On Modeling and Solving the Boltzmann Equation

TL;DR

This paper reviews recent advances in solving the linear Boltzmann equation, emphasizing the discrete ordinates method and its applications in neutron transport, optical tomography, and gas dynamics, highlighting the versatility of the ADO method.

Contribution

It provides a comprehensive overview of analytical and numerical techniques for the linear Boltzmann equation, focusing on the ADO method's effectiveness across various applications.

Findings

Discrete ordinates approximation effectively models neutron and photon transport.

The ADO method offers concise and accurate solutions for complex phenomena.

Connections between different physical phenomena and the Boltzmann model are established.

Abstract

The Boltzmann equation has been a driving force behind significant mathematical research over the years. Its challenging theoretical complexity, combined with a wide variety of current scientific and technological problems that require numerical simulations based on this model, justifies such interest. This work provides a brief overview of studies and advances on the solution of the linear Boltzmann equation in one- and two-dimensional spatial dimensions. In particular, relevant aspects of the discrete ordinates approximation of the model are highlighted for neutron and photon transport applications, including nuclear safeguards, nuclear reactor shielding problems, and optical tomography. In addition, a short discussion of rarefied gas dynamics problems, relevant, for instance, to the study of micro-electro-mechanical systems, and their connection with the Linearized Boltzmann Equation, is presented. A primary goal of the work is to establish as much as possible the connections between the different phenomena described by the model and the versatility of the analytical methodology, the ADO method, in providing concise and accurate solutions, which are fundamental for numerical simulations.