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Parameter-uniform finite difference schemes for singularly perturbed parabolic diffusion-convection-reaction problems


Authors: E. O'Riordan, M. L. Pickett and G. I. Shishkin
Journal: Math. Comp. 75 (2006), 1135-1154
MSC (2000): Primary 65M06, 65M15; Secondary 65M12
DOI: https://doi.org/10.1090/S0025-5718-06-01846-1
Published electronically: April 3, 2006
MathSciNet review: 2219022
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Abstract: In this paper, parameter-uniform numerical methods for a class of singularly perturbed parabolic partial differential equations with two small parameters on a rectangular domain are studied. Parameter-explicit theoretical bounds on the derivatives of the solutions are derived. The solution is decomposed into a sum of regular and singular components. A numerical algorithm based on an upwind finite difference operator and an appropriate piecewise uniform mesh is constructed. Parameter-uniform error bounds for the numerical approximations are established. Numerical results are given to illustrate the parameter-uniform convergence of the numerical approximations.


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Additional Information

E. O'Riordan
Affiliation: School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland
Email: eugene.oriordan@dcu.ie

M. L. Pickett
Affiliation: School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland
Email: maria.pickett2@mail.dcu.ie

G. I. Shishkin
Affiliation: Institute for Mathematics and Mechanics, Russian Academy of Sciences, Ekaterinburg, Russia
Email: shishkin@imm.uran.ru

DOI: https://doi.org/10.1090/S0025-5718-06-01846-1
Keywords: Two parameter, reaction-convection-diffusion, piecewise-uniform mesh
Received by editor(s): September 22, 2004
Published electronically: April 3, 2006
Additional Notes: This research was supported in part by the National Center for Plasma Science and Technology Ireland, by the Enterprise Ireland research scholarship BR-2001-110 and by the Russian Foundation for Basic Research under grant No. 04-01-00578.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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