电磁粒子模拟中电荷守恒的电流分配方案满足的统一公式

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电磁粒子模拟中电荷守恒的电流分配方案满足的统一公式

付梅艳,卢朓,朱湘疆

Unified Formulation of Charge-conserving Current Assignment in Electromagnetic Particle-in-Cell Simulation

Meiyan Fu,Tiao Lu,Xiangjiang Zhu

表 1 一维典型形状函数与形因子

$\mbox{Order}$	$D(x; x_p)$	$S_{(i)}(x_p)$
$0$	$\delta(x-x_p) \label{shapefunction_1}$ ,	$\left\{\begin{array}{ll} 1, \; \mbox{if }\; \ x_p \in\left(x_i -\frac{\Delta x}{2}, x_i + \frac{\Delta x}{2} \right), \\0 \; \mbox{else}, \end{array}\right. \label{form-factor_1}$
$1$	$\left\{\begin{array}{ll} \frac{1}{\Delta x}, \mbox{if }\ x\in\left(x_p - \frac{\Delta x}{2}, \; x_p + \frac{\Delta x}{2} \right), \\0, \; \mbox{else}, \end{array}\right. \label{shapefunction_2}$	$\left\{\begin{array}{ll} 1 - \frac{\|x_p-x_i\|}{\Delta x}, & \mbox{if }\ \|x_p - x_i\| \leq \Delta x, \\0, &\mbox{else}, \end{array}\right. \label{form-factor_2}$
$2$	$\left\{\begin{array}{ll} \frac{1}{\Delta x}\left[1- \frac{\|x-x_p\|}{\Delta x}\right], &\mbox{if }\ \|x-x_p\| < \Delta x, \\0, &\mbox{else}, \end{array}\right. \label{shapefunction_3}$	$\left\{\begin{array}{ll} \frac{3}{4}-\left(\frac{x_i-x_p}{\Delta x} \right)^2, &\mbox{if }\ \|\xi-x_i\| < \frac{1}{2}\Delta x, \\\frac{1}{2}\left[\frac{3}{2}+\frac{x_i-x_p}{\Delta x} \right]^2, &\mbox{if } \frac{\Delta x}{2} \leq x_p-x_i \leq \frac{3\Delta x}{2}, \\ \frac{1}{2}\left[\frac{3}{2}-\frac{x_i-x_p}{\Delta x} \right]^2, &\mbox{if } \frac{\Delta x}{2} \leq x_i-x_p \leq \frac{3\Delta x}{2}, \\ 0, &\mbox{else}. \end{array}\right. \label{form-factor_3}$