The Solution of the Multigroup Neutron Transport Equation Using Spherical Harmonics

A solution of the multigroup neutron transport equation in one, two, or three space dimensions is presented. The flux φ_g(r, Ω) at point r in direction for energy group g takes the form of an expansion in unnormalized spherical harmonics. Thus, where θ and φ are the axial and azimuthal angles of Ω, the associated Legendre polynomials, and N an arbitrary odd number. Using the various recurrence formulas for , a linked set of first-order differential equations in the moments results. Terms with odd 1 are eliminated yielding a second-order system to be solved by two methods. First, a finite difference formulation using an iterative procedure is given, and second, in XYZ and XY geometry, a finite element solution is presented. Results for a test problem using both methods are exhibited and compared.