On piecewise-uniform meshes for upwind- and central-difference operators for solving singularly perturbed problems

Numerical methods composed of upwind-difference operators on uniform meshes are shown analytically to be defective for solving singularly perturbed differential equations, in the sense that, as the mesh is refined, the error in the numerical approximation increases until the mesh parameter has decreased to the same order of magnitude as the singular perturbation parameter. It is also shown that the same is true for upwind-difference operators on piecewise uniform meshes having a transition point that is dependent solely on the singular perturbation parameter. It is then shown, with specific analytic examples, that both upwind- and central-difference operators on specially designed piecewise-uniform meshes give numerical methods which do not suffer from this defect. Conditions are also given on the structure of a piecewise uniform mesh that are necessary if the numerical method, composed of this mesh and an upwind-difference operator, is to be convergent uniformly with respect to the singular perturbation parameter.