Moment Methods for the 3D Radiative Transfer Equation Based on $\varphi$-Divergences

Abstract

The method of moments is widely used for the reduction of kinetic equations into fluid models. It consists in extracting the moments of the kinetic equation with respect to a velocity variable, but the resulting system is a priori underdetermined and requires a closure relation. In this paper, we adapt the $\varphi$-divergence based closure, recently developed for rarefied gases i.e. with a velocity variable describing $\mathbb{R}^d$, to the radiative transfer equation where velocity describes the unit sphere $\mathbb{S}^2$. This closure is analyzed and a numerical method to compute it is provided. Eventually, it provides the main desirable properties to the resulting system of moments: Similarily to the entropy minimizing closure ($M_N$), it dissipates an entropy and captures exactly the equilibrium distribution. However, contrarily to $M_N$, it remains computationnally tractable even at high order and it relies on an exact quadrature formula which preserves exactly symmetry properties, i.e. it does not trigger ray effects. The purely anisotropic regimes (beams) are not captured exactly but they can be approached as close as desired and the closures remains again tractable in this limit.