case_fdtd_2d_te_cylindrical_wave

测试目标

case_fdtd_2d_te_cylindrical_wave 是 Python-only 的 2D TE 圆形波前可视化案例。 run.sh 运行 fdtd_2d_te_cylindrical_wave.py 生成图片，不编译、也不调用 E_Maxwell 目录中的 Fortran 子程序。这里的“柱面波”指 x-y 平面内由中心源向外传播的圆形波前，主要用于文档图和直观检查。

对应算法

该脚本的内部 Yee 更新与 E03 3D Cartesian 中 sub_E03_fdtd_3d_cartesian_H 和 sub_E03_fdtd_3d_cartesian_E 的二维 TEz 截面一致：取 ∂/∂z = 0，只保留 Ex, Ey, Hz。中心软源和一阶 Mur 边界是 Python 脚本里的展示设置，不属于 E03 Fortran 子程序本身。它不对应 E01 的 r-z 柱坐标 TEz 子程序。

文件说明

文件	作用
`fdtd_2d_te_cylindrical_wave.py`	Python FDTD 演示脚本，生成 TE 柱面波。
`fdtd_2d_te_cylindrical_wave.png`	生成的参考图。
`run.sh`	运行 Python 脚本并重新生成图片。

运行方式

source ~/.venv/bin/activate
cd tests/005_maxwell/case_fdtd_2d_te_cylindrical_wave
bash run.sh

主流程

建立 2D Cartesian x-y Yee 网格。
在中心位置加入 Hz 正弦源。
按二维 TEz 截面公式更新 Ex/Ey 和 Hz。
施加一阶 Mur 边界，保存最终 |Hz| 幅值图。
用图像检查圆形波前形状和边界附近是否有明显异常。

典型例子：中心 Hz 源的 TEz 波前

fdtd_2d_te_cylindrical_wave.py 与 TMz 可视化脚本成对出现，但场分量换成 Ex/Ey/Hz。它同样使用 201 x 201 的 x-y 网格和 278 MHz 正弦源，只是源项加在 Hz 上。这个例子适合学生对比 TEz/TMz：TMz 用 Ez 看波前，TEz 用 Hz 看波前。

nx, ny = 201, 201
dx = dy = 0.054
dt = dx / (c0 * np.sqrt(2.0))
nsteps = 190

hz = np.zeros((nx, ny), dtype=np.float64)
ex = np.zeros((nx, ny - 1), dtype=np.float64)
ey = np.zeros((nx - 1, ny), dtype=np.float64)

时间推进时先用 Hz 更新 Ex/Ey，再用 Ex/Ey 的 curl 更新内部 Hz。源项和一阶 Mur 边界放在同一个主循环里。需要注意，一阶 Mur 对近似垂直入射的平面波吸收较好；对柱面波中的斜入射分量，边界反射会更明显。因此这页主要说明“波前是否合理”，不作为严格的 Fortran 内核误差判据。

ex += (dt / (eps0 * dy)) * (hz[:, 1:] - hz[:, :-1])
ey -= (dt / (eps0 * dx)) * (hz[1:, :] - hz[:-1, :])

curl_e = (ey[1:, 1:-1] - ey[:-1, 1:-1]) / dx - \
         (ex[1:-1, 1:] - ex[1:-1, :-1]) / dy
hz[1:-1, 1:-1] -= (dt / mu0) * curl_e

hz[src_i, src_j] += np.sin(w0 * t)

重点调用方式

update Ex and Ey from Hz
update Hz from Ex and Ey
add center Hz source and Mur boundaries
save |Hz| image

结果和图像

这张图显示 TEz 可视化脚本中的场幅值。读图时主要检查三点：中心源是否产生连续波前，波前在二维平面内是否平滑向外传播，边界附近是否出现比主体波前更强的反射结构。由于图中显示的是幅值，正负相位信息被隐藏；它是波形和边界行为的直观检查图。

../../_images/fdtd_2d_te_cylindrical_wave.png — 2D TE 圆形波前参考图。图像应显示从源区传播出去的平滑波前。

常见误读

这里的 TEz 是 Cartesian 二维截面的 Ex,Ey,Hz 分量组，不是 E01 的 Ephi,Hr,Hz 柱坐标 RZ 分量组。这个 Python 图不是 Fortran 内核的数值精度证明；精度仍看 single-step 和 MMS。

Test Goal

case_fdtd_2d_te_cylindrical_wave is a Python-only 2D TE circular-wavefront visualization case. run.sh runs fdtd_2d_te_cylindrical_wave.py to generate the figure; it does not compile or call any Fortran routine under E_Maxwell. Here “cylindrical wave” means an outward circular wavefront on an x-y plane, for documentation and qualitative inspection.

Related Algorithm

The interior Yee update is consistent with the 2D TEz reduction of E03 3D Cartesian: set ∂/∂z = 0 and keep only Ex, Ey, Hz from sub_E03_fdtd_3d_cartesian_H and sub_E03_fdtd_3d_cartesian_E. The centered soft source and first-order Mur boundary are Python-side display choices, not parts of the E03 Fortran routines. This case is not the E01 r-z cylindrical-coordinate TEz routine.

Files

File	Role
`fdtd_2d_te_cylindrical_wave.py`	Python FDTD demo for a TE cylindrical wave.
`fdtd_2d_te_cylindrical_wave.png`	Generated reference image.
`run.sh`	Run the Python script and regenerate the image.

Run Command

source ~/.venv/bin/activate
cd tests/005_maxwell/case_fdtd_2d_te_cylindrical_wave
bash run.sh

Main Flow

Build a 2D Cartesian x-y Yee grid.
Add a centered sinusoidal Hz source.
Update Ex/Ey and Hz using the 2D TEz reduction.
Apply first-order Mur boundaries and save the final |Hz| image.
Inspect wavefront shape and boundary artifacts.

Example: Centered Hz Source for a TEz Wavefront

fdtd_2d_te_cylindrical_wave.py is paired with the TMz visualization script, but the active field group is Ex/Ey/Hz. It uses the same 201 x 201 x-y grid and 278 MHz sinusoidal source, with the source added to Hz. This makes it a useful side-by-side comparison for students: TMz visualizes Ez, while TEz visualizes Hz.

nx, ny = 201, 201
dx = dy = 0.054
dt = dx / (c0 * np.sqrt(2.0))
nsteps = 190

hz = np.zeros((nx, ny), dtype=np.float64)
ex = np.zeros((nx, ny - 1), dtype=np.float64)
ey = np.zeros((nx - 1, ny), dtype=np.float64)

Each time step updates Ex/Ey from Hz and then updates interior Hz from the electric curl. The source and first-order Mur boundary conditions live in the same loop. First-order Mur works best for nearly normal plane-wave incidence; oblique components in a cylindrical wave can reflect more strongly. This page is therefore for qualitative wavefront inspection rather than strict Fortran-kernel error measurement.

ex += (dt / (eps0 * dy)) * (hz[:, 1:] - hz[:, :-1])
ey -= (dt / (eps0 * dx)) * (hz[1:, :] - hz[:-1, :])

curl_e = (ey[1:, 1:-1] - ey[:-1, 1:-1]) / dx - \
         (ex[1:-1, 1:] - ex[1:-1, :-1]) / dy
hz[1:-1, 1:-1] -= (dt / mu0) * curl_e

hz[src_i, src_j] += np.sin(w0 * t)

Core Pattern

update Ex and Ey from Hz
update Hz from Ex and Ey
add center Hz source and Mur boundaries
save |Hz| image

Result and Figure

The figure shows the field magnitude from the TEz visualization script. Use it to check whether the centered source produces a coherent wavefront, whether the wavefront remains smooth as it expands, and whether the boundary region shows reflection structures stronger than the outgoing packet. Since the plotted value is a magnitude, signed phase information is hidden.

Common Pitfall

TEz here is the Cartesian 2D Ex,Ey,Hz field group, not the E01 cylindrical-coordinate Ephi,Hr,Hz group. This Python figure is not a Fortran accuracy proof; use single-step and MMS tests for accuracy.