Standard shock-capturing schemes discretizing the compressible Euler equations cannot usually provide an accurate solution for low Mach number flows. This is related to the fact the numerical dissipation produces wrong order pressure fluctuations in the discrete solution [1]. Since the 2000s, a large number of approximate Riemann solvers have been developed in order to compute consistent solutions in the incompressible limit. However, low Mach number corrections modify the asymptotic scaling of the numerical dissipation and may yield severe stability conditions, which are not frequently addressed. This study aimed at analyzing in the low Mach number limit the Roe’s matrix dissipation corrected by the artificial speed of sound approach according to Rossow [2]. To our knowledge, this point has not been investigated from an asymptotic analysis point of view. This correction has a significant advantage to be less prone to checkerboard mode problems, but, suffers from a stringent stability condition. The numerical results presented in this work are obtained thanks to a Quasi-Newton method using a Von Neumann stability condition based on the spectral radius of the matrix dissipation, according to a stability analysis conducted in [3]. The exact Jacobian matrix of the flux balance required for the time integration, is calculated using Automatic Differentiation [4]. This produces a highly stable numerical method allowing to use large CFL numbers for steady and unsteady flows. Moreover for steady low-Mach problems, an adaptive CFL is added, resulting in a faster convergence to the steady state in a few hundred of iterations, even for very low Mach number flows.