Skip to main content
SpringerLink
Log in

A Hybridized Discontinuous Galerkin Method for 2D Fractional Convection–Diffusion Equations

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

A hybridized discontinuous Galerkin method is proposed for solving 2D fractional convection–diffusion equations containing derivatives of fractional order in space on a finite domain. The Riemann–Liouville derivative is used for the spatial derivative. Combining the characteristic method and the hybridized discontinuous Galerkin method, the symmetric variational formulation is constructed. The stability of the presented scheme is proved. Theoretically, the order of \(\mathscr {O}(h^{k+1/2}+\varDelta t)\) is established for the corresponding models and numerically the better convergence rates are detected by carefully choosing the numerical fluxes. Extensive numerical experiments are performed to illustrate the performance of the proposed schemes. The first numerical example is to display the convergence orders, while the second one justifies the benefits of the schemes. Both are tested with triangular meshes.

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

Similar content being viewed by others

References

  1. Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: The fractional-order governing equation of Lévy motion. Water Resour. Res. 36(6), 1413–1423 (2000)

    Article  Google Scholar 

  2. Carreras, B.A., Lynch, V.E., Zaslavsky, G.M.: Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model. Phys. Plasmas 8(12), 5096–5103 (2001)

    Article  Google Scholar 

  3. Castilo, P., Cockburn, B., Perugia, I., Sötzau, D.: An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38(5), 1676–1706 (2000)

    Article  MathSciNet  Google Scholar 

  4. Chen, Z.X.: Characteristic mixed discontinuous finite element methods for advection-dominated diffusion problems. Comput. Methods Appl. Mech. Eng. 191, 2509–2538 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, Z.X., Ewing, R.E., Jiang, Q.Y., Spagnuolo, M.: Error analysis for characteristic-based methods for degenerate parabolic problems. SIAM J. Numer. Anal. 40, 1491–1515 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, M.H., Deng, W.H.: A second-order numerical method for two-dimensional two-sided space fractional convection-diffusion equation. Appl. Math. Model. 38(13), 3244–3259 (2014)

    Article  MathSciNet  Google Scholar 

  7. 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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cockburn, B., Shu, C.-W.: Runge–Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16(3), 173–261 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cockburn, B., Mustapha, K.: A hybridizable discontinuous Galerkin method for fractional diffusion problems. Numer. Math. 130(2), 293–314 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cui, M.R.: A high-order compact exponential scheme for fractional convection–diffusion equations. J. Comput. Appl. Math. 255, 404–416 (2014)

    Article  MathSciNet  MATH  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  MATH  Google Scholar 

  12. Douglas Jr, J., Russell, T.F.: Numerical method for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures. SIAM J. Numer. Anal. 19(5), 871–885 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ervin, V.J., Roop, J.P.: Variational formulation for the stational fractional advection dispersion equation. Numer. Methods Partial Differ. Equ. 22(3), 558–576 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, New York (2008)

    Book  MATH  Google Scholar 

  15. 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. Methods Appl. 5(3), 333–358 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kirchner, J.W., Feng, X.H., Neal, C.: Fractal stream chemistry and its implication for contaminant transport in catchments. Nature 403, 524–526 (2002)

    Article  Google Scholar 

  17. Lin, Y., Xu, C.J.: Finite difference/spectral approximations for the time-fractional diffusion equations. J. Comput. Phys. 225, 1533–1552 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Liu, F., Zhuang, P., Anh, V., Turner, I., Burrage, K.: Stability and convergence of the difference methods for the space–time fractional advection–diffusion equation. Appl. Math. Comput. 191, 12–20 (2007)

    MathSciNet  MATH  Google Scholar 

  19. Magin, R.L.: Fractional Calculus in Bioengineering. Begell House, Redding (2006)

    Google Scholar 

  20. Mclean, W., Mustapha, K.: Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation. Numer. Algor. 52(1), 69–88 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mclean, W., Mustapha, K.: Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations. SIAM J. Numer. Anal. 51(1), 491–515 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mclean, W.: Fast summation by interval clustering for an evolution equation with memory. SIAM J. Sci. Comput. 34(6), A3039–A3056 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mustapha, K., Mclean, W.: Piece-linear, discontinuous Galerkin method for a fractional diffusion equation. Numer. Algor. 56(2), 159–184 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mustapha, K.: Time-stepping discontinuous Galerkin methods for fractional diffusion problems. Numer. Math. 130(3), 497–516 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)

    MATH  Google Scholar 

  26. Qiu, L.L., Deng, W.H., Hesthaven, J.S.: Nodal discontinuous Galerkin methods for fractional diffusion equations on 2D domain with triangualr meshes. J. Comput. Phys. 298, 678–694 (2015)

    Article  MathSciNet  Google Scholar 

  27. Rivière, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. SIAM, Philadelphia (2008)

    Book  MATH  Google Scholar 

  28. Shlesinger, M.F., West, B.J., Klafter, J.: Lévy dynamics of enhanced diffusion: application to turbulence. Phys. Rev. Lett. 58(11), 1100–1103 (1987)

    Article  MathSciNet  Google Scholar 

  29. Wang, Z.B., Vong, S.W.: A high-order exponential ADI scheme for two dimensional time fractional convection–diffusion equations. Comput. Math. Appl. 68, 185–196 (2014)

    Article  MathSciNet  Google Scholar 

  30. Xu, Q., Hesthaven, J.S.: Discontinuous Galerkin method for fractional convection–diffusion equations. SIAM J. Numer. Anal. 52(1), 405–423 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhai, S.Y., Feng, X.L., He, Y.N.: An unconditionally stable compact ADI method for three-dimensional time-fractional convection–diffusion equations. J. Comput. Phys. 269, 138–155 (2014)

Download references

Acknowledgments

This work was partially supported by the National Basic Research (973) Program of China under Grant 2011CB706903, the National Natural Science Foundation of China under Grant 11271173 and 11471150, and the CAPES and CNPq in Brazil.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Shuqin Wang, Weihua Deng & Yujiang Wu

  2. Department of Mathematics, Centro Politécnico, Federal University of Paraná, CP: 19.089, Curitiba, PR, CEP: 81531-980, Brazil

    Shuqin Wang & Jinyun Yuan

Authors
  1. Shuqin Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jinyun Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Weihua Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yujiang Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jinyun Yuan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, S., Yuan, J., Deng, W. et al. A Hybridized Discontinuous Galerkin Method for 2D Fractional Convection–Diffusion Equations. J Sci Comput 68, 826–847 (2016). https://doi.org/10.1007/s10915-015-0160-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-015-0160-y

Keywords

Mathematics Subject Classification

Search

Navigation