Finite Grid Instability Research Articles

Gauss's Law satisfying Energy-Conserving Semi-Implicit Particle-in-Cell method

The Energy Conserving Semi-Implicit Method (ECSIM) introduced by Lapenta (2017) has many advantageous properties compared to the classical semi-implicit and explicit PIC methods. Most importantly, energy conservation eliminates the growth of the finite grid instability. We have implemented ECSIM in a different and more efficient manner than the original approach. More importantly, we have addressed two major shortcomings of the original ECSIM algorithm: there is no mechanism to enforce Gauss's law and there is no mechanism to reduce the numerical oscillations of the electric field. A classical approach to satisfy Gauss's law is to modify the electric field and its divergence using either an elliptic or a parabolic/hyperbolic correction based on the Generalized Lagrange Multiplier method. This correction, however, violates the energy conservation property, and the oscillations related to the finite grid instability reappear in the modified ECSIM scheme. We invented a new alternative approach: the particle positions are modified instead of the electric field in the correction step. Displacing the particles slightly does not change the energy conservation property, while it can satisfy Gauss's law by changing the charge density. We found that the new Gauss's Law satisfying Energy Conserving Semi-Implicit Method (GL-ECSIM) produces superior results compared to the original ECSIM algorithm. In some simulations, however, there are still some numerical oscillations present in the electric field. We attribute this to the simple finite difference discretization of the energy conserving implicit electric field solver. We modified the spatial discretization of the field solver to reduce these oscillations while only slightly violating the energy conservation properties. We demonstrate the improved quality of the GL-ECSIM method with several tests.

Slurm: Fluid particle-in-cell code for plasma modeling

With the approach of exascale computing era, particle-based models are becoming the focus of research due to their excellent scalability. We present a new code, Slurm, which implements the classic particle-in-cell algorithm for modeling magnetized fluids and plasmas. It features particle volume evolution which damps the numerical finite grid instability, and allows modeling of key physical instabilities such as Kelvin–Helmholtz and Rayleigh–Taylor. The magnetic field in Slurm is handled via the electromagnetic vector potential carried by particles. Numerical diffusion of the magnetic flux is extremely low, and the solenoidality of the magnetic field is preserved to machine precision. A double-linked list is used to carry particles, thus implementation of open boundary conditions is simple and efficient. The code is written in C++ with OpenMP multi-threading, and has no external dependencies except for Boost. It is easy to install and use on multi-core desktop computers as well as on large shared-memory machines. Slurm is an ideal tool for its primary goal, modeling of space weather events in the heliosphere. This article walks the reader through the physical model, the algorithm, and all important details of implementation. Ideally, after finishing this paper, the reader should be able to either use Slurm for solving the desired problem, or create a new fluid PIC code.

Performance analysis and implementation details of the Energy Conserving Semi-Implicit Method code (ECsim)

We present in this work the implementation of the Energy Conserving Semi-Implicit Method in a parallel code called ECsim. This new code is a three-dimensional, fully electromagnetic particle in cell (PIC) code. It is written in C/C++ and uses MPI to allow massive parallelization. ECsim is unconditionally stable in time, eliminates the finite grid instability, and has the same cycle scheme as the explicit code with a computational cost comparable to other semi-implicit PIC codes. All these features make it a very valuable tool to address situations which have not been possible to analyze until now with other PIC codes. In this work, we show the details of the algorithm implementation and we study its performance in different systems. ECsim is compared with another semi-implicit PIC code with different time and spectral resolution, showing its ability to address situations where other codes fail.

Finite grid instability and spectral fidelity of the electrostatic Particle-In-Cell algorithm

The origin of the Finite Grid Instability (FGI) is studied by resolving the dynamics in the 1D electrostatic Particle-In-Cell (PIC) model in the spectral domain at the single particle level and at the collective motion level. The spectral fidelity of the PIC model is contrasted with the underlying physical system or the gridless model. The systematic spectral phase and amplitude errors from the charge deposition and field interpolation are quantified for common particle shapes used in the PIC models. It is shown through such analysis and in simulations that the lack of spectral fidelity relative to the physical system due to the existence of aliased spatial modes is the major cause of the FGI in the PIC model.

On the numerical dispersion of electromagnetic particle-in-cell code: Finite grid instability

The Particle-In-Cell (PIC) method is widely used in relativistic particle beam and laser plasma modeling. However, the PIC method exhibits numerical instabilities that can render unphysical simulation results or even destroy the simulation. For electromagnetic relativistic beam and plasma modeling, the most relevant numerical instabilities are the finite grid instability and the numerical Cherenkov instability. We review the numerical dispersion relation of the Electromagnetic PIC model. We rigorously derive the faithful 3-D numerical dispersion relation of the PIC model, for a simple, direct current deposition scheme, which does not conserve electric charge exactly. We then specialize to the Yee FDTD scheme. In particular, we clarify the presence of alias modes in an eigenmode analysis of the PIC model, which combines both discrete and continuous variables. The manner in which the PIC model updates and samples the fields and distribution function, together with the temporal and spatial phase factors from solving Maxwell's equations on the Yee grid with the leapfrog scheme, is explicitly accounted for. Numerical solutions to the electrostatic-like modes in the 1-D dispersion relation for a cold drifting plasma are obtained for parameters of interest. In the succeeding analysis, we investigate how the finite grid instability arises from the interaction of the numerical modes admitted in the system and their aliases. The most significant interaction is due critically to the correct representation of the operators in the dispersion relation. We obtain a simple analytic expression for the peak growth rate due to this interaction, which is then verified by simulation. We demonstrate that our analysis is readily extendable to charge conserving models.

The energy conserving particle-in-cell method

A new particle-in-cell (PIC) method, that conserves energy exactly, is presented. The particle equations of motion and the Maxwell’s equations are differenced implicitly in time by the midpoint rule and solved concurrently by a Jacobian-free Newton Krylov (JFNK) solver. Several tests show that the finite grid instability is eliminated in energy conserving PIC simulations, and the method correctly describes the two-stream and Weibel instabilities, conserving exactly the total energy. The computational time of the energy conserving PIC method increases linearly with the number of particles, and it is rather insensitive to the number of grid points and time step. The kinetic enslavement technique can be effectively used to reduce the problem matrix size and the number of JFNK solver iterations.

Evolution of the fastest-growing relativistic mixed mode instability driven by a tenuous plasma beam in one and two dimensions

Particle-in-cell simulations confirm here that a mixed plasma mode is the fastest growing when a highly relativistic tenuous electron-proton beam interacts with an unmagnetized plasma. The mixed modes grow faster than the filamentation and two-stream modes in simulations with beam Lorentz factors Γ of 4, 16, and 256, and are responsible for thermalizing the electrons. The mixed modes are followed to their saturation for the case of Γ=4 and electron phase space holes are shown to form in the bulk plasma, while the electron beam becomes filamentary. The initial saturation is electrostatic in nature in the considered one- and two-dimensional geometries. Simulations performed with two different particle-in-cell simulation codes evidence that a finite grid instability couples energy into high-frequency electromagnetic waves, imposing simulation constraints.

Finite-Grid Instability in Quasineutral Hybrid Simulations

Analysis and simulation of numerical instability caused by the spatial grid is presented for the case of quasineutral hybrid plasma simulation. The simplest case of particle ions advanced in the ambipolar electric field due to an isothermal, charge neutralizing electron fluid is examined. Stationary plasma is destabilized by finite ion temperature with the threshold temperature T i characterized by the ratio ZT e/T i , where Z is the charge state of the ions and T e is the electron temperature. A cold drifting plasma is found to be unstable with typical growth rate γ ≈ 0.4-0.6 C s/Δx ( Δx is the grid spacing and C s is the ion acoustic speed); simulations show that unstable growth is saturated by particle trapping leading to heated ions. For nearest grid point weighting, saturation, and threshold for growth are observed to be T i ≈ 0.6 ZT e while for linear particle weighting, T i ≈ 0.1 ZT e . These results are relevant to hybrid particle simulations of laser generated plasmas, a regime where ZT e/T i ⪢ 1 is encountered.

The ringing instability in particle-in-cell calculations of low-speed flow

The cause of the ringing instability in particle-in-cell (PIC) fluid calculations is identified, and its properties are studied. The ringing instability is caused by aliasing, just as is the finite grid instability previously identified in PIC plasma simulation, and results in large amplitude fluctuations in the particle density. The instability occurs when the flow speed drops below a critical value, which is lower with higher order interpolation and zero with Gaussian interpolation. The instability growth rate is only weakly dependent on the number of particles per cell, but appears to be suppressed by implicit differencing in time.

An implicit method for electromagnetic plasma simulation in two dimensions

A new method for modeling low-frequency plasma phenomena is presented. The method uses an implicit formulation of the Vlasov-Maxwell equations to relax restrictions on the time-step and mesh spacing so that larger values which correspond to the frequencies and wavelengths of interest can be used. As a result, the range of length and time scales accessible to plasma simulation is increased by orders of magnitude. The algorithm, as embodied in a new code VENUS for electromagnetic plasmas in two dimensions, is described, its stability and accuracy analyzed through linear and nonlinear analysis, and its properties, including suppression of the finite grid instability, illustrated through its application to the Weibel instability.