Positive asymptotic preserving approximation of the radiation transport equation
This work provides a linear, positivity-preserving scheme for radiation transport that is discretization-agnostic and avoids overshoots, addressing a known issue in standard methods for computational physics.
The paper introduces a linear, positive, and asymptotic preserving method for solving the one-group radiation transport equation, which is first-order accurate in space and avoids overshoots at interfaces between optically thin and thick regions. The method converges with rate O(h) in L2-norm on manufactured solutions and O(h^2) in the diffusion regime.
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.