In this paper the CSCM type upwind flux difference splitting Navier-Stokes method has been applied to study the ARL-SL19 transoni $c^ersonic compressor cascade flow. First, the general characteristics of baseline cascade flow were analyzed. At freestream Mach n.1.612 and exit/inlet pressure ratio 2.15, the results from current laminar flow were compared well in suction surface with the experiment; however, not well in pressure surface. Second, numerical study of the transoni $c^ersonic compressor cascade flow demonstrated the effectiveness of a passive control by the various size cavities. A cavity under the shock foot point at the suction surface of the blades was used as a passive control. The passive control of shock-boundary layer interaction by a cavity reduced total pressure losses. The effect of cavity length and depth was studied. The total pressure loss was reduced by about 10% and the isentropic efficiency was improved slightly. The effect of cavity depth in current study(d/l = 0.05, 0.02) was not found strong. Further adequate turbulence modeling and TVD schemes would help to capture the shock more accurately and increase the effectiveness of the current shock-boundary layer interaction study using upwind flux difference splitting computational methods.thods.