Third order Compact Upwind Difference (CUD) Schemes are applied to compute steady-state solutions to the inviscid and viscous Burgers equation. A simple estimate shows and numerical experiments confirm, that the error in the maximum norm is inversely proportional to the square of (he number of grid points in the viscous layer. The discretization error of the viscous term appears to dominate the overall performance of the scheme in the convection-diffusion problem.