Spectral formulations to the solution of particle transport problems

In this work, a deterministic approach to the solution of the integro differential linear Boltzmann equation, in one and twodimensional media, is presented. Explicit solutions, in terms of the spatial variables, for the discrete ordinates approximation of the original model are obtained from a spectral formulation. A relevant feature of the methodology is the reduced order of the eigenvalue problems, which is given as half of the number of the discrete directions. Such aspect, along with the use of arbitrary quadrature schemes to represent the integral term of the equation, are fundamental to provide fast, concise and accurate solutions to several problems of interest. In particular, the solutions here are derived for models associated with two different applications: rarefied gas dynamics and neutron transport. Mathematical aspects present in the solution of the two models are emphasized and extensions of the formulation to other applications are commented and referenced.