Moment methods for the radiative transfer equation based on φ-divergences

On the Renormalization Maps for the φ -Divergence Moment Closures Applied in Radiative Transfer

“…Our aim in this paper is to provide and study approximations of a function I with respect to the X-variable that preserve the properties presented in this section at the moment level and that are adapted to a selected convex entropy. Dynamical simulations for Equation (1a) with a first version of the present approximation were performed in Abdelmalik, Cai, and Pichard (2023). Extensions to Equation (3) when considering the frequency dependency would require further techniques, such as multi-group approximations, and are left as future work.…”

“…We recall here the construction of the moment closure from Abdelmalik, Cai, and Pichard (2023) and the problems tackled in the present work.…”

“…In our work, we aim at imposing b 2 P K ðRÞ with K � 1 for numerical quadrature (up to a sufficient order) to be exact, this provided a rotationally invariant algorithm in Abdelmalik, Cai, and Pichard (2023). We chose renormalizations of the form…”

“…This paper is a follow-up to Abdelmalik, Cai, and Pichard (2023) and is concerned about the angular approximation of the solution to the radiative transfer equation (RTE). This kinetic equation consists in a transport at a velocity X with a constant norm combined with a collision operator.…”

“…Recently, the entropy-based method of moment was simplified by exploiting u-divergence techniques (Csisz� ar 1972) while preserving its main properties, originally in the context of rarefied gases (Abdelmalik 2017;Abdelmalik and van Brummelen 2016) and for the RTE in Abdelmalik, Cai, and Pichard (2023). This method consists in a Galerkin approximation where the test function space chosen to be polynomials set and the approximation set is chosen to be a non-linear renormalization applied to the same polynomial set.…”