© American Institute of Aeronautics and Astronautics, Inc., 1989. All rights reserved. A method is proposed to determine directly the amount of artificial viscosity needed for stability using an eigenvalue analysis for a finite difference representation of the Navier-Stokes equations. The stability and growth of small perturbations about a steady flow over the airfoils are analyzed for various amounts of artificial viscosity. The eigenvalues were determined for a small perturbation about a steady inviscid flow over a NACA 0012 airfoil at a Mach number of 0.8 and angle of attack of 0 degrees. The movement of the eigenvalue constellation with respect to the amount of artificial viscosity is studied. The stability boundaries as a function of the amount of artificial viscosity from both the eigenvalue analysis and the time marching scheme are also presented. This procedure not only allows for determining the effect of varying amounts of artificial viscosity, but also for the effects of different forms of terms for artificial viscosity.