Top

Journal of Applied Mathematics and Computing

Published in:

02-03-2018 | Original Research

Parameter-uniform numerical method for singularly perturbed 2D delay parabolic convection–diffusion problems on Shishkin mesh

Authors: Abhishek Das, Srinivasan Natesan

Published in: Journal of Applied Mathematics and Computing | Issue 1-2/2019

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

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.

previous article A parameter-uniform numerical scheme for the parabolic singularly perturbed initial boundary value problems with large time delay

next article Computational algorithm for solving singular Fredholm time-fractional partial integrodifferential equations with error estimates

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

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)MathSciNetCrossRefMATH

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)MathSciNetCrossRefMATH

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)MathSciNetCrossRef

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)MathSciNetCrossRefMATH

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)MathSciNetMATH

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)MathSciNetMATH

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)MathSciNetCrossRefMATH

Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press Inc, Boston, MA (1993)MATH

Ladyženskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society, Providence, RI (1968)CrossRef

10.

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)MathSciNetCrossRefMATH

11.

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)MathSciNetCrossRefMATH

12.

Roos, H.G., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations. Springer, Berlin (2008)MATH

13.

Shishkin, G.I., Shishkina, L.P.: Difference Methods for Singular Perturbation Problems. CRC Press, Boca Raton, FL (2009)MATH

14.

Wang, P.K.C.: Asymptotic stability of a time-delayed diffusion system. J. Appl. Mech. 30, 500–504 (1963)MathSciNetCrossRefMATH

Title: Parameter-uniform numerical method for singularly perturbed 2D delay parabolic convection–diffusion problems on Shishkin mesh
Authors: Abhishek Das
Srinivasan Natesan
Publication date: 02-03-2018
Publisher: Springer Berlin Heidelberg
Published in: Journal of Applied Mathematics and Computing / Issue 1-2/2019
Print ISSN: 1598-5865
Electronic ISSN: 1865-2085
DOI: https://doi.org/10.1007/s12190-018-1175-y

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Other articles of this Issue 1-2/2019

A quadrat neighborhood estimator for intensity function of point processes

Study on negacyclic codes over the ring

A priori error analysis of the stabilized Lagrange multiplier method for elliptic problems with natural norms

Effect of one redundant parity-check equation on the stopping distance

Drug therapy model with time delays for HIV infection with virus-to-cell and cell-to-cell transmissions

Simplified full Nesterov–Todd step infeasible interior-point algorithm for semidefinite optimization based on a kernel function

Premium Partner