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.