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

摘要

In this work, we demonstrate theoretically and numerically the equivalence between the SPN and P-N approximation for time-dependent problems in homogeneous media with anisotropic scattering. Our theory extends the solid harmonics derivation, which was used by Ackroyd et al. to derive the steady-state SPN 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. 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. Numerical results show that the difference between the SPN and PN scalar flux solution is at the roundoff level.