# Solution of the Generalized Linear Boltzmann Equation for Transport in Multidimensional Stochastic Media

## TL;DR

This paper extends the generalized linear Boltzmann equation framework to multidimensional stochastic media with finite boundaries, using discrete ordinates and approximations, verified through numerical experiments.

## Contribution

It introduces a multidimensional solution approach for the GLBE with finite boundaries, including discretization techniques and approximations, advancing beyond previous one-dimensional models.

## Key findings

- Verified the solution against Gaussian process media
- Demonstrated the effectiveness of sweeping methods in multidimensional settings
- Introduced practical approximations for non-classical cross-sections

## Abstract

The generalized linear Boltzmann equation (GLBE) is a recently developed framework based on non-classical transport theory for modeling the expected value of particle flux in an arbitrary stochastic medium. Provided with a non-classical cross-section for a given statistical description of a medium, any transport problem in that medium may be solved. Previous work has only considered one-dimensional media without finite boundary conditions and discrete binary mixtures of materials. In this work the solution approach for the GLBE in multidimensional media with finite boundaries is outlined. The discrete ordinates method with an implicit discretization of the pathlength variable is used to leverage sweeping methods for the transport operator. In addition, several convenient approximations for non-classical cross-sections are introduced. The solution approach is verified against random realizations of a Gaussian process medium in a square enclosure.