# Direct Kinetic Simulation

### Introduction

In the electric propulsion community, two computational methods have been previously developed to predict thruster performance and model plasma behavior. Fluid methods assume a distribution function close to a Maxwellian. The use of macroscopic quantities such as number density, temperature, and mean velocity allows fast computation, but non-equilibrium ects are often neglected. On the other hand, particle simulations, such as particle-in-cell (PIC) and direct simulation Monte Carlo (DSMC), are able to simulate the non-equilibrium nature of the discharge plasma. However, statistical noise due to the use of macro-particles is unavoidable. In particular, low density regions such as near anode region and the tail of VDFs (i.e. the high energy electrons) that mainly contribute to ionization suffer from poor resolution due to statistical noise.

### Method

A high-fidelity kinetic simulation without statistical noise is required. This project is to develop a direct kinetic (DK) simulation where the kinetic equations, such as the Boltzmann or Vlasov equation, are solved directly to achieve better resolution of VDFs than using discrete particles that inherently introduce numerical fluctuations. However, collision terms in the Boltzmann equation are often highly nonlinear and difficult to solve explicitly. Thus, it is useful to verify our direct kinetic simulation method in a collisionless case where the plasma transport is simpler. The collisionless Boltzmann equation, called the Vlasov equation, is given by

$\frac{\partial f}{\partial t}+\vec{v}\cdot\frac{\partial f}{\partial \vec{x}}+\frac{q}{m}\left(\vec{E}+\vec{v}\times\vec{B}\right)\cdto \frac{\partial f}{\partial \vec{v}}=0$

For self-consistent unmagnetized plasmas, the Poisson equation determines the profile of the electric field (or the potential) based on the boundary conditions. Maxwell's equation can also be coupled for a magnetized plasma in order to obtain the electric and magnetic fields.

The first step of developing a DK simulation is to develop a 1D Vlasov solver. Mathematically, the 1D Vlasov equation is a 2D hyperbolic partial differetial equation: 1D in physical space and 1D in velocity space. Thus, numerical methods that are developed in the computational fluid dynamics community can be used. Numerical schemes that have been used for Vlasov solvers include cubic spline interpolation, WENO schemes, discontinuous Galerkin methods, and finite volume methods using high-order Runge-Kutta method for time integration.

We developed a finite volume Vlasov solver using a MUSCL scheme, developed by Van Leer, for flux reconstruction. This method has good properties such as (1) positivity preservation of the VDFs, (2) conservativeness, (3) robustness, and (4) efficiency.

There are several test cases that can be used to evaluate Vlasov solvers. Here, we show simulations of a collisionless sheath and a nonlinear Landau damping.

### Results

Figure 1 shows the ion VDFs, electron VDFs, and potential in a collisionless sheath between a wall and the sheath edge.[1] The wall is an absorbing wall, where ions and electrons accumulate, which results in a floating potential. The electric field at the wall is determined by the surface charge, which is accumulated by the ions and electrons. The other boundary condition is the sheath edge where the ions follow a Bohm-velocity shifted Maxwellian and electrons are Maxwellian.

Ions accelerate towards the wall while preserving the maximum value of the VDFs. On the other hand, electrons decelerate due to the potential field. It can be seen that the magnitude of VDFs decreases. In addition, the electron VDFs are truncated at positive velocities. This is because high energy electrons that approach the wall are absorbed whereas the slow electrons are reflected back to the plasma. The numerical results are in good agreement with the analytic solutions derived by Kolobov et al [2].

 (a) Ion VDF (b) Electron VDF
 (c) Potential

Figure 1: Collisionless sheath

Figure 2 shows the evolution of the electric energy for nonlinear Landau damping. Landau damping is a collisionless nonlinear phenomenon that prevents instability from developing. Ions are assumed to be immobile and an initial perturbation of electrons is imposed. A periodic boundary condition is used in the physical space.

Figure 2: Nonlinear Landau damping

### Acknowledgments

The financial support is provided by the U. S. Department of Energy Office of Science, Fusion Energy Sciences Program and the Air Force Research Laboratory.

### Reference

[1] Hara, K., Boyd, I. D., and Kolobov, V. I., "Investigation of Presheath and Sheath Using a Full-Vlasov Simulation", 65th Annual Gaseous Conference, Austin, TX, October 2012.

[2] Kolobov, V. I. and Arslanbekov, R. R., "Towards Adaptive Kinetic-Fluid Simulations of Weakly Ionized Plasmas," Journal of Computational Physics, Vol. 231, No. 3, 2012, pp. 839-869.