Multigroup Neutron Transport and Diffusion Computations

  • Alain Hébert
Reference work entry


The transport equation is introduced to describe a population of neutral particles such as neutrons or photons, in a close domain, under steady-state (i.e., stationary) conditions. Its derivation is based on the principle of particle conservation. The transport equation describes the statistical behavior of a large population of particles. The exact number of particles per unit volume is continuously varying with time, even at steady-state conditions. Under steady-state conditions, the number density of particles oscillates about an average value related to the solution of the steady-state transport equation. A solution of the transport equation is required in many fields of nuclear engineering, notably in reactor physics, in safety and criticality, and in radiation shielding and protection. We review legacy approaches for solving the steady-state transport equation, namely, the method of spherical harmonics, the collision probability method, the discrete ordinates method, and the method of characteristics. The full-core calculation consists of solving a simplified transport equation, either the diffusion equation or the simplified P n equation.


Neutron Flux Collision Probability Source Density Discrete Ordinate Linear Boltzmann Equation 
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  1. Abramowitz M and Stegun IA (1970) Handbook of mathematical functions. Dover Publications, Inc., New YorkGoogle Scholar
  2. Alcouffe RE (1977) Diffusion synthetic acceleration methods for the diamond differenced discrete ordinates equations. Nucl Sci Eng 64:344Google Scholar
  3. Alcouffe RE, O’Dell D (1986) Florida, transport calculations for nuclear reactors. In: Ronen Y (ed) CRC Handbook of nuclear reactors calculations, vol I. CRC Press, Boca RatonGoogle Scholar
  4. Askew JR (1972) A characteristics formulation of the neutron transport equation in complicated geometries. Report AEEW-M 1108, United Kingdom atomic energy establishment, WinfrithGoogle Scholar
  5. Carlson BG, Bell GI (1958) Solution of the transport equation by the S N Method. Proceedings of the United Nations international conference on peaceful uses of atomic energy, 2nd Geneva P/2386Google Scholar
  6. Carlson BG (1971) On a more precise definition of discrete ordinates methods. Proceedings of the 2nd conference on transport theory, April 1971, Los Alamos, N. M., pp. 348–390, CONF–710107, U. S. Atomic Energy CommissionGoogle Scholar
  7. Carlvik I (1965) A Method for Calculating Collision Probabilities in General Cylindrical Geometry and Applications to Flux Distributions and Dancoff Factors, Proceedings of the United Nations international conference on peaceful uses of atomic energy, vol. 2. Geneva, p 225Google Scholar
  8. Chandrasekhar S (1960) Radiative transfer. Dover, New YorkGoogle Scholar
  9. Eddington A (1926) The internal constitution of the stars. Dover, New YorkMATHGoogle Scholar
  10. Gelbard EM (1960) Application of spherical harmonics method to reactor problems. WAPD-BT-20, Bettis Atomic Power LaboratoryGoogle Scholar
  11. Gelbard EM (1968) Computing methods in reactor physics (chap. 4). In: Greenspan H, Kelber CK, Okrent D (eds). Gordon and Breach, New YorkGoogle Scholar
  12. Gunther G, Kinnebrock W (1973) SNOW: a two-dimensional SN program for solving the neutron-transport equation in rectangular and cylindrical geometry, Report EURFNR-1130 (KFK 1826)Google Scholar
  13. Hébert A (1981) Développement de la méthode SPH: Homogénéisation de cellules dans un réseau non uniforme et calcul des paramètres de réflecteur, Note CEA-N-2209, Commissariat à l’Énergie Atomique, FranceGoogle Scholar
  14. Hébert A (1987) Development of the nodal collocation method for solving the neutron diffusion equation Ann Nucl Energy 14(10): 527Google Scholar
  15. Hébert A (1987) TRIVAC, a modular diffusion code for fuel management and design applications. Nucl J Canada 1(4):325. Available at
  16. Hébert A (1993) Application of a dual variational formulation to finite element reactor calculations. Ann Nucl Energy 20:823CrossRefGoogle Scholar
  17. Hébert A (2006) The search for superconvergence in spherical harmonics approximations Nucl Sci Eng 154:134Google Scholar
  18. Hébert A (2006) High order diamond differencing schemes. Ann nucl Energy 33:1479CrossRefGoogle Scholar
  19. Hébert A (2008) A Raviart–Thomas–Schneider solution of the diffusion equation in hexagonal geometry Ann Nucl Energy, 35, 363–376Google Scholar
  20. Hébert A (2009) Applied reactor physics. Presses internationales polytechnique, Montréal (See the home page at
  21. Halsall MJ (1980) CACTUS, a characteristics solution to the neutron transport equation in complicated geometries. Report AEEW-R 1291, United Kingdom Atomic Energy Establishment, WinfrithGoogle Scholar
  22. Jenal JP, Erickson PJ, Rhoades WA, Simpson DB, Williams ML (1977) The generation of a computer library for discrete ordinates quadrature sets. Report ORNL/TM-6023, Oak Ridge National Laboratory, TennesseeGoogle Scholar
  23. Kavenoky A (1969) Calcul et utilisation des probabilités de première collision pour les milieux hétérogènes à une dimension: Les programmes ALCOLL et CORTINA, Note CEA-N-1077, Commissariat à l’Énergie Atomique, FranceGoogle Scholar
  24. Kelley CT (1995) Iterative methods for linear and nonlinear equations. SIAM, PhiladelphiaMATHCrossRefGoogle Scholar
  25. Knott D, Edenius M (1993) Validation of the CASMO-4 transport solution. In: Kusters H, Stein E, Werner W (eds) Proceedings of the international conference on mathematical methods and supercomputing in nuclear applications, vol 2, Kernforschungszentrum Karlsruhe GmbH, Karlsruhe, Germany, p 547, April 19–23.Google Scholar
  26. Lathrop KD (1968) Ray effects in discrete ordinates equations. Nucl Sci Eng 32:357Google Scholar
  27. Leonard A, McDaniel CT (1995) Optimal polar angles and weights for the characteristics method. Trans Am Nucl Soc 73:172Google Scholar
  28. Lewis EE, Miller WF, Jr (1984) Computational methods of neutron transport. John Wiley, New York. (Republished by American Nuclear Society, Inc., 1993)Google Scholar
  29. Marleau G, Hébert A, Roy R (1992) New computational methods used in the lattice code dragon. Proceedings of the international topl meeting on advances in reactor physics, Charleston, USA, March 8–11. American Nuclear Society. Available at
  30. Raviart P, Thomas J (1977) A mixed finite element method for second order elleptic equations. In Mathematical aspects of the finite element method. Lecture notes in mathematics, vol 606. Springer, Berlin, pp 292–315Google Scholar
  31. Roy R (1998) The cyclic characteristics method International conference on physics of nuclear science and technology, Long Island, New York, October 5–8Google Scholar
  32. Saad Y, Schultz MH (1986) GMRES: a Generalized Minimal RESidual algorithm for solving nonsymmetric linear systems. SIAM J Sci Stat Comput 7:856–869MathSciNetMATHCrossRefGoogle Scholar
  33. Sanchez R, McCormick NJ (1982) A review of neutron transport approximations. Nucl Sci Eng 80:481Google Scholar
  34. Sanchez R, Chetaine A (2000) A synthetic acceleration for a two-dimensional characteristic method on unstructured meshes Nucl Sci Eng 136:122Google Scholar
  35. Santandrea S, Sanchez R (2002) Acceleration techniques for the characteristic method in unstructured meshes. Ann Nucl Energy 29:323CrossRefGoogle Scholar
  36. Santandrea S, Sanchez R (2005) Analysis and improvements of the DPN acceleration technique for the method of characteristics in unstructured meshes. Ann Nucl Energy 32:163CrossRefGoogle Scholar
  37. Santandrea S, Mosca P (2006) Linear surface characteristic scheme for the neutron transport equation in unstructured geometries. Proceedings of the international meeting on advances in nuclear analysis and simulation, September 10 – 14, Vancouver, CanadaGoogle Scholar
  38. Semenza LA, Lewis EE, Rossow EC (1972) The application of the finite element method to the multigroup neutron diffusion equation. Nucl Sci Eng 47:302Google Scholar
  39. Shober RA, Henry AF (1976) Nonlinear methods for solving the diffusion equation. M.I.T. Report MITNE–196Google Scholar
  40. Smith KS (1979) An analytic nodal method for solving the two-group, multidimensional, static and transient neutron diffusion equation. Nuclear Engineer’s Thesis, Massachusetts Institute of Technology, Department of Nuclear EngineeringGoogle Scholar
  41. Suslov IR (1993) Solution of transport equation in 2- and 3-dimensional irregular geometry by the method of characteristics. In: Kusters H, Stein E, Werner W (eds) Proceedings of the international conference on mathematical methods and supercomputing in nuclear applications, vol 1, KernforschungszentrumGoogle Scholar
  42. Suslov IR (1997) An improved transport theory scheme based on the quasi-stationary derivatives principle International conference on mathematical methods and supercomputing in nuclear applications, Saratoga Springs, New York, October 5–9Google Scholar
  43. Le Tellier R (2006) Développement de la méthode des caractéristiques pour le calcul de réseau, Ph. D. thesis presented at the École Polytechnique de Montréal, CanadaGoogle Scholar
  44. Le Tellier R, Hébert A (2006) On the integration scheme along a trajectory for the characteristics method. Ann Nucl Energy 33:1260CrossRefGoogle Scholar
  45. Le Tellier R, Hébert A (2007) An improved algebraic collapsing acceleration with general boundary conditions for the characteristics method. Nucl Sci Eng 156:121Google Scholar
  46. Le Tellier R, Hébert A (2008) Anisotropy and particle conservation for trajectory–based deterministic methods. Nucl Sci Eng 158:28Google Scholar
  47. Le Tellier R, Hébert A, Santamarina A, Litaize O (2008) A modeling of BWR-MOX assemblies based on the characteristics method combined with advanced self-shielding models Nucl Sci Eng 158:231Google Scholar
  48. Villarino EA, Stamm’ler RJJ, Ferri AA, Casal JJ (1992) HELIOS: angularly dependent collision probabilities. Nucl Sci Eng 112:16Google Scholar
  49. Wu GJ, Roy R (2003) A new characteristics algorithm for 3D transport calculations. Ann Nucl Energy 30:1CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Institut de Génie NucléaireÉcole Polytechnique de MontréalQuébecCanada

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