Ray Effect Mitigation for the Discrete Ordinates Method Using Artificial Scattering

Abstract

Solving the radiative transfer equation with the discrete ordinates (S${ }_N$) method leads to a nonphysical imprint of the chosen quadrature set on the solution. To mitigate these so-called ray effects, we propose a modification of the S${ }_N$ method that we call artificial scattering S${ }_N$ (as-S${ }_N$). The method adds an artificial forward-peaked scattering operator that generates angular diffusion to the solution and thereby mitigates ray effects. Similar to artificial viscosity for spatial discretizations, the additional term vanishes as the number of ordinates approaches infinity. Our method allows an efficient implementation of explicit and implicit time integration according to standard S${ }_N$ solver technology. For two test cases, we demonstrate a significant reduction of error for the as-S${ }_N$ method when compared to the standard S${ }_N$ method, both for explicit and implicit computations. Furthermore, we show that a prescribed numerical precision can be reached with less memory due to the reduction in the number of ordinates.

Keywords:

• Discrete ordinates method
• ray effects
• radiative transfer, quadrature

Acknowledgments

The authors wish to thank Ryan G. McClarren (University of Notre Dame) for many fruitful discussions.

This material is based upon work supported by the National Science Foundation under Grant No. 1913277 and by UT-Battelle, LLC, under Contract No. DE-AC0500OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for the United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).

Notes

^a While the idea of artificial scattering works with any Dirac sequence, the asymptotic analysis that is performed later imposes stronger requirements to obtain a Fokker-Planck operator in the respective limit.