Evaluation of deterministic and stochastic methods for the subcritical linear Boltzmann equation: PN, double-PN and Monte Carlo frameworks

Abstract

This article evaluates and compares deterministic and stochastic numerical methods for solving the Radiative Transfer Equation (RTE), with application to the modeling of light propagation in participating media such as fog—key challenges for automotive sensor performance in adverse weather. The study focuses on two deterministic frameworks: the 1D Double-PN method (DPN) and the 3D PN method under spherical symmetry with an isotropic point source. These are benchmarked against analytical solutions in isotropic and extreme scattering cases (purely forward/backward) to analyze convergence, numerical stability, and error behavior. In parallel, a probabilistic formulation of the RTE is introduced, and an explicit expression for the variance of the backward Monte Carlo method is derived, enabling rigorous assessment of stochastic simulations. Results show first-order convergence in space for the DPN method and convergence difficulties for the PN method for distances close to the point source and for strong forward or backward scattering. Furthermore, numerical simulations validate the analytical expressions of the backward Monte Carlo variances obtained in the 1D case under isotropic scattering. These findings provide essential tools for verifying Monte Carlo codes or other physically-based simulator, supporting virtual testing of autonomous driving systems under extreme weather conditions.