Two-dimensional convective flows in shallow cavities with conducting horizontal boundaries and driven by differential heating of the two vertical end walls, are studied numerically over a range of Rayleigh numbers and Prandtl numbers. As the Rayleigh number increases, nonlinearity first affects the flow structure in the turning regions near the ends of the cavity. These ’end-zone problems’ have been investigated by a combined computational and analytical approach. Numerical solutions are found using a Dufort-Frankel-Multigrid method, and appear to be in good agreement with stability analysis.