In this work, four types of quadrature schemes are used to define discrete directions in the solution of a two-dimensional fixed-source discrete ordinates problem in Cartesian geometry. Such schemes enable generating numerical results for averaged scalar fluxes over specified regions of the domain with high number (up to 105) of directions per octant. Two different nodal approaches, the ADO and AHOT-N0 methods, are utilized to obtain the numerical results of interest. The AHOT-N0 solutions on a sequence of refined meshes are then used to develop an asymptotic analysis of the spatial discretization error in order to derive a reference solution. It was more clearly observed that the spatial discretization error converges asymptotically with second order for the source region with all four quadratures employed, while for the other regions refined meshes along with tighter convergence criterion must be applied to evidence the same behavior. However, in that case, some differences among the four quadrature schemes results were found.