Theoretical Aspects of the Simplified P_n Equations

References

• Ackroyd , R. T. , de Oliveira , C. R. E. , Zolfaghari , A. and Goddard , A. J. H. 1999 . On a rigorous resolution of the transport equation into a system of diffusion-like equations. . Progress in Nuclear Energy , 35 ( 1 ) : 1 – 64 .
   Web of Science ®Google Scholar
• Backofen , R. , Bilz , T. , Ribalta , A. and Voigt , A. 2004 . SPN-approximations of internal radiation in crystal growth of optical materials. . Journal of Crystal Growth , 266 : 264 – 270 .
   Web of Science ®Google Scholar
• Banda , M. , Seaïd , M. and Teleaga , I. 2008 . Large-eddy simulation of thermal flows based on discrete-velocity models. . SIAM Journal on Scientific Computing , 30 ( 4 ) : 1756 – 1777 .
   Web of Science ®Google Scholar
• Beckert , C. and Grundmann , U. 2008 . Development and verification of a nodal approach for solving the multigroup SP 3 equations. . Annals of Nuclear Energy , 35 : 75 – 86 .
   Web of Science ®Google Scholar
• Bell , G. I. and Glasstone , S. 1970 . Nuclear Reactor Theory , Malabar , Florida : Robert E. Kreiger Publishing .
   Google Scholar
• Boyd , J. P. 2001 . Chebyshev and Fourier Spectral Methods , Mineola , New York : Dover Publications .
   Google Scholar
• Brantley , P. S. and Larsen , E. W. 1997 . “ A variational derivation of the simplified P 3 approximation ” . In Proceedings of ANS Topcial Meeting on Mathematical Models and Supercomputing for Nuclear Applications , New York : Saratoga Springs .
   Google Scholar
• Brantley , P. and Larsen , E. 2000 . The simplified P 3 approximation. . Nuclear Science and Engineering , 134 ( 1 ) : 1 – 21 .
   Web of Science ®Google Scholar
• Brunner , T. A. 2001 . One-dimensional Riemann solvers and the maximum entropy closure. . J. Quant. Spec. Rad. Transf. , 69 ( 5 ) : 543 – 566 .
   Web of Science ®Google Scholar
• Buet , C. and Despres , B. 2006 . Asymptotic preserving and positive schemes for radiation hydrodynamics. . Journal of Computational Physics , 215 ( 2 ) : 717 – 740 .
   Web of Science ®Google Scholar
• Capeillere , J. , Ségur , P. , Bourdon , A. , Célestin , S. and Pancheshnyi , S. 2008 . The finite volume method solution of the radiative transfer equation for photon transport in non-thermal gas discharges: application to the calculation of photoionization in streamer discharges. . J. Phys. D: Appl. Phys , 41 ( 234018 ) : 234018
   Web of Science ®Google Scholar
• Case , K. M. and Zweifel , P. F. 1967 . Linear Transport Theory , Reading , Massachusetts : Addison-Wesley .
   Google Scholar
• Chu , M. , Klose , A. and Dehghani , H. 2008 . “ Light transport in soft tissue based on simplified spherical harmonics approximation to radiative transport equation ” . In Biomedical Optics Topical Meeting St. Petersburg , FL
   Google Scholar
• Ciolini , R. , Coppa , G. G. M. , Montagnini , B. and Ravetto , P. 2002 . Simplified PN and AN methods in neutron transport. . Prog. Nucl. Energy , 40 ( 2 ) : 237 – 264 .
   Web of Science ®Google Scholar
• Ciolini , R. , Montagnini , B. , Ravetto , P. and Sumini , M. 2006 . Solution of some 2d transport problems by a high order an-sp2n-1 method. . Annals of Nuclear Energy , 33 : 1010 – 1029 .
   Web of Science ®Google Scholar
• Colombo , V. 1988 . “ Calcolo numerico di integrali singolari per la valutazione delle condizioni al contorno rigorose del metodo approssimato AN in trasporto neutronico pluridimensionale ” . In Atti del Politecnico di Torino PT–DE–IN 172.
   Google Scholar
• Coppa , G. and Ravetto , P. 1982 . An approximate method to study the one-velocity neutron integral transport equation. . Annals of Nuclear Energy , 9 : 169 – 174 .
   Web of Science ®Google Scholar
• Davison , B. 1960 . On the rate of convergence of the spherical harmonics method. . Can. J. Phys. , 38 : 1526 – 1545 .
   Web of Science ®Google Scholar
• Domínguez , J. and Bérubé-Lauzière , Y. 2010 . Diffuse light propagation in biological media by a time-domain parabolic simplified spherical harmonics approximation with ray-divergence effects. . Applied Optics , 49 ( 8 ) : 1414 – 1429 .
   PubMed Web of Science ®Google Scholar
• Frank , M. , Klar , A. , Larsen , E. W. and Yasuda , S. 2007 . Time-dependent simplified PN approximation to the equations of radiative transfer. . Journal of Computational Physics , 226 ( 2 ) : 2289 – 2305 .
   Web of Science ®Google Scholar
• Gelbard , E. M. 1960 . “ Applications of spherical harmonics method to reactor problems ” . In Technical Report WAPD-BT-20 , Bettis Atomic Power Laboratory .
   Google Scholar
• Gelbard , E. M. 1961 . “ Simplified spherical harmonics equations and their use in shielding problems ” . In Technical Report WAPD-T-1182 , Bettis Atomic Power Laboratory .
   Google Scholar
• Gelbard , E. M. 1962 . “ Applications of simplified spherical harmonics equations in spherical geometry ” . In Technical Report WAPD-TM-294 , Bettis Atomic Power Laboratory .
   Google Scholar
• Gelbard , E. M. 1968 . “ Spherical harmonics methods: PL and double-PL approximations ” . In Computing Methods in Reactor Physics , Edited by: Greenspan , H. , Kelber , C. N. and Okrent , D. New York : Gordon and Breach Science Publishers .
   Google Scholar
• Guo , B. 1995 . A spectral method for the vorticity equation on the surface. . Mathematics of Computation , 64 ( 211 ) : 1067 – 1079 .
   Web of Science ®Google Scholar
• Hauck , C. D. 2011 . High-order entropy-based closures for linear transport in slab geometries. . Communications in Mathematical Sciences , 9 ( 1 ) : 187 – 205 .
   Web of Science ®Google Scholar
• Hauck , C. D. and McClarren , R. G. 2010 . Positive PN closures. . SIAM Journal on Scientific Computing , 32 ( 5 ) : 2603 – 2626 .
   Web of Science ®Google Scholar
• Hébert , A. 2010 . Mixed-dual implementations of the simplified PN method. . Annals of Nuclear Energy , 37 : 498 – 511 .
   Web of Science ®Google Scholar
• Josef , J. and Morel , J. 1998 . Simplified spherical harmonic method for coupled electron-photon transport calculations. . Phys. Rev. E , 57 ( 5 ) : 6161 – 6171 .
   Web of Science ®Google Scholar
• Klose , A. and Larsen , E. 2006 . Light transport in biological tissue based on the simplified spherical harmonics equations. . Journal of Computational Physics , 220 ( 1 ) : 441 – 470 .
   Web of Science ®Google Scholar
• Kofink , W. 1958 . Studies of the spherical harmonics method in neutron transport theory. . Il Nuovo Cimento , 9 : 497 – 541 .
   Google Scholar
• Kotiluoto , P. , Pyyry , J. and Helminen , H. 2007 . Multitrans sp3 code in coupled photon-electron transport problems. . Radiation Physics and Chemistry , 76 ( 1 ) : 9 – 14 .
   Web of Science ®Google Scholar
• Larsen , E. , Morel , J. and McGhee , J. 1996 . Asymptotic derivation of the multigroup P 1 and simplified PN equations with anisotropic scattering. . Nuclear Science and Engineering , 123 ( 3 ) : 328 – 342 .
   Web of Science ®Google Scholar
• Larsen , E. W. , Morel , J. E. and McGhee , J. 1993 . “ Asymptotic derivation of the simplified Pn equations ” . In Proceedings of Joint International Conference on Mathematical Methods and Supercomputing in Nuclear Applications , Oregon : Portland .
   Google Scholar
• Lewis , E. and Miller , W. 1984 . Computational Methods of Neutron Transport , New York : John Wiley and Sons .
   Google Scholar
• McClarren , R. G. , Evans , T. M. , Lowrie , R. B. and Densmore , J. D. 2008 . Semi-implicit time integration for PN thermal radiative transfer. . J. Comput. Phys. , 227 ( 16 ) : 7561 – 7586 .
   Web of Science ®Google Scholar
• Montagnini , B. and Ravetto , P. 2004 . SP ∞-A ∞ elementary solutions in general geometry. . Annals of Nuclear Energy , 31 ( 6 ) : 619 – 646 .
   Web of Science ®Google Scholar
• Noh , T. and Miller , W. F. 1996 . The effectiveness of P 2 and simplified P 2 synthetic accelerations in the solutions of discrete ordinates transport equations. . Nucl. Sci. Eng. , 124 : 18 – 30 .
   Web of Science ®Google Scholar
• Oh , K. S. and Holloway , J. P. 2008 . “ A quasi-static closure for 3rd order spherical harmonics time-dependent radiation transport in 2-D ” . In Proceedings of the 2009 International Conference on Mathematics and Computational Methods and Reactor Physics , American Nuclear Society .
   Google Scholar
• Pinnau , R. and Seaïd , M. 2008 . Simplified pn models and natural convection—radiation. . Progress in Industrial Mathematics at ECMI 2006 , : 397
   Google Scholar
• Pomraning , G. 1993 . Asymptotic and variational derivations of the simplified PN equations. . Annals of Nuclear Energy , 20 ( 9 ) : 623 – 637 .
   Web of Science ®Google Scholar
• Pomraning , G. C. 1964 . A generalized PN approximation for neutron transport problems. . Nukleonik , 6
   Google Scholar
• Rulko , R. P. and Larsen , E. W. 1993 . Variational derivation and numerical analysis of P 2 theory in planar geometry. . Nucl. Sci. Eng. , 114 : 271 – 285 .
   Web of Science ®Google Scholar
• Sanchez , R. 2008 . “ PN -SPN ” . personal communication.
   Google Scholar
• Schaefer , M. , Frank , M. and Levermore , C. 2009 . Diffusive corrections to PN approximations. . Arxiv preprint arXiv , 0907 : 2099
   Google Scholar
• Schneider , E. , Seaid , M. , Janicka , J. and Klar , A. 2008 . Validation of simplified P-N models for radiative transfer in combustion systems. . Communications in Numerical Methods in Engineering , 24 ( 2 ) : 85
   Web of Science ®Google Scholar
• Seaīd , M. , Frank , M. , Klar , A. , Pinnau , R. and Thömmes , G. 2004 . Efficient numerical methods for radiation in gas turbines. . Journal of Computational and Applied Mathematics , 170 ( 1 ) : 217 – 239 .
   Web of Science ®Google Scholar
• Ségur , P. , Bourdon , A. , Marode , E. , Bessieres , D. and Paillol , J. 2006 . The use of an improved eddington approximation to facilitate the calculation of photoionization in streamer discharges. . Plasma Sources Science and Technology , 15 ( 4 ) : 648 – 660 .
   Web of Science ®Google Scholar
• Selengut , D. S. 1970 . A new form of the P 3 approximation. . Trans. Am. Nucl. Soc. , 13 : 625
   Google Scholar
• Smith , K. S. 1986a . Multidimensional nodal transport using the Simplified PL method. . Trans. Am. Nucl. Soc. , 52 : 427
   Google Scholar
• Smith , K. S. 1986b . “ Multidimensional nodal transport using the Simplified PL method ” . In Proc. Topl. Mtg. Reactor Physics and Safety , 223 New York : Saratoga Springs .
   Google Scholar
• Teleaga , I. and Seaīd , M. 2008 . Simplified radiative models for low-mach number reactive flows. . Applied Mathematical Modelling , 32 : 971 – 991 .
   Web of Science ®Google Scholar
• Tomasevic , D. I. and Larsen , E. W. 1996 . The simplified p2 approximation. . Nuclear Science and Engineering , 122 ( 3 ) : 309 – 325 .
   Web of Science ®Google Scholar
• Yaun , G. 2000 . Studies of the spherical harmonics method in neutron transport theory. . J. Math. Phys. , 41 : 867 – 874 .
   Web of Science ®Google Scholar