  The discrete ordinates discontinuous Galerkin ($S_N$-DG) method is a
well-established and practical approach for solving the radiative transport
equation. In this paper, we study a low-memory variation of the upwind $S_N$-DG
method. The proposed method uses a smaller finite element space that is
constructed by coupling spatial unknowns across collocation angles, thereby
yielding an approximation with fewer degrees of freedom than the standard
method. Like the original $S_N$-DG method, the low memory variation still
preserves the asymptotic diffusion limit and maintains the characteristic
structure needed for mesh sweeping algorithms. While we observe second-order
convergence in scattering dominated, diffusive regime, the low-memory method is
in general only first-order accurate. To address this issue, we use upwind
reconstruction to recover second-order accuracy. For both methods, numerical
procedures based on upwind sweeps are proposed to reduce the system dimension
in the underlying Krylov solver strategy.
