In extracorporeal shock wave lithotripsy (ESWL), weak converging shock waves are focused to shatter kidney stones. Accurate knowledge of the flow field in the focal region is required to understand the source of tissue damage and the primary mechanisms of stone fragmentation. A finite-difference numerical method for geometrical shock dynamics has been developed and used to evaluate the accuracy of this approximate theory for the complex nonlinear processes which occur in shock focusing. The shock dynamics results duplicate the strong, moderate, and weak shock behaviors seen in experiments [B. Sturtevant and V.A. Kulkarny, JFM, 73, 651--671], with good agreement for focal pressures and triple-point path. Similar behavior is observed for the ESWL problem of axisymmetric shock focusing in water. The primary source of error is the fact that shock dynamics always yields Mach reflection at the centerline, so regular reflection in the theory is impossible. Adequate resolution of the focal region is necessary to assess the accuracy of the shock dynamics theory, or of any other theory. The shock dynamics theory and numerical method have also been extended to the more general case of shock propagation into a nonuniform media with free-stream velocity.