Models and numerical methods for time- and energy-dependent particle transport

Particles passing through a medium can be described by the Boltzmann transport equation. Therein, all physical interactions of particles with matter are given by cross sections. We compare different analytical models of cross sections for photons, electrons and protons to state-of-the-art databases. The large dimensionality of the transport equation and its integro-differential form make it analytically difficult and computationally costly to solve. In this work, we focus on the following approximative models to the linear Boltzmann equation: (i) the time-dependent simplified PN (SPN) equations, (ii) the M1 model derived from entropy-based closures and (iii) a new perturbed M1 model derived from a perturbative entropy closure. In particular, an asymptotic analysis for SPN equations is presented and confirmed by numerical computations in 2D. Moreover, we design an explicit Runge-Kutta discontinuous Galerkin (RKDG) method to the M1 model of radiative transfer in slab geometry and construct a scheme ensuring the realizability of the moment variables. Among other things, M1 numerical results are compared with an analytical solution in a Riemann problem and the Marshak wave problem is considered. Additionally, we rigorously derive a new hierarchy of kinetic moment models in the context of grey photon transport in one spatial dimension. Numerical examples, such as the two beam instability or the analytical benchmark due to Su and Olson [173], are shown for the perturbed M1 model and compared to the standard M1 as well as transport solutions.

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