Theoretical Aspects of the Simplified P_n Equations

Abstract

In celebration of the 50th anniversary of the simplified P_n equations (SP_n), this work reviews the theory underpinning the SP_nequations for neutral particle transport. We recount the derivation of these equations by Gelbard’s formal procedure and by more recent asymptotic and variational analyses. The relation between the SP_n equations and several other low order approximations is discussed. Also, the conditions under which the SP_n equations are equivalent to the full P_n equations of the same order are discussed as well as the accuracy of the SP_nequations. Several open problems in the theory of the SP_n equations are posed.

Keywords:

• Simplified P_n method
• Transport theory

Acknowledgments

Thanks to Dave Griesheimer at Bettis Laboratory for scouring the corporate library for copies of the original Gelbard reports, Ed Larsen for calling my attention to the variational derivations, and James Paul Holloway for encouraging this special issue celebrating the golden anniversary of the SP_n equations.

Notes

¹The original Latin phrase is “potuit, decuit, ergo fecit.”

²The derivation when there is anisotropic scattering requires some tensor analysis that we would rather avoid resorting to in this review.