自由空间的时间相关麦克斯韦旋度方程为
$$ \begin{aligned}\frac{\partial \boldsymbol{E}}{\partial t} &=\frac{1}{\varepsilon_{0}} \nabla \times \boldsymbol{H} \\\frac{\partial \boldsymbol{H}}{\partial t} &=-\frac{1}{\mu_{0}} \nabla \times \boldsymbol{E} .\end{aligned} $$
E 和 H 是三个维度的向量,因此,一般来说,等式 (1.1a) 和 (1.1b) 分别代表三个方程。 我们将从仅使用 Ex 和 Hy 的简单一维案例开始,因此 等式(1.1a) 和 (1.1b) 变为
$$ \begin{aligned}&\frac{\partial E_{x}}{\partial t}=-\frac{1}{\varepsilon_{0}} \frac{\partial H_{y}}{\partial z}, \\&\frac{\partial H_{y}}{\partial t}=-\frac{1}{\mu_{0}} \frac{\partial E_{x}}{\partial z} .\end{aligned} $$
这些是沿z方向行驶的平面波的方程,其电场在X方向上定向,并以y方向定向磁场。
对时间和空间导数进行中心差分近似,给出
$$ \begin{aligned}\frac{E_{x}^{n+1 / 2}(k)-E_{x}^{n-1 / 2}(k)}{\Delta t} &=-\frac{1}{\varepsilon_{0}} \frac{H_{y}^{n}\left(k+\frac{1}{2}\right)-H_{y}^{n}\left(k-\frac{1}{2}\right)}{\Delta x} \\\frac{H_{y}^{n+1}\left(k+\frac{1}{2}\right)-H_{y}^{n}\left(k+\frac{1}{2}\right)}{\Delta t} &=-\frac{1}{\mu_{0}} \frac{E_{x}^{n+1 / 2}(k+1)-E_{x}^{n+1 / 2}(k)}{\Delta x}\end{aligned} $$
import numpy as np
from math import exp
from matplotlib import pyplot as plt
ke = 200
ex = np.zeros(ke)
hy = np.zeros(ke)
# Pulse parameters
kc = int(ke / 2)
t0 = 40
spread = 12
nsteps = 100
# Main FDTD Loop
for time_step in range(1, nsteps + 1):
# Calculate the Ex field
for k in range(1, ke):
ex[k] = ex[k] + 0.5 * (hy[k - 1] - hy[k])
# Put a Gaussian pulse in the middle
pulse = exp(-0.5 * ((t0 - time_step) / spread) ** 2)
ex[kc] = pulse
# Calculate the Hy field
for k in range(ke - 1):
hy[k] = hy[k] + 0.5 * (ex[k] - ex[k + 1])
# Plot the outputs as shown in Fig. 1.2
plt.rcParams['font.size'] = 12
plt.figure(figsize=(8, 3.5))
plt.subplot(212)
plt.plot(hy, color='k', linewidth=1)
plt.ylabel('H$_y$', fontsize='14')
plt.xlabel('FDTD cells')
plt.xticks(np.arange(0, 201, step=20))
plt.xlim(0, 200)
plt.yticks(np.arange(-1, 1.2, step=1))
plt.ylim(-1.2, 1.2)
plt.subplots_adjust(bottom=0.2, hspace=0.45)
plt.show()
https://embed.notionlytics.com/wt/ZXlKd1lXZGxTV1FpT2lKalpXWmhZVGN3TW1Sa01EWTBaVE00T1Rjd1pUYzNNR1l5T0RRd01XTmhZU0lzSW5kdmNtdHpjR0ZqWlZSeVlXTnJaWEpKWkNJNklsZHNTR2hsVEZSUFdXeHpaVmRhUW1ZNU1YQmxJbjA9