[@NJOY] program. More precisely, we compute the exact energy-integrated solution, $$\int_{E_{i}}^{E_{i+1}}\psi_{\varepsilon}\left(x_{i},\mu_{j},E\right)dE,$$ for $10$ equispaced spatial points $x_{i}\in\left[0,1\right]$ and for $8$ Gauss-Legendre nodes $\mu_{j}$, as well as the homogenized energy-integrated solution. We compare the “exact” scalar flux (that is, exact to within angular discretization errors), $$\varphi_{\varepsilon}\left(x_{k}\right)=\sum_{j}\sum_{i}\int_{E_{i}}^{E_{i+1}}\psi_{\varepsilon}\left(x_{k},\mu_{j},E\right)dE,\label{eq:discrete scalar flux, exact}$$ against its homogenized version $$\varphi_{0}\left(x_{k}\right)=\sum_{j}\sum_{i}\int_{E_{i}}^{E_{i+1}}\psi_{0}\left(x_{k},\mu_{j},E\right)dE.\label{eq:discrete scalar flux, approx}$$

We compare the results to the standard multigroup method using Planck-weighted and Rosseland-weighted opacities. In particular, we integrate (\[eq:transport equation, intro\]) over $\left[E_{i},E_{i+1}\right]$, $$\mu\partial_{x}\int_{E_{g}}^{E_{g+1}}\psi_{\varepsilon}\left(x,\mu,E\right)dE=\int_{E_{g}}^{E_{g+1}}\sigma_{\varepsilon}\left(E,T\right)\left(\frac{S\left(E\right)}{\sigma_{\varepsilon}\left(E\right)}-\psi_{\varepsilon}\left(x,\mu,E\right)\right)dE.$$ We write $$\int_{E_{g}}^{E_{g+1}}\sigma_{\varepsilon}\left(E\right)\left(\frac{S\left(E\right)}{\sigma_{\varepsilon}\left(E\right)}-\psi_{\varepsilon}\left(x,\mu,E\right)\right)dE\approx\sigma_{g}\left(\int_{E_{g}}^{E_{g+1}}\frac{S\left(E\right)}{\sigma_{\varepsilon}\left(E\right)}dE-\int_{E_{g}}^{E_{g+1}}\psi_{\varepsilon}\left(x,\mu,E\right)dE\right),$$ where $$\sigma_{g}=\frac{\int_{E_{g}}^{E_{g+1}}S\left(E\right)dE}{\int_{E_{g}}^{E_{g+1}}S\left(E\right)/\sigma_{\varepsilon}\left(E\right)dE}.$$ Thus, we need to solve $$\mu\partial_{x}\psi_{g}\left(x,\mu\right)+\sigma_{g}\psi_{g}\left(x,\mu\right)=S_{g}\left(T\right),$$ where $$S_{g}\left(T\right)=\int_{E_{g}}^{E_{g+1}}\frac{S\left(E\right)}{\sigma_{\varepsilon}\left(E\right)}dE,\,\,\,\,\sigma_{g}=\frac{\int_{E_{g}}^{E_{g+1}}S\left(E\right)dE}{\int_{E_{g}}^{E_{g+1}}S\left(E\right)/\sigma_{\varepsilon}\left(E\right)dE}.$$

A regular band model example\[sub:Example-1:-the\]
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In order to assess the accuracy of the discrete approximation of the Young measure $\lambda_{E}$ associated with (\[eq:Elsasser band model, intro\]), we first compute $\lambda_{E}$ analytically (note that, since (\[eq:Elsasser band model, intro\]) does not depend on temperature $T$, we drop the superscript $T$ on **$\lambda_{E}^{T}$**).

Using that $\sigma_{\varepsilon}\left(E\right)=\sigma_{1}\left(E/\varepsilon\right)$, with $\sigma_{1}\left(E\right)$ a $1$-periodic function, it is a standard result (see e.g. [@BESICO:1954]) that $$\lim_{\varepsilon\rightarrow0}\int_{E_{0}}^{E_{1}}f\left(\sigma_{\varepsilon}\left(E'\right)\right)dE'=\int_{0}^{1}f\left(\sigma_{1}\left(E'\right)\right)dE',$$ for any values $0\leq E<E_{1}\leq1$. Now, consider the change of variables, $$\xi=\frac{\cosh\left(\beta\right)+1}{\cosh\left(\beta\right)-\cos\left(2\pi E\right)}.$$ Then $$d\xi=-\frac{2\pi}{\cosh\left(\beta\right)+1}\xi^{2}\sqrt{1-\frac{\left(\xi\cosh\left(\beta\right)-\cosh\left(\beta\right)-1\right)^{2}}{\xi^{2}}}dE.$$ It follows that, with $c_{\beta}=\left(\cosh\left(\beta\right)+1\right)/\left(\cosh\left(\beta\right)-1\right)$, $$\int_{0}^{1}f\left(\sigma_{1}\left(E'\right)\right)dE'=\frac{\cosh\left(\beta\right)+1}{2\pi}\int_{1}^{c_{\beta}}\frac{f\left(\xi\right)}{\xi^{2}\sqrt{1-\xi^{-2}\left(\sigma\cosh\left(\beta\right)-\cosh\left(\beta\right)-1\right)^{2}}}d\xi.$$ Therefore, the Young measure is given by $$d\lambda_{E}\left(\xi\right)=\frac{1}{\xi^{2}\sqrt{1-\xi^{-2}\left(\sigma\cosh\left(\beta\right)-\cosh\left(\beta\right)-1\right)^{2}}}\chi_{\left[1,c_{\beta}\right]}\left(\xi\right)d\xi.\label{eq:analytic Young measure}$$ As expected, $\lambda_{E}\left(\xi\right)$ is independent of 