Skip to main content
SpringerLink
Log in

A robust numerical algorithm on harmonic mesh for parabolic singularly perturbed convection-diffusion problems with time delay

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

Abstract

This article deals with the class of singularly perturbed convection-diffusion problem with time delay. A parameter uniform numerical method is developed, and its detailed analysis is done. To discretize the spatial domain, a harmonic mesh H(ℓ) is used, which gives more accurate results in comparison with Shishkin, S(ℓ) and Bakhvalov mesh. Numerical experiments are carried out to validate the proposed method. The computational results on H(ℓ) mesh have been compared with the other existing meshes like B-mesh, Shishkin mesh and S(ℓ) mesh.

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
Fig. 5
Fig. 6

Similar content being viewed by others

Data availability

All data generated or analyzed during this study are included in this published article.

References

  1. Ansari, A.R., Bakr, S.A., Shishkin, G.I.: A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations. J. Comput. Appl. Math. 205, 552–566 (2007)

    Article  MathSciNet  Google Scholar 

  2. Kumar, D., Kumari, P.: A parameter-uniform numerical scheme for the parabolic singularly perturbed initial boundary value problems with large time delay. J. Appl. Math. Comput. 59, 179–206 (2018)

    Article  MathSciNet  Google Scholar 

  3. Bashier, E.B.M., Patidar, K.C.: A second-order fitted operator finite difference method for a singularly perturbed delay parabolic partial differential equation. J. Differ. Equ. Appl. 17, 779–794 (2011)

    Article  MathSciNet  Google Scholar 

  4. Kumar, D., Kumari, P.: A parameter-uniform scheme for singularly perturbed partial differential equations with a time lag. Numer. Methods Partial Differ. Eq. 36, 868–886 (2020)

    Article  MathSciNet  Google Scholar 

  5. Ramesh, V.P., Kadalbajoo, M.K.: Upwind and midpoint upwind difference methods for time-dependent differential difference equations with layer behavior. Appl. Math. Comput. 202, 453–471 (2008)

    Article  MathSciNet  Google Scholar 

  6. Kumar, S., Kumar, M.: High order parameter-uniform discretization for singularly perturbed parabolic partial differential equations with time delay. Comput. Math. Appl. 68, 1355–1367 (2014)

    Article  MathSciNet  Google Scholar 

  7. Linß, T., Madden, N.: Layer adapted meshes for a linear system of coupled singularly perturbed reaction diffusion problems. IMA J. Numer. Anal. 29, 109–125 (2009)

    Article  MathSciNet  Google Scholar 

  8. Bakhvalov, N.S.: Towards optimization of methods for solving boundary value problems in the presence of boundary layers. Zh. Vychisl. Mat. Mat. Fiz. 9, 841–859 (1969). (in Russian)

    Google Scholar 

  9. Roos, H.G., Stynes, M., Tobiska, L.: Robust numerical mathods for singularly perturbed differential equations. Springer Ser. Comput. Math., 24 (2008)

  10. Das, A., Natesan, S.: Second-order uniformly convergent numerical method for singularly perturbed delay parabolic partial differential equations. Int. J. Comput. Math. 95, 490–510 (2018)

    Article  MathSciNet  Google Scholar 

  11. Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Fitted Mesh Methods for Singular Perturbation Problems Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions. World Scientific, Singapore (1996)

    MATH  Google Scholar 

  12. Kaushik, A., Sharma, K.K., Sharma, M.: A parameter uniform difference scheme for parabolic partial differential equation with a retarded argument. Appl. Math. Model. 34, 4232–4242 (2010)

    Article  MathSciNet  Google Scholar 

  13. Solonnikov, V.A., Ladyzhenskaya, O.A., Ural’tseva, N.N.: Linear and Quasilinear Equation of Parabolic Type. American Mathematical Society Providence, Rhode Island (1988)

    Google Scholar 

  14. Singh, J., Kumar, S., Kumar, M.: A domain decomposition method for solving singularly perturbed perturbed parabolic parabolic reaction-diffusion reaction-diffusion problems singularly problems with time delay. Numer. Methods Partial Differ. Eq. 34, 1849–1866 (2018)

    Article  Google Scholar 

  15. Ramesh, V.P., Kadalbajoo, M.K., Priyanga, B., Prithvi, M.: An a priori harmonic mesh for singularly perturbed boundary value problems. Int. J. Appl. Comput. Math. 4, 139–153 (2018)

    Article  MathSciNet  Google Scholar 

  16. Kellogg, R.B., Tsan, A.: Analysis of some difference approximations for a singular perturbation problem without turning points. Math. Comp. 32, 1025–1039 (1978)

    Article  MathSciNet  Google Scholar 

  17. Kadalbajoo, M.K., Awasthi, A.: The midpoint upwind finite difference scheme for time-dependent singularly perturbed convection-diffusion equations on non-uniform mesh. Int. J. Comput. Methods Eng. Sci. Mech. 12, 150–159 (2011)

    Article  MathSciNet  Google Scholar 

  18. Gowrisankar, S., Natesan, S.: A robust numerical scheme for singularly perturbed delay parabolic initial-boundary-value problems on equidistributed grids. Elec. Trans. Numer. Anal. 41, 376–395 (2014)

    MathSciNet  MATH  Google Scholar 

  19. Doolan, E.P., Miller, J.J.H., Schilders, W.H.A.: Uniform Numerical Methods for Problems with Initial and Boundary Layers. Boole Press, Dublin (1980)

    MATH  Google Scholar 

  20. Gowrisankar, S., Natesan, S.: ε-Uniformly convergent numerical scheme for singularly perturbed delay parabolic partial differential equations. Int. J. Comput. Math. 94, 902–921 (2017)

    Article  MathSciNet  Google Scholar 

  21. Farrell, P.A., Hegarty, A.F.: On the determination of the order of uniform convergence. In: Procs. 13th IMACS World Congress on Computational and Applied Mathematics, vol. 91, pp 501–502 (1991)

  22. Vulanovic, R.: A priori meshes for singularly perturbed quasilinear two-point boundary value problems. IMA J. Numer. Anal. 21, 349–366 (2001)

    Article  MathSciNet  Google Scholar 

  23. Shishkin, G.I.: Grid approximation of singularly perturbed elliptic and parabolic equations, second doctoral thesis. Keldysh Institute, Moscow [in Russian] (1990)

  24. Linß, T.: Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems. Lecture Notes in Mathematics, vol. 1985. Springer, Berlin (2010)

    Book  Google Scholar 

  25. Cimen, E., Amiraliyev, G.M.: Uniform convergence method for a delay differential problem with layer behaviour. Mediterr. J. Math. 16, 1–15 (2019)

    Article  MathSciNet  Google Scholar 

  26. Govindarao, L., Mohapatra, J., Das, A.: A fourth-order numerical scheme for singularly perturbed delay parabolic problem arising in population dynamics. J. Appl. Math. Comput. 63, 171–195 (2020)

    Article  MathSciNet  Google Scholar 

  27. Vulanovic, R.: On a numerical solution of a type of singularly perturbed boundary value problem by using a special discretization mesh. Univ. u Novom Sadu, Zb. Rad. Prirod.- Mat. Fak., Ser. Mat 13, 187–201 (1983)

    MathSciNet  MATH  Google Scholar 

Download references

Funding

The third author gratefully acknowledged the support of the Science and Engineering Research Board, Government of India, through the SERB Project No. MTR/2017/001014.

Author information

Authors and Affiliations

  1. Department of Mathematics, IAH, GLA University, Mathura, U.P, India

    Gajendra Babu

  2. Department of Mathematics, South Asian University, New Delhi, India

    Gajendra Babu & Kapil K. Sharma

  3. Department of Mathematics, Central University of Tamil Nadu, Thiruvarur, India

    M. Prithvi & V.P. Ramesh

Authors
  1. Gajendra Babu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Prithvi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kapil K. Sharma
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. V.P. Ramesh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kapil K. Sharma.

Ethics declarations

Conflict of interest

The third author, Kapil K. Sharma, received research support from the Science and Engineering Research Board, Government of India, through the SERB Project No. MTR/2017/001014.

Additional information

Publisher’s note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Babu, G., Prithvi, M., Sharma, K.K. et al. A robust numerical algorithm on harmonic mesh for parabolic singularly perturbed convection-diffusion problems with time delay. Numer Algor 91, 615–634 (2022). https://doi.org/10.1007/s11075-022-01275-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-022-01275-1

Keywords

Search

Navigation