On the P_N Method in Spherical Geometry: A Stable Solution for the Exterior of a Sphere

References

• Antosiewicz, H. A. 1964. Bessel functions of fractional order. In Handbook of mathematical functions, ed. M. Abramowitz, and I. A. Stegun, 435–478. Washington, DC: National Bureau of Standards.
   Google Scholar
• Aronson, R. 1984a. Subcritical problems in spherical geometry. Nucl. Sci. Eng. 86:136–149.
   Web of Science ®Google Scholar
• Aronson, R. 1984b. Critical problems for bare and reflected slabs and spheres. Nucl. Sci. Eng. 86:150–156.
   Web of Science ®Google Scholar
• Brown, F., and N. Barnett. 2007. A tutorial on using MCNP for 1-group transport calculations. LA-UR-07-4594, Los Alamos National Laboratory, Los Alamos, NM.
   Google Scholar
• Case, K. M., F. de Hoffmann, and G. Placzek. 1953. Introduction to the theory of neutron diffusion, Vol. I. Washington, DC: U. S. Government Printing Office.
   Google Scholar
• Dave, J. V., and B. H. Armstrong. 1974. Smoothing of the intensity curve obtained from a solution of the spherical harmonics approximation to the transfer equation. J. Atmos. Sci. 31:1934–1937.
   Web of Science ®Google Scholar
• Davison, B. 1958. Neutron transport theory. London: Oxford University Press.
   Google Scholar
• Dongarra, J. J., J. R. Bunch, C. B. Moler, and G. W. Stewart. 1979. LINPACK users’ guide. Philadelphia, PA: SIAM.
   Google Scholar
• Garcia, R. D. M. 2000. An analysis of the source-function integration technique for postprocessing PN angular fluxes. Ann. Nucl. Energy 27:1217–1226.
   Web of Science ®Google Scholar
• Garcia, R. D. M. 2017. A PN particular solution for the radiative transfer equation in spherical geometry. J. Quant. Spectrosc. Radiat. Transf. 196:155–158.
   Web of Science ®Google Scholar
• Garcia, R. D. M., C. E. Siewert, and J. R. Thomas Jr. 2017. A computationally viable version of the PN method for spheres. Nucl. Sci. Eng. 186:103–119.
   Web of Science ®Google Scholar
• Gautschi, W., and W. F. Cahill. 1964. Exponential integrals and related functions, In Handbook of mathematical functions, ed. M. Abramowitz and I. A. Stegun, 227–251. Washington, DC: National Bureau of Standards.
   Google Scholar
• Goorley, T., M. James, T. Booth, F. Brown, J. Bull, L. J. Cox, J. Durkee, J. Elson, M. Fensin, R. A. Forster, J. Hendricks, H. G. Hughes, R. Johns, B. Kiedrowski, R. Martz, S. Mashnik, G. McKinney, D. Pelowitz, R. Prael, J. Sweezy, L. Waters, T. Wilcox, and T. Zukaitis. 2012. Initial MCNP6 release overview. Nucl. Technol. 180:298–315.
   Web of Science ®Google Scholar
• Kadak, A. C. 2005. A future for nuclear energy: Pebble bed reactors. Int. J. Crit. Infrastruct. 1:330–345.
   Google Scholar
• Kaper, H. G., J. K. Shultis, and J. G. Veninga. 1970. Numerical evaluation of the slab albedo problem solution in one-speed anisotropic transport theory. J. Comput. Phys. 6:288–313.
   Google Scholar
• Karp, A. H. 1981. Computing the angular dependence of the radiation of a planetary atmosphere. J. Quant. Spectrosc. Radiat. Transf. 25:403–412.
   Web of Science ®Google Scholar
• Kourganoff, V. 1952. Basic methods in transfer problems. Oxford: Clarendon Press.
   Google Scholar
• Mark, J. C. 1945. The spherical harmonics method, II. MT-97, Montreal Laboratory, Atomic Energy Project, National Research Council of Canada, Ottawa.
   Google Scholar
• Marshak, R. E. 1947. Note on the spherical harmonic method as applied to the Milne problem for a sphere. Phys. Rev. 71:443–446.
   Web of Science ®Google Scholar
• McCormick, N. J., and R. Sanchez. 1981. Inverse problem transport calculations for anisotropic scattering coefficients. J. Math. Phys. 22:199–208.
   Web of Science ®Google Scholar
• McCormick, N. J., and C. E. Siewert. 1991. Particular solutions for the radiative transfer equation. J. Quant. Spectrosc. Radiat. Transf. 46:519–522.
   Web of Science ®Google Scholar
• Pomraning, G. C., and C. E. Siewert. 1982. On the integral form of the equation of transfer for a homogeneous sphere. J. Quant. Spectrosc. Radiat. Transf. 28:503–506.
   Web of Science ®Google Scholar
• Siewert, C. E. 1993a. On intensity calculations in radiative transfer. J. Quant. Spectrosc. Radiat. Transf. 50:555–560.
   Web of Science ®Google Scholar
• Siewert, C. E. 1993b. A spherical-harmonics method for multi-group or non-gray radiation transport. J. Quant. Spectrosc. Radiat. Transf. 49:95–106.
   Web of Science ®Google Scholar
• Siewert, C. E., and N. J. McCormick. 1997. Some identities for Chandrasekhar polynomials. J. Quant. Spectrosc. Radiat. Transf. 57:399–404.
   Web of Science ®Google Scholar
• Siewert, C. E., and J. R. Thomas Jr. 1990. A particular solution for the PN method in radiative transfer. J. Quant. Spectrosc. Radiat. Transf. 43:433–436.
   Web of Science ®Google Scholar
• Smith, B. T., J. M. Boyle, J. J. Dongarra, B. S. Garbow, Y. Ikebe, V. C. Klema, and C. B. Moler. 1976. Matrix eigensystem routines – EISPACK guide. 2nd ed. Berlin: Springer-Verlag.
   Google Scholar
• Verfondern, K., H. Nabielek, and J. M. Kendall. 2007. Coated particle fuel for high temperature gas cooled reactors. Nucl. Eng. Technol. 39:603–616.
   Web of Science ®Google Scholar
• Weniger, E. J. 1989. Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series. Comput. Phys. Rep. 10:189–371.
   Google Scholar
• Wynn, P. 1956a. On a device for computing the em(Sn) transformation. Math. Tables Aids Comput. 10:91–96.
   Google Scholar
• Wynn, P. 1956b. On a Procrustean technique for the numerical transformation of slowly convergent sequences and series. Math. Proc. Camb. Phil. Soc. 52:663–671.
   Google Scholar
• Yvon, J. 1957. La diffusion macroscopique des neutrons – Une methode d’approximation. J. Nucl. Energy I 4:305–318.
   Google Scholar