Solution and Study of the Two-Dimensional Nodal Neutron Transport Equation

In the last decade Vilhena and coworkers¹⁰ reported an analytical solution to the two-dimensional nodal discrete-ordinates approximations of the neutron transport equation in a convex domain. The key feature of these works was the application of the combined collocation method of the angular variable and nodal approach in the spatial variables. By nodal approach we mean the transverse integration of the S_N equations. This procedure leads to a set of one-dimensional S_N equations for the average angular fluxes in the variables x and y. These equations were solved by the old version of the LTS_N method⁹, which consists in the application of the Laplace transform to the set of nodal S_N equations and solution of the resulting linear system by symbolic computation. It is important to recall that this procedure allow us to increase N the order of S_N up to 16. To overcome this drawback we step forward performing a spectral painstaking analysis of the nodal S_N equations for N up to 16 and we begin the convergence of the S_N nodal equations defining an error for the angular flux and estimating the error in terms of the truncation error of the quadrature approximations of the integral term. Furthermore, we compare numerical results of this approach with those of other techniques used to solve the two-dimensional discrete approximations of the neutron transport equation⁶.