Positive and Asymptotic Preserving Approximation of the Radiation Transport Equation
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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.
SIAM Journal on Numerical Analysis
author list (cited authors)
Guermond, J., Popov, B., & Ragusa, J.
complete list of authors
Guermond, Jean-Luc||Popov, Bojan||Ragusa, Jean