Electrostatic PIC with adaptive Cartesian mesh

📝 Abstract

We describe an initial implementation of an electrostatic Particle-in-Cell (ES-PIC) module with adaptive Cartesian mesh in our Unified Flow Solver framework. Challenges of PIC method with cell-based adaptive mesh refinement (AMR) are related to a decrease of the particle-per-cell number in the refined cells with a corresponding increase of the numerical noise. The developed ES-PIC solver is validated for capacitively coupled plasma, its AMR capabilities are demonstrated for simulations of streamer development during high-pressure gas breakdown. It is shown that cell-based AMR provides a convenient particle management algorithm for exponential multiplications of electrons and ions in the ionization events.

💡 Analysis

We describe an initial implementation of an electrostatic Particle-in-Cell (ES-PIC) module with adaptive Cartesian mesh in our Unified Flow Solver framework. Challenges of PIC method with cell-based adaptive mesh refinement (AMR) are related to a decrease of the particle-per-cell number in the refined cells with a corresponding increase of the numerical noise. The developed ES-PIC solver is validated for capacitively coupled plasma, its AMR capabilities are demonstrated for simulations of streamer development during high-pressure gas breakdown. It is shown that cell-based AMR provides a convenient particle management algorithm for exponential multiplications of electrons and ions in the ionization events.

📄 Content

Electrostatic PIC with adaptive Cartesian mesh Vladimir Kolobov1,2 and Robert Arslanbekov1 1 CFD Research Corporation, Huntsville, AL, USA 2 The University of Alabama in Huntsville, Huntsville, AL 35899, USA
E-mail: vladimir.kolobov@cfdrc.com Abstract. We describe an initial implementation of an electrostatic Particle-in-Cell (ES-PIC) module with adaptive Cartesian mesh in our Unified Flow Solver framework. Challenges of PIC method with cell-based adaptive mesh refinement (AMR) are related to a decrease of the particle-per-cell number in the refined cells with a corresponding increase of the numerical noise. The developed ES-PIC solver is validated for capacitively coupled plasma, its AMR capabilities are demonstrated for simulations of streamer development during high-pressure gas breakdown. It is shown that cell-based AMR provides a convenient particle management algorithm for exponential multiplications of electrons and ions in the ionization events.

1. Introduction Particle-in-Cell (PIC) is a mature technology widely used for plasma simulations. There are three versions of PIC: an electrostatic (EC) PIC uses Poisson equation for calculating electric field, an electromagnetic (EM) PIC employs full-wave Maxwell solver for calculations of electric and magnetic fields, and a quasi-static (QS) PIC uses a quasi-static version of the Maxwell equations (such as Darwin model) for calculation of the EM fields. Surprisingly, PIC solvers with adaptive mesh refinement (AMR) are relatively rare. It is worth mentioning the WARP code for accelerator applications [1], and an explicit EM-PIC code of Fujimoto [2] for astrophysics applications, both using block-structured AMR. In this paper, we describe an initial implementation of an ES-PIC module in our Unified Flow Solver (UFS) framework [3]. UFS is designed for hybrid simulations of partially-ionized collisional plasmas using adaptive Cartesian mesh and automatic selection of kinetic and fluid solvers for transport processes. This is called Adaptive Mesh and Algorithm Refinement (AMAR) methodology. A cell-by-cell selection of most suitable solvers for different plasma species is based on continuum- breakdown criteria that can be specified in advance by the user.
   The PIC module in UFS supplements already developed fluid models, mesh-based kinetic solvers (Boltzmann, Vlasov, Fokker-Plank) and the particle-based Direct Simulation Monte Carlo (DSMC) and Photon Monte Carlo (PMC) solvers previously developed for simulations of mixed rarefied- continuum flows, radiation transport and plasma dynamics.
   Compared to neutral gases, plasma simulations pose extra challenges due to the disparity of time and spatial scales associated with small electron mass [4]. Criteria for selecting kinetic and fluid models for electrons, ions and neutrals are different, and strongly depend on plasma conditions. In collisionless magnetized plasmas, ions are often treated kinetically and electrons are assumed to be a fluid. In gas discharge physics, electrons are commonly treated kinetically, and fluid models applied

1 To whom any correspondence should be addressed.

for ions and neutrals [5]. First steps towards adaptive kinetic-fluid plasma simulations have been made. Two-way coupling of a global Hall magnetohydrodynamics with a local implicit PIC model (MHD with Embedded PIC regions (MHD-EPIC)) has been recently demonstrated for space plasmas.6 In the present paper, we first provide some details of the UFS methodology and discuss parallel strategy for the AMAR codes. Then, we describe an initial implementation of the ES-PIC module in UFS focusing on challenges associated with AMR. Finally, validation of the ES-PIC solver for capacitively coupled plasma (CCP) and demonstration of its AMR capabilities for simulations of streamer breakdown are described. We show that cell-based AMR technique provides a convenient adaptive particle management algorithm for exponential multiplication of charged particles in PIC codes. 2. Unified Flow Solver Basic architecture of UFS is shown in Figure 1. The AMAR core is implemented on top of Gerris Flow Solver (GFS) - an open source computing environment for solving partial differential equations with AMR [7]. GFS provides automatic mesh generation for complex geometries, portable parallel support using the MPI library, dynamic load balancing, and parallel offline visualization. GFS physics includes time-dependent incompressible variable-density Euler, Stokes or Navier-Stokes equations with volume of fluid method for interfacial flows. A coarse-grained parallelization algorithm is based on the “Forest of Trees” methodology [8]. UFS enables the higher degree of adaptation by using different physical models in different parts of computational domain. In particular, the computational domain could be decomposed into kinetic and fluid cells using

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