Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

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

Uniformly convergent numerical method for solving modified Burgers' equations on a non-uniform mesh 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Numerical Mathematics de Gruyter

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

Download PDF
19 pages

Loading next page...
 
/lp/de-gruyter/uniformly-convergent-numerical-method-for-solving-modified-burgers-yUxSGHKEqn

References (12)

Publisher
de Gruyter
Copyright
© de Gruyter 2008
ISSN
1570-2820
eISSN
1569-3953
DOI
10.1515/JNUM.2008.010
Publisher site
See Article on Publisher Site

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.

Journal

Journal of Numerical Mathematics – de Gruyter

Published: Nov 1, 2008

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

There are no references for this article.