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Journal of Numerical Mathematics

Editor-in-Chief: Hoppe, Ronald H. W. / Kuznetsov, Yuri

Managing Editor: Olshanskii, Maxim

Editorial Board Member: Benzi, Michele / Brenner, Susanne C. / Carstensen, Carsten / Dryja, M. / Feistauer, Miloslav / Glowinski, R. / Lazarov, Raytcho / Nataf, Frédéric / Neittaanmaki, P. / Bonito, Andrea / Quarteroni, Alfio / Guzman, Johnny / Rannacher, Rolf / Repin, Sergey I. / Shi, Zhong-ci / Tyrtyshnikov, Eugene E. / Zou, Jun / Simoncini, Valeria / Reusken, Arnold

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Uniformly convergent numerical method for solving modified Burgers' equations on a non-uniform mesh

M. K. Kadalbajoo
  • *Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur 208016, India
/ A. Awasthi
  • Department of Mathematics and Humanities, National Institute of Technology, Warangal 506004, India
Published Online: 2008-11-24 | DOI: https://doi.org/10.1515/JNUM.2008.010

Abstract

In this paper, we consider the one-dimensional modified Burgers' equation in the finite domain. This type of problem arises in the field of sonic boom and explosions theory. At the high Reynolds' number there is a boundary layer in the right side of the domain. From the numerical point of view, one of the difficulties in dealing with this problem is that even smooth initial data can give rise to solution varying regions, i.e., boundary layer regions. To tackle this situation, we propose a numerical method on non-uniform mesh of Shishkin type, which works well at high as well as low Reynolds number. The proposed numerical method comprises of Euler implicit and upwind finite difference scheme. First we discretize in the temporal direction by means of Euler implicit method which yields the set of ordinary differential equations at each time level. The resulting set of differential equations are approximated by upwind scheme on Shishkin mesh. The proposed method has been shown to be parameter uniform and of almost first order accurate in the space and time. An extensive amount of analysis has been carried out in order to prove parameter uniform convergence of the method. some test examples have been solved to verify the theoretical results.

Keywords:: Modified Burgers' equation; Euler implicit method; Shishkin mesh; upwind scheme and uniform convergence


Received: 2008-01-28

Revised: 2008-04-13

Published Online: 2008-11-24

Published in Print: 2008-11-01


Citation Information: Journal of Numerical Mathematics. Volume 16, Issue 3, Pages 217–235, ISSN (Online) 1569-3953, ISSN (Print) 1570-2820, DOI: https://doi.org/10.1515/JNUM.2008.010, November 2008

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