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

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