Mini-Workshop: Innovative Trends in the Numerical Analysis and Simulation of Kinetic Equations

In multiscale modeling hierarchy, kinetic theory plays a vital role in connecting microscopic Newtonian mechanics and macroscopic continuum mechanics. As computing power grows, numerical simulation of kinetic equations has become possible and undergone rapid development over the past decade. Yet the unique challenges arising in these equations, such as highdimensionality, multiple scales, random inputs, positivity, entropy dissipation, etc., call for new advances of numerical methods. This mini-workshop brought together both senior and junior researchers working on various fastpaced growing numerical aspects of kinetic equations. The topics include, but were not limited to, uncertainty quantification, structure-preserving methods, phase transitions, asymptotic-preserving schemes, and fast methods for kinetic equations

Nonlinear gyrokinetic PIC simulations in stellarators with the code EUTERPE

In this work, the first nonlinear particle-in-cell simulations carried out in a stellarator with the global gyrokinetic code EUTERPE using realistic plasma parameters are reported. Several studies are conducted with the aim of enabling reliable nonlinear simulations in stellarators with this code. First, EUTERPE is benchmarked against ORB5 in both linear and nonlinear settings in a tokamak configuration. Next, the use of noise control and stabilization tools, a Krook-type collision operator, markers weight smoothing and heating sources is investigated. It is studied in detail how these tools influence the linear growth rate of instabilities in both tokamak and stellarator geometries and their influence on the linear zonal flow evolution in a stellarator. Then, it is studied how these tools allow improving the quality of the results in a set of nonlinear simulations of electrostatic turbulence in a stellarator configuration. Finally, these tools are applied to a W7-X magnetic configuration using experimental plasma parameters.Comment: 24 pages, 19 figure

On the definition of a kinetic equilibrium in global gyrokinetic simulations

Nonlinear electrostatic global gyrokinetic simulations of collisionless ion temperature gradient (ITG) turbulence and ExB zonal flows in axisymmetric toroidal plasmas are examined for different choices of the initial distribution function. Using a local Maxwellian leads to the generation of axisymmetric ExB flows that can be so strong as to prevent ITG mode growth. A method using a canonical Maxwellian is shown to avoid this spurious generation of ExB flows. In addition, a revised delta f scheme is introduced and compared to the standard delta f method. (c) 2006 American Institute of Physics

High-order stochastic integration schemes for the Rosenbluth-Trubnikov collision operator in particle simulations

In this study, we consider a numerical implementation of the nonlinear Rosenbluth-Trubnikov collision operator for particle simulations in plasma physics in the framework of the finite element method (FEM). The relevant particle evolution equations are formulated as stochastic differential equations, both in the Stratonovich and Itô forms, and are then solved with advanced high-order stochastic numerical schemes. Due to its formulation as a stochastic differential equation, both the drift and diffusion components of the collision operator are treated on an equal footing. Our investigation focuses on assessing the accuracy of these schemes. Previous studies on this subject have used the Euler-Maruyama scheme, which, although popular, is of low order, and requires small time steps to achieve satisfactory accuracy. In this work, we compare the performance of the Euler-Maruyama method to other high-order stochastic methods known in the stochastic differential equations literature. Our study reveals advantageous features of these high-order schemes, such as better accuracy and improved conservation properties of the numerical solution. The main test case used in the numerical experiments is the thermalization of isotropic and anisotropic particle distributions

Simulations of an ASA flow crystallizer with a coupled stochastic-deterministic approach

A coupled solver for population balance systems is presented, where the flow, temperature, and concentration equations are solved with finite element methods, and the particle size distribution is simulated with a stochastic simulation algorithm, a so-called kinetic Monte-Carlo method. This novel approach is applied for the simulation of an axisymmetric model of a tubular flow crystallizer. The numerical results are compared with experimental data

Simulations of an ASA flow crystallizer with a coupled stochastic-deterministic approach

A coupled solver for population balance systems is presented, where the flow, temperature, and concentration equations are solved with finite element methods, and the particle size distribution is simulated with a stochastic simulation algorithm, a so-called kinetic Monte-Carlo method. This novel approach is applied for the simulation of an axisymmetric model of a tubular flow crystallizer. The numerical results are compared with experimental data

Accelerating DSMC data extraction.

In many direct simulation Monte Carlo (DSMC) simulations, the majority of computation time is consumed after the flowfield reaches a steady state. This situation occurs when the desired output quantities are small compared to the background fluctuations. For example, gas flows in many microelectromechanical systems (MEMS) have mean speeds more than two orders of magnitude smaller than the thermal speeds of the molecules themselves. The current solution to this problem is to collect sufficient samples to achieve the desired resolution. This can be an arduous process because the error is inversely proportional to the square root of the number of samples so we must, for example, quadruple the samples to cut the error in half. This work is intended to improve this situation by employing more advanced techniques, from fields other than solely statistics, for determining the output quantities. Our strategy centers on exploiting information neglected by current techniques, which collect moments in each cell without regard to one another, values in neighboring cells, nor their evolution in time. Unlike many previous acceleration techniques that modify the method itself, the techniques examined in this work strictly post-process so they may be applied to any DSMC code without affecting its fidelity or generality. Many potential methods are drawn from successful applications in a diverse range of areas, from ultrasound imaging to financial market analysis. The most promising methods exploit relationships between variables in space, which always exist in DSMC due to the absence of shocks. Disparate techniques were shown to produce similar error reductions, suggesting that the results shown in this report may be typical of what is possible using these methods. Sample count reduction factors of approximately three to five were found to be typical, although factors exceeding ten were shown on some variables under some techniques