solving the time-dependent TRT equation, in the absence of scattering, has also been pursued in [@ET-GO-SA:2010] and [@MAT-SAL:2013], where the authors develop homogenized equations on an enlarged phase space. This approach is similar in spirit to that taken in the current paper; however, one advantage of that given here is the ability to handle scattering (in angle and energy), as well as to reduce the numerical computation to a standard multigroup formulation. Finally, we remark that, when the opacity is of the form $\sigma_{\varepsilon}\left(E,T\right)=\sigma_{0}\left(E/\varepsilon,E,T\right)$, where $\sigma_{0}\left(\kappa,E,T\right)$ is almost periodic in $\kappa$, then (\[eq:transport equation on extended phase space, intro\]) and (\[eq:psi\_0 as weighting against Young measure, intro\]) is analogous to the two-scale homogenization theory developed in [@ALLAIR:1992].

Outline of the new multigroup method
====================================

Here we outline a computational method for the transport equation $$\begin{aligned}
\boldsymbol{\Omega}\cdot\nabla_{\mathbf{x}}\psi_{\varepsilon}+\sigma_{\varepsilon}\left(T\left(\mathbf{x}\right),E\right)\psi_{\varepsilon} & = & \sigma_{\varepsilon}^{a}\left(T\left(\mathbf{x}\right),E\right)S\left(\mathbf{x},E\right)+Q_{0}\left(\mathbf{x},\boldsymbol{\Omega},E\right)+\nonumber \\
 &  & \int_{\mathbb{S}^{2}}\int_{0}^{\infty}\Sigma^{s}\left(\mathbf{x},E,E',\boldsymbol{\Omega}\cdot\boldsymbol{\Omega}'\right)\psi_{\varepsilon}dE'd\boldsymbol{\Omega}',\label{eq:transport equation, intro}\end{aligned}$$ where $$\Sigma^{s}\left(\mathbf{x},E,E',\boldsymbol{\Omega}\cdot\boldsymbol{\Omega}'\right)=\sigma^{s}\left(T\left(\mathbf{x}\right),E'\right)K\left(E,E',\boldsymbol{\Omega}\cdot\boldsymbol{\Omega}'\right),$$ $$\sigma_{\varepsilon}\left(T\left(\mathbf{x}\right),E\right)=\sigma_{\varepsilon}^{a}\left(T\left(\mathbf{x}\right),E\right)+\sigma^{s}\left(T\left(\mathbf{x}\right),E\right),$$ and the absorption opacity (cross-section) $\sigma_{\varepsilon}^{a}\left(T\left(\mathbf{x}\right),E\right)$ rapidly varies in energy $E$ on the microscale $0<\varepsilon\ll1$; here $\varepsilon$ denotes the characteristic spacing between spectral lines or resonances. For simplicity, our discussion ignores density and pressure dependence in the opacity $\sigma_{\varepsilon}$, but its incorporation into the proposed algorithm is straightforward; in fact, see 