# On the convergence of the multigroup, discrete-ordinates solutions for subcritical transport media

- 42 Downloads
- 3 Citations

## Summary

*In the multigroup, discrete-ordinates approximations 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, with the energy dependence discretized by replacing the cross section data by weighted averages over each energy interval. The stability, consistency, and convergence rely fundamentally on 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, and as the number of angular nodes becomes infinite. Our analysis is based on using a natural Nyström method of extending the discrete-ordinates, multigroup approximates to all values of the angular and energy variables. Such an extension enables us to employ generalizations of the collectively compact operator approximation theory of P. M. Anselone to deduce stability and convergence of the approximates*.

## Keywords

Total Cross Section Compact Operator Operator Approximation Secondary Particle Energy Variable## Preview

Unable to display preview. Download preview PDF.

## References

- [1]E. J. Allen,
*A finite element approach for treating the energy variable in the numerical solution of the neutron transport equation*, Trans. Th. Stat. Phys.,**15**(1986), pp. 449–478.MathSciNetCrossRefMATHGoogle Scholar - [2]P. M. Anselone,
*Collectivity Compact Operator Approximation Theory*, Prentice-Hall, Englewood Cliffs, N. J., 1971.Google Scholar - [3]A. Belleni-Morante -G. Busoni,
*Multigroup neutron transport*, J. Math. Phys.,**13**(1972), pp. 1146–1150.MathSciNetCrossRefGoogle Scholar - [4]M. Borysiewicz -N. Kruszynska,
*Smoothness of the solution of the 2-D neutron trnsport equation in the polygon region*, Atomkemenergie-Kerntechnik,**34**(1979), pp. 11–15.Google Scholar - [5]J. Dieudonné,
*Foundations of Modern Analysis*, 1st ed., New York, Academic Press, 1960.MATHGoogle Scholar - [6]R. B. Kellogg,
*Numerical analysis of the neutron transport ebuations*, in: Numerical Solution of Partial Differential Equations - III, SYNSPADE 1975, B. Hubbard ed., New York, Academic Press, Inc., 1976.Google Scholar - [7]E. Michael,
*Some extension theorems for continuous functions*, Pacific J. Math.,**3**(1953), pp. 789–806.MathSciNetCrossRefMATHGoogle Scholar - [8]P. Nelson -H. D. Victory, Jr.,
*Convergence of two-dimensional Nyström discrete-ordinates in solving the linear transport equation*, Numer. Math.,**34**(1980), pp. 353–370.MathSciNetCrossRefMATHGoogle Scholar - [9]P. Nelson -H. D. Victory, Jr.,
*On the convergence of the multigroup approximations for submultiplying slab media*, Math. Methods Appl. Sci.,**4**(1982), pp. 206–229.MathSciNetCrossRefMATHGoogle Scholar - [10]J. Pitkäranta -L. Ridgway Scott,
*Error estimates for the combined spatial and angular approximations of the transport equation for slab geometry*, SIAM J. Numer. Anal.,**20**(1983), pp. 922–950.MathSciNetCrossRefMATHGoogle Scholar - [11]
- [12]
- [13]J. D. Tamarkin,
*On the compactness of the space L*_{p}, Bull. Amer. Math. Soc.,**38**(1932), pp. 79–84.MathSciNetCrossRefMATHGoogle Scholar - [14]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.MathSciNetCrossRefMATHGoogle Scholar - [15]H. D. Victoey, Jr.,
*Convergence of the multigroup approximations for subcritical slab media with applications to shielding calculations*, Adv. Appl. Math.,**5**(1984), pp. 227–259.MathSciNetCrossRefGoogle Scholar - [16]H. D. Victory, Jr.,
*On the convergence of the multigroup approximations for multidimensional media*, Ann. Math. Pura ed Applicata, (IV),**140**(1985), pp. 197–207.MathSciNetMATHGoogle Scholar - [17]Yang Mingzhu -Zhu Guangtian,
*Multigroup theory for neutron transport*, Scientia Sinica,**22**(1979), pp. 1114–1127.MathSciNetMATHGoogle Scholar