Volume 6, Issue 5
A High-Accuracy Finite Difference Scheme for Solving Reaction-Convection-Diffusion Problems with a Small Diffusivity

Adv. Appl. Math. Mech., 6 (2014), pp. 637-662.

Published online: 2014-06

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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 $\varepsilon$. With a novel treatment for the reaction term, we first derive a difference scheme of accuracy $\mathcal{O}(\varepsilon^2 h + \varepsilon h^2 + h^3)$ 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 better stability.

• Keywords

Reaction-convection-diffusion equation, incompressible Navier-Stokes equations, boundary layer, interior layer, finite difference scheme.

65N06, 65N12, 65N15, 76M20

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@Article{AAMM-6-637, author = {}, title = {A High-Accuracy Finite Difference Scheme for Solving Reaction-Convection-Diffusion Problems with a Small Diffusivity}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2014}, volume = {6}, number = {5}, pages = {637--662}, abstract = {

This paper is devoted to a new high-accuracy finite difference scheme for solving reaction-convection-diffusion problems with a small diffusivity $\varepsilon$. With a novel treatment for the reaction term, we first derive a difference scheme of accuracy $\mathcal{O}(\varepsilon^2 h + \varepsilon h^2 + h^3)$ 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 better stability.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.2014.5.s4}, url = {http://global-sci.org/intro/article_detail/aamm/40.html} }
TY - JOUR T1 - A High-Accuracy Finite Difference Scheme for Solving Reaction-Convection-Diffusion Problems with a Small Diffusivity JO - Advances in Applied Mathematics and Mechanics VL - 5 SP - 637 EP - 662 PY - 2014 DA - 2014/06 SN - 6 DO - http://doi.org/10.4208/aamm.2014.5.s4 UR - https://global-sci.org/intro/article_detail/aamm/40.html KW - Reaction-convection-diffusion equation, incompressible Navier-Stokes equations, boundary layer, interior layer, finite difference scheme. AB -

This paper is devoted to a new high-accuracy finite difference scheme for solving reaction-convection-diffusion problems with a small diffusivity $\varepsilon$. With a novel treatment for the reaction term, we first derive a difference scheme of accuracy $\mathcal{O}(\varepsilon^2 h + \varepsilon h^2 + h^3)$ 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 better stability.

Po-Wen Hsieh, Suh-Yuh Yang & Cheng-Shu You. (1970). A High-Accuracy Finite Difference Scheme for Solving Reaction-Convection-Diffusion Problems with a Small Diffusivity. Advances in Applied Mathematics and Mechanics. 6 (5). 637-662. doi:10.4208/aamm.2014.5.s4
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