Mathematical and numerical validation of the simplified spherical harmonics approach for time-dependent anisotropic-scattering transport problems in homogeneous media

TL;DR

This paper extends the SP$_N$ method to transient anisotropic-scattering problems, demonstrating through numerical tests that SP$_N$ and P$_N$ solutions are nearly identical in homogeneous media.

Contribution

The work provides a mathematical derivation and numerical validation of the SP$_N$ approach for time-dependent anisotropic scattering, showing its equivalence to P$_N$ in homogeneous media.

Findings

SP$_N$ and P$_N$ solutions are numerically indistinguishable.

The derivation confirms the equivalence between SP$_N$ and P$_N$ approximations.

Results are consistent even with higher-order scattering moments.

Abstract

In this work, we extend the solid harmonics derivation, which was used by Ackroyd et al to derive the steady-state SP$_N$ equations, to transient problems. The derivation expands the angular flux in ordinary surface harmonics but uses harmonic polynomials to generate additional surface spherical harmonic terms to be used in Galerkin projection. The derivation shows the equivalence between the SP$_N$ and the P$_N$ approximation. Also, we use the line source problem and McClarren's "box" problem to demonstrate such equivalence numerically. Both problems were initially proposed for isotropic scattering, but here we add higher-order scattering moments to them. Results show that the difference between the SP$_N$ and P$_N$ scalar flux solution is at the roundoff level.