The feasibility of using a vectorized banded direct solver for the compressible Euler and Navier-Stokes equations is examined for both single-grid and multi-grid strategies. A procedure is developed for comparing the computational effort required for the direct method with that of the vertical line Gauss-Seidel iteration scheme in order to provide a criteria for choosing between the two techniques. The direct method is shown to have a relatively wide range of application on a vector processor with large memory. Indeed, the primary limitation of the direct method at this time is machine memory. Results for both inviscid and viscous test problems over a range of Mach numbers and Reynolds numbers are examined for two dimensions. / Ph. D.