AI 中文总结
提出一种基于无网格方法分析PIC代码中网格不稳定性的新方法，通过线性化与离散化得到动力学矩阵，揭示混叠效应，并与模拟高度一致。
AI 中文摘要
本文描述了一种在用于等离子体动力学理论的粒子网格（PIC）代码中分析网格（或混叠）不稳定性的新方法。该方法首先采用无网格方法，包含$N_{p}$个宏粒子，其中$N_{p}<\infty$。这些方程的线性化是在由宏粒子在均匀格点上规定的均匀密度、无漂移平衡态附近进行的。宏粒子位置扰动在该格点附近被引入。所得的线性方程被分析其稳定性和色散关系。然后，线性化方程在具有$N_{g}$个点（$N_{g} \le N_{p}$）的网格上进行离散化，分别采用动量守恒（MCP）和能量守恒（ECP）两种公式。对于冷静止等离子体，这导致一个动力学矩阵。对于ECP，该矩阵是对称正定（SPD）的，并立即显示稳定性。对于MCP，动力学矩阵既不对称也不正定，阻碍了对稳定性的直接结论。对于MCP和ECP，对于冷静止等离子体，所得的矩阵元素相对于无网格值发生变化，这是混叠的结果。这种偏差随粒子格点相对于网格的位移呈周期性变化。在MCP离散化中，对于粒子格点相对于网格的一般放置，特征值以复共轭对出现。这些共轭对表明非负增长率。研究了混叠的性质，并表明其与网格上的梯形规则误差有关。研究了线性增长率相对于$N_{g}$，尤其是相对于$N_{ppc}$的标度。这些解析结果与PIC模拟进行了比较，发现它们高度一致。最后简要讨论了与冷漂移束的联系，特别是MCP和ECP的网格诱导不稳定性。
英文摘要
A new method of analyzing grid, or aliasing, instabilities is described, in particle-in-cell (PIC) codes for plasma kinetic theory. It starts with a meshfree approach with $N_{p}$ macroparticles, with $N_{p}<\infty$. Linearization of these equations is done about a uniform density non-drifting equilibrium prescribed by macroparticles on a uniform \emph{lattice}. Macroparticle positions perturbations are induced about this lattice. The resulting linear equations are analyzed for stability and dispersion relations. The linearized equations are then discretized on a grid of $N_{g}$ points with $N_{g} \le N_{p}$, in both momentum conserving (MCP) and energy conserving (ECP) formulations. For a cold stationary plasma, this leads a \emph{dynamical matrix}. For ECP, the matrix is symmetric and positive definite (SPD) and immediately shows stability. For MCP, the dynamical matrix is neither symmetric nor positive definite, preventing immediate conclusions on stability. For both MCP and ECP and for a cold stationary plasma, the resulting matrix elements vary relative to the meshfree value, a result of aliasing. This deviation is periodic in the displacement of the particle lattice relative to the grid. In the MCP discretization, eigenvalues occur in complex conjugate pairs for general placement of the particle lattice relative to the grid. These conjugate pairs indicate nonnegative growth rates. The nature of the aliasing is studied and shown to be related to the trapezoidal rule error over the grid. The scaling of the linear growth rates with respect to $N_{g}$ and especially with respect to $N_{ppc}$ is studied. These analytical results are compared with PIC simulations and found to be in excellent agreement. The connection with a cold drifting \emph{beam} is discussed briefly, specifically, grid-induced instabilities for both the MCP and ECP.
