Skip to main content
Springer Nature Link
Log in

Filtered Discrete Ordinates Equations for Radiative Transport

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

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 \(\Vert u(t) \Vert _B = \Vert u \Vert _B(t)\).

References

  1. Abramowitz, M., Stegun, I. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards, Gaithersburg (1964)

    MATH  Google Scholar 

  2. Alexander, R.: Diagonally implicit Runge–Kutta method for stiff O.D.E.’s. SIAM J. Numer. Anal. 14, 1006–1021 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  3. Atkinson, K.: Numerical intergration on the sphere. J. Aust. Math. Soc. 23, 332–347 (1982)

    Article  MATH  Google Scholar 

  4. Atkinson, K., Han, W.: Spherical Harmonics and Approximations on the Unit Sphere: An Introduction. Springer, Berlin (2012)

    Book  MATH  Google Scholar 

  5. Aubin, T.: Nonlinear Analysis on Manifolds, Monge–Ampere Equations. Grundlehren der mathematischen Wissenschaften, vol. 252. Springer, Berlin (1982)

    Google Scholar 

  6. Ben-Yu, G.: Spectral Methods and Their Applications. World Scientific, Singapore (1998)

    Book  MATH  Google Scholar 

  7. Briggs, L., Miller Jr., W.F., Lewis, E.: Ray-effect mitigation in discrete ordinate-like angular finite element approximations in neutron transport. Nucl. Sci. Eng. 57, 205–217 (1975)

    Article  Google Scholar 

  8. Brunner, T., Holloway, J.: Two-dimensional time-dependent riemann solvers for neutron transport. J. Comput. Phys. 210, 386–399 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Buras, R., Rampp, M., Janka, H.-T., Kifonidis, K.: Two-dimensional hydrodynamic core-collapse supernova simulations with spectral neutrino transport. Astron. Astrophys. 447, 1049–1092 (2005)

    Article  Google Scholar 

  10. Canuto, C., Hussaini, M., Quarteroni, A., Zang, T.: Spectral Methods: Fundamentals in Single Domain. Springer, Berlin (2006)

    Book  MATH  Google Scholar 

  11. Carlson, B.: Transport theory: discrete ordinates quadrature over the unit sphere. Technical report, Los Alamos Scientific Lab. (1970)

  12. Case, K., Zweifel, P.: Linear Transport Theory. Addison-Wesley, Reading (1967)

    MATH  Google Scholar 

  13. Chandrasekhar, S.: Radiative Transfer. Dover Publications Inc., Mineola (1960)

    MATH  Google Scholar 

  14. Crockatt, M., Christlieb, A., Garrett, C.K., Hauck, C.: An arbitrary-order, fully implicit, hybrid kinetic solver for linear radiative transport using integral deferred correction. J. Comput. Phys. 346, 212–241 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dautray, R., Lions, J.-L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 6. Springer, Berlin (1984)

    MATH  Google Scholar 

  16. Dolejsi, V., Feistauer, M.: Discontinuous Galerkin Method: Analysis and Applications to Compressible Flow. Springer, Berlin (2015)

    Book  MATH  Google Scholar 

  17. Frank, M., Hauck, C., Kupper, K.: Convergence of filtered spherical harmonic equations for radiation transport. Commun. Math. Sci. 14, 1443–1465 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ganapol, B., Baker, R., Dahl, J., Alcouffe, R.: Homogeneous infinite media time-dependent analytical benchmarks. Technical report, Los Alamos National Laboratory (2001)

  19. Garrett, C.K., Hauck, C.: A comparison of moment closures for linear kinetic transport equations: the line source benchmark. Transp. Theory Stat. Phys. 42, 203–235 (2014)

    Article  MATH  Google Scholar 

  20. Han, W., Huang, J., Eichholz, J.A.: Discrete-ordinate discontinuous galerkin methods for solving the radiative transfer equation. SIAM J. Sci. Comput. 32, 477–497 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hansen, P.C., O’Leary, D.P.: The use of the L-curve in the regularization of discrete ill-posed problems. SIAM J. Sci. Comput. 14, 1487–1503 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hesthaven, J.S., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time-Dependent Problems, vol. 21. Cambridge University Press, Cambridge (2007)

    Book  MATH  Google Scholar 

  23. Kaplan, S.: A new derivation of discrete ordinate approximations. Nucl. Sci. Eng. 34, 76–82 (1968)

    Article  Google Scholar 

  24. Keller, H.B.: Approximate solutions of transport problems. II. Convergence and applications of the discrete ordinate method. J. Soc. Ind. Appl. Math. 8, 43–73 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  25. Larsen, E.W., Morel, J.E.: Nuclear Computional Science, Chapter 1: Advances in Discrete-Ordinates Methodology. Springer, Dordrecht (2010)

    Google Scholar 

  26. Lathrop, K.: Remedies for ray effects. Nucl. Sci. Eng. 45, 255–268 (1971)

    Article  Google Scholar 

  27. Lathrop, K., Carlson, B.: Discrete Ordinates Angular Quadrature of the Neutron Transport Equation. Los Alamos Scentific Laboratory, Los Alamos (1964)

    Book  Google Scholar 

  28. Lebedev, V.: Quadratures on a sphere. USSR Comput. Math. Math. Phys. 16, 10–24 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lewis, E.E., Miller, J.W.F.: Computational Methods of Neutron Transport. American Nuclear Society, La Grange Park (1993)

    MATH  Google Scholar 

  30. Li, H.-S., Flamant, G., Lu, J.-D.: Mitigation of ray effects in the discrete ordinates method. Numer. Heat Transf. B Fundam. 43, 445–466 (2003)

    Article  Google Scholar 

  31. McClarren, R.G., Hauck, C.D.: Robust and accurate filtered spherical harmonics expansions for radiative transfer. J. Comput. Phys. 229, 5597–5614 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Mezzacappa, A., Calder, A., Bruenn, S., Blondin, J., Guidry, M., Strayer, M., Umar, A.: An investigation of neutrino-driven convection and the core collapse supernova mechanism using multigroup neutrino transport. Astrophys. J. 495, 911–926 (1998)

    Article  Google Scholar 

  33. Mihalis, D., Weibel-Mihalis, B.: Foundations of Radiation Hydrodynamics. Dover, Mineola (1999)

    Google Scholar 

  34. Morel, J.E.: A hybrid collocation-galerkin-\({S}_{N}\) method for solving the boltzmann transport equation. Nucl. Sci. Eng. 101, 72–87 (1989)

    Article  Google Scholar 

  35. Muller, C.: Spherical Harmonics. Springer, Berlin (1966)

    Book  MATH  Google Scholar 

  36. Pomraning, G.C.: Radiation Hydrodynamics. Pergamon Press, New York (1973)

    Google Scholar 

  37. Radice, D., Abdikamalov, E., Rezzolla, L., Ott, C.D.: A new spherical harmonics scheme for multi-dimensional radiation transport I. Static matter configurations. J. Comput. Phys. 242, 648–669 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  38. Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  39. Seibold, B., Frank, M.: StaRMAP: a second order staggered grid method for spherical harmonics moment equation of radiative transfer. ACM Trans. Math. Softw. 41, 4 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  40. Thurgood, C., Pollard, A., Becker, H.: The \(T_{N}\) quadrature set for the discrete ordinates method. J. Heat Transf. 117, 1068–1070 (1995)

    Article  Google Scholar 

  41. Vandeven, H.: Family of spectral filters for discontinuous problems. J. Sci. Comput. 6, 159–192 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  42. Victory Jr., H.D.: Convergence properties of discrete-ordinates solutions for neutron transport in three-dimensional media. SIAM J. Numer. Anal. 17, 71–83 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zheng-Ming, L., Brahme, A.: An overview of the transport theory of charged particles. Radiat. Phys. Chem. 41, 673–703 (1993)

    Article  Google Scholar 

Download references

Author information

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

Authors
  1. Cory Hauck
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Vincent Heningburg
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Vincent Heningburg.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This manuscript has been authored, in part, 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). This material is based, in part, upon work supported by the National Science Foundation under Grant No. 1217170 and by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-019-00950-1

Keywords