A hybrid difference scheme on a Shishkin mesh for linear convection–diffusion problems

Abstract

We consider a new difference scheme on a special piecewise equidistant tensor-product mesh (a Shishkin mesh) for a model singularly perturbed convection–diffusion problem in two dimensions. Our hybrid method chooses between upwinding and central differencing, depending on the local mesh width in each coordinate direction. We prove that this method is first-order convergent in the discrete L ∞ norm, independently of the diffusion parameter. Thus the new scheme is more accurate than simple upwinding (which is the standard difference method used on Shishkin meshes), even though it requires exactly the same computational effort. Numerical experiments support these theoretical results.