Discontinuous Galerkin for the Radiative Transport Equation

  • Jean-Luc GuermondEmail author
  • Guido Kanschat
  • Jean C. Ragusa
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 157)


This note presents some recent results regarding the approximation of the linear radiative transfer equation using discontinuous Galerkin methods. The locking effect occurring in the diffusion limit with the upwind numerical flux is investigated and a correction technique is proposed.


Finite elements Discontinuous Galerkin Neutron transport Diffusion limit 



This material is based upon work supported by the Department of Homeland Security under agreement 2008-DN-077-ARI018-02, National Science Foundation grants DMS-0713829, DMS-0810387, and CBET-0736202, and is partially supported by award KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST).


  1. [1]
    M. L. Adams. Discontinuous finite element methods in thick diffusive problems. Nucl. Sci. Eng., 137:298–333, 2001.Google Scholar
  2. [2]
    B. Ayuso and L. D. Marini. Discontinuous Galerkin methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal., 47(2): 1391–1420, 2009.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    I. Babuška and M. Suri. On locking and robustness in the finite element method. SIAM J. Numer. Anal., 29(5):1261–1293, 1992.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    S. Chandrasekhar. Radiative Transfer. Oxford University Press, 1950.Google Scholar
  5. [5]
    R. Dautray and J.-L. Lions. Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 5. Evolution problems, I. Springer-Verlag, Berlin, Germany, 1992. ISBN 3-540-50205-X; 3-540-66101-8.Google Scholar
  6. [6]
    A. Ern and J.-L. Guermond. Discontinuous Galerkin methods for Friedrichs’ systems. I. General theory. SIAM J. Numer. Anal., 44(2): 753–778, 2006.Google Scholar
  7. [7]
    J.-L. Guermond and G. Kanschat. Asymptotic analysis of upwind discontinuous Galerkin approximation of the radiative transport equation in the diffusive limit. SIAM J. Numer. Anal., 48(1):53–78, 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    E. W. Larsen. Deterministic transport methods. Technical Report LA-9533-PR, Los Alamos Scientific Laboratory, Los Alamos, NM, 1982.Google Scholar
  9. [9]
    E. W. Larsen. On numerical solutions of transport problems in the diffusion limit. Nucl. Sci. Engr., 83:90–99, 1983.Google Scholar
  10. [10]
    E. W. Larsen and J. B. Keller. Asymptotic solution of neutron transport problems for small mean free paths. J. Mathematical Phys., 15:75–81, 1974.MathSciNetCrossRefGoogle Scholar
  11. [11]
    E. W. Larsen and J. E. Morel. Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes. II. J. Comput. Phys., 83(1):212–236, 1989.Google Scholar
  12. [12]
    E. W. Larsen, J. E. Morel, and W. F. Miller, Jr. Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes. J. Comput. Phys., 69(2):283–324, 1987.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    P. Lesaint and P.-A. Raviart. On a finite element method for solving the neutron transport equation. In Mathematical Aspects of Finite Elements in Partial Differential Equations, pages 89–123. Publication No. 33. Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974.Google Scholar
  14. [14]
    F. Malvagi and G. C. Pomraning. Initial and boundary conditions for diffusive linear transport problems. J. Math. Phys., 32(3):805–820, 1991.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    J. C. Ragusa, J.-L. Guermond, and G. Kanschat. A robust S N-DG-approximation for radiation transport in optically thick and diffusive regimes. J. Comput. Phys., 231(4):1947–1962, 2012.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    W. Reed and T. Hill. Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, Los Alamos, NM, 1973.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Jean-Luc Guermond
    • 1
    Email author
  • Guido Kanschat
    • 2
  • Jean C. Ragusa
    • 3
  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA
  2. 2.Heidelberg UniversityHeidelbergUSA
  3. 3.Department of Nuclear EngineeringTexas A&M UniversityCollege StationUSA

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