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 /phi/ /SUB g/ (r,..cap omega..) at point r in direction ..cap omega.. for energy group g takes the form of an expansion in unnormalized spherical harmonics whose solution takes into account the axial and azimuthal angles of ..cap omega.., the associated Legendre polynomials, and an arbitrary odd number. Using various recurrence formulas, a linked set of first-order differential equations results. Terms with odd limits 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.