Modeling of Ionized Gas Flows with a Velocity-space Hybrid Boltzmann Solver

Abstract

In the present work, the Tensor-Train decomposition algorithm is applied to reduce the memory footprint of a stochastic discrete velocity solver for rarefied gas dynamics simulation. An energyconserving modification to the algorithm is proposed, along with an interleaved collision/convection routine which allows for easy application of higher-order convection schemes. The performance of the developed algorithm is analyzed for several 0-and 1-dimensional model problems in terms of solution error and reduction in memory use requirements.Keywords Boltzmann equation • discrete velocity method • tensor decomposition • tensor train • rarefied gas • low-rank approximation IntroductionThe motion of gas or fluid can be successfully described by a set of partial different equations under a wide range of conditions of interest. However, in rarefied regimes, when the ratio of the molecular mean free path to the characteristic length scale becomes large, the continuum approaches no longer apply, and instead, the full Boltzmann equation describing the evolution of the velocity distribution function due to advection, external forces, and collisions, has to be solved. It is a complicated integro-differential equation, with a 6-dimensional integral (the collision operator) appearing on the right-hand side. Numerous approaches have been developed over the years to tackle the equation. These include stochastic approaches, such as the Direct Simulation Monte Carlo (DSMC) method [1], and deterministic approaches, such as discrete velocity methods [2] and spectral methods [3,4].The family of discrete velocity methods (DVM), which is the main focus of the present work, operates by discretizing the velocity distribution function on a grid in velocity space, and obtains a system of partial differential equations for the values of the distribution function at each grid node. A significant advantage offered by discrete velocity methods is the noticeable reduction in noise compared to particle-based methods, which is needed to better understand the small time-scale dynamics of complex rarefied flows, especially in unsteady scenarios [5,6,7].Within the discrete velocity framework, the complex collision operator is often replaced with a simple relaxation term [8,9]. Such relaxation terms include the Bhatnagar-Gross-Krook (BGK) model [10] and its extensions, the ellipsoidal-statistical BGK model (ES-BGK) [11] and the Shakhov model (S-model) [12]. However, the correct incorporation of complex collisional physical phenomena, such as chemical reactions, transitions of internal energy, and complicated scattering laws within the framework of these model equations is still an open question and a topic of active research [13,14,15]. Despite these drawbacks of the linear models, they offer advantages in terms of being scalable to dense regimes [16,17].Another subfamily of the discrete velocity methods does not resort to substituting the collision operator with a simplified model relaxation term, and instead models the full collision process, utilizing...

Improving PIC-DSMC Simulations of Electrical Breakdown via Event Splitting

“…Our proposed algorithm, which we will refer to as "event splitting" [8,12], consists of the following steps: an additional particle split is performed for the two equal-weight particles based on the probabilities of the possible collision events that can occur for the considered collision. After the initial split step (during which the particle with the larger computational weight is split into two smaller particles) the (equal weight) particles are split further into smaller particles, with weights 2 , = 1, .…”

“…However, in cases where multiple inelastic events can occur during a collision, the particle splitting can be leveraged: instead of choosing a single collision process (using acceptance-rejection) to be modelled, particles can be split further proportional to the probabilities of these processes, and all processes modelled simultaneously. We have previously shown that such an approach significantly reduces the level of stochastic noise in the computed ionization rate in an unsteady spatially homogeneous setup [8]. The resulting reduction in stochastic noise may be beneficial to unsteady PIC-DSMC simulations such as streamer propagation [9,10], as low-probability events can significantly affect the flow physics, perhaps even qualitatively.…”