The flow field of a transverse jet in a supersonic airstream subjected to shock wave-turbulent boundary layer interactions is simulated numerically by adaptive mixed explicit-implicit generalized-Galerkin finite element methods. In this scheme, convection and diffusion implicitness parameters are introduced to resolve shock wave discontinuities and widely disparate time and length scales of turbulence and finite rate chemistry. These parameters are flow field dependent, calculated from local Mach, Reynolds, and Damkohler numbers for each element. Effects of turbulence are taken into account with a two-equation (k-ϵ) model with a compressibility correction. Various cases of mixing, slot widths, and total pressure ratios with and without chemical reactions are examined. Favorable comparisons with experimental measurements are demonstrated.