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abstract: |
    We present a numerical method for solving the time-independent thermal radiative transfer (TRT) equation or the neutron transport (NT) equation when the opacity or cross-section varies rapidly in energy (frequency) on the microscale $\varepsilon$; $\varepsilon$ corresponds to the characteristic spacing between absorption lines or resonances, and is much smaller than the macroscopic energy (frequency) variation of interest. The approach is based on a rigorous homogenization of the TRT/NT equation in the energy (frequency) variable. Discretization of the homogenized TRT/NT equation results in a multigroup-type system, and can therefore be solved by standard methods.

    We demonstrate the accuracy and efficiency of the approach on three model problems. First we consider the Elsasser band model with constant temperature and a line spacing $\varepsilon=10^{-4}$. Second, we consider a neutron transport application for fast neutrons incident on iron, where the characteristic resonance spacing $\varepsilon$ necessitates $\approx16,000$ energy discretization parameters if Planck-weighted cross sections are used. Third, we consider an atmospheric TRT problem for an opacity corresponding to water vapor over a frequency range $1000-2000\,\mathrm{cm}^{-1}$, where we take $12$ homogeneous layers between $1\,$km - $15\,$km, and temperature/pressure values in each layer from the standard US atmosphere. For all three problems, we demonstrate 