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Technol. 27 (2018) 065004 (14pp) https://doi.org/10.1088/1361-6595/aac6b9 Test cases for grid-based direct kinetic modeling of plasma flows Kentaro Hara1 and Kyle Hanquist2 1 Texas A&M University, College Station, Texas, United States of America 2 University of Michigan, Ann Arbor, Michigan, United States of America E-mail: [email protected] Received 19 January 2018, revised 1 May 2018 Accepted for publication 22 May 2018 Published 18 June 2018 Abstract Grid-based kinetic models are promising in that the numerical noise inherent in particle-based methods is essentially eliminated. Here, we call such grid-based techniques a direct kinetic (DK) model. Velocity distribution functions are directly obtained by solving kinetic equations, such as the Vlasov equation, in discretized phase space, i.e., both physical and velocity space. In solving the kinetic equations that are hyperbolic partial differential equations, we employ a conservative, positivity-preserving numerical scheme, which is necessary for robust calculations of problems particularly including ionization. Test cases described in this paper include plasma sheaths with electron emission and injection and expansion of neutral atom flow in a two-dimensional configuration. A unifying kinetic theory of space charge limited sheaths for both floating and conducting surfaces is presented. The improved theory is verified using the collisionless DK simulation, particularly for small sheath potentials that particle-based kinetic simulations may struggle due to statistical noise. For benchmarking of the grid-based and particle-based kinetic simulations, hybrid simulations of Hall thruster discharge plasma are performed. While numerical diffusion occurs in the phase space in the DK simulation, ionization oscillations are well resolved since ionization events can be taken into account deterministically at every time step. Keywords: kinetic simulation, plasma sheaths, plasma instability, nonlinear plasma waves, Hall thruster, space charge limited sheath, Vlasov simulation 1. Introduction particle methods are easy to implement, the statistical noise due to the use of macroparticles in particle simulations may Understanding time-varying (dynamic) and kinetic (non- become problematic, particularly if the high frequency signals Maxwellian) effects is important when investigating tempo- are altered by the statistical noise. Hence, the DK method can rally and spatially small-scale plasma phenomena and their serve as an alternative to the particle methods. effects on macroscopic dynamics. Some examples include Since first-principles gas kinetic equations are a multi- plasma-wave, plasma-material, and plasma-beam interactions, dimensional first-order hyperbolic partial differential equation where the velocity distribution functions (VDFs) may become (PDE), the numerical methods for advection problems non-Maxwellian. Such plasma dynamics are relevant in low developed in the computational fluid dynamics (CFD) com- temperature plasmas, including ion acoustic waves in cathode munity can be used effectively for DK methods. Development plumes [1–3], plasma waves in cross-field devices [4–6], and of high-order accurate Vlasov simulation techniques has also plasma sheath dynamics [7–11]. Although particle methods been a popular research topic, mainly in the applied mathe- have been widely used for low temperature plasmas [12–14], matics community. The first Vlasov simulation developed an alternative method is to directly solve the kinetic was a finite difference scheme using cubic spline interpolation equations, which we call a grid-based direct kinetic (DK) [15]. Since then, many researchers have developed other method. This approach is also called a continuum kinetic methods including semi-Lagrangian methods [16, 17], method, Vlasov method, and discrete velocity method. While Weighted Essentially Non-Oscillatory (WENO) schemes 0963-0252/18/065004+14$33.00 1 © 2018 IOP Publishing Ltd Plasma Sources Sci. Technol. 27 (2018) 065004 K Hara and K Hanquist [18], Discontinuous Galerkin methods [19], and finite volume ordinary differential equations (ODEs). The macroscopic methods using Runge-Kutta (RK) methods for time integra- quantities are obtained by sampling the particle information. tion [20]. Collision processes are often taken into account probabil- While the DK methods have popularly been used in istically using random numbers. There exists statistical noise collisionless plasma physics, the noiseless kinetic simulation in the system that may affect estimation of the collision int- can be applied to low-temperature plasmas and other fluid egral and alter high frequency plasma oscillations. On the dynamics. It is, however, important to test a new kinetic other hand, grid-based DK methods eliminate the statistical model via verification, validation, and benchmarking [21]. noise inherent in particle simulations, as the VDFs are directly First, verification refers to checking whether a numerical code solved for in discretized phase space. The collision integral is correctly implemented. This is often done by comparing the can be calculated from the discretized VDFs. numerical results with analytic (exact) solutions. In addition Memory requirement and computational cost has been a to the code verification, a theory can also be verified by drawback for grid-based kinetic simulations since the phase numerics. Second, benchmarking of multiple codes can serve space needs to be discretized in all dimensions. Let us assume as an important testing. Different solvers can be used to solve that the discretization in physical coordinate is identical in the exact same equations to investigate the performance of a grid-based and particle-based kinetic simulations (N and d [ ] x x numerical method 22 . For instance, comparing grid-based are the number of cells and dimensions in physical space). and particle-based kinetic methods is essential in under- dv The total number of velocity bins per one physical cell is Nv , standing the similarities and differences of different kinetic where N is the number of velocity bins and d is the number [ ] v v models 23, 24 . Finally, validation of a computer simulation of dimensions in the velocity space. For particle simulations, can be performed by comparing the results with measure- each macroparticle has (dx+dv)-dimensional information, so ments or experimental observations. It is important to keep in the total number of degrees of freedom per one computational mind that experiments also include uncertainties through cell is (dx+dv) Np, where Np is the number of computational measurements, making them not exact, which is the differ- particles per cell. The computational memory required is fi ence between validation and veri cation. directly associated with the resolution of the VDFs. As In this paper, we summarize the test cases suitable for computational cost is linearly proportional to the degrees of kinetic simulations relevant to low temperature plasmas. The freedom in the calculation, grid-based DK methods are often test cases discussed in this paper include electron emission more computationally expensive than particle methods. from plasma-immersed materials and a multidimensional The most critical comparison may be the error associated collisionless flow that accounts for injection, wall reflection, with the kinetic methods. Truncation error is a type of error and expansion of neutral atoms. Benchmarking of a grid- that corresponds to omitting high order terms when numeri- based and particle-based kinetic model in a Hall thruster cally solving a discretized differential equation [25]. The discharge [23] is also discussed, with a particular emphasis on + local truncation error is O(h p 1) for a p-th order method, the time-varying ion VDFs. In addition to development of DK where h is the discretization step, e.g., time step or grid size. method, a general theory of plasma sheaths in the presence of The global truncation error is the accumulative error through strong electron emission from a surface is proposed. The improved theory takes into account both floating and con- multiple integration steps. For a total length of L, the total number of integration steps will be L/h. Hence, the global ducting materials and is consistent with existing theories. p truncation error, òT,ofap-th order accurate scheme is O(h ). In table 1, the numerical error in the grid-based method is shown to be proportional to O()N-p as the cell size h can be 2.