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

摘要/Abstract

摘要：

该文针对电磁粒子模拟中满足电荷守恒的电流分配方案，给出了适用于二维和三维Yee网格以及宏粒子的电荷分布为常数的统一公式，同时列举并分析了常用的、满足电荷守恒的三种电流分配方案.根据电荷守恒定律，带有某种电荷分布的宏粒子在一个时间步内运动所引起的元胞上的电流密度，满足一个超定的线性代数方程组.这个线性代数方程组的每一个解对应一种电流分配方案.该文对电荷分布为常数的宏粒子在二维Yee网格中的运动列出了三种可能的情形、在三维Yee网格中的运动选了最常见、最简单的情形.对每一种情形，建立相应的线性代数方程组、求出对应的通解公式.将三种常用的电荷守恒的电流分配方案作为每种情形下线性代数方程组的特解，分别给出其对应于通解公式的参数.

关键词: 电磁粒子模拟, 电荷守恒, 电流分配

Abstract:

Unified formulation of charge-conserving current assignment in the Electromagnetic Particle-in-Cell simulation in two and three dimensional Cartesian geometry is proposed. This formulation is adapt for macroparticle with constant charge distribution. From the law of charge conservation, it is revealed that the involving current densities caused by the movement of a macroparticle with certain charge distribution satisfy a linear algebraic system which is usually underdetermined. By solving the linear system, all the possible current assignments could be obtained as the general solution of the system and any charge-conserving current assignment scheme is a special solution of the system. When a macroparticle with constant shape function moving in two dimensional Cartesian geometry three possible situations are presented, and when the macroparticle moving in three dimensional Cartesian geometry, the general and simplest situation is picked. The corresponding linear algebraic systems are listed and solved, and the concrete solution expressions are given. Three prevailing charge-conserving methods as three special solutions of these expressions are listed and analysed to give the corresponding parameters.

Key words: Electromagnetic Particle-in-Cell(PIC), Charge-conserving, Current assignment

中图分类号:

O242.2

付梅艳,卢朓,朱湘疆. 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[J]. 数学物理学报, 2020, 40(5): 1393-1408.

Meiyan Fu,Tiao Lu,Xiangjiang Zhu. Unified Formulation of Charge-conserving Current Assignment in Electromagnetic Particle-in-Cell Simulation[J]. Acta mathematica scientia,Series A, 2020, 40(5): 1393-1408.

图/表 12

图 1

图 2

二维Yee网格上一个中心坐标为 $(x_p, \ y_p)$ 的宏粒子"

图 2

表 1

一维典型形状函数与形因子"

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

表 1

图 3

表 2

在四个元胞情形下, V方法、E方法和U方法给出的电流密度公式"

$J$	V method $(\times \frac{q}{2\Delta t})$	E method $(\times \frac{q}{2\Delta t})$	U method $(\times \frac{q}{2\Delta t})$
$\xi_1$	$\Delta x(1-y^{n}-y^{n+1})$	$\Delta x(1-y^{n}-y^{n+1})$	$\Delta x(1-y^{n}-y^{n+1})$
$\xi_2$	$\Delta y(1-x^{n}-x^{n+1})$	$\Delta y(1-x^{n}-x^{n+1})$	$\Delta y(1-x^{n}-x^{n+1})$
$\xi_3$	$\Delta y(1+x^{n}+x^{n+1})$	$\Delta y(1+x^{n}+x^{n+1})$	$\Delta y(1+x^{n}+x^{n+1})$
$\xi_4$	$\Delta x(1+y^{n}+y^{n+1})$	$\Delta x(1+y^{n}+y^{n+1})$	$\Delta x(1+y^{n}+y^{n+1})$
其中: $\Delta x = x^{n+1}-x^{n}, \ \Delta y = y^{n+1}-y^{n}.$

表 2

表 3

在六个元胞情形下, V方法、E方法和U方法给出的电流密度公式"

$J$	V method $(\times \frac{q}{2\Delta t})$	E method $(\times \frac{q}{2\Delta t})$	U method $(\times \frac{q}{2\Delta t})$
$\xi_1$	$\Delta x_1 (1 - 2y^{n} - \Delta y\frac{\Delta x_1}{\Delta x})$	$\Delta x_1 (1 - 2y^{n} - \Delta y)$	$\Delta x_1 (1 - 2y^{n} - \frac{1}{2}\Delta y)$
$\xi_2$	$\Delta x_1 \Delta y \frac{\Delta x_1}{\Delta x}$	$\Delta x_1 \Delta y$	$\Delta x_1 \frac{1}{2}\Delta y$
$\xi_3$	$\Delta y[\frac{\Delta x_1}{\Delta x}(2-\Delta x_1) + \frac{\Delta x_2}{\Delta x}(2-\Delta x_2)]$	$\Delta y (2-\Delta x_1-\Delta x_2)$	$\Delta y (2 - \frac{1}{2}\Delta x_1 - \frac{1}{2}\Delta x_2)$
$\xi_4$	$\Delta x_1 (1 + 2y^{n} + \Delta y\frac{\Delta x_1}{\Delta x})$	$\Delta x_1 (1 + 2y^{n} + \Delta y)$	$\Delta x_1 (1 + 2y^{n} + \frac{1}{2}\Delta y)$
$\xi_5$	$\Delta x_2 (1 - 2y^{n} - \Delta y - \Delta y\frac{\Delta x_1}{\Delta x})$	$\Delta x_2 (1 - 2y^{n} - \Delta y)$	$\Delta x_2 (1 - 2y^{n} - \frac{3}{2}\Delta y)$
$\xi_6$	$\Delta x_2 \frac{\Delta x_2}{\Delta x} \Delta y$	$\Delta x_2 \Delta y$	$\Delta x_2 \frac{1}{2} \Delta y$
$\xi_7$	$\Delta x_2 (1 + 2y^{n} + \Delta y + \Delta y\frac{\Delta x_1}{\Delta x})$	$\Delta x_2 (1 + 2y^{n} + \Delta y)$	$\Delta x_2 (1 + 2y^{n} + \frac{3}{2}\Delta y)$
其中: $\Delta x = x^{n+1}-x^{n}, \ \Delta y = y^{n+1}-y^{n}, \ \Delta x_1 = 0.5-x^{n}, \ \Delta x_2 = x^{n+1}-0.5.$

表 3

表 4

在七个元胞情形下, V方法, E方法和U方法给出的电流密度公式"

$J$	V method $(\times \frac{q}{2\Delta t})$	E method $(\times \frac{q}{2\Delta t})$	U method $(\times \frac{q}{2\Delta t})$
$\xi_1$	$\Delta x_1 (1-2y^{n}-\Delta y \frac{\Delta x_1}{\Delta x})$	$(0.5-x^n)(0.5-y^n)$	$(0.5-x^n)(0.5-y^n)$
$\xi_2$	$\Delta x_1 \Delta y {\Delta x_1}/ (\Delta x)$	$(0.5-x^n)(0.5-y^n)$	$(0.5-x^n)(0.5-y^n)$
$\xi_3$	$[\Delta y (2-\Delta x_1)\frac{\Delta x_1}{\Delta x} + (2-\Delta x \frac{\Delta y_2}{\Delta y})(0.5-y^{n}-\Delta y \frac{\Delta x_1}{\Delta x})]$	$(2+x^n-x^{n+1})(0.5-y^n)$	$(1.5+x^n)(0.5-y^n)$
$\xi_4$	$\Delta x_1 (1+2y^{n}+\Delta y \frac{\Delta x_1}{\Delta x})$	$(0.5-x^{n})(2+y^{n}-y^{n+1})$	$(0.5-x^n)(1.5+y^n)$
$\xi_5$	$\Delta x_2 (0.5-y^{n}-\Delta y \frac{\Delta x_1}{\Delta x})$	$(x^{n+1}-0.5)(0.5-y^{n})$	$0$
$\xi_6$	$\Delta x \frac{\Delta y_2}{\Delta y}(0.5-y^{n}-\Delta y \frac{\Delta x_1}{\Delta x})$	$(x^{n+1}-0.5)(0.5-y^{n})$	$0$
$\xi_7$	$[\Delta x_2 (1.5+y^{n}+\Delta y \frac{\Delta x_1}{\Delta x}) + (x^{n+1}-0.5-\Delta x \frac{\Delta y_2}{\Delta y})(2.5-y^{n+1})]$	$(x^{n+1}-0.5)(2+y^{n}-y^{n+1})$	$(x^{n+1}-0.5)(2.5-y^{n+1})$
$\xi_8$	$(y^{n+1}-0.5) (2.5-\Delta x \frac{\Delta y_2}{\Delta y}-x^{n+1})$	$(2+x^{n}-x^{n+1})(y^{n+1}-0.5)$	$(2.5-x^{n+1})(y^{n+1}-0.5)$
$\xi_9$	$(y^{n+1}-0.5) (-0.5+\Delta x \frac{\Delta y_2}{\Delta y}+x^{n+1})$	$(x^{n+1}-0.5)(y^{n+1}-0.5)$	$(x^{n+1}-0.5)(y^{n+1}-0.5)$
$\xi_{10}$	$(y^{n+1}-0.5) (-0.5-\Delta x \frac{\Delta y_2}{\Delta y}+x^{n+1})$	$(x^{n+1}-0.5)(y^{n+1}-0.5)$	$(x^{n+1}-0.5)(y^{n+1}-0.5)$
$\xi_{11}$	$0$	$(0.5-x^{n})(y^{n+1}-0.5)$	$0$
$\xi_{12}$	$0$	$(0.5-x^{n})(y^{n+1}-0.5)$	$0$
其中: $\Delta x = x^{n+1}-x^{n}$ , $\Delta y = y^{n+1}-y^{n}$ , $\Delta x_1 = 0.5-x^{n}$ , $\Delta y_1 = \Delta y {\Delta x_1}/{\Delta x},$
$x_{M_1} = 0.5$ , $y_{M_1} = y^{n}+\Delta y_1$ , $\Delta y_2 = 0.5-y_{M_1}$ , $\Delta x_2 = \Delta x {\Delta y_2}/{\Delta y}.$

表 4

图 4

表 1

涉及四个元胞时相关形因子(二维)"

Form-factor	$-0.5 \leq x < 0.5, -0.5 \leq y < 0.5$
$S_{(i, j)}(x, y)$	$\left(0.5-x\right)\left(0.5-y\right)$
$S_{(i, j+1)}(x, y)$	$\left(0.5-x\right)\left(0.5+y\right)$
$S_{(i+1, j)}(x, y)$	$\left(0.5+x\right)\left(0.5-y\right)$
$S_{(i+1, j+1)}(x, y)$	$\left(0.5+x\right)\left(0.5+y\right)$

表 1

表 2

涉及六个元胞时相关形因子(二维)"

Form-factor	$-0.5 \leq x < 0.5, -0.5 \leq y < 0.5$	$0.5 \leq x < 1.5, -0.5 \leq y < 0.5$
$S_{(i, j)}(x, y)$	$\left(0.5-x\right)\left(0.5-y\right)$	$0$
$S_{(i, j+1)}(x, y)$	$\left(0.5-x\right)\left(0.5+y\right)$	$0$
$S_{(i+1, j)}(x, y)$	$\left(0.5+x\right)\left(0.5-y\right)$	$\left(1.5-x\right)\left(0.5-y\right)$
$S_{(i+1, j+1)}(x, y)$	$\left(0.5+x\right)\left(0.5+y\right)$	$\left(1.5-x\right)\left(0.5+y\right)$
$S_{(i+2, j) }(x, y)$	$0$	$\left(x-0.5\right)\left(0.5-y\right)$
$S_{(i+1, j+1)}(x, y)$	$0$	$\left(x-0.5\right)\left(0.5+y\right)$

表 2

表 3

涉及七个元胞时相关形因子(二维)"

Form-factor	$-0.5 \leq x < 0.5$		$0.5 \leq x < 1.5$
	$-0.5 \leq y < 0.5$	$0.5 \leq y < 1.5$	$-0.5 \leq y < 0.5$	$0.5 \leq y < 1.5$
$S_{(i, j)}(x, y)$	$\left(0.5-x\right)\left(0.5-y\right)$	$0$	$0$	$0$
$S_{(i, j+1)}(x, y)$	$\left(0.5-x\right)\left(0.5+y\right)$	$\left(0.5-x\right)\left(1.5-y\right)$	$0$	$0$
$S_{(i+1, j)}(x, y)$	$\left(0.5+x\right)\left(0.5-y\right)$	$0$	$\left(1.5-x\right)\left(0.5-y\right)$	$0$
$S_{(i+1, j+1)}(x, y)$	$\left(0.5+x\right)\left(0.5+y\right)$	$\left(0.5+x\right)\left(1.5-y\right)$	$\left(1.5-x\right)\left(0.5+y\right)$	$\left(1.5-x\right)\left(1.5-y\right)$
$S_{(i+2, j+1)}(x, y)$	$0$	$0$	$0$	$\left(x-0.5\right)\left(1.5-y\right)$
$S_{(i+1, j+2)}(x, y)$	$0$	$\left(0.5+x\right)\left(y-0.5\right)$	$0$	$\left(1.5-x\right)\left(y-0.5\right)$
$S_{(i+2, j+2)}(x, y)$	$0$	$0$	$0$	$\left(x-0.5\right)\left(y-0.5\right)$

表 3

表 4

涉及八个元胞时相关形因子(三维,最常见情形)"

Form-factor(1st order)	$-0.5 \leq x < 0.5, \ -0.5 \leq y < 0.5, \ -0.5 \leq z < 0.5$
$S_{(i, j, k)}(x, y, z)$	$\left(0.5-x\right)\left(0.5-y\right)\left(0.5-z\right)$
$S_{(i+1, j, k)}(x, y, z)$	$\left(0.5+x\right)\left(0.5-y\right)\left(0.5-z\right)$
$S_{(i, j+1, k)}(x, y, z)$	$\left(0.5-x\right)\left(0.5+y\right)\left(0.5-z\right)$
$S_{(i+1, j+1, k)}(x, y, z)$	$\left(0.5+x\right)\left(0.5+y\right)\left(0.5-z\right)$
$S_{(i, j, k+1)}(x, y, z)$	$\left(0.5-x\right)\left(0.5-y\right)\left(0.5+z\right)$
$S_{(i+1, j, k+1)}(x, y, z)$	$\left(0.5+x\right)\left(0.5-y\right)\left(0.5+z\right)$
$S_{(i, j+1, k+1)}(x, y, z)$	$\left(0.5-x\right)\left(0.5+y\right)\left(0.5+z\right)$
$S_{(i+1, j+1, k+1)}(x, y, z)$	$\left(0.5+x\right)\left(0.5+y\right)\left(0.5+z\right)$

表 4

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