Perturbed, Entropy-Based Closure for Radiative Transfer

We derive a hierarchy of closures based on perturbations of well-known entropy-based closures; we therefore refer to them as perturbed entropy-based models. Our derivation reveals final equations containing an additional convective and diffusive term which are added to the flux term of the standard closure. We present numerical simulations for the simplest member of the hierarchy, the perturbed M1 or PM1 model, in one spatial dimension. Simulations are performed using a Runge-Kutta discontinuous Galerkin method with special limiters that guarantee the realizability of the moment variables and the positivity of the material temperature. Improvements to the standard M1 model are observed in cases where unphysical shocks develop in the M1 model.Comment: 35 pages, 8 figure

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. For the perturbed M1 model, we present numerical results known as the two beam instability or the analytical benchmark due to Su and Olson and compare them to the standard M1 as well as transport solutions

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. For the perturbed M1 model, we present numerical results known as the two beam instability or the analytical benchmark due to Su and Olson and compare them to the standard M1 as well as transport solutions