A numerical scheme has been developed for solving tho two-dimensional incompressible Navier -Stokes equations on a domain that is infinite in the vertical (y) direction and finite in the streamwise (x) direction. The fourth-order equation for the streamwise velocity (u) is advanced in time explicitly using a compact third-order Runge-Kutta scheme. A standard Fourier method is used in the x direction and a mapped spectral method in the y direction. The various parts of the code are tested by solving two problems with analytical or independently-established known solutions. The nonlinear Stuart solution is employed to test the convective parts, the Poisson part of the code and the time advancement. Finally, we find that the disturbance obtained from linear stability theory grows in the linear regime of a free shear flow and that the nonlinear growth of the amplitude corresponds to the vortex roll-up.