Several higher order finite difference schemes have been proposed in the literature for the solution of a discrete ordinates transport equation. The performance charcteristics of these methods have been studied through numerical and mathematical analyses. In these studies, attention was restricted to a single, homogeneous medium and to uniform meshes only. However, in practice one has to employ nonuniform meshes such as, for instance, near the interface of any two media. A second criterion that needs examination is the influence of the cross section of the medium on the behavior of these schemes. Finally, the mathematical analysis is, in principle, restricted to a single energy group. Although it is believed that there should be no significant differences in the conclusions with respect to the multigroup problem, it appears that the order of convergence is not as high as estimated when the higher order schemes are applied to a multigroup neutron transport. The results of test cases are presented and discussed, where some of the finite difference schemes, when applied to an interface and multigroup problems, exhibit different behavior than reported earlier.