Supersonic slot injection problems were studied by a finite volume method. The numerical technique used is the upwind method of Roe’s flux difference splitting (FDS) with vertical line Gauss-Seidel relaxation applied to the thin layer Navier-Stokes equations. To test the accuracy of the numerical methods without the complications and uncertainties of turbulence modeling, two sample cases were chosen with laminar flows. The sample problems were the compressible laminar boundary layer flow over a flat plate and the laminar boundary layer - shock interaction problem. For these problems, both the results from Roe’s FDS and van Leer’s flux vector splitting (FVS) are compared with exact solutions and experimental data. For the sample problems, comparisons showed that Roe’s FDS method is more accurate than van Leer’s FVS method. Because of the very complicated wave patterns and strong viscous-inviscid interaction produced by supersonic slot injection, an adaptive grid based on the equidistribution law was combined with the solution algorithm. The results from Roe’s FDS method with the adaptive grid showed good results for the supersonic slot injection over a flat plate. For the slot injection over a 10-degree wedge surface case, there is a significant difference between the numerical and experimental wall pressure distribution. Some potential reasons for the discrepancy including 3D effects and/or transition in the reattachment region in the experiments and possibly a need for a much finer grid in the calculations are discussed. / Ph. D.