Large eddy numerical simulations of supersonic flows over an axisymmetric backward-facing step have been completed using a recently developed axisymmetric version of the finite element method-flux corrected transport algorithm, FEM-FCT. The code is based on the mixing layer and recompression premises of the Chapman-Korst model. It solves the time-accurate Euler equations utilizing 1) adaptive unstructured gridding to resolve flow feature details and 2) a conservative nonlinear scheme to capture features in the compressible flowfield. This approach may be employed since the location of the separation point is fixed at the sharp corner. The simulation allows the large-scale structures present in the mixing layer to interact dynamically with the recompression region. Comparisons have been made with available experimental data for a range of Mach numbers and step height to step radius ratios for which boundary-layer effects are not pronounced. In general, good agreement between predictions and measurements was found for 1) the time-averaged surface pressure distributions along the backstep and the reattachment wall, 2) the flowfield structure in general, and 3) the downstream reattachment lengths. Effects of axisymmetry were noted in flowfield characteristics such as increased base pressure, curvature of expansion fans, stronger recompression shocks, and higher Mach number levels in the subsonic recirculating region.