A Preliminary Study of the Influence of Approximate Boundary Conditions on the Accuracy of Nonclassical Spherical Harmonic Solutions

Nonclassical neutral particle transport theory provides a generalized framework for modeling transport in complex, heterogeneous, and correlated media. In the deterministic setting, the classical linear Boltzmann equation is extended to the nonclassical transport equation (NTE), which incorporates nonexponential free-path distributions and memory effects, enabling the representation of a broader range of transport phenomena. The Nonclassical Spherical Harmonic Approximation (NSHA), derived from the NTE through spherical harmonic expansions analogous to the classical PN equations, offers a more efficient direction-of-motion formulation than the full phase-space NTE. A central challenge in applying the NSHA is the consistent formulation of boundary conditions under nonclassical assumptions, since classical conditions such as Marshak, Mark, Asymptotic/Variational (A-V), and Federighi-Pomraning (F-P) must be rederived to reflect nonexponential free-path statistics. This work provides a comprehensive derivation of these approximate boundary conditions for the NSHA in slab geometry, establishing a unified basis for nonclassical transport modeling. Numerical results are presented to evaluate the influence of each boundary condition on the accuracy of NSHA solutions across a range of transport scenarios. Among the conditions examined, the Mark boundary condition consistently produced the most accurate NSHA solutions over the scattering ratios and approximation orders considered.