The first-Order scattering P 1 method

Highlights: • A numerical method is proposed to solve the Radiative Transfer Equation. • Higher accuracy than any method free of a directional mesh or statistical noise. • The method is as versatile and accurate as MC method and as efficient as P1 method. • The method is accurate for moderately anisotropic phase functions (|g|$<$0.6). • For forward-scattering range, accuracy is problem dependent. This work proposes a numerical method to solve the Radiative Transfer Equation by sequentially coupling the First-Order Scattering (FOS) and the $P_1$ approximations. The FOS method represents highly-anisotropic radiative intensity distributions for zero and one scattering events; the $P_1$ approximation considers the smoother distributions obtained in subsequent scattering events. As long as the phase function is not highly anisotropic ($\left|g\right|<0.6$), the numerical method proposed proved accurate (averaged local errors below 10%) practically for any source, albedo ($\omega$), and domain optical thickness ($\tau$). Moreover, in the highly forward scattering range ($0.6<g<0.9$), the FOS$P_1$ method remains accurate as long as one of the following holds: 1) sources are not highly anisotropic (radiative cone angle, ${\theta }_c>\pi /4$), 2) $\tau \le 1$ or 3) $\omega \le 0.5$. For the cases covered in this study, the FOS$P_1$ method achieves 10 to 100-fold speed up (w.r.t. Monte Carlo simulations with a similar accuracy). The method is free of a directional mesh and statistical noise, providing an interesting combination between accuracy, computational efficiency, and ease of implementation in arbitrary domains.