An Efficient Algorithm to Solve the GITT-Transformed 2-D Neutron Diffusion Equation

  • M. Schramm
  • C. Z. Petersen
  • M. T. Vilhena
  • B. E. J. Bodmann


In the last few years special attention has been devoted to searching analytic solutions for the diffusion equation. We are aware of literature for this sort of solution for specialized topics dealing with the simulation of pollutant dispersion in the atmosphere. For illustration we cite the works of (Buske et al. in Atmosphere Environment 2007), (Buske et al. in Environmental Fluid Mechanics 2007), (Moreira et al. 2009). On the other hand, the literature is scarce regarding analytical solutions for the neutron diffusion equation, except for very specialized problems (Maiorino et al. 2007). Work on analytical solutions to the one-dimensional and two-dimensional two-group neutron diffusion equation for either homogeneous or heterogeneous sheets by the well-known GITT technique (Moreira et al. 2009) has recently emerged in the literature. The key feature of this methodology is that is it uses an expansion of the fast and thermal fluxes in a series written in terms of a set of orthogonal eigenfunctions. Replacing these expansions in the original equation and taking moments, yields a second-order matrix differential equation, known in the framework of this methodology as the GITT transformed problem.


Diffusion Equation Thermal Neutron Fast Neutron Pollutant Dispersion Interpolation Theory 
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  1. [AbHa08]
    Aboander, A.E., Hamada, Y.M.: Generalized Runge–Kutta method for two- and three-dimensional space-time diffusion equations with a variable time step. Annals of Nuclear Energy, 35, 1024–1040 (2008). CrossRefGoogle Scholar
  2. [BoViFeBa10]
    Bodmann, B., Vilhena, M.T., Ferreira, L.S., Bardaji, J.B.: An analytical solver for the multi-group two dimensional neutron-diffusion equation by integral transform techniques. Il Nuovo Cimento della Società Italiana di Fisica, C. Geophysics and Space Physics, 1, 1–10 (2010). Google Scholar
  3. [BuViMoTi07]
    Buske, D., Vilhena, M.T., Moreira, D.M., Tirabassi, T.: Simulation of pollutant dispersion for low wind in stable and convective Planetary Boundary Layer. Atmosphere Environment, 41, 5496–5501 (2007). CrossRefGoogle Scholar
  4. [BuViMoTi07a]
    Buske, D., Vilhena, M.T., Moreira, D.M., Tirabassi, T.: An analytical solution of the advection–diffusion equation considering non-local turbulence closure. Environmental Fluid Mechanics, 41, 5496–5501 (2007). Google Scholar
  5. [Ce10]
    Ceolin, C.: Solução Analítica da Equação Cinética de Difusão Multigrupo de Nêutrons em Geometria Cartesiana Unidimensional pela Técnica da Transformada Integral, M.Sc. Dissertation, PROMEC/UFRGS (in Portuguese), Brazil (2010). Google Scholar
  6. [Co93]
    Cotta, R.M.: Integral Transforms in Computational Heat and Fluid Flow, CRC Press, Florida, USA (1993). MATHGoogle Scholar
  7. [CoMi97]
    Cotta, R., Mikhaylov, M.: Heat Conduction Lumped Analysis, Integral Transforms, Symbolic Computation, John Wiley & Sons, Lane, Chinchester, England (1997). Google Scholar
  8. [Co98]
    Cotta, R.M.: The Integral Transform Method in Thermal and Fluids Science and Engineering, Begell House Inc., New York, USA (1998). Google Scholar
  9. [DuHa76]
    Duderstadt, J.J., Hamilton, L.J.: Nuclear Reactor Analysis, John Wiley & Sons, New York, USA (1976). Google Scholar
  10. [Go05]
    Goodman, J.W.: Introduction to Fourier Optics, Inc., Hardcover, USA (2005). Google Scholar
  11. [He75]
    Henry, A.: Nuclear-Reactor Analysis, Chap. 7, The MIT Press, Cambridge Massachusetts, USA (1975). Google Scholar
  12. [La65]
    Lamarsh, J.R.: Introduction to Nuclear Reactor Theory, Addison-Wesley Publishing Company, Inc., Massachusetts, USA (1965). Google Scholar
  13. [MaSaCaRoAnOlLe07]
    Maiorino, J.R., Santos, A., Carluccio, T., Rossi, P.C.R., Antunes, A., Oliveira, F., Lee, S.M., The participation of IPEN in the AIEA coordinate research projects on accelerator driven systems (ADS), in: International Nuclear Atlantic Conference – INAC (2007). Google Scholar
  14. [MoViBuTi09]
    Moreira, D.M., Vilhena, M.T., Buske, D., Tirabassi, T.: The state-of-art of the GILTT method to simulate pollutant dispersion in the atmosphere. Atmospheric Research, 92, 1–17 (2009). CrossRefGoogle Scholar
  15. [MoVa78]
    Moler, C., Van Loan, C., Nineteen dubious ways of computing the exponential of a matrix. SIAM Review, 45, 801–836 (1978). CrossRefGoogle Scholar
  16. [ViCoMoTi08]
    Vilhena, M.T., Costa, C.P., Moreira, D., Tirabassi, T.: A semi-analytical solution for the three-dimensional advection diffusion equation considering non-local turbulence closure. Atmospheric Research, 63 (2008). Google Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • M. Schramm
    • 1
  • C. Z. Petersen
    • 1
  • M. T. Vilhena
    • 1
  • B. E. J. Bodmann
    • 1
  1. 1.Universidade Federal do Rio Grande do SulPorto AlegreBrazil

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