Skip to main content
Log in

A Numerical Method for Solving Boundary and Interior Layers Dominated Parabolic Problems with Discontinuous Convection Coefficient and Source Terms

  • Original Research
  • Published:
Differential Equations and Dynamical Systems Aims and scope Submit manuscript

Abstract

In this article, a parameter uniform numerical method is developed for a two-parameter singularly perturbed parabolic partial differential equation with discontinuous convection coefficient and source term. The presence of perturbation parameter and the discontinuity in the convection coefficient and source term lead to the boundary and interior layers in the solution. On the spatial domain, an adaptive mesh is introduced before discretizing the continuous problem. The present method observes a uniform convergence in maximum norm which is almost first-order in space and time irrespective of the relation between convection and diffusion parameters. Numerical experiment is carried out to validate the present scheme.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /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.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Chandru, M., Prabha, T., Shanthi, V.: A parameter robust higher order numerical method for singularly perturbed two parameter problems with non-smooth data. J. Comput. Appl. Math. 309, 11–27 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  2. Clavero, C., Gracia, J.L., Shishkin, G.I., Shishkina, L.P.: An efficient numerical scheme for 1d parabolic singularly perturbed problems with an interior and boundary layers. J. Comput. Appl. Math. 318, 634–645 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Das, P.: Comparison of a priori and a posteriori meshes for singularly perturbed nonlinear parameterized problems. J. Comput. Appl. Math. 290, 16–25 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Das, P., Mehrmann, V.: Numerical solution of singularly perturbed convection-diffusion-reaction problems with two small parameters. BIT Numer. Math. 56(1), 51–76 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Das, P., Natesan, S.: Higher order parameter uniform convergent schemes for robin type reaction diffusion problems using adaptively generated grid. Int. J. Comput. Methods (2012). doi:10.1142/S0219876212500521

  6. Das, P., Natesan, S.: Numerical solution of a system of singularly perturbed convection-diffusion boundary-value problems using mesh equidistribution technique. Aust. J. Math. Anal. Appl. 10(1), 1–17 (2013)

    MathSciNet  MATH  Google Scholar 

  7. Das, P., Natesan, S.: Richardson extrapolation method for singularly perturbed convection-diffusion problems on adaptively generated mesh. CMES. Comput. Model. Eng. Sci. 90(6), 463–485 (2013)

    MathSciNet  MATH  Google Scholar 

  8. Das, P., Natesan, S.: Adaptive mesh generation for singularly perturbed fourth order ordinary differential equations. Int. J. Comput. Math. 92, 562–578 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  10. Gander, M.J., Kwok, F., Mandal, B.C.: Dirichlet–Neumann and Neumann–Neumann waveform relaxation for the Wave equation. Domain Decompos. Sci. Eng. XXII, LNCSE 104, 501–509 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gander, M.J., Kwok, F., Mandal, B.C.: Dirichlet-Neumann and Neumann-Neumann waveform relaxation algorithms for parabolic problems. Electron. Trans. Numer. Anal. 45, 424–456 (2016)

    MathSciNet  MATH  Google Scholar 

  12. Gracia, J.L., O’Riordan, E., Pickett, M.L.: A parameter robust second order numerical method for a singularly perturbed two-parameter problem. Appl. Numer. Math. 56(7), 962–980 (2006). doi:10.1016/j.apnum.2005.08.002

    Article  MathSciNet  MATH  Google Scholar 

  13. Jha, A., Kadalbajoo, M.K.: A robust layer adapted difference method for singularly perturbed two-parameter parabolic problems. Int. J. Comput. Math. 92(6), 1204–1221 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kadalbajoo, M.K., Jha, A.: Exponentially fitted cubic spline for two parameter singularly perturbed boundary value problems. Int. J. Comput. Math. 89(6), 836–850 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kadalbajoo, M.K., Yadaw, A.S.: Parameter-uniform finite element method for two-parameter singularly perturbed parabolic reaction-diffusion problems. Int. J. Comput. Methods 9(04) (2012). doi:10.1142/S0219876212500478.

  16. Kaushik, A.: Singular perturbation analysis of bistable differential equation arising in the nerve pulse propagation. Nonlinear Anal. Real World Appl. 9(5), 2106–2127 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. 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(12), 4232–4242 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kaushik, A., Sharma, M.: An optimal asymptotic-numerical method for convection dominated systems having exponential boundary layers. J. Differ. Equ. Appl. 22(9), 1307–1324 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’tseva, N.N.: Linear and Quasi-Linear Equations of Parabolic Type, Translations of Mathematical Monographs, vol. 23. American Mathematical Society (1968)

  20. Linß, T.: A posteriori error estimation for a singularly perturbed problem with two small parameters. Int. J. Numer. Anal. Model. 7(3), 491–506 (2010)

    MathSciNet  MATH  Google Scholar 

  21. Munyakazi, J.B.: A robust finite difference method for two-parameter parabolic convection-diffusion problems. Appl. Math 9(6), 2877–2883 (2015)

    MathSciNet  Google Scholar 

  22. O’Malley, R.E.: Two-parameter singular perturbation problems for second-order equations(constant and variable coefficient initial and boundary value problems for second order differential equations). J. Math. Mech. 16, 1143–1164 (1967)

    MathSciNet  MATH  Google Scholar 

  23. O’Riordan, E.: Opposing flows in a one dimensional convection-diffusion problem. Central Eur. J. Math. 10(1), 85–100 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. O’Riordan, E.: Interior layers in singularly perturbed problems. In: Differential Equations and Numerical Analysis. Springer, New York (2016)

  25. O’Riordan, E., Pickett, M.L., Shishkin, G.I.: Singularly perturbed problems modeling reaction-convection-diffusion processes. Comput. Methods Appl. Math. 3(3), 424–442 (2003). doi:10.2478/cmam-2003-0028

    Article  MathSciNet  MATH  Google Scholar 

  26. O’Riordan, E., Pickett, M.L., Shishkin, G.I.: Parameter-uniform finite difference schemes for singularly perturbed parabolic diffusion-convection-reaction problems. Math. Comput. 75(255), 1135–1154 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. O’Riordan, E., Shishkin, G.I.: Singularly perturbed parabolic problems with non-smooth data. J. Comput. Appl. Math. 166(1), 233–245 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  28. Patidar, K.C.: A robust fitted operator finite difference method for a two-parameter singular perturbation problem. J. Differ. Equ. Appl. 14(12), 1197–1214 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Schlichting, H., Gersten, K., Krause, E., Oertel, H., Mayes, K.: Boundary-Layer Theory, vol. 7. Springer, New York (1960)

    Google Scholar 

  30. Shanthi, V., Ramanujam, N., Natesan, S.: Fitted mesh method for singularly perturbed reaction-convection-diffusion problems with boundary and interior layers. J. Appl. Math. Comput. 22(1–2), 49–65 (2006). doi:10.1007/BF02896460

    Article  MathSciNet  MATH  Google Scholar 

  31. Shishkin, G.I.: Approximation of singularly perturbed parabolic reaction-diffusion equations with non-smooth data. Computational Methods in Applied Mathematics Comput. Methods. Appl. Math. 1(3), 298–315 (2001)

    MathSciNet  Google Scholar 

  32. Vulanović, R.: A higher-order scheme for quasilinear boundary value problems with two small parameters. Computing 67(4), 287–303 (2001)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to P. Das.

Additional information

The first and fourth authors wish to thank Department of Science and Technology(SERB), Government of India, New Delhi for financial support of project SR/FTP/MS-039/2012.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chandru, M., Prabha, T., Das, P. et al. A Numerical Method for Solving Boundary and Interior Layers Dominated Parabolic Problems with Discontinuous Convection Coefficient and Source Terms. Differ Equ Dyn Syst 27, 91–112 (2019). https://doi.org/10.1007/s12591-017-0385-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12591-017-0385-3

Keywords

Mathematics Subject Classification

Navigation