Hilbert expansion based fluid models for kinetic equations describing neutral particles in the plasma edge of a fusion device

Abstract

Neutral particles in the plasma edge of fusion devices based on magnetic confinement are described by a transient kinetic equation incorporating ionization, recombination, and charge-exchange collisions. In charge-exchange dominated regimes, the neutral particle velocity distribution approaches the drifting Maxwellian defined by the mean velocity and temperature of the plasma. This enables model order reduction from the kinetic equation to approximate fluid models. We derive transient fluid models consistent with the kinetic equation by exploring a splitting based approach. We split the kinetic equation in sources and sinks on the one hand, and transport combined with charge-exchange on the other hand. Combining transport with charge-exchange collisions allows for deriving Hilbert expansion based fluid models. The retrieved fluid models depend on the assumed importance (scaling) of the different terms in the split equation describing transport and charge-exchange. We explore two scalings: the hydrodynamic scaling and the diffusive scaling. The diffusive scaling fluid model closely resembles phenomenological fluid models for describing neutral particles in the plasma edge that have been derived in the past. Therefore, the Hilbert expansion based fluid models can serve as a theoretical basis for such phenomenological fluid models and elucidate some of their properties. The performance of the fluid models with respect to a discrete velocity model and a Monte Carlo reference solver is assessed in numerical experiments. The code used to perform the numerical experiments is openly available.

Cell Escape Probabilities for Markov Processes on a Grid

“…For several applications, it is desirable to have a uniform framework for calculating the cell escape probability P (x k+1 / ∈ Ω j |x k ∈ Ω j ) for some grid cell Ω j ⊂ Ω, specifically in the context where both the step ∆x k , but also the initial position of the particle x k are random. A first application is variance prediction for quantity-of-interest estimation using particle tracing Monte Carlo methods [9]. In that field, the cell escape probability is an important parameter for deriving a local Markov process-based variance predictor for a cell.…”

“…In that field, the cell escape probability is an important parameter for deriving a local Markov process-based variance predictor for a cell. Second, the framework for calculating the escape probability from a grid cell can be extended to the derivation of transition probabilities between two grid cells P (x k+1 ∈ Ω i |x k ∈ Ω j ), which can be used for hidden Markov process-based variance calculations [9]. A third application is to use the escape probabilities or the related transition probabilities for the construction of diffusion Monte Carlo-like algorithms [10,6,3,5,4] where these probabilities allow to quantify the leakage of particles from their current cell to one of the other (neighbouring) cells in the grid.…”