The Non-Classical Boltzmann Equation, and Diffusion-Based Approximations to the Boltzmann Equation

TL;DR

This paper demonstrates that diffusion-based approximations to the Boltzmann equation can be exactly represented by a non-classical transport equation, enabling Monte Carlo solutions with only statistical errors.

Contribution

It introduces a method to solve diffusion-based approximations to the Boltzmann equation using Monte Carlo without truncation errors.

Findings

Diffusion approximations can be represented by a non-classical transport equation.

Monte Carlo methods can solve these approximations with purely statistical errors.

The approach applies to an infinite, homogeneous medium.

Abstract

We show that several diffusion-based approximations (classical diffusion or SP1, SP2, SP3) to the linear Boltzmann equation can (for an infinite, homogeneous medium) be represented exactly by a non-classical transport equation. As a consequence, we indicate a method to solve diffusion-based approximations to the Boltzmann equation via Monte Carlo, with only statistical errors - no truncation errors.