A UNIFORMLY CONVERGENT NUMERICAL METHOD FOR A WEAKLY COUPLED SYSTEM OF SINGULARLY PERTURBED CONVECTION-DIFFUSION PROBLEMS WITH BOUNDARY AND WEAK INTERIOR LAYERS

- Journal title : Journal of applied mathematics & informatics
- Volume 33, Issue 5_6, 2015, pp.635-648
- Publisher : The Korean Society of Computational and Applied Mathematics
- DOI : 10.14317/jami.2015.635

Title & Authors

A UNIFORMLY CONVERGENT NUMERICAL METHOD FOR A WEAKLY COUPLED SYSTEM OF SINGULARLY PERTURBED CONVECTION-DIFFUSION PROBLEMS WITH BOUNDARY AND WEAK INTERIOR LAYERS

CHAWLA, SHEETAL; RAO, S. CHANDRA SEKHARA;

CHAWLA, SHEETAL; RAO, S. CHANDRA SEKHARA;

Abstract

We consider a weakly coupled system of singularly perturbed convection-diffusion equations with discontinuous source term. The diffusion term of each equation is associated with a small positive parameter of different magnitude. Presence of discontinuity and different parameters creates boundary and weak interior layers that overlap and interact. A numerical method is constructed for this problem which involves an appropriate piecewise uniform Shishkin mesh. The numerical approximations are proved to converge to the continuous solutions uniformly with respect to the singular perturbation parameters. Numerical results are presented which illustrates the theoretical results.

Keywords

Singular perturbation;Weakly coupled system;Discontinuous source term;Uniformly convergent;Shishkin mesh;Interior layers;

Language

English

Cited by

References

1.

J.J.H. Miller, E.O. Riordan and G.I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions, World Scientific, Singapore, 1996.

2.

P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E.O. Riordan and G.I. Shishkin, Robust Computational Techniques for Boundary Layers, Chapman & Hall, CRC Press, Boca Raton, Florida, 2000.

3.

H.G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Springer, Berlin, 1996.

4.

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

5.

P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E.O Riordan and G.I. Shishkin, Singularly perturbed convection-diffusion problems with boundary and weak interior layers, J. Comput. Appl. Math. 166 (2004), 133-151.

6.

P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E.O. Riordan, and G.I. Shishkin, Global Maximum Norm Parameter-Uniform Numerical Method for a Singularly Perturbed Convection-Diffusion Problem with Discontinuous Convection Coefficient, Math. Comput. Modelling 40 (2004), 1375-1392.

7.

Z. Cen, Parameter-uniform finite difference scheme for a system of coupled singularly perturbed convection-diffusion equations, Intern. J. Comput. Math. 82 (2005), 177-192.

8.

T. Linss, Analysis of a system of singularly perturbed convection-diffusion equations with strong coupling, SIAM J. Numer. Anal. 47 (2009), 1847-1862.

9.

C. Clavero, J.L. Gracia and F. Lisbona, High order methods on Shishkin meshes for singular perturbation problems of convection-diffusion type, Numer. Algorithms 22 (1999), 73-97.

10.

E.O. Riordan and M. Stynes, Numerical analysis of a strongly coupled system of two singularly preturbed convection-diffusion problems, Adv. Comput. Math. 30 (2009), 101-121.

11.

S. Bellew and E.O. Riordan, A parameter robust numerical method for a system of two singularly perturbed convection-diffusion equations, Appl. Numer. Math. 51 (2004), 171-186.

12.

J.B. Munyakazi, A uniformly convergent nonstandard finite difference scheme for a system of convection-diffusion equations, Comp. Appl. Math. doi:10.1007/s40314-014-0171-6, (2014).

13.

R.M. Priyadharshini and N. Ramanaujam, Uniformly-convergent numerical methods for a system of coupled singularly perturbed convection-diffusion equations with mixed type boundary conditions, Math. Model. Anal. 18 (2013), 577-598.

14.

A. Tamilselvan, N. Ramanujam, R.M. Priyadharshini and T. Valanarasu, Parameter - uniform numerical method for a system of coupled singularly perturbed convection-diffusion equations with mixed type boundary conditions, J. Appl. Math. Informatics 28 (2010), 109-130.

15.

A. Tamilselvan and N. Ramanujam, A numerical method for singularly perturbed system of second order ordinary differential equations of convection diffusion type with a discontinuous source term, J. Appl. Math. Informatics 27 (2009), 1279-1292.