Abstract: Spatial discretization of N-S equations was first deduced based on the classical Galerkin method. Taking the above semi-discretized formulation as the basic scheme, a diffusion and an associated anti-diffusion terms were then added, keeping the scheme to possess a local extremum diminishing (LED) property. After taking the time discretization of the above derived semi-discretized scheme, the resulting system of sparse linear algebraic equations was solved by utilizing generalized minimal residual(GMRES). In order to validate the developed scheme with associated code, numerical computations for the shock tube problems, as well as supersonic flows over a cylinder and a double ellipsoid were performed, which revealed that the developed scheme did provide high resolution results.