On the solution of systems of linear equations associated to the ADO method in particle transport problems

Abstract

In this work, a study is carried out on the solution of large linear systems of algebraic equations relevant to establish a general solution, based on a spectral formulation, to the discrete ordinates approximation of the two-dimensional particle transport equation in Cartesian geometry. The number of discrete ordinates (discrete directions of the particles) is determined by the order of the quadrature scheme on the unity sphere used to approximate the integral term of the linear Boltzmann equation (also called the transport equation). A nodal technique is applied to the discrete ordinates approximation of this equation, yielding to a system of first order ordinary differential equations for average unknowns along the directions x and y. The developed formulation is explicit for the spatial variables. The order of the linear system is defined by the number of discrete directions as well as the number of the spatial nodes. High-quality solutions are expected as both, the number of discrete directions and the refinement of the spatial mesh, increase. Here, the performance of direct and iterative methods, for the solution of the linear systems, are discussed, along with domain decomposition techniques and parallel implementation. Alternative arrangements in the configuration of the equations allowed solutions to higher order systems. A dependence on the type of the quadrature scheme as well as the class of problems to be solved (neutron or radiation problems, for instance) directly affect the final choice of the numerical algorithm.