Mathematical and Numerical Validation of the Simplified Spherical Harmonics Approach for Time-Dependent Anisotropic-Scattering Transport Problems in Homogeneous Media

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 higherorder scattering moments to them. Results show that the difference between the SP N and P N scalar flux solution is at the roundoff level.

The second-order formulation of the $P_N$ equations with Marshak boundary conditions

“…This is different from the classical ad-hoc SP N derivation in 1D slab geometry, e.g., [11][12][13][14][15], as we perform all calculations on the full 3D system. One advantage of this, depending on a few mild assumptions on the coefficients and the regularity of the solutions, is the equivalence of the solutions to those of the original P N method what is discussed as one of the main issues of the classical SP N approach [16]. This approach comes along with two issues that we would like to address in this work.…”

“…Furthermore we suggest a certain selection of boundary half-moments for which we prove the existence of a weak formulation of the second-order system. Here we would like to note that the proper treatment of boundary conditions has been seen as one of the major issues in this context in [16].…”