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Filtered Discrete Ordinates Equations for Radiative Transport

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Abstract

We present and analyze a discrete ordinates (\(\text {S}_N\)) discretization of a filtered radiative transport equation (RTE). Under certain conditions, \(\text {S}_N\) discretizations of the standard RTE create numeric artifacts, known as “ray-effects”; the goal of the filter is to remove such artifacts. We analyze convergence of the filtered discrete ordinates solution to the solution of the non-filtered RTE, taking into account the effect of the filter as well as the usual quadrature and truncation errors that arise in discretize ordinate methods. We solve the filtered \(\text {S}_N\) equations numerically with a discontinuous Galerkin spatial discretization and implicit time stepping. The form of the filter enables the resulting linear systems to be solved in an established Krylov framework. We demonstrate, via the simulation of two benchmark problems, the effectiveness of the filtering approach in reducing ray effects. In addition, we also examine efficiency of the method, in particular the balance between improved accuracy and additional cost of including the filter.

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Notes

  1. Throughout the paper we use superscripts to denote the order of an approximation. We reserve subscripts to denote components.

  2. Given a generic time-dependent function \(u :t \mapsto u(t) \in B\), with B a normed vector space, we abuse notation slightly by writing \(\Vert u(t) \Vert _B = \Vert u \Vert _B(t)\).

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Authors and Affiliations

  1. Computational Mathematics Group, Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN, 37831, USA

    Cory Hauck

  2. Mathematics Department, University of Tennessee, Knoxville, TN, 37996, USA

    Vincent Heningburg

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  1. Cory Hauck
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  2. Vincent Heningburg
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Correspondence to Vincent Heningburg.

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Hauck, C., Heningburg, V. Filtered Discrete Ordinates Equations for Radiative Transport. J Sci Comput 80, 614–648 (2019). https://doi.org/10.1007/s10915-019-00950-1

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