A combined method of local Green's functions and central difference method for singularly perturbed convection–diffusion problems

Abstract

The paper presents a combined finite difference method with local 1D Green's functions along the characteristic lines for the numerical solution of singularly perturbed convection–diffusion problems. The advantages of this technique compared with some commonly used methods are illustrated by numerical experiments with outflow boundary layers. Numerical experiments also show that the method on a Shishkin-type mesh has discretization error of second order uniformly in the perturbation parameter. Moreover, they indicate that for the singularly perturbed problems, unstable methods (like central difference method) can be stabilized in combination with a stable method on a few mesh points.