# The radiative transport equation with heterogeneous cross-sections

## TL;DR

This paper analyzes the radiative transport equation with heterogeneous cross-sections, providing explicit bounds on the integral operator, convergence rates, and solution estimates, with applications to nuclear reactor safety.

## Contribution

It offers explicit estimates for the integral operator and solution of the RTE with heterogeneous coefficients, and applies these results to eigenvalue problems in reactor safety.

## Key findings

- Bound on the integral operator norm explicit in cross-sections
- Convergence rate estimate for source iteration
- Eigenvalues in reactor safety are real and positive

## Abstract

We consider the classical integral equation reformulation of the radiative transport equation (RTE) in a heterogeneous medium, assuming isotropic scattering. We prove an estimate for the norm of the integral operator in this formulation which is explicit in the (variable) coefficients of the problem (also known as the cross-sections). This result uses only elementary properties of the transport operator and some classical functional analysis. As a corollary, we obtain a bound on the convergence rate of source iteration (a classical stationary iterative method for solving the RTE). We also obtain an estimate for the solution of the RTE which is explicit in its dependence on the cross-sections. The latter can be used to estimate the solution in certain Bochner norms when the cross-sections are random fields. Finally we use our results to give an elementary proof that the generalised eigenvalue problem arising in nuclear reactor safety has only real and positive eigenvalues.