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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.

  • Keywords

Singularly perturbed, convection-diffusion, coupled system, piecewise-uniform mesh.

  • AMS Subject Headings

65L10, 65L12, 65L20, 65L70

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-1-176, author = {Eugene and O'Riordan and and 9729 and and Eugene O'Riordan and Jeanne and Stynes and and 9730 and and Jeanne Stynes and Martin and Stynes and and 9731 and and Martin Stynes}, 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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