A study of spatial discretization schemes for the multigroup discrete-ordinates transport equations in slab geometry is described. The purpose of the study is to determine the most computationally efficient method, defined as the one that produces the minimum error for a given cost. We define cost as the total amount of computer time required to complete one inner iteration, given a limit on storage, and we use three error norms to measure the accuracies of edge fluxes, cell average fluxes, and integral parameters. We study three test problems; the first is a model one-group problem we examine in detail, while the second and third are more realistic multigroup problems. Our conclusion is that a new method, labeled linear characteristic, significantly outperforms all other methods that have been implemented up to the present time.