At low Mach number, the Roe scheme presents an excess of artificial viscosity. A correction of this scheme, which uses the preconditioning of Turkel, leads to an improvement of the solution. We refer to this new scheme as the Roe-Turkel scheme. We show that for the Roe-Turkel scheme the convergence of the numerical solution towards the exact solution depends only on the mesh size parameter whereas that of the Roe scheme depends on the ratio between mesh size parameter and Mach number. We also show that for the computation of steady state solutions, when the Roe-Turkel scheme is introduced in the physical phase and in the mathematical phase, the iterative convergence of the implicit Defect-Correction method is only moderately affected at low Mach number as compared with the transonic regime. The proposed formulation is consistent in time (applicable to unsteady flows). It enables a solution at Mach $10^{-6}$ of the same quality on a same mesh as that obtained at Mach $10^{-1}$ for flow around a NACA0012 airfoil. The results obtained at Mach $10^{-3}$ are close to those given by an incompressible Navier-Stokes code on comparable mesh for shear-driven cavity flow.