开发一个使用理想匹配层吸收边界条件求解时域麦克斯韦方程组的时域有限差分程序

以下是使用理想匹配层吸收边界条件求解时域麦克斯韦方程组的时域有限差分程序的Python实现：

import numpy as np
import matplotlib.pyplot as plt

# Constants
c = 299792458.0    # Speed of light (m/s)
mu0 = np.pi*4e-7   # Permeability of free space (H/m)
eps0 = 8.854e-12   # Permittivity of free space (F/m)

# Simulation parameters
Lx = 0.5           # Domain size (m)
Ly = 0.5
dx = 0.002         # Spatial resolution (m)
dy = 0.002
dt = dx/c          # Time step size (s)
t_end = 1e-8       # End time of simulation (s)
f0 = 10e9          # Center frequency of source (Hz)
w0 = 2*np.pi*f0    # Angular frequency of source (rad/s)
n_steps = int(t_end/dt)  # Number of time steps

# Grid
Nx = int(Lx/dx)    # Number of grid points in x-direction
Ny = int(Ly/dy)    # Number of grid points in y-direction
xa = np.linspace(0, Lx, Nx, endpoint=False)
ya = np.linspace(0, Ly, Ny, endpoint=False)
xm, ym = np.meshgrid(xa, ya)

# Source
xc = Lx/2          # Center of source
yc = Ly/2
sigma = 30*dx      # Width of source
src = np.exp(-((xm-xc)**2 + (ym-yc)**2) / (2*sigma**2)) * np.sin(w0*np.arange(n_steps)*dt)

# Fields
Hz = np.zeros((Ny, Nx))
Ex = np.zeros((Ny, Nx))
Ey = np.zeros((Ny, Nx))

# PML parameters
n_pml = 10         # Number of grid points in PML
kappa_max = 5      # Maximum conductivity of PML
alpha_max = 0.05   # Maximum absorption coefficient of PML

# PML functions
def sigma_x(k, n):
    if k < n:
        return kappa_max * ((n - k)/n)**3
    elif k >= Nx - n:
        return kappa_max * ((k - Nx + n + 1)/n)**3
    else:
        return 0.0

def sigma_y(k, n):
    if k < n:
        return kappa_max * ((n - k)/n)**3
    elif k >= Ny - n:
        return kappa_max * ((k - Ny + n + 1)/n)**3
    else:
        return 0.0

def alpha_x(k, n):
    if k < n:
        return alpha_max * ((n - k)/n)**2
    elif k >= Nx - n:
        return alpha_max * ((k - Nx + n + 1)/n)**2
    else:
        return 0.0

def alpha_y(k, n):
    if k < n:
        return alpha_max * ((n - k)/n)**2
    elif k >= Ny - n:
        return alpha_max * ((k - Ny + n + 1)/n)**2
    else:
        return 0.0

# Update coefficients
def update_coeff():
    c1 = dt/(dx*mu0)
    c2 = dt/(dy*mu0)
    c3 = dt/(dx*eps0)
    c4 = dt/(dy*eps0)
    for j in range(Ny):
        for i in range(Nx):
            sigma_x_val = sigma_x(i, n_pml)
            sigma_y_val = sigma_y(j, n_pml)
            alpha_x_val = alpha_x(i, n_pml)
            alpha_y_val = alpha_y(j, n_pml)
            bx = (1 - alpha_y_val)/(1 + sigma_y_val*c2)
            by = (1 - alpha_x_val)/(1 + sigma_x_val*c1)
            ax = c1/(1 + sigma_x_val*c1)
            ay = c2/(1 + sigma_y_val*c2)
            Ex[i,j] = bx*Ex[i,j] + ax*(Hz[i,j] - Hz[i-1,j])
            Ey[i,j] = by*Ey[i,j] + ay*(Hz[i,j] - Hz[i,j-1])

# Main loop
for n in range(n_steps):
    # Update Hz
    Hz[:,:] += dt/(dx*mu0) * (Ey[:,1:] - Ey[:,:-1]) - dt/(dy*mu0) * (Ex[1:,:] - Ex[:-1,:])
    
    # Add source
    Hz[1:-1,1:-1] += src[n]
    
    # Update Ex, Ey with PML
    update_coeff()
    
    # Update Ex
    Ex[:,:-1] -= dt/(dy*eps0) * (Hz[:,1:] - Hz[:,:-1])
    
    # Update Ey
    Ey[:-1,:] += dt/(dx*eps0) * (Hz[1:,:] - Hz[:-1,:])
    
    # Plot fields
    if n % 10 == 0:
        plt.clf()
        plt.imshow(Hz, cmap='RdBu_r', vmin=-1, vmax=1, extent=[0, Lx, 0, Ly])
        plt.colorbar()
        plt.pause(0.001)

plt.show()

在此程序中，我们使用了二维的时域有限差分方法来求解麦克斯韦方程组。我们使用了中心差分法来离散化空间导数，并使用了迎风格式来离散化时间导数。我们还使用了理想匹配层吸收边界条件来避免反射波的产生。具体来说，我们在模拟区域的边界上添加了一层PML（完美匹配层），并根据PML的位置使用不同的吸收参数来计算更新系数，以达到吸收边界反射波的目的。在每个时间步长中，我们首先更新Hz的值，然后根据源的位置和形状将源项添加到Hz中。接下来，我们使用更新系数来更新Ex和Ey的值，并最后再次更新Hz的值。在每个时间步长结束后，我们还通过plt.imshow()函数绘制了Hz的图像，以便我们可以观察波的传播情况。注意，由于我们使用了PML，因此模拟区域的边界上可能会有一些数字噪声，但这不会影响波的传播。