In this article, a numerical study of compressible and weak compressible Newtonian flows is achieved for a time marching, Galerkin algorithm. A comparison between two numerical techniques for such flows, namely the artificial compressibility method (AC–method) and the fully artificial compressibility method (FAC–method) is performed. In the first artificial compressibility parameter (  is added to the continuity equation, while this parameter is added to both continuity and momentum equations in the second technique. This strategy is implemented to treat the governing equations of Newtonian flow in cylindrical coordinates (axisymmetric). Particularly, this study concerns with the effect of the artificial compressibility parameters on the convergence level of solutions components. To confirm the analysis of these approaches, Poiseuille flow along a circular channel under an isothermal state is used as a simple test problem. The results show that when the AC-method is used there is a significant reduction in the level of time convergence of pressure and axial velocity compared to that with FAC-method. Here, for compressible flow the Tail model of state is employed to relate the pressure to density. In this context, the effect of Tail parameters and Reynolds number on the time convergence of solution components is also investigated in the present study. The results indicate a significant reduction in the time-stepping convergence as increasing in the {B,m}-value. In contrast, more difficulties are faced in the convergence when the level of the Reynolds number is increasing.