An algorithm for the simulation of unsteady, viscous, stratified compressible flows, which remains valid at all speeds, is presented. The method is second-order accurate in both space and time and is independent of the Mach number. In order to remove the stiffness of the numerical problem due to the large disparity between the flow speed and the acoustic wave speed at low Mach number, an approximate Newton method, based on artificial compressibility, is proposed. Additionally, a modified advection upstream splitting method (AUSM +) scheme is used, which permits accurate computations of both compressible and incompressible flows. A detailed description of the method and an efficiency comparison with other approximate Newton methods described in the literature are given. Furthermore, it is shown that the accuracy of the algorithm is not dependent on the Mach number through the computations of various benchmark test cases. Copyright (C) 2000 John Wiley & Sons, Ltd.