@Article{NMTMA-1-2,
author = {Stynes, Jeanne and O'Riordan, Eugene and Stynes, Martin and Stynes, 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 = {<p style="text-align: justify;">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.</p>},
issn = {2079-7338},
doi = {https://doi.org/2008-NMTMA-10111},
url = {https://global-sci.com/article/90780/a-parameter-uniform-finite-difference-method-for-a-coupled-system-of-convection-diffusion-two-point-boundary-value-problems}
}