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

Abstract

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 P N approximation, the S N method (discrete ordinates method) and their variants. In the discrete ordinates (S N) formulations of the transport equation, it is assumed that the linearized Boltzmann equation only holds for a set of distinct numerical values of the direction-of-motion variables. In this work, looking forward to confirm the capabilities of deterministic methods in obtaining accurate results, we present a general overview of deterministic methods to solve the Boltzmann transport equation for neutral and charged particles. First, we describe a review in the Laplace transform technique applied to S N 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 monoenergetic electron beam in a rectangular domain. The main idea relies on applying the P N 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.

Técnicas de Matriz Resposta com Aproximações de Baixa Ordem Aplicadas a Problemas de Transporte de Fótons

“…Por outro lado, os métodos determinísticos baseiam-se em resolver numericamente a equação de transporte para geometrias não muito complexas, requerem de recursos computacionais menores e as simulações são mais rápidas quando comparados com os métodos probabilísticos. O uso de métodos determinísticos de difusão, harmônicos esféricos, e ordenadas discretas aplicados a problemas de transporte e transferência radiativa é amplamente abordado na literatura científica [5][6][7][8][9].…”

“…Dentre os desafios que ainda existem na resolução de problemas de transporte de fótons e transferência radiativa está o desenvolvimento de algoritmos eficientes para resolver a equação de transporte [6,9,10]. O objetivo deste trabalho é apresentar os esquemas numéricos de tipo matriz resposta com aproximações de baixa ordem, constante nodal e linear nodal pleno, para resolver problemas de transporte de fótons em atmosfera cinza, geometria bidimensional Cartesiana, espalhamento isotrópico e fótons de baixas energias usando como modelo matemático a formulação em ordenadas discretas (𝑆 𝑁 ) da equação de transporte.…”

A fast algorithm for radiative transport in isotropic media

“…Numerical methods for solving the equation of radiative transfer has been extensively studied, see for instance [16,27,44,49,60] and references therein for an overview. Besides Monte Carlo type of methods that are based on stochastic representation of the ERT [9,18,22,32,66], many different deterministic discretization schemes have been proposed [2,3,5,11,17,20,24,25,30,31,34,37,38,41,39,43,47,48,50,53,56,65] and numerous iterative schemes, as well as preconditioning strategies, have been developed to solve the discretized systems; see for instance [1,12,26,54,55] and references therein.…”