In this paper the CSCM type upwind flux difference splitting Navier-Stokes method has been applied to study the ARL-SL19 supersonic/transonic compressor cascade flow. H-type grid was chosen for its simplicity in applying cyclic tridiagonal matrix algorithm along with conventional slip/no-slip boundary conditions. The thin-layer algebraic model of Baldwin-Lomax was employed for the calculation of turbulent flows. The test case inlet Mach No. was 1.612 and inlet/exit pressure ratio($P_2/P_1$) was 2.15. The results were compared with experimental results from current method were compared well in suction surface with the experiments and other computational results; however, not well in pressure surface. It might be due to the complex flowfields such as shock/boundary layer interaction, turbulence, and flow separation, etc. In the future, a proper turbulence modelling and adaptive grid system will be studied to improve the solution quality.