An analytical approach for solving a nodal formulation of two-dimensional fixed-source neutron transport problems with linearly anisotropic scattering

In this work, the formulation of the Analytical Discrete Ordinates (ADO) method for two-dimensional fixed-source neutron transport problems is extended to problems with anisotropic scattering. The methodology is developed from the discrete ordinates approximation of the two-dimensional transport, whose equations are integrated transversally within homogenized regions of the domain, yielding one-dimensional equations for the average angular fluxes. Such one-dimensional equations are solved by the ADO method, after considering approximations for the transverse leakage terms, and their solutions are written in terms of eigenvalues and eigenfunctions. The relevant feature of the ADO method for the isotropic scattering cases, which reduced the order of the eigenvalue problems to half of the number of discrete directions, is also preserved for the present anisotropic cases. Explicit spatial variables expressions are derived for the average angular fluxes in regions of interest defined in the domain. The full domain solution is defined by the coupling of local solutions in different regions, without the use of sweeping procedures. As usual in nodal schemes, auxiliary equations are necessary and, in this case, the unknown region-edge angular fluxes are approximated by constants embedded in the source term. Numerical results for region-average scalar fluxes are obtained and compared with the ones available in the literature to illustrate the feasibility of keeping the computational efficiency already verified in treating problems with isotropic scattering, with the use of the ADO method.