A new and rigorous SPN theory – Part II: Generalization to GSPN. (December 2017) - Chao, Yung-An
Highlights: The rigorous SPN theory is generalized to multi-groups and anisotropic scattering of all orders. The GSPN theory, valid for piecewise homogeneous regions, is rigorously equivalent to PN . All differential equations in GSPN are of diffusion type involving only the Laplacian operator. Generalized Helmholtz decomposition of tensors and Davison's lemma for solid harmonics are used. All relevant equations are explicitly given for GSP3 . Abstract: A new SPN theory valid for piecewise homogeneous regions was rigorously derived in a paper published last year by the author. The new theory does not need the assumption of locally 1D behavior near a surface. The resulting interface and the boundary conditions, different from the ad hoc conditions conventionally assumed, contain higher order derivatives of the solution functions to the SPN equations. This new SPN theory was presented in the previous paper for the mono-energetic case with generic isotropic neutron sources in piecewise homogeneous regions. In this sequel paper the theory is generalized to the generic case of multi-energy groups including anisotropic neutron sources as well. Although the multi-group extension is straightforward, the inclusion of anisotropic sources poses subtle problems. To be valid for any arbitrary linear anisotropic external source a Generalized SP3 (GSP3 ) theory is proposed. GSP3 clarifies how to decompose a linearly anisotropic source via Helmholtz decomposition so that its irrotational component can be properly incorporated into SP3 . GSP3 is more accurate than SP3 . It includes two additional diffusion type equations decoupled from the two original SP3 equations, with in total eight instead of four interface or boundary conditions. The GSP3 equations are all of diffusion type and the additional interface and boundary conditions are manageable for practical engineering applications. By generalizing the Helmholtz decomposition to higher order tensors and using a powerful lemma embedded in a Boris Davison's 1958 paper, we can formulate a comprehensive theory for the generic GSPN, which can accommodate any spatially differentiable anisotropic neutron sources of any order in anisotropy. The only restriction to GSPN is piecewise homogeneity. GSPN theory provides various levels of approximations to PN, from the simplest SPN all the way to the rigorous equivalency to PN . GSPN contains decomposed independent sets of diffusion type second order differential equations without any crossed variable derivatives, which are very easy to solve. But the interface and boundary conditions are more complicated where the solutions to the independent sets of equations do get coupled. An interesting byproduct of GSPN is that it provides the recipe of how to decompose and incorporate arbitrary anisotropic neutron sources into the SPN equations via the use of the generalized Helmholtz decomposition. The general formulation for the GSPN theory is presented, although specific details are given only for GSP3 .