An overview of the Boltzmann transport equation solution for neutrons, photons and electrons in Cartesian geometry

Questions regarding accuracy and efficiency of deterministic transport methods are still on our mind today, even with modern supercomputers. The most versatile and widely used deterministic methods are the PN approximation, the SN method (discrete ordinates method) and their variants. In the discrete ordinates (SN) formulations of the transport equation, it is assumed that the linearised Boltzmann equation only holds for a set of distinct numerical values of the direction-of-motion variables. In this paper, looking forward to confirm the capabilities of deterministic methods in obtaining accurate results, we describe the recent advances in the class of deterministic methods applied to one and two dimensional transport problems for photons and electrons in Cartesian Geometry. First, we describe the Laplace transform technique applied to SN two dimensional transport equation in a rectangular domain considering Compton scattering. Next, we solved the Fokker–Planck (FP) equation, an alternative approach for the Boltzmann transport equation, assuming a mono-energetic electron beam in a rectangular domain. The main idea relies on applying the PN approximation, a recent advance in the class of deterministic methods, in the angular variable, to the two dimensional Fokker–Planck equation and then applying the Laplace Transform in the spatial x variable. Numerical results are given to illustrate the accuracy of deterministic methods presented.

► Advances for mono-energetic particle transport in two dimensions are described.
► The Fokker–Planck solution is analytical without introducing approximations.
► The LTSN nodal solution is validated against GEANT4 simulations.
► The closed form solution serves as a theoretical model for future simulations.