A Discretization Scheme for the Three-Dimensional Angular Fokker-Planck Operator

A new S_n discretization of the angular Fokker-Planck operator used in three-dimensional calculations is derived for product quadrature sets. It is straightforward to define discretizations that preserve the null space and zeroth angular moment of the analytic operator and are self-adjoint, monotone, and nonpositive-definite. Our new discretization differs from more straightforward discretizations in that it also preserves the three first angular moments of the analytic operator when applied in conjunction with product quadrature sets constructed with Chebychev azimuthal quadrature. Otherwise, it preserves only two of the three first angular moments. Computational results are presented that demonstrate the superiority of this new discretization relative to a straightforward discretization. Two-dimensional versions of the new discretization are also given for x-y and r-z geometries.