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Volume 1, Issue 2
A Parameter-Uniform Finite Difference Method for a Coupled System of Convection-Diffusion Two-Point Boundary Value Problems

Eugene O'Riordan, Jeanne Stynes & Martin Stynes

Numer. Math. Theor. Meth. Appl., 1 (2008), pp. 176-197.

Published online: 2008-01

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  • Abstract

A system of $m$ (≥ 2) linear convection-diffusion two-point boundary value problems is examined, where the diffusion term in each equation is multiplied by a small parameter $ε$ and the equations are coupled through their convective and reactive terms via matrices $B$ and $A$ respectively. This system is in general singularly perturbed. Unlike the case of a single equation, it does not satisfy a conventional maximum principle. Certain hypotheses are placed on the coupling matrices $B$ and $A$ that ensure existence and uniqueness of a solution to the system and also permit boundary layers in the components of this solution at only one endpoint of the domain; these hypotheses can be regarded as a strong form of diagonal dominance of $B$. This solution is decomposed into a sum of regular and layer components. Bounds are established on these components and their derivatives to show explicitly their dependence on the small parameter $ε$. Finally, numerical methods consisting of upwinding on piecewise-uniform Shishkin meshes are proved to yield numerical solutions that are essentially first-order convergent, uniformly in $ε$, to the true solution in the discrete maximum norm. Numerical results on Shishkin meshes are presented to support these theoretical bounds.

  • AMS Subject Headings

65L10, 65L12, 65L20, 65L70

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COPYRIGHT: © Global Science Press

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@Article{NMTMA-1-176, author = {O'Riordan , EugeneStynes , Jeanne and Stynes , Martin}, title = {A Parameter-Uniform Finite Difference Method for a Coupled System of Convection-Diffusion Two-Point Boundary Value Problems}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2008}, volume = {1}, number = {2}, pages = {176--197}, abstract = {

A system of $m$ (≥ 2) linear convection-diffusion two-point boundary value problems is examined, where the diffusion term in each equation is multiplied by a small parameter $ε$ and the equations are coupled through their convective and reactive terms via matrices $B$ and $A$ respectively. This system is in general singularly perturbed. Unlike the case of a single equation, it does not satisfy a conventional maximum principle. Certain hypotheses are placed on the coupling matrices $B$ and $A$ that ensure existence and uniqueness of a solution to the system and also permit boundary layers in the components of this solution at only one endpoint of the domain; these hypotheses can be regarded as a strong form of diagonal dominance of $B$. This solution is decomposed into a sum of regular and layer components. Bounds are established on these components and their derivatives to show explicitly their dependence on the small parameter $ε$. Finally, numerical methods consisting of upwinding on piecewise-uniform Shishkin meshes are proved to yield numerical solutions that are essentially first-order convergent, uniformly in $ε$, to the true solution in the discrete maximum norm. Numerical results on Shishkin meshes are presented to support these theoretical bounds.

}, issn = {2079-7338}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/nmtma/10111.html} }
TY - JOUR T1 - A Parameter-Uniform Finite Difference Method for a Coupled System of Convection-Diffusion Two-Point Boundary Value Problems AU - O'Riordan , Eugene AU - Stynes , Jeanne AU - Stynes , Martin JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 176 EP - 197 PY - 2008 DA - 2008/01 SN - 1 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/nmtma/10111.html KW - Singularly perturbed, convection-diffusion, coupled system, piecewise-uniform mesh. AB -

A system of $m$ (≥ 2) linear convection-diffusion two-point boundary value problems is examined, where the diffusion term in each equation is multiplied by a small parameter $ε$ and the equations are coupled through their convective and reactive terms via matrices $B$ and $A$ respectively. This system is in general singularly perturbed. Unlike the case of a single equation, it does not satisfy a conventional maximum principle. Certain hypotheses are placed on the coupling matrices $B$ and $A$ that ensure existence and uniqueness of a solution to the system and also permit boundary layers in the components of this solution at only one endpoint of the domain; these hypotheses can be regarded as a strong form of diagonal dominance of $B$. This solution is decomposed into a sum of regular and layer components. Bounds are established on these components and their derivatives to show explicitly their dependence on the small parameter $ε$. Finally, numerical methods consisting of upwinding on piecewise-uniform Shishkin meshes are proved to yield numerical solutions that are essentially first-order convergent, uniformly in $ε$, to the true solution in the discrete maximum norm. Numerical results on Shishkin meshes are presented to support these theoretical bounds.

Eugene O'Riordan, Jeanne Stynes & Martin Stynes. (2020). A Parameter-Uniform Finite Difference Method for a Coupled System of Convection-Diffusion Two-Point Boundary Value Problems. Numerical Mathematics: Theory, Methods and Applications. 1 (2). 176-197. doi:
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