A 4th-Order Particle-in-Cell Method with Phase-Space Remapping for the Vlasov--Poisson Equation

Abstract

Numerical solutions to the Vlasov-Poisson system of equations have important applications to both plasma physics and cosmology. In this paper, we present a new Particle-in-Cell (PIC) method for solving this system that is 4th-order accurate in both space and time. Our method is a high-order extension of one presented previously [B. Wang, G. Miller, and P. Colella, SIAM J. Sci. Comput., 33 (2011), pp. 3509-3537]. It treats all of the stages of the standard PIC updatecharge deposition, force interpolation, the field solve, and the particle push -with 4th-order accuracy, and includes a 6th-order accurate phase-space remapping step for controlling particle noise. We demonstrate the convergence of our method on a series of one-and two-dimensional electrostatic plasma test problems, comparing its accuracy to that of a 2nd-order method. As expected, the 4thorder method can achieve comparable accuracy to the 2nd-order method with many fewer resolution elements.

An unsupervised machine-learning checkpoint-restart algorithm using Gaussian mixtures for particle-in-cell simulations

“…The implication in the context of GM is that the estimated continuum PDF may be able distinguishing between noise and signal, and, if so, provide a measure of noise reduction (i.e., variance control), such that the GM PDF, once resampled, may lead to an improved PIC solution vs. the unrestarted one. The subject of noise control in PIC algorithms has received significant attention recently [37,38,39], but it has mostly been circumscribed to the remapping of the particle PDF via interpolation to a (semi-)structured phase-space mesh (i.e., bins), and subsequent resampling within bins. Some of these approaches [39] explicitly embed arbitrary moment conservation in their formulation, which is a desirable property.…”

Sparse Grids based Adaptive Noise Reduction strategy for Particle-In-Cell schemes

“…In general the charge density e after sparse grids transformation is not guaranteed to be positive everywhere. This is not unique to our approach and also happens in other noise reduction strategies such as high-order shape functions [20], compensating filters [6] and wavelet-based density estimation [37]. In our numerical results in section 5 we do not observe any problems caused by this.…”

“…It scales as (N p h d ) −1/2 [1], where d is the spatial dimension of the problem. The initial distribution is sampled using one of the standard sampling techniques such as the naive Monte-Carlo strategy [11], importance sampling [11] or by means of the quiet start [18,19,20]. The choice of initial sampling plays an important role in determining the constant associated with the particle noise.…”

“…Noise reduction can be achieved in several ways in the context of PIC simulations and they can be categorized as: (i) variance reduction techniques such as the δf method [10,11,12] and quiet start [12]; (ii) phase space remapping [18,19,20]; (iii) filtering in physical domain [6,21,13,14,22], Fourier domain [6,15] and wavelet domain [23,16,24]. The list is not exhaustive and there are many other contributions in this area.…”

“…In our numerical results in section 5 we do not observe any problems caused by this. However, we could adopt the density redistribution procedure used in [20] Algorithm 1 tauEstimator: An algorithm for estimating optimal τ . 1: Compute Fourier transform of the charge density ρe = F (ρ e ).…”