Positive asymptotic preserving approximation of the radiation transport equation

TL;DR

This paper presents a linear, positive, asymptotic preserving numerical method for the radiation transport equation that is discretization-agnostic, converges at optimal rates, and avoids overshoot issues at material interfaces.

Contribution

It introduces a novel linear method that is positive, asymptotic preserving, and compatible with various spatial discretizations for the radiation transport equation.

Findings

Converges at rate O(h) in L^2-norm for manufactured solutions.

Achieves O(h^2) convergence in the diffusion regime.

Does not suffer from overshoots at interfaces between different optical regions.

Abstract

We introduce a (linear) positive and asymptotic preserving method or solving the one-group radiation transport equation. The approximation in space is discretization agnostic: the space approximation can be done with continuous or discontinuous finite elements (or finite volumes, or finite differences). The method is first-order accurate in space. This type of accuracy is coherent with Godunov's theorem since the method is linear. The two key theoretical results of the paper are Theorem~4.4 and Theorem~4.8. The method is illustrated with continuous finite elements. It is observed to converge with the rate $\calO(h)$ in the $L^2$-norm on manufactured solutions, and it is $\calO(h^2)$ in the diffusion regime. Unlike other standard techniques, the proposed method does not suffer from overshoots at the interfaces of optically thin and optically thick regions.