A multigrid method for solving the one-dimensional slab-geometry ${\text{S}}_N $ equations with isotropic scattering and absorption is presented. The case with no absorption was treated in part I of this paper [Manteuffel, McCormick, Morel, Oliveira, and Yang, SIAM J. Sci. Comput., 16 (1995), pp. 601–635]. Relaxation is based on a two-cell inversion, which is very efficient because it takes advantage of the structure of the two-cell problem. For interpolation we use kinked linear elements. The kink is based on the amount of absorption present. The restriction operator is full weighting. Numerical results show this algorithm to be faster than diffusion synthetic acceleration (DSA) in all regimes. This scheme is also well suited for massively parallel computer architectures.

MSC codes

  1. 65N20
  2. 65F10

MSC codes

  1. multigrid
  2. particle transport

Get full access to this article

View all available purchase options and get full access to this article.


M. L. Adams, W. R. Martin, Diffusion-synthetic acceleration of discontinuous finite-element transport iterations, Nuclear Sci. Eng, to appear
R. E. Alcouffe, Diffusion synthetic acceleration methods for the diamond-differenced discrete-ordinates equations, Nuclear Sci. Eng., 64 (1997), 344–
Achi Brandt, Multi-level adaptive solutions to boundary-value problems, Math. Comp., 31 (1977), 333–390
V. Faber, Thomas A. Manteuffel, P. Nelson, A look at transport theory from the point of view of linear algebraTransport theory, invariant imbedding, and integral equations (Santa Fe, NM, 1988), Lecture Notes in Pure and Appl. Math., Vol. 115, Dekker, New York, 1989, 37–61
E. W. Larsen, Unconditionally stable diffusion-synthetic acceleration methods for the slab geometry discrete ordinates equations, Nuclear Sci. Eng., 82 (1982), 47–
Edward W. Larsen, J. E. Morel, Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes. II, J. Comput. Phys., 83 (1989), 212–236
E. E. Lewis, W. F. Miller, Jr., Computational Methods of Neutron Transport, 1984
T. A. Manteuffel, S. McCormick, J. Morel, S. Oliveira, G. Yang, Parallel multigrid methods for transport equations, Proc. Copper Mountain Conference on Iterative Methods, Copper Mountain, CO, 1992, April 9–14
T. A. Manteuffel, S. McCormick, J. Morel, S. Oliveira, G. Yang, A parallel version of a multigrid algorithm for isotropic transport equations, SIAM J. Sci. Comput., 15 (1994), 474–493
T. A. Manteuffel, S. McCormick, J. Morel, S. Oliveira, G. Yang, A fast multigrid algorithm for isotropic transport problems. I. Pure scattering, SIAM J. Sci. Comput., 16 (1995), 601–635
T. A. Manteuffel, S. McCormick, J. Morel, G. Yang, A Fast Multigrid Algorithm for Isotropic Transport Problems II: With Absorption, Report, LA-UR-94-2361, Los Alamos National Laboratories, Los Alamos, NM, 1994, June
Stephen F. McCormick, Multilevel adaptive methods for partial differential equations, Frontiers in Applied Mathematics, Vol. 6, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989x+162
J. E. Morel, E. W. Larsen, A multiple balance approach for differencing the $S_{N}$ equations, Nuclear Sci. Eng., 105 (1990), 1–
J. E. Morel, T. A. Manteuffel, An angular multigrid acceleration technique for the $S_{N}$ equations with highly forward-peaked scattering, Nuclear Sci. Eng., 107 (1991), 330–
S. Oliveira, Ph.D. Thesis, Parallel Multigrid Methods for Transport Equations, Department of Mathematics, University of Colorado at Denver, Denver, CO, 1993

Information & Authors


Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: 1449 - 1474
ISSN (online): 1095-7197


Submitted: 1 July 1993
Accepted: 19 June 1995
Published online: 17 February 2012

MSC codes

  1. 65N20
  2. 65F10

MSC codes

  1. multigrid
  2. particle transport



Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.







Copy the content Link

Share with email

Email a colleague

Share on social media