A uniformly convergent nonstandard finite difference scheme for a system of convection-diffusion equations

We propose a nonstandard finite difference method to integrate coupled systems of singularly perturbed convection-diffusion equations. We prove that the underlying discrete operator satisfies a stability property in the maximum norm. This fact is used to prove that the proposed method converges uniformly with respect to the singular perturbation parameters and the convergence rate is linear with respect to the step size. Numerical investigations are presented to confirm our theoretical findings. We also show that the proposed approach compares well with existing methods.