Annali di Matematica Pura ed Applicata

, Volume 140, Issue 1, pp 179–207 | Cite as

On the convergence of the multigroup approximations for multidimensional media

  • H. D. VictoryJr.


In this paper, we investigate the stability, convergence, and consistency properties of steady-state multigroup models for submultiplying media with spatial dimensions greater than one. We define these concepts in a Banach space whose norm measures the collision density integrated over phase space. Stability and consistency occur under the conditions that the maximum fluctuations in the total cross section, and in the expected number of secondary particles arising from each energy level, tend to zero as the energy mesh becomes finer. A concluding example and discussion deal with pathologies of the multigroup model in situations where these fluctuations do not tend to zero as the norms of the energy partitions. The results in this paper complement the time-dependent results of both Belleni-Morante and Busoni for isotropic slabs and of Yang Mingzhu and Zhu Guangtian for bounded, three-dimensional media. This work directly extends the steady-state results of Paul Nelson, Jr. and H. D. Victory, Jr. for slab media to steady-state transport in multidimensional media.


Banach Space Phase Space Total Cross Section Spatial Dimension Secondary Particle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    M. Becker,Influence of Deep Minima on Multigroup Cross-Section Generation, Nucl. Sci. Eng.,57 (1975), pp. 75–93.Google Scholar
  2. [2]
    A. Belleni-Morante -G. Busoni,Multigroup Neutron Transport, J. Math. Phys.,13 (1972), pp. 1146–1150.Google Scholar
  3. [3]
    I.I.Bondarenko,Group Constants for Nuclear Reactor Calculations, Consultants Bureau, 1964.Google Scholar
  4. [4]
    D. Cullen,Applications of the Probability Table Method to Multigroup Calculations of Neutron Transport, Nucl. Sci. Eng.,55 (1974), pp. 387–400.Google Scholar
  5. [5]
    G. I. Marchuk,Computational Methods in Transfer Theory, Am. Math. Soc. Transl., (Series 2),70 (1968), pp. 116–136.Google Scholar
  6. [6]
    P. Nelson,An Error Estimate for the Multigroup Method with Application to Shielding Calculations, Ann. Nucl. Energy,7 (1980), pp. 297–306.Google Scholar
  7. [7]
    P. Nelson, Jr -H. D. Victory, Jr.,Convergence of Two-Dimensional Nyström Discrete-Ordinates in Solving the Linear Transport Equation, Numer. Math.,34 (1980), pp. 353–370.Google Scholar
  8. [8]
    P. Nelson -H. D. Victory, Jr.,On the Convergence of the Multigroup Approximations for Submultiplying Slab Media, Math. Meth. in the Appl. Sci.,4 (1982), pp. 206–229.Google Scholar
  9. [9]
    M. N. Nikolaev,Comments on the Probability Table Method, Nucl. Sci. Eng.,61 (1976), pp. 286–287.Google Scholar
  10. [10]
    W. Rudin,Real and Complex Analysis, McGraw-Hill, New York, 1968.Google Scholar
  11. [11]
    H. D. Victory, Jr.,Convergence Properties of Discrete-Ordinates Solutions for Neutron Transport in Three-Dimensional Media, SIAM J. Numer. Anal.,17 (1980), pp. 71–83.Google Scholar
  12. [12]
    V. S.Vladimirov,Mathematical Problems in the One-Velocity Theory of Particle Transport, Trudy. Math. Inst. Steklov,61 (1961). English translation report no. AECL-1661, Atomic Energy of Canada Limited, Chalk Biver, Ontario, 1963.Google Scholar
  13. [13]
    Yang Mingzhu -Zhu Guangtian,Multigroup Theory for Neutron Transport, Scientia Sinica, XXII (1979), pp. 1114–1127.Google Scholar

Copyright information

© Nicola Zanichelli Editore 1985

Authors and Affiliations

  • H. D. VictoryJr.
    • 1
  1. 1.TexasUSA

Personalised recommendations