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ANZIAM J. 47(EMAC2005) pp.C633--C648, 2007.

A Reynolds uniform scheme for singularly perturbed parabolic differential equation

X. Cai

F. Liu

(Received 28 October 2005; revised 21 February 2007)

Abstract

Time dependent convection diffusion problems with large Reynolds number are considered. Such a problem has been considered by using Shishkin's scheme, which was uniformly convergent with respect to large Reynolds number in order O(N^-1log²N+M^-1), where N and M are number of intervals in x direction and t direction respectively. A three-transition points scheme, four piecewise-uniform mesh, is introduced. The mesh partition, the barrier function, the estimate of truncation error and the techniques of proof are different from others. The new scheme is non-equidistant. It is proved uniformly convergent with respect to large Reynolds number in order O(N^-1+M^-1). Our work is better than Shishkin's traditional scheme, while the computational procedure is as simple as Shishkin's scheme. This novel method also has the same accurate result as Bakhvalov--Shishkin's scheme, while the computational procedure is simpler than Bakhvalov--Shishkin's scheme. Shishkin's scheme and Bakhvalov--Shishkin's scheme are compared with the new method. Finally, numerical results support the theoretical results.

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Authors

X. Cai

School of Sciences, Jimei University, Xiamen 361021, China. mailto:cx85@263.net, cxxm05@126.com

F. Liu

School of Mathematical Sciences, Queensland University of Technology, Queensland 4001, Australia.

Published April 9, 2007. ISSN 1446-8735

References

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3. P. Farrell, A. F. Hegarty, J. J. H. Miller, E. O'Riordan and G. I. Shishkin, Robust Computational Techniques for Boundary layers, Chapman and Hall/CRC, Boca Raton, 2000.
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