Skip to main content
Springer Nature Link
Log in

A high-order numerical scheme for multidimensional convection-diffusion-reaction equation with time-fractional derivative

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

This paper considers a high-order numerical method for a computed solution of multidimensional convection-diffusion-reaction equation with time-fractional derivative subjected to appropriate initial and boundary conditions. The stability and error estimates of the proposed numerical approach are analyzed using the \(L^{\infty }(0,T;L^{2})\)-norm. The theoretical study suggests that the new technique is unconditionally stable and temporal accurate with order O(τ2+α), where τ denotes the time step and 0 < α < 1. This result shows that the developed algorithm is faster and more efficient than a broad range of numerical techniques widely studied in the literature for the considered problem. Numerical experiments confirm the theory and they indicate that the proposed numerical scheme converges with accuracy O(τ2+α + h4), where h represents the space step.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Data availability

Not applicable.

References

  1. Alikhanov, A.A.: A new difference scheme for time fractional diffusion equation. J. Comput. Phys. 280(1), 424–438 (2015)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  3. Daftardar-Geji, V., Jafari, H.: Adomain decomposition: a tool for solving a system of fractional differential equations. J. Comput. Appl. Math. 301, 508–518 (2005)

    MATH  Google Scholar 

  4. Dehghan, M.: Weighted finite difference techniques for the one-dimensional advection-diffusion equation. Appl. Math. Comput. 147(2), 307–319 (2004)

    MathSciNet  MATH  Google Scholar 

  5. Deng, W.H.: Finite element method for the space and time fractional Fokker-Planck equation. SIAM J. Numer. Anal. 47, 204–226 (2008)

    MathSciNet  MATH  Google Scholar 

  6. Donatelli, M., Mazza, M., Capizzano, S.S.: Spectral analysis and multigrid methods for finite volume approximations of space-fractional diffusion equations. SIAM J. Scientific Comput. 40(6), A4007–A4039 (2018)

    MathSciNet  MATH  Google Scholar 

  7. Donatelli, M., Mazza, M., Capizzano, S.S.: Spectral analysis and structure preserving preconditioners for fractional diffusion equations. J. Comput. Phys. 307, 262–279 (2016)

    MathSciNet  MATH  Google Scholar 

  8. Gupta, M.M., Manohar, R.P., Stephenson, J.W.: A single cell high order scheme for the convection-diffusion equation with variable coefficients. Int. J. Numer. Meth. Fluids 4, 641–651 (1984)

    MathSciNet  MATH  Google Scholar 

  9. Guvanasen, V., Volker, R.E.: Numerical solutions for solute transport in unconfined aquifers. Int. J. Num. Meth. Fluids 3, 103–123 (1983)

    MATH  Google Scholar 

  10. Jiang, Y.J., Ma, J.T.: High-order finite element methods for time-fractional partial differential equations. J. Comput. Appl. Math. 235, 3285–3290 (2011)

    MathSciNet  MATH  Google Scholar 

  11. Karaa, S., Zhang, J.: Higher order ADI method for solving unsteady convection-diffusion problems. J. Comput. Phys. 198, 1–9 (2004)

    MathSciNet  MATH  Google Scholar 

  12. Li, C.P., Zhao, Z.G., Chen, Y.Q.: Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion. Comput. Appl. Math. 62, 855–875 (2011)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  14. Li, M., Tang, T., Fornberg, B.: A compact fourth-order finite-difference scheme for the incompressible Navier-Stokes equations. Int. J. Num. Meth. Fluids 20, 1137–1151 (1995)

    MathSciNet  MATH  Google Scholar 

  15. Liu, E., Yang, C., Burrage, K.: Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term. J. Comput. Appl. Math. 231, 160–176 (2009)

    MathSciNet  MATH  Google Scholar 

  16. Liu, Y., Du, Y.W., Li, H., He, S., Gao, W.: Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction-diffusion problem. Comput. Math. Appl. 70, 573–591 (2015b)

    MathSciNet  MATH  Google Scholar 

  17. Liu, Y., Du, Y.W., Li, H., Li, J.C., He, S.: A two-grid mixed finite element method for a nonlinear fourth-order reaction diffusion problem with time-fractional derivative. Comput. Math. Appl. 70, 2474–2492 (2015a)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  19. Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172, 65–77 (2004)

    MathSciNet  MATH  Google Scholar 

  20. Momani, S., Odibat, Z.M.: Fractional green function for linear time-fractional inhomogeneous partial differential equations in fluid mechanics. J. Comput. Appl. Math. 24, 167–178 (2007)

    MathSciNet  MATH  Google Scholar 

  21. Ngondiep, E.: A novel three-level time-split approach for solving two-dimensional nonlinear unsteady convection-diffusion-reaction equation. J. Math. Computer Sci. 26(3), 222–248 (2022)

    Google Scholar 

  22. Namio, F.T., Ngondiep, E., Ntchantcho, R., Ntonga, J.C.: Mathematical models of complete shallow water equations with source terms, stability analysis of Lax-Wendroff scheme. J. Theor. Comput. Sci., vol. 2(132) (2015)

  23. Ngondiep, E.: Stability analysis of MacCormack rapid solver method for evolutionary Stokes-Darcy problem. J. Comput. Appl. Math. 345, 269–285 (2019)

    MathSciNet  MATH  Google Scholar 

  24. Ngondiep, E.: A fourth-order two-level factored implicit scheme for solving two-dimensional unsteady transport equation with time-dependent dispersion coefficients. Int. J. Comput. Meth. Engrg. Sci. Mech., https://doi.org/10.1080/15502287.2020.1856972 (2020)

  25. Ngondiep, E.: A novel three-level time-split MacCormack scheme for two-dimensional evolutionary linear convection-diffusion-reaction equation with source term. Int. J. Comput. Math. 98(1), 47–74 (2021)

    MathSciNet  MATH  Google Scholar 

  26. Ngondiep, E.: A three-level explicit time-split MacCormack method for 2D nonlinear reaction-diffusion equations. Preprint available online from arXiv:1903.10877 (2019)

  27. Ngondiep, E.: A six-level time-split Leap-Frog/Crank-Nicolson approach for two-dimensional nonlinear time-dependent convection-diffusion-reaction equation Int. J. Comput. Meth., https://doi.org/10.1142/S0219876222500645 (2023)

  28. Ngondiep, E.: An efficient explicit approach for predicting the Covid-19 spreading with undetected infectious: the case of Cameroon. preprint available online from arXiv:2005.11279 (2020)

  29. Ngondiep, E.: Long time stability and convergence rate of MacCormack rapid solver method for nonstationary Stokes-Darcy problem. Comput. Math. Appl. 75, 3663–3684 (2018)

    MathSciNet  MATH  Google Scholar 

  30. Ngondiep, E.: An efficient three-level explicit time-split approach for solving 2D heat conduction equations. Appl. Math. Inf. Sci. 14(6), 1075–1092 (2020)

    MathSciNet  Google Scholar 

  31. Ngondiep, E.: An efficient three-level explicit time-split scheme for solving two-dimensional unsteady nonlinear coupled Burgers equations. Int. J. Numer. Methods Fluids 92(4), 266–284 (2020)

    MathSciNet  Google Scholar 

  32. Ngondiep, E.: A two-level fourth-order approach for time-fractional convection-diffusion-reaction equation with variable coefficients. Commun. Nonlinear Sci. Numer. Simul. 111, 106444 (2022). https://doi.org/10.1016/j.cnsns.2022.106444

    MathSciNet  MATH  Google Scholar 

  33. Ngondiep, E.: A robust three-level time-split MacCormack scheme for solving two-dimensional unsteady convection-diffusion equation. J. Appl. Comput. Mech. 7(2), 559–577 (2021)

    MATH  Google Scholar 

  34. Ngondiep, E.: Long time unconditional stability of a two-level hybrid method for nonstationary incompressible Navier-Stokes equations. J. Comput. Appl. Math. 345, 501–514 (2019)

    MathSciNet  MATH  Google Scholar 

  35. Ngondiep, E.: Asymptotic growth of the spectral radii of collocation matrices approximating elliptic boundary problems. Int. J. Appl. Math. Comput. 4, 199–219 (2012)

    Google Scholar 

  36. Ngondiep, E.: A two-level factored Crank-Nicolson method for two-dimensional nonstationary advection-diffusion equation with time dependent dispersion coefficients and source sink/term. Adv. Appl. Math. Mech. 13(5), 1005–1026 (2021)

    MathSciNet  MATH  Google Scholar 

  37. Ngondiep, E.: A robust numerical two-level second-order explicit approach to predict the spread of covid-2019 pandemic with undetected infectious cases. J. Comput. Appl. Math., vol. 403. https://doi.org/10.1016/j.cam.2021.113852 (2022)

  38. Ngondiep, E.: Error estimate of MacCormack rapid solver method for 2D incompressible Navier-Stokes problems. Preprint available online from arXiv:1903.10857 (2019)

  39. Ngondiep, E., Kerdid, N., Abaoud, M.A.M., Aldayel, I.A.I.: A three-level time-split MacCormack method for two-dimensional nonlinear reaction-diffusion equations. Int. J. Numer. Meth. Fluids 92(12), 1681–1706 (2020)

    MathSciNet  Google Scholar 

  40. Ngondiep, E., Alqahtani, R., Ntonga, J.C.: Stability analysis and convergence rate of MacCormack scheme for complete shallow water equations with source terms. Preprint available online from arXiv:1903.11104 (2019)

  41. Ngondiep, E.: Unconditional stability over long time intervals of a two-level coupled MacCormack/Crank-Nicolson method for evolutionary mixed Stokes-Darcy model. J. Comput. Appl. Math. 409, 114148 (2022). https://doi.org/10.1016/j.cam.2022.114148

    MathSciNet  MATH  Google Scholar 

  42. Nochetto, R.H., Arola, E.O., Sagado, A.J.: A PDE approach to space-time fractional parabolic problems. SIAM J. Num. Anal. 54, 848–873 (2016)

    MathSciNet  Google Scholar 

  43. Noye, B.J., Tan, H.H.: Finite difference methods for solving two-dimensional advection-diffusion equation. Int. J. Numer. Methods Fluids 9(1), 75–98 (1989)

    MATH  Google Scholar 

  44. Radhakrishna Pillai, A.C.: Fourth-order exponential finite difference methods for boundary value problems of convective diffusion type. Int. J. Numer. Methods Fluids 37, 87–106 (2001)

    MathSciNet  MATH  Google Scholar 

  45. Samarskii, A.A., Andreev, V.B.: Difference methods for elliptic equation. Moscow, Nauka (1976). (in Russian)

    MATH  Google Scholar 

  46. Tian, Z.F., Ge, Y.B.: A fourth-order compact ADI method for solving two-dimensional unsteady convection-diffusion problem. J. Comput. Appl. Math. 198, 268–286 (2007)

    MathSciNet  MATH  Google Scholar 

  47. Zhang, Y.-N., Sun, Z.-Z., Zhao, X.: Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation. SIAM J. Numer. Anal. 50(3), 1535–1555 (2012)

    MathSciNet  MATH  Google Scholar 

  48. Zhang, J., Liu, F., Phanikumar, M.S., Meerschaert, M.M.: A novel numerical method for the time variable fractional order mobile-immobile advection-dispersion model. Comput. Appl. Math. 66, 693–701 (2013)

    MathSciNet  MATH  Google Scholar 

  49. Zhang, J., Zhang, X., Yang, B.: An approximation scheme for the time fractional convection-diffusion equation. Appl. Math. Comput. 335, 305–312 (2018)

    MathSciNet  MATH  Google Scholar 

  50. Zheng, M.L., Liu, F., Turner, I., Anh, V.: A novel high-order space-time spectral method for the time fractional Fokker-Planck equation. SIAM J. Sci. Comput. 37, 701–724 (2015)

    MathSciNet  MATH  Google Scholar 

  51. Zhuang, P., Gu, Y., Liu, F., Turner, I., Yarlagadda: Time dependent fractional advection-diffusion equations by an implicit mls meshless method. Int. J. Numer. Meth. Engrg 88(13), 1346–1362 (2011)

    MathSciNet  MATH  Google Scholar 

  52. Zlatev, Z., Berkowicz, R., Prahm, L.P.: Implementation of a variable stepsize variable formula in the time-integration part of a code for treatment of long-range transport of air polluants. J. Comput. Phys. 55, 278–301 (1984)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author appreciates the valuable comments of anonymous referees which helped to improve the quality of this paper.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), 90950, Riyadh, 11632, Saudi Arabia

    Eric Ngondiep

  2. Hydrological Research Centre, Institute for Geological and Mining Research, 4110, Yaounde, Cameroon

    Eric Ngondiep

Authors
  1. Eric Ngondiep
    View author publications

    You can also search for this author inPubMed Google Scholar

Contributions

The whole work has been carried out by the author.

Corresponding author

Correspondence to Eric Ngondiep.

Ethics declarations

Ethics approval

Not applicable.

Conflict of interest

The author declares no competing interests.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ngondiep, E. A high-order numerical scheme for multidimensional convection-diffusion-reaction equation with time-fractional derivative. Numer Algor 94, 681–700 (2023). https://doi.org/10.1007/s11075-023-01516-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-023-01516-x

Keywords

Mathematics Subject Classification (2010)