The development of the patched grid solution methodology for both the Euler and the Navier-Stokes equations in two dimensions is presented. The governing equations are written in the integral form and the basic numerical algorithm is finite volume. The method is capable of first through third order accuracy in space. The flux vectors associated with the Euler equations are split into two sub-vectors (based on the signs of the characteristic speeds) and discretized separately. The viscous and heat flux contributions are treated with central differences. Patched grid results are demonstrated on shock reflection, subsonic boundary layer, and shock-boundary layer interaction flow problems. The results are compared with non-patched or single zone grids. The patched grid approach shows an improvement in resolution while minimizing storage and computer time. / Master of Science