Variational Nodal Formulation for the Spherical Harmonics Equations

The variational nodal formulation of the neutron transport equation is generalized to provide spherical harmonics approximations of arbitrary odd order. The even angular parity trial functions within the nodes are complemented by new odd angular parity trial functions at the node interfaces. These are derived from the spherical harmonic continuity conditions presented in the classical work of Rumyantsev. The Y_n±n terms are absent for all odd n in the resulting odd-parity trial function sets. This result is shown to be equivalent to requiring the variational nodal matrix that couples even- and odd-parity angular trial functions to be of full rank and yields vacuum and reflected boundary conditions as well as nodal interface conditions within the framework of the variational formulation. Nodal P₁, P₃, and P₅ approximations are implemented in the Argonne National Laboratory code VARIANT, utilizing the existing spatial trial functions in x-y geometry. The accuracy of the approximations is demonstrated on model fixed source and few-group eigenvalue problems. The new interface trial functions have no effect on P₁ approximations and yield P₃ results that differ very little from those obtained with existing trial functions, even where the P₅ approximation leads to further improvement. More significantly, the new trial functions allow P₅ or higher order algorithms to be implemented in a consistent straightforward manner.