Numerische Mathematik

, Volume 34, Issue 4, pp 353–370 | Cite as

Convergence of two-dimensional Nyström discrete-ordinates in solving the linear transport equation

  • P. NelsonJr.
  • H. D. VictoryJr.


In the discrete-ordinates approximation to the linear transport equation, the integration over the directional variable is replaced by a numerical quadrature rule involving a weighted sum over functional values at selected directions. The purpose of this paper is to show that the Nyström technique of defining the angular flux in directions other than the quadrature points, as outlined by P.M. Anselone and A. Gibbs and utilized by P. Nelson for anisotropically scattering slabs, produces an approximation scheme which is stable, consistent with, and convergent to the transport equation in two-dimensional geometry.

Subject Classifications

65R05 (Primary) 82A75 


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  1. 1.
    Abu-Shumays, I.K.: Compatible product angular quadrature for neutron transport inxy geometry. Nucl. Sci. Eng.64, 299–316 (1977).Google Scholar
  2. 2.
    Anselone, P.M.: Convergence of Chandrasekhar's method for inhomogeneous transfer problems. J. Math. Mech.10, 537–546 (1961)Google Scholar
  3. 3.
    Anselone, P.M.: Collectively compact operator approximation theory. Englewood Cliffs, N.J.: Prentice-Hall 1971Google Scholar
  4. 4.
    Anselone, P.M., Gibbs, A.G.: Convergence of the discrete ordinates method for the transport equation. In: Constructive and computational methods for differential and integral equations, D.L. Colton, R.P. Gilbert, eds. Lecture Notes in Mathematics, Vol. 430, pp. 1–27, Berlin Heidelberg New York: Springer 1974Google Scholar
  5. 5.
    Chuyanov, V. A.: On the convergence of the approximate solution of the kinetic equation (by a quadrature method of Gaussian type. In: Some mathematical problems of neutron physics, pp. 199–200. Moscow: Izd-vo, MGU 1960Google Scholar
  6. 6.
    Dieudonne, J.: Foundations of modern analysis, 1st ed. New York: Academic Press 1960Google Scholar
  7. 7.
    Dugundji, J.: Topology, 1st ed. Boston: Allyn and Bacon, 1966Google Scholar
  8. 8.
    Keller, H.B.: Approximate solutions of transport problems II. Convergence and applications of the discrete ordinate method. J. Soc. Indus. Appl. Math.8, 43–73 (1960)Google Scholar
  9. 9.
    Keller, H.B.: On the pointwise convergence of the discrete ordinate method.ibid.8, 560–567 (1960)Google Scholar
  10. 10.
    Keller, H.B.: Convergence of the discrete ordinate method for the anisotropic-scattering transport equation. In: Symposium, Provisional International Computation Centre. Basel: Birkhauser 1960Google Scholar
  11. 11.
    Kraft, Richard: Convergence of semidiscrete approximations of linear transport equations. J. Math. Anal. Appl.37, 412–431 (1972)Google Scholar
  12. 12.
    Lathrop, K.D. Remedies for ray effects, Nucl. Sci. Eng.45, 255–268 (1971), esp. pp. 259–260Google Scholar
  13. 13.
    Lathrop, K.D.: Discrete-ordinates methods for the numerical solution of the transport equation. Reactor Technology15, 107–35 (1972)Google Scholar
  14. 14.
    Lathrop, K.D., Brinkley, F.W.: TWOTRAN II: An interfaced, exportable version of the TWOTRAN code for two-dimensional transport. Los Alamos Scientific Laboratory: LA-4848-MS (1975)Google Scholar
  15. 15.
    Lathrop, K.D., Carlson, B.G.: Discrete ordinates angular quadrature of the neutron transport equation. Los Alamos Scientific Laboratory: LA-3186 (1965)Google Scholar
  16. 16.
    Lathrop, K.D., Carlson, B.G.: Properties of new numerical approximations to the transport equation. J. Quant. Spectrosc. Radiat. Transfer11, 921–48 (1971)Google Scholar
  17. 17.
    Madsen, N.K.: Pointwise convergence of the three-dimensional discrete-ordinates method. SIAM J. Numer. Anal.8, 266–269 (1971)Google Scholar
  18. 18.
    Miller, W.F., Jr., Dudziak, D., O'Dell, R.P., Gomez, M.F.: Transport and reactor theory. Los Alamos Scientific Laboratory: LA-7271-PR (Progress Report), January 1–March 31, 1978Google Scholar
  19. 19.
    Miller, W.F. Jr., Reed, W.H.: Ray-effect mitigation methods for two-dimensional neutron transport theory. Nucl. Sci. Eng.62, 391–411 (1977)Google Scholar
  20. 20.
    Natanson, I.P.: Constructive function theory, 1st ed. New York: Frederick Ungar 1965Google Scholar
  21. 21.
    Nelson, P., Jr.: Subcriticality for transport of multiplying particles in a slab. J. Math. Anal. Appl.35, 90–104 (1971)Google Scholar
  22. 22.
    Nelson, P., Jr.: Convergence of the discrete-ordinates method for anisotropically scattering multiplying particles in a subcritical slab. SIAM J. Numer. Anal.10, 175–181 (1973)Google Scholar
  23. 23.
    Nelson, P., Jr., Victory, H.D., Jr.: Theoretical properties of one-dimensional discrete-ordinates. SIAM J. Numer. Anal.16, 270–283 (1979)Google Scholar
  24. 24.
    Nestell, M.K.: The convergence of the discrete-ordinates method for integral equations of anisotropic radiative transfer. Techn. Rep. 23. Department of Mathematics, Oregon State University, Corvallis, 1965Google Scholar
  25. 25.
    Rockafellar, R.T.: Convex analysis, 1st ed. Princeton University Press 1970Google Scholar
  26. 26.
    Rudin, W.: Real and complex analysis, 1st ed. New York: McGraw Hill 1968Google Scholar
  27. 27.
    Stroud, A.H.: Approximate calculation of multiple integrals. Englewood Cliffs, N.J.: Prentice Hall 1971Google Scholar
  28. 28.
    Wendroff, B.: On the convergence of the discrete-ordinates method. J. Soc. Indus. Appl. Math.8, 508–513 (1960)Google Scholar
  29. 29.
    Wilson, D.G.: The time dependent linear transport equation in a multi-dimensional parallepiped with partially reflecting walls. Doctoral Thesis: University of Maryland, University Park, Md. 1971Google Scholar
  30. 30.
    Wilson, D.G.: An alternating direction implicit scheme for discrete-ordinate transport equations. J. Math. Anal. Appl.48, 95–113 (1974)Google Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • P. NelsonJr.
    • 1
  • H. D. VictoryJr.
    • 1
  1. 1.Institute for Numerical Transport Theory, Department of MathematicsTexas Tech UniversityLubbockUSA

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