دانلود رایگان مقاله بی ثباتی شبکه محدود و راحت الگوریتم الکترواستاتیک ذره در سلول

Abstract

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.

5. Summary

In this paper, the origin of the FGI is studied by employing particle and spectral resolutions into the dynamics of the 1D electrostatic PIC model and by contrasting the spectral fidelity of the PIC model with respect to the underlying physical system (or the gridless model). The particle resolution can be either adopted for individual particle or for the collective motion. The use of the particle and spectral resolutions are not the only options for this purpose, but are suitable ones as the particle dynamics consist of pair-wise interactions and can also be viewed as collective wave–particle interactions. At the individual particle level, the charge deposition and field interpolation operations of the PIC models exhibit systematic spectral errors relative to the physical system (or gridless model) due to the existence of the spatial alias modes from the use of the discrete grid in conjunction with Lagrangian particles in continuous space. In principle, these errors can be calculated for arbitrary particle shape, and surprisingly, they have relatively compact analytic forms shown in Eq. (3.1) for the B-spline particle shapes and in Eq. (3.2) for the Gaussian particle shape. These forms allow us to understand the effects in the spectral domain from using shaped particles in the PIC models. It is observed that these systematic spectral errors depend on both k∆x and the particle’s normalized position in the cell. These errors are directly related to the instability growth rate but they improve slowly with the increase of the order (smoothness) of the B-spline particle shape. Another benefit of using higher order particle shape is the stronger damping to the short-wavelength modes in the fundamental Brillouin zone which can enlarge the stability domain. Such a damping effect also applies to the physical modes and may be achieved using a spatial mode filter on the grid which has a lower computation cost and does not depend on individual particle’s position.