Fractional Step Method for Singularly Perturbed 2D Delay Parabolic Convection Diffusion Problems on Shishkin Mesh

  • Abhishek Das
  • Srinivasan Natesan
Original Paper


In this article, we are interested to approximate the solution of a singularly perturbed 2D delay parabolic convection–diffusion initial-boundary-value problem. To discretize the continuous problem in the temporal direction, we use a fractional step method which results a set of two 1D problems. Next, we apply classical finite difference scheme on a special mesh to discretize those 1D problems in the spatial directions. Fractional step method for the time variable permits the computational cost reduction and the special mesh is used to capture the boundary layers. We derive the truncation errors for the scheme to obtain the error estimates, which shows that the scheme is uniformly convergent of almost first-order (up to a logarithmic factor) in space and first-order in time. Numerical examples are presented to support the theoretical results.


Singularly perturbed 2D delay parabolic convection–diffusion problems Boundary layers Finite difference scheme Piecewise-uniform Shishkin meshes Fractional step method Uniform convergence 

Mathematics Subject Classification

65M06 65M12 65M15 


  1. 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(1), 552–566 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Clavero, C., Gracia, J.L., Jorge, J.C.: A uniformly convergent alternating direction HODIE finite difference scheme for 2D time-dependent convection–diffusion problems. IMA J. Numer. Anal. 26(1), 155–172 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Clavero, C., Jorge, J.C., Lisbona, F., Shishkin, G.I.: A fractional step method on a special mesh for the resolution of multidimensional evolutionary convection–diffusion problems. Appl. Numer. Math. 27(3), 211–231 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Das, A., Natesan, S.: Uniformly convergent hybrid numerical scheme for singularly perturbed delay parabolic convection–diffusion problems on Shishkin mesh. Appl. Math. Comput. 271, 168–186 (2015)MathSciNetGoogle Scholar
  5. 5.
    Gowrisankar, S., Natesan, S.: A robust numerical scheme for singularly perturbed delay parabolic initial-boundary-value problems on equidistributed grids. Electron. Trans. Numer. Anal. 41, 376–395 (2014)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Gowrisankar, S., Natesan, S.: \(\varepsilon \)-Uniformly convergent numerical scheme for singularly perturbed delay parabolic partial differential equations. Int. J. Comput. Math. 94(5), 902–921 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kellogg, R.B., Tsan, A.: Analysis of some difference approximations for a singular perturbation problem without turning points. Math. Comput. 32(144), 1025–1039 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press Inc, Boston (1993)zbMATHGoogle Scholar
  9. 9.
    Linß, T., Madden, N.: Analysis of an alternating direction method applied to singularly perturbed reaction–diffusion problems. Int. J. Numer. Anal. Model. 7(3), 507–519 (2010)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Nelson, P.W., Perelson, A.S.: Mathematical analysis of delay differential equation models of HIV-1 infection. Math. Biosci. 179, 73–94 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Villasana, M., Radunskaya, A.: A delay differential equation model for tumor growth. J. Math. Biol. 47, 270–294 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Wang, P.K.C.: Asymptotic stability of a time-delayed diffusion system. Trans. ASME Ser. E. J. Appl. Mech. 30, 500–504 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Zhao, T.: Global periodic-solutions for a differential delay system modeling a microbial population in the chemostat. J. Math. Anal. Appl. 193, 329–352 (1995)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer (India) Private Ltd., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Science and TechnologyICFAI UniversityAgartalaIndia
  2. 2.Department of MathematicsIndian Institute of TechnologyGuwahatiIndia

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