Summary: A new method for calculating the two-dimensional, laminar Navier-Stokes equations is presented. The method uses Newton’s method for nonlinear systems of equations to find steady-state solutions. The Navier-Stokes equations are approximated by finite differences using Roe’s flux difference splitting. Second-order accuracy is attained by using Spekreijse’s interpolation with Van Albada’s limiter. The exact Newton’s method Jacobian matrix is inverted by using recent sparse matrix routines. The symbolic manipulation package MACSYMA is used to develop and write the FORTRAN code. Numerical results are presented for flat plate and wedge type attached and separated viscous flows at high supersonic Mach numbers.