59. 59.
    B. Cockburn, High-Order Methods for Computational Physics (Springer, Berlin, Heidelberg, 1999) pp. 69--224,  https://doi.org/10.1007/978-3-662-03882-6_2
  60. 60.
    S. Gottlieb, C.-W. Shu, Math. Comput. 67, 73 (1998)ADSCrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsAssiut UniversityAssiutEgypt

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