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A High-Accuracy Finite Difference Scheme for Solving Reaction-Convection-Diffusion Problems with a Small Diffusivity

Published online by Cambridge University Press:  03 June 2015

Po-Wen Hsieh ,
Suh-Yuh Yang  and
Cheng-Shu You
Po-Wen Hsieh*
Affiliation:
Department of Applied Mathematics, Chung Yuan Christian University, Jhongli City, Taoyuan County 32023, Taiwan
Suh-Yuh Yang*
Affiliation:
Department of Mathematics, National Central University, Jhongli City, Taoyuan County 32001, Taiwan
Cheng-Shu You*
Affiliation:
Department of Mathematics, National Central University, Jhongli City, Taoyuan County 32001, Taiwan
*
Email: pwhsieh0209@gmail.com
Corresponding author. Email: syyang@math.ncu.edu.tw
Email: csdyou@gmail.com
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Abstract

This paper is devoted to a new high-accuracy finite difference scheme for solving reaction-convection-diffusion problems with a small diffusivity ε. With a novel treatment for the reaction term, we first derive a difference scheme of accuracy O(εh2+εh2+h3) for the 1-D case. Using the alternating direction technique, we then extend the scheme to the 2-D case on a nine-point stencil. We apply the high-accuracy finite difference scheme to solve the 2-D steady incompressible Navier-Stokes equations in the stream function-vorticity formulation. Numerical examples are given to illustrate the effectiveness of the proposed difference scheme. Comparisons made with some high-order compact difference schemes show that the newly proposed scheme can achieve good accuracy with a better stability.

Keywords

65N0665N1265N1576M20Reaction-convection-diffusion equationincompressible Navier-Stokes equationsboundary layerinterior layerfinite difference scheme
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 6 , Issue 5 , October 2014 , pp. 637 - 662
Copyright
Copyright © Global-Science Press 2014

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