A new algorithm to compute parabolized axisymmetric flows is presented. Axi-symmetric flows are computed using Χ, distance along the centerline line of the duct, and ξ, the stream function, as the independent variables. The dependent variables used are U(Χ, ξ), the streamwise velocity, and R(Χ, ξ), the cross-stream coordinate of the stream surface with value of the stream function equal to ξ. Since the value of stream function is constant along the duct walls, this choice of variables makes it easy to satisfy the boundary conditions; and, thus eliminates the need for an externally generated boundary fitted grid. Sample computations are presented for incompressible parabolized developing laminar flow through pipes and diffusers. The results of the computations are compared with the available experimental data.