case_fdtd_2d_tm_cylindrical_wave

测试目标

case_fdtd_2d_tm_cylindrical_wave 是 Python-only 的 2D TM 柱面波前可视化案例。 run.sh 运行 fdtd_2d_tm_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；它主要用于文档展示和直观检查，不是严格的 Fortran 子程序回归测试。

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 TMz 截面一致：取 ∂/∂z = 0，只保留 Hx, Hy, Ez。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 TMz 子程序。

文件说明

文件	作用
`fdtd_2d_tm_cylindrical_wave.py`	Python FDTD 演示脚本，使用中心正弦源生成 TM 柱面波图。
`fdtd_2d_tm_cylindrical_wave.png`	生成的 `\|Ez\|` 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_tm_cylindrical_wave
bash run.sh

主流程

建立 2D Cartesian x-y Yee 网格。
在中心位置加入 Ez 正弦源。
按二维 TMz 截面公式更新 Hx/Hy 和 Ez。
施加一阶 Mur 边界，运行到指定步数后输出 |Ez| 图。
用图像检查圆形波前是否从源附近向外传播。

典型例子：中心 Ez 源的 TMz 波前

fdtd_2d_tm_cylindrical_wave.py 里最值得看的部分是场数组和更新顺序。脚本使用 201 x 201 的二维网格，dx=dy=0.054，时间步长取二维 CFL 上限附近的 dt = dx/(c0*sqrt(2))，源项频率为 278 MHz。这里直接使用 x/y 网格做二维 Cartesian Yee 更新；文件名里的 cylindrical_wave 只描述圆形波前形态，不表示柱坐标 r-z 离散。

nx, ny = 201, 201
dx = dy = 0.054
dt = dx / (c0 * np.sqrt(2.0))
nsteps = 190
f0 = 278.0e6

ez = np.zeros((nx, ny), dtype=np.float64)
hx = np.zeros((nx, ny - 1), dtype=np.float64)
hy = np.zeros((nx - 1, ny), dtype=np.float64)

每个时间步先用 Ez 更新 Hx/Hy，再用 Hx/Hy 的 curl 更新内部 Ez。中心源在电场更新之后加入，这样图上可以看到从源附近向外传播的波前。四条边使用一阶 Mur absorbing boundary condition。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。因此它适合这个轻量展示，但不是本仓库 CPML 测试的替代品。

hx -= (dt / (mu0 * dy)) * (ez[:, 1:] - ez[:, :-1])
hy += (dt / (mu0 * dx)) * (ez[1:, :] - ez[:-1, :])

curl_h = (hy[1:, 1:-1] - hy[:-1, 1:-1]) / dx - \
         (hx[1:-1, 1:] - hx[1:-1, :-1]) / dy
ez[1:-1, 1:-1] += (dt / eps0) * curl_h

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

重点调用方式

update Hx and Hy from Ez
update Ez from Hx and Hy
add center Ez source and Mur boundaries
save |Ez| image

结果和图像

这张图显示单个输出时刻的 |Ez| 幅值。读图时先看源点附近是否形成平滑波前，再看波前是否大体向外传播，最后看外边界附近是否出现明显反射条纹。由于画的是幅值而不是 signed Ez，相位正负被隐藏；它适合做形态检查，不适合当作误差收敛证据。

../../_images/fdtd_2d_tm_cylindrical_wave.png — 2D TM `|Ez|` 圆形波前图。波前应围绕源位置形成清晰的外传结构。

常见误读

这张图适合发现明显的源项、边界或绘图错误；不要把它理解为 E01 柱坐标 RZ 算例。是否真正通过 Fortran 回归测试，应看 single-step、MMS 和稳定性测试。

Test Goal

case_fdtd_2d_tm_cylindrical_wave is a Python-only 2D TM circular-wavefront visualization case. run.sh runs fdtd_2d_tm_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. The case is for documentation and qualitative inspection, not for strict Fortran routine regression.

Related Algorithm

The interior Yee update is consistent with the 2D TMz reduction of E03 3D Cartesian: set ∂/∂z = 0 and keep only Hx, Hy, Ez 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 TMz routine.

Files

File	Role
`fdtd_2d_tm_cylindrical_wave.py`	Python FDTD demo with a centered sinusoidal source.
`fdtd_2d_tm_cylindrical_wave.png`	Generated `\|Ez\|` 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_tm_cylindrical_wave
bash run.sh

Main Flow

Build a 2D Cartesian x-y Yee grid.
Add a centered sinusoidal Ez source.
Update Hx/Hy and Ez using the 2D TMz reduction.
Apply first-order Mur boundaries and save a |Ez| image.
Visually inspect the outgoing circular wavefront.

Example: Centered Ez Source for a TMz Wavefront

The most useful part of fdtd_2d_tm_cylindrical_wave.py is the field layout and update order. The script uses a 201 x 201 grid, dx=dy=0.054, a two-dimensional CFL time step dt = dx/(c0*sqrt(2)), and a 278 MHz sinusoidal source. The script directly uses an x/y Cartesian Yee grid; cylindrical_wave in the file name describes the circular wavefront shape, not a cylindrical-coordinate r-z discretization.

nx, ny = 201, 201
dx = dy = 0.054
dt = dx / (c0 * np.sqrt(2.0))
nsteps = 190
f0 = 278.0e6

ez = np.zeros((nx, ny), dtype=np.float64)
hx = np.zeros((nx, ny - 1), dtype=np.float64)
hy = np.zeros((nx - 1, ny), dtype=np.float64)

Each step updates Hx/Hy from Ez, updates interior Ez from the magnetic curl, and then adds the centered soft source. The four edges use a first-order Mur absorbing boundary condition. First-order Mur works best for nearly normal plane-wave incidence; oblique components in a cylindrical wave can reflect more strongly. It is sufficient for this compact visual demo, but it is not a substitute for the CPML regression cases.

hx -= (dt / (mu0 * dy)) * (ez[:, 1:] - ez[:, :-1])
hy += (dt / (mu0 * dx)) * (ez[1:, :] - ez[:-1, :])

curl_h = (hy[1:, 1:-1] - hy[:-1, 1:-1]) / dx - \
         (hx[1:-1, 1:] - hx[1:-1, :-1]) / dy
ez[1:-1, 1:-1] += (dt / eps0) * curl_h

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

Core Pattern

update Hx and Hy from Ez
update Ez from Hx and Hy
add center Ez source and Mur boundaries
save |Ez| image

Result and Figure

The figure shows |Ez| at one output time. Read it by checking whether a smooth wavefront forms near the source, whether it propagates outward, and whether obvious reflected stripes appear near the outer edges. Because the plotted quantity is a magnitude, signed phase information is hidden; use this figure for qualitative shape inspection, not convergence evidence.

Common Pitfall

This image can reveal obvious source, boundary, or plotting mistakes. Do not read it as the E01 cylindrical-coordinate RZ case. Formal pass/fail evidence comes from the single-step, MMS, and stability tests.