摘要
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 parameterεand 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 princi- ple.Certain hypotheses are placed on the coupling matrices B and A that ensure exis- tence 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 compo- nents and their derivatives to show explicitly their dependence on the small parameterε.Finally,numerical methods consisting of upwinding on piecewise-uniform Shishkin meshes are proved to yield numerical solutions that are essentially first-order conver- gent,uniformly inε,to the true solution in the discrete maximum norm.Numerical results on Shishkin meshes are presented to support these theoretical bounds.
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.
