Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Modeling and Solving the Boltzmann Equation

Liliane Basso Barichello

TL;DR

This work surveys the Analytical Discrete Ordinates (ADO) method for solving the linear Boltzmann equation in one and two spatial dimensions, with emphasis on photon and neutron transport and connections to rarefied gas dynamics. It outlines the integro-differential RTE formulation, develops the 1D ADO solution via eigenvalue problems of half-order (ν) and explicit homogeneous/particular solutions, and extends to 2D through ADO-Nodal formulations that combine multidimensional quadratures with nodal methods to produce tractable ODE/PDE systems. A key contribution is the unification of transport and kinetic-model analyses, including synthetic kernels for LBE (CES/CEBS), the G-problem framework, and inverse problems in optical tomography, all yielding accurate solutions on coarse meshes and offering computational efficiency advantages. The approach’s practical impact spans nuclear engineering, optical tomography, and MEMS, where fast, reliable benchmarks and analytic insights are valuable for code validation, parameter reconstruction, and design optimization; ongoing work targets domain decomposition and rigorous angular-discretization error analysis to further enhance scalability and accuracy.

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.

On Modeling and Solving the Boltzmann Equation

TL;DR

This work surveys the Analytical Discrete Ordinates (ADO) method for solving the linear Boltzmann equation in one and two spatial dimensions, with emphasis on photon and neutron transport and connections to rarefied gas dynamics. It outlines the integro-differential RTE formulation, develops the 1D ADO solution via eigenvalue problems of half-order (ν) and explicit homogeneous/particular solutions, and extends to 2D through ADO-Nodal formulations that combine multidimensional quadratures with nodal methods to produce tractable ODE/PDE systems. A key contribution is the unification of transport and kinetic-model analyses, including synthetic kernels for LBE (CES/CEBS), the G-problem framework, and inverse problems in optical tomography, all yielding accurate solutions on coarse meshes and offering computational efficiency advantages. The approach’s practical impact spans nuclear engineering, optical tomography, and MEMS, where fast, reliable benchmarks and analytic insights are valuable for code validation, parameter reconstruction, and design optimization; ongoing work targets domain decomposition and rigorous angular-discretization error analysis to further enhance scalability and accuracy.

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.

Paper Structure

This paper contains 15 sections, 120 equations, 10 figures.

Table of Contents

  1. Introduction
  2. The Integro-differential Formulation
  3. A "Simplest" Problem
  4. The ADO Solution for One-dimensional Geometry Problems
  5. Numerical Results and Contributions
  6. The ADO Solution for Two-dimensional Geometry Problems
  7. Numerical Quadrature Schemes
  8. The ADO-Nodal Formulation
  9. Neutron Transport
  10. Numerical Results and Contributions
  11. A Taste of the Linearized Boltzmann Equation
  12. A Synthetic Kernel
  13. Kinetic Models and the G-problem
  14. Inverse Problems
  15. Concluding Remarks and Ongoing Research

Figures (10)

  • Figure 1: Coordinates in the phase space
  • Figure 2: The slab geometry
  • Figure 3: Level Symmetric Quadrature Scheme
  • Figure 4: Legendre-Chebyshev Quadrangular Scheme
  • Figure 5: Domain subdivided into a mesh of HxK rectangular regions
  • ...and 5 more figures