×

A parameter uniform fitted mesh method for a weakly coupled system of two singularly perturbed convection-diffusion equations. (English) Zbl 1427.65124

Summary: In this paper, a boundary value problem for a singularly perturbed linear system of two second order ordinary differential equations of convection-diffusion type is considered on the interval [0, 1]. The components of the solution of this system exhibit boundary layers at 0. A numerical method composed of an upwind finite difference scheme applied on a piecewise uniform Shishkin mesh is suggested to solve the problem. The method is proved to be first order convergent in the maximum norm uniformly in the perturbation parameters. Numerical examples are provided in support of the theory.

MSC:

65L11 Numerical solution of singularly perturbed problems involving ordinary differential equations
65L12 Finite difference and finite volume methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations
PDFBibTeX XMLCite
Full Text: arXiv Link

References:

[1] S. Bellow, 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. · Zbl 1059.65063
[2] Z. Cen, Parameter-uniform finite difference scheme for a system of coupled singularly perturbed convection-diffusion equations, Int. J. Comput. Math. 82(2005), 177-192. · Zbl 1068.65101
[3] C. Clavero, J. L. Gracia, F. J. Lisbona, An almost third order finite difference scheme for singularly perturbed reaction-diffusion system, J. Comput. Appl. Math. 234(2010), 2501-2515. · Zbl 1195.65101
[4] E. P. Doolan, J. J. H. Miller, W. H. A. Schilders, Uniform numerical methods for problems with initial and boundary layers, Boole Press, Dublin, 1980. · Zbl 0459.65058
[5] P. A. Farrell, A. Hegarty, J. J. H. Miller, E O’Riordan, G. I. Shishkin, Robust computational techniques for boundary layers, Chapman and Hall/CRC Press, Boca Raton, 2000. · Zbl 0964.65083
[6] T. Linss, Analysis of an upwind finite difference scheme for a system of coupled singularly perturbed convection-diffusion equations, Computing 79(2007), 23-32. · Zbl 1115.65084
[7] T. Linss, N. Madden, Layer-adapted meshes for a linear system of coupled singularly perturbed reaction-diffusion problems, IMA J. Numer. Anal. 29(2009), 109-125. · Zbl 1168.65046
[8] N. Madden, M. Stynes, A uniformly convergent numerical for a coupled system of two singularly perturbed linear reaction-diffusion problems, IMA J. Numer. Anal. 23(2003), 627-644. · Zbl 1048.65076
[9] J. J. H. Miller, E. O’Riordan, G. I. Shishkin, Fitted numerical methods for singular perturbation problems, World Scientific Publishing Co., Singapore, 1996. · Zbl 0915.65097
[10] E. O’Riordan, M. Stynes, Numerical analysis of a strongly coupled system of two singularly perturbed convection-diffusion problems, Adv. Comput. Math. 30(2009), 101-121. · Zbl 1167.65044
[11] M. Paramasivam, S. Valarmathi, J. J. H. Miller, Second order parameter-uniform convergence for a finite difference method for a singularly perturbed linear reactiondiffusion system, Math. Commun. 15(2010), 587-612. · Zbl 1223.65062
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.