In this thesis, we develop and test a fundamentally new linear-discontinuous least-squares (LDLS) method for spatial discretization of the one-dimensional (1-D) discrete-ordinates (SN) equations. This new scheme is based upon a least-squares method with a discontinuous trial space. We implement our new method, as well as the lineardiscontinuous Galerkin (LDG) method and the lumped linear-discontinuous Galerkin (LLDG) method. The implementation is in FORTRAN. We run a series of numerical tests to study the robustness, L2 accuracy, and the thick diffusion limit performance of the new LDLS method. By robustness we mean the resistance to negativities and rapid damping of oscillations. Computational results indicate that the LDLS method yields a uniform second-order error. It is more robust than the LDG method and more accurate than the LLDG method. However, it fails to preserve the thick diffusion limit. Consequently, it is viable for neutronics but not for radiative transfer since radiative transfer problems can be highly diffusive.