Abstract
In this article, we study the numerical solution of a singularly perturbed 2D delay parabolic convection–diffusion problem. First, we discretize the domain with a uniform mesh in the temporal direction and a special mesh in the spatial directions. The numerical scheme used to discretize the continuous problem, consists of the implicit-Euler scheme for the time derivative and the classical upwind scheme for the spatial derivatives. Stability analysis is carried out, and parameter-uniform error estimates are derived. The proposed scheme is of almost first-order (up to a logarithmic factor) in space and first-order in time. Numerical examples are carried out to verify the theoretical results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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)
Britton, N.F.: Spatial structures and periodic travelling waves in an integro-differential reaction–diffusion population model. SIAM J. Appl. Math. 50(6), 1663–1688 (1990)
Clavero, C., Jorge, J.C.: Another uniform convergence analysis technique of some numerical methods for parabolic singularly perturbed problems. Comput. Math. Appl. 70(3), 222–235 (2015)
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)
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)
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)
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)
Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press Inc, Boston, MA (1993)
Ladyženskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society, Providence, RI (1968)
Linß, T., Stynes, M.: A hybrid difference scheme on a Shishkin mesh for linear convection–diffusion problems. Appl. Numer. Math. 31(3), 255–270 (1999)
Linß, T., Stynes, M.: Asymptotic analysis and Shishkin-type decomposition for an elliptic convection–diffusion problem. J. Math. Anal. Appl. 261(2), 604–632 (2001)
Roos, H.G., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations. Springer, Berlin (2008)
Shishkin, G.I., Shishkina, L.P.: Difference Methods for Singular Perturbation Problems. CRC Press, Boca Raton, FL (2009)
Wang, P.K.C.: Asymptotic stability of a time-delayed diffusion system. J. Appl. Mech. 30, 500–504 (1963)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Das, A., Natesan, S. Parameter-uniform numerical method for singularly perturbed 2D delay parabolic convection–diffusion problems on Shishkin mesh. J. Appl. Math. Comput. 59, 207–225 (2019). https://doi.org/10.1007/s12190-018-1175-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-018-1175-y
Keywords
- Singularly perturbed 2D delay parabolic problems
- Boundary layers
- Upwind scheme
- Piecewise-uniform Shishkin mesh
- Uniform convergence