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Differential Equations and Dynamical Systems

, Volume 25, Issue 2, pp 301–325 | Cite as

Parameter-Robust Numerical Method for Time-Dependent Weakly Coupled Linear System of Singularly Perturbed Convection-Diffusion Equations

  • S. Chandra Sekhara Rao
  • Varsha Srivastava
Original Research

Abstract

We present a parameter-robust numerical method for a time-dependent weakly coupled linear system of singularly perturbed convection-diffusion equations. A small perturbation parameter multiplies the second order spatial derivative in all the equations. The proposed numerical method uses backward Euler method in time direction on an uniform mesh together with a suitable combination of HODIE scheme and the central difference scheme in spatial direction on a Shishkin mesh. It is proved that the numerical method is parameter-robust of first order in time and almost second order in space. Numerical results are given in support of theoretical findings.

Keywords

Time-dependent convection-diffusion problems Weakly coupled systems Shishkin mesh Backward Euler method HODIE scheme Parameter-uniform convergence 

Mathematics Subject Classification

65M06 65M12 65M15 

Notes

Acknowledgments

The authors gratefully acknowledge the valuable comments and suggestions from the anonymous referees. The research work of the second author is supported by Council of Scientific and Industrial Research, India.

References

  1. 1.
    Clavero, C., Gracia, J.L., Lisbona, F.: High order methods on Shishkin meshes for singular perturbation problems of convection-diffusion type. Numer. Algorithms 22, 73–97 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Clavero, C., Gracia, J.L., Jorge, J.C.: High-order numerical methods for one-dimensional parabolic singularly perturbed problems with regular layers. Numer. Meth. Part. Diff. Equat. 21, 149–169 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Clavero, C., Gracia, J.L., Stynes, M.: A simpler analysis of a hybrid numerical method for time-dependent convection-diffusion problems. J. Comput. Appl. Math. 235, 5240–5248 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kadalbajoo, M.K., Awasthi, A.: A parameter uniform difference scheme for singularly perturbed parabolic problem in one space dimension. Appl. Math. Comput. 183, 42–60 (2006)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Kellogg, R.B., Tsan, A.: n Analysis of some difference approximations for a singular perturbation problem without turning points. Math. Comput. 32, 1025–1039 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kopteva, N.: Uniform pointwise convergence of difference schemes for convection-diffusion problems on layer-adapted meshes. Computing 66, 179–197 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Lenferink, W.: A second order scheme for a time-dependent singularly perturbed convection-diffusion equation. J. Comput. Appl. Math. 143, 49–68 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Linss, T.: Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems. Springer-Verlog, Berlin (2010)CrossRefzbMATHGoogle Scholar
  9. 9.
    Marchuk, G.I.: Methods of Numerical Mathematics. Springer, New York (1982)CrossRefzbMATHGoogle Scholar
  10. 10.
    Roos, H.G., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations, 2nd edn. Springer-Verlag, Berlin (2008)zbMATHGoogle Scholar
  11. 11.
    Samarskii, A.A.: Theory of Difference Schemes. Published in English by Marcel Dekker, CRC Press (2001)Google Scholar
  12. 12.
    Shishkin, G.I.: Grid approximation of singularly perturbed boundary value problems for systems of elliptic and parabolic equations. Comp. Maths. Math. Phys. 35, 429–446 (1995)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Shishkin, G.I.: Grid approximation of singularly perturbed systems of elliptic and parabolic equations with convective terms. Differ. Equ. 34, 1693–1704 (1998)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Stynes, M., Tobiska, L.: A finite difference analysis of a streamline diffusion method on a Shishkin mesh. Numer. Algorithms 18, 337–360 (1998)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2016

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology DelhiNew DelhiIndia

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