A boundary value problem is considered for a singularly perturbed parabolic convection-diffusion equation; we construct a finite difference scheme on α priori （sequentially） adapted meshes and study its convergence...A boundary value problem is considered for a singularly perturbed parabolic convection-diffusion equation; we construct a finite difference scheme on α priori （sequentially） adapted meshes and study its convergence. The scheme on α priori adapted meshes is constructed using a majorant function for the singular component of the discrete solution, which allows us to find α priori a subdomain where the computed solution requires a further improvement. This subdomain is defined by the perturbation parameter ε, the step-size of a uniform mesh in χ, and also by the required accuracy of the discrete solution and the prescribed number of refinement iterations K for improving the solution. To solve the discrete problems aimed at the improvement of the solution, we use uniform meshes on the subdomains. The error of the numerical solution depends weakly on the parameter ε. The scheme converges almost ε-uniformly, precisely, under the condition N^-1 = o （ε^v）, where N denotes the number of nodes in the spatial mesh, and the value v = v（K） can be chosen arbitrarily small for suitable K.展开更多
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 c...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.展开更多
文摘A boundary value problem is considered for a singularly perturbed parabolic convection-diffusion equation; we construct a finite difference scheme on α priori （sequentially） adapted meshes and study its convergence. The scheme on α priori adapted meshes is constructed using a majorant function for the singular component of the discrete solution, which allows us to find α priori a subdomain where the computed solution requires a further improvement. This subdomain is defined by the perturbation parameter ε, the step-size of a uniform mesh in χ, and also by the required accuracy of the discrete solution and the prescribed number of refinement iterations K for improving the solution. To solve the discrete problems aimed at the improvement of the solution, we use uniform meshes on the subdomains. The error of the numerical solution depends weakly on the parameter ε. The scheme converges almost ε-uniformly, precisely, under the condition N^-1 = o （ε^v）, where N denotes the number of nodes in the spatial mesh, and the value v = v（K） can be chosen arbitrarily small for suitable K.
文摘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.
