Complex Eigenvalues Analysis of the SN equations for deterministic coarse-mesh methods development applied in one-dimensional neutron shielding calculation

Abstract

When using spectral nodal methods in the solution of fixed-source problems, one of the steps involves obtaining the intranodal homogeneous solution of the neutron transport equations in the discrete ordinates formulation (SN), where an eigenvalue problem is solved. Up until now, this process involved the emergence of N (even order for Gauss-Legendre quadrature) real and symmetric eigenvalues. However, in some cases, complex conjugates may appear in this step. Thus, we present a significant innovation in this type of computational modelling, by using the Euler's Formula to manipulate the local analytical solution and achieve a possible application of coarse-mesh methods in these cases. In order to showcase this technique, we use the spectral deterministic method to solve a model-problem with different sets of Gaussian quadrature, which came to compute hundreds of complex eigenvalues in its analytical solution, where a good precision was achieved when comparing the obtained numerical results with the reference.