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 - <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>