Particle-in-Cell (PIC) simulation is an interpolation-based method on the Newton–Maxwell (N–M) system. Its well-known drawback is its shape/interpolation functions often causing the violation of continuity equations (CEs) at mesh nodes and that of Maxwell equations (MEs) at particles' positions. Whether this drawback can be overcome by choosing/solving suitable shape/interpolation functions is of fundamental importance for the PIC simulation. Until now, these shape/interpolation functions are usually subjectively chosen and, hence, always invoke the drawback. Here, we first investigate whether these shape/interpolation functions can be self-consistently solved by considering under what condition the CEs and the MEs can be satisfied anywhere. Strict mathematical analysis reveals that strict self-consistent shape/interpolation functions are unavailable. Only few approximately self-consistent shape/interpolation functions are luckily found by some authors. This fact drives us to present another universal interpolation-free strict method on the N–M system.

中文翻译：

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无插值粒子模拟

像元内粒子（PIC）模拟是牛顿-麦克斯韦（N-M）系统上基于插值的方法。它的众所周知的缺点是其形状/插值函数经常会导致网格节点处的连续性方程（CE）和粒子位置处的麦克斯韦方程（ME）违反。通过选择/求解合适的形状/插值函数是否可以克服此缺点，对于PIC仿真至关重要。到目前为止，这些形状/插值函数通常是主观选择的，因此总是会带来缺点。在这里，我们首先通过考虑在任何条件下都可以满足CE和ME的条件来研究这些形状/插值函数是否可以自洽地求解。严格的数学分析表明，没有严格的自洽形状/插值函数。幸运的是，一些作者只发现了很少几个近似自洽的形状/插值函数。这一事实促使我们提出了另一种在NM系统上通用的无插值严格方法。