Skip to main content
SpringerLink
Log in

Second-Order Neutron Transport Methods

  • Chapter
  • First Online:
Book cover Nuclear Computational Science

Abstract

Among the approaches to obtaining numerical solutions for neutral particle transport problems, those classified as second-order or even-parity methods have found increased use in recent decades. First-order and second-order methods differ in a number of respects. Following discretization of the energy variable, invariably through some form of the multigroup approximation, the time-independent forms of both are differential in the spatial variable and integral in angle. They differ in that the more conventional first-order equation includes only first derivatives in the spatial variables, but requires solution over the entire angular domain. Conversely, the second-order form includes second derivatives but requires solution over one half of the angular domain. The two forms in turn lead to contrasting approaches to reducing the differential–integral equations to sets of linear equations and in the formulation of iterative methods suitable for the numerical solution of large engineering design problems. In what follows, we explore the state of methods used to solve the second-order transport equation, comparing them, where possible, to first-order methods.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. HASSITT (1968) Diffusion theory in two and three dimensions. In: Greenspan H, Kelber CN, Okrent D (eds) Computing methods in reactor physics, Chap. 2. Gordon & Breach, New York

    Google Scholar 

  2. Semenza LA, Lewis EE, Rossow EC (1972) Application of the finite element method to the multigroup neutron diffusion equation. Nucl Sci Eng 47:302

    Google Scholar 

  3. Hansen KF, Kang CM (1975) Finite element methods in reactor physics analysis. Adv Nucl Sci Tech 8:173

    Google Scholar 

  4. Kavenoky A, Lautard JJ (1977) A finite element depletion diffusion calculations method with space-dependent cross-sections. Nucl Sci Eng 64:563

    Google Scholar 

  5. Vladimirov VS (1961) Mathematical problems in the one-velocity theory of particle transport, Atomic Energy of Canada Ltd., Ontario (1963) (trans: V. A. Steklov Mathematical Institute) 61.

    Google Scholar 

  6. Kaplan S, Davis JA (1967) Canonical and involutory transformations of the variational problems of transport theory. Nucl Sci Eng 28:166–176

    Google Scholar 

  7. Zienkiewicz OC (1989) The finite element method, 4th edn. McGraw-Hill, London

    Google Scholar 

  8. Brezzi F, Fortin M (1991) Mixed and hybrid finite element methods. Springer-Verlag, New York

    MATH  Google Scholar 

  9. Strang G, Fix GJ (1973) An analysis of the finite element method. Prentice-Hall, Englewood Cliffs, NJ

    MATH  Google Scholar 

  10. Blomquist RN, Lewis EE (1980) A rigorous treatment of transverse buckling effects in two-dimensional neutron transport computations. Nucl Sci Eng 73:125

    Google Scholar 

  11. Dilber I, Lewis EE (1985) Variational nodal methods for neutron transport. Nucl Sci Eng 91:132

    Google Scholar 

  12. deOliveira CRC (1986) An arbitrary geometry finite element method for multigroup neutron transport with anisotropic scattering. Prog Nucl Energy 18:227

    Article  Google Scholar 

  13. Carrico CB, Lewis EE, Palmiotti G (1992) Three-dimensional variational nodal transport methods for Cartesian, triangular and hexagonal criticality calculations. Nucl Sci Eng 111:223

    Google Scholar 

  14. Lewis EE, Carrico CB, Palmiotti G (1996) Variational nodal formulation for the spherical harmonics equations. Nucl Sci Eng 122:194

    Google Scholar 

  15. Lillie RA, Robinson JC (1976) A linear triangle finite element formulation for multigroup neutron transport analysis with anisotropic scattering, ORNL/TM-5281. Oak ridge National Laboratory

    Google Scholar 

  16. Jung J, Kobayashi NO, Nishihara N (1973) Second-order discrete ordinate P l equations in multi-dimensional geometry. J Nucl Energy 27:577

    Article  Google Scholar 

  17. Morel JE, McGhee JM (1995) A diffusion-synthetic acceleration technique for the even-parity Sn equations with anisotropic scattering. Nucl Sci Eng 120:147–164

    Google Scholar 

  18. Morel JE, McGhee JM (1999) A self-adjoint angular flux equation. Nucl Sci Eng 132:312–325

    Google Scholar 

  19. Lautard JJ, Schneider D, Baudron AM (1999) Mixed dual methods for neutronic reactor core calculations in the CRONOS system. In: Proc. Int. Conf. Mathematics and Computation, Reactor Physics and Environmental Analysis of Nuclear Systems, 27–30 Sept 1999, Madrid

    Google Scholar 

  20. Fedon-Magnaud C (1999) Pin-by-pin transport calculations with CRONOS reactor code. In: Proc. Int. Conf. Mathematics and Computation, Reactor Physics and Environmental Analysis of Nuclear Systems, 27–30 Sept 1999, Madrid

    Google Scholar 

  21. Akherraz B, Fedon-Magnaud C, Lautard JJ, Sanchez R (1995) Anisotropic scattering treatment for the neutron transport equation with primal finite elements. Nucl Sci Eng 120:187–198

    Google Scholar 

  22. Miller WF Jr, Lewis EE, Rossow EC (1973) The application of phase-space finite elements to the two-dimensional neutron transport equation in X-Y geometry. Nucl Sci Eng 52:12

    Google Scholar 

  23. Briggs LL, Miller WF Jr, Lewis EE (1975) Ray-effect mitigation in discrete ordinate-like angular finite element approximations in neutron transport. Nucl Sci Eng 57:205–217

    Google Scholar 

  24. Carlson BG, Lathrop KD (1968) Transport theory: the method of discrete ordinates. In: Greenspan H, Kelber CN, Okrent D (eds) Computing methods in reactor physics, Chap. 3. Gordon & Breach, New York

    Google Scholar 

  25. Gelbard EM (1968) Spherical harmonics methods: PL and double PL approximations. In: Greenspan H, Kelber CN, Okrent D (eds) Computing methods in reactor physics, Chap. 4. Gordon & Breach, New York

    Google Scholar 

  26. Fletcher JK (1994) The solution of the multigroup neutron transport equation using spherical harmonics. Nucl Sci Eng 116:73

    Google Scholar 

  27. Fedon-Magnaud C, Lautard JJ, Akherraz B, Wu GJ (1995) Coarse mesh methods for the transport calculations in the CRONOS reactor code. In: Proc. int. conf. mathematics, computations, reactor physics and environmental analysis, 30 Apr–4 May 1995, Portland, Oregon

    Google Scholar 

  28. Lewis EE, Miller WF Jr (1984) Computational methods of neutron transport. Wiley, New York

    Google Scholar 

  29. Greenbalm A (1977) Iterative methods for solving linear systems. SIAM, Philadelphia, PA

    Google Scholar 

  30. Saad Y (1996) Iterative methods for sparse linear systems. PWS Publishing Co, Boston, MA

    MATH  Google Scholar 

  31. Bemmel JD (1997) Applied Numerical Linear Algebra. SIAM, Philadelphia, PA

    Google Scholar 

  32. Palmiotti G, Carrico CB, Lewis EE (1966) Variational nodal transport methods with anisotropic scattering. Nucl Sci Eng 122:194

    Google Scholar 

  33. Morse PM, Feshbach H (1953) Methods of theoretical physics. McGraw-Hill, New York

    MATH  Google Scholar 

  34. Gelbard EM (1960) Application of spherical harmonics method to reactor problems, WARD-BT-20. Bettis Atomic Power Laboratory

    Google Scholar 

  35. Gelbard EM (1961) Simplified spherical harmonics equations and their use in shielding problems, WAPD-T 1182 (Rev. 1). Bettis Atomic Power Laboratory

    Google Scholar 

  36. Smith KM (1986) Multidimensional nodal transport using the simplified PL method. In: Proc. topl. mtg. reactor physics and safety, 17–19 Sept 1986, Saratoga Springs, New York, p 223

    Google Scholar 

  37. Smith KS (1991) Multi-dimensional nodal transport using the simplified PL method. In: Proc. ANS topl. mtg. advances in mathematics, computations, and reactor physics, 29 Apr–2 May 1991, Pittsburgh, PA

    Google Scholar 

  38. Pomraning CG (1993) Asymptotic and variational derivations of the simplified Pn equations. Ann Nucl Energy 20:623

    Article  Google Scholar 

  39. Larsen EW, McGhee JM, Morel JE (1993) Asymptotic derivation of the simplified Pn equations. In: Proc. topl. mtg. mathematical methods and supercomputers in nuclear applications, M&C + SNA’93, 19–23 Apr 1993, Karlsruhe, Germany

    Google Scholar 

  40. Larsen EW, Morel JE, McGhee JM (1995) Asymptotic derivation of the multigroup P1 and simplified Pn equations. In: Proc. int. conf. mathematics and computations, reactor physics and environmental analysis, 30 Apr– 4 May 1995, Portland, Oregon

    Google Scholar 

  41. Lewis EE, Palmiotti G (1997) Simplified spherical harmonics in the variational nodal method. Nucl Sci Eng 126:48

    Google Scholar 

  42. Noh T, Miller WF Jr, Morel JE (1996) The even-parity and simplified even-parity transport equations in two-dimensional x-y geometry. Nucl Sci Eng 123:38–56

    Google Scholar 

  43. Azmy YY (1988) The weighted diamond-difference form of the nodal transport methods. Nucl Sci Eng 98:29

    Google Scholar 

  44. Lewis EE, Miller WF Jr, Henry TP (1975) A two-dimensional finite element method for integral neutron transport calculations. Nucl Sci Eng 58:202

    Google Scholar 

  45. Lawrence RD (1986) Progress in nodal methods for the solution of the neutron diffusion and transport equations. Prog Nucl Energy 17:271

    Article  Google Scholar 

  46. Palmiotti G, Lewis EE, Carrico CB (1995) VARIANT: VARIational Anisotropic Nodal Transport for multidimensional Cartesian and hexagonal geometry calculation, ANL-95/40. Argonne National Laboratory

    Google Scholar 

  47. Yang WS, Palmiotti G, Lewis EE (2001) Numerical optimization of computing algorithms for the variational nodal method. Nucl Sci Eng 139:74–185

    Google Scholar 

  48. Smith MA, Tsoulfanidis N, Lewis EE, Palmiotti G, Taiwo TA (2003) A finite subelement generalization of the variational nodal method. Nucl Sci Eng 144:36

    Google Scholar 

  49. Smith MA, Palmiotti G, Lewis EE, Tsoulfanidis N (2004) An integral form of the variational nodal method. Nucl Sci Eng 146:141

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Northwestern University, Evanston, IL, 60208, USA

    E. E. Lewis

Authors
  1. E. E. Lewis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. E. Lewis .

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer Science+Business Media B.V.

About this chapter

Cite this chapter

Lewis, E.E. (2010). Second-Order Neutron Transport Methods. In: Nuclear Computational Science. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3411-3_2

Download citation

  • DOI: https://doi.org/10.1007/978-90-481-3411-3_2

  • Published:

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-3410-6

  • Online ISBN: 978-90-481-3411-3

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation