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A Novel Technique for Constructing Difference Schemes for Systems of Singularly Perturbed Equations

Part of: Basic methods in fluid mechanics Partial differential equations, boundary value problems

Published online by Cambridge University Press:  17 May 2016

Po-Wen Hsieh ,
Yin-Tzer Shih ,
Suh-Yuh Yang  and
Cheng-Shu You
Po-Wen Hsieh*
Affiliation:
Department of Applied Mathematics, Chung Yuan Christian University, Jhongli District, Taoyuan City 32023, Taiwan
Yin-Tzer Shih*
Affiliation:
Department of Applied Mathematics, National Chung Hsing University, Taichung 40227, Taiwan
Suh-Yuh Yang*
Affiliation:
Department of Mathematics, National Central University, Jhongli District, Taoyuan City 32001, Taiwan
Cheng-Shu You*
Affiliation:
Department of Mathematics, National Central University, Jhongli District, Taoyuan City 32001, Taiwan
*
*Corresponding author. Email addresses:pwhsieh0209@gmail.com (P.-W. Hsieh), yintzer_shih@nchu.edu.tw (Y.-T. Shih), syyang@math.ncu.edu.tw (S.-Y. Yang), csdyou@gmail.com (C.-S. You)
*Corresponding author. Email addresses:pwhsieh0209@gmail.com (P.-W. Hsieh), yintzer_shih@nchu.edu.tw (Y.-T. Shih), syyang@math.ncu.edu.tw (S.-Y. Yang), csdyou@gmail.com (C.-S. You)
*Corresponding author. Email addresses:pwhsieh0209@gmail.com (P.-W. Hsieh), yintzer_shih@nchu.edu.tw (Y.-T. Shih), syyang@math.ncu.edu.tw (S.-Y. Yang), csdyou@gmail.com (C.-S. You)
*Corresponding author. Email addresses:pwhsieh0209@gmail.com (P.-W. Hsieh), yintzer_shih@nchu.edu.tw (Y.-T. Shih), syyang@math.ncu.edu.tw (S.-Y. Yang), csdyou@gmail.com (C.-S. You)

Abstract

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In this paper, we propose a novel and simple technique to construct effective difference schemes for solving systems of singularly perturbed convection-diffusion-reaction equations, whose solutions may display boundary or interior layers. We illustrate the technique by taking the Il'in-Allen-Southwell scheme for 1-D scalar equations as a basis to derive a formally second-order scheme for 1-D coupled systems and then extend the scheme to 2-D case by employing an alternating direction approach. Numerical examples are given to demonstrate the high performance of the obtained scheme on uniform meshes as well as piecewise-uniform Shishkin meshes.

Keywords

System of singularly perturbed equationssystem of viscous Burgers' equationsboundary layerinterior layerIl'in-Allen-Southwell schemeformally second-order scheme

MSC classification

Secondary: 65N06: Finite difference methods 65N12: Stability and convergence of numerical methods 76M20: Finite difference methods

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 5 , May 2016 , pp. 1287 - 1301
Copyright
Copyright © Global-Science Press 2016 

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