New results are presented for inviscid, supersonic and hypersonic blunt-body flow fields obtained with a numerical time-dependent method patterned after that of Moretti and Abbett. In addition, important comments are made with regard to the physical and numerical nature of the method. Specifically, numerical results are presented for two-dimensional and axisymmetric parabolic and cubic blunt bodies as well as blunted wedges and cones; these results are presented for zero degrees angle of attack and for a calorically perfect gas with gamma = 1.4. The numerical results are compared with other existing theoretical and experimental data. Also, the effects of initial conditions and boundary conditions are systematically examined with regard to the convergence of the time-dependent numerical solutions, and the point is made that the initial conditions can not be completely arbitrary. Finally, in order to learn more about the performance of the time-dependent method, a numerical experiment is conducted to examine the unsteady propagation and region of influence of a slight pressure disturbance introduced at a point on the surface of a blunt body.