case_fdtd_mms_convergence

测试目标

case_fdtd_mms_convergence 用 manufactured solution 验证 E01/E02/E03 核心 FDTD 内核是否在网格加密时达到接近二阶的误差收敛。MMS 中的源项是为了维持解析解而构造的，不是物理天线或电流源。

覆盖子程序

mod_E01_fdtd_2d_rz_tez 和 mod_E01_fdtd_2d_rz_tmz。
mod_E02_fdtd_3d_cylindrical，覆盖 m=0 和带 phi 变化的 m=1。
mod_E03_fdtd_3d_cartesian。

文件说明

文件	作用
`test_mms_*_convergence.f90`	五个主收敛测试，对不同几何和模式跑三套分辨率。
`mms_exact_sources.f90`	解析场、解析源项和各分量采样位置。
`mms_convergence_utils.f90`	误差范数、收敛阶和 PASS/FAIL 汇总。
`visualize_mms_*.py`	可选后处理脚本，用于画误差或数值/解析对比图。

运行方式

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

Note

这是 005 Maxwell 中较重的全量 MMS 收敛验证。当前开发机上一次完整运行约 15 分钟，主要耗时来自 3D Cartesian、3D cylindrical m=0 和 3D cylindrical m=1。程序运行时会打印类似 progress 100/4000 steps 的进度行；如果这些数字仍在增长，说明细网格层还在正常推进。

主流程

对每个几何选择三套网格，并保持相同物理终止时间和 CFL 比例。
用 mms_exact_sources.f90 初始化 staggered 位置上的解析场。
每步调用生产 FDTD H 更新，再注入对应 MMS 磁场源项。
调用生产 FDTD E 更新，再注入对应 MMS 电场源项。
在更新区间的原生 Yee 位置计算 L2、Linf 和轴线带误差。
用三套分辨率的误差估计 observed order。

典型例子：3D Cartesian MMS 收敛

test_mms_3d_cartesian_convergence.f90 是最标准的 MMS 例子。它使用三套网格 16^3、32^3、64^3，每套网格根据 CFL 重新计算 dt，并保持相同物理终止时间。这样比较误差时，网格间距和时间步长一起减半，二阶空间/时间离散的主误差应接近四倍下降。

integer, parameter :: nx_list(nlev) = (/16, 32, 64/)

dt_levels(ilev) = compute_dt(nx_list(ilev),ny_list(ilev),nz_list(ilev),c0)
call run_level(nx_list(ilev),ny_list(ilev),nz_list(ilev), &
    dt_levels(ilev),nsteps(ilev),omega, ...)

在每个 run_level 中，生产 FDTD 内核只负责 Maxwell curl 更新；MMS 源项由测试文件随后加入，用来让离散解追踪预设解析解。最后 compute_errors 在 Yee 原生采样位置比较数值场和解析场， observed_order 再用相邻两级误差给出收敛阶。

call sub_E03_fdtd_3d_cartesian_H(..., Ex,Ey,Ez,Hx,Hy,Hz,dt,dx,dy,dz,mu)
call add_h_source(nx,ny,nz,dx,dy,dz,dt,t_n,omega0,Hx,Hy,Hz)

call sub_E03_fdtd_3d_cartesian_E(..., Ex,Ey,Ez,Hx,Hy,Hz,dt,dx,dy,dz,ep)
call add_e_source(nx,ny,nz,dx,dy,dz,dt,t_n+0.5*dt,omega0,Ex,Ey,Ez)

call compute_errors(..., Ex,Ey,Ez,Hx,Hy,Hz, l2c, linfc, ...)

重点调用方式

call sub_*_H(...)
call add_h_source(..., t_n, ...)
call sub_*_E(...)
call add_e_source(..., t_n + 0.5*dt, ...)
call compute_error_against_mms_exact(...)

结果判断

代码内置判据：L2 observed order 至少 1.8，Linf observed order 至少 1.5。当前五个 case 均为 PASS；典型摘要：

3D Cartesian periodic MMS: L2 orders 2.017, 2.009, result PASS
2D RZ TMz MMS:             L2 orders 2.014, 2.007, result PASS
3D cylindrical m=0 MMS:    L2 orders 2.015, 2.007, result PASS
2D RZ TEz MMS:             L2 orders 2.016, 2.008, result PASS
3D cylindrical m=1 MMS:    L2 orders 2.019, 1.998, result PASS

常见误读

MMS 检查的是离散算子和源项采样的一致性；它不代表自由空间波包传播，也不测试 CPML。对 RZ/cylindrical，误差范数带有 r 权重，轴线附近还会单独报告 axis_band_Linf。

Test Goal

case_fdtd_mms_convergence uses manufactured solutions to verify that the E01/E02/E03 core FDTD kernels approach second-order convergence under mesh refinement. The MMS source terms are constructed to sustain the analytic solution; they are not physical antennas or current sources.

Covered Routines

mod_E01_fdtd_2d_rz_tez and mod_E01_fdtd_2d_rz_tmz.
mod_E02_fdtd_3d_cylindrical for m=0 and m=1 with phi variation.
mod_E03_fdtd_3d_cartesian.

Files

File	Role
`test_mms_*_convergence.f90`	Five convergence drivers, each using three resolutions.
`mms_exact_sources.f90`	Analytic fields, analytic sources, and component sampling locations.
`mms_convergence_utils.f90`	Error norms, observed orders, and PASS/FAIL helpers.
`visualize_mms_*.py`	Optional postprocessing for error or numerical/exact comparison plots.

Run Command

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

Note

This is a relatively heavy full MMS convergence check in the 005 Maxwell test set. On the current development machine, a complete run takes about 15 minutes. Most of the time is spent in 3D Cartesian, 3D cylindrical m=0, and 3D cylindrical m=1. During execution, the drivers print lines such as progress 100/4000 steps; if these numbers continue to increase, the fine-grid level is still advancing normally.

Main Flow

Choose three grids for each geometry with the same physical end time and CFL scale.
Initialize analytic fields at native staggered locations.
Call the production H update, then inject the MMS magnetic source.
Call the production E update, then inject the MMS electric source.
Compute L2, Linf, and axis-band errors on active update regions.
Estimate observed order from the three resolutions.

Example: 3D Cartesian MMS Convergence

test_mms_3d_cartesian_convergence.f90 is the most standard MMS example. It uses three grids, 16^3, 32^3, and 64^3. Each grid recomputes dt from the CFL condition while keeping the same physical end time. With both spacing and time step halved, a second-order discretization should show roughly a fourfold error reduction.

integer, parameter :: nx_list(nlev) = (/16, 32, 64/)

dt_levels(ilev) = compute_dt(nx_list(ilev),ny_list(ilev),nz_list(ilev),c0)
call run_level(nx_list(ilev),ny_list(ilev),nz_list(ilev), &
    dt_levels(ilev),nsteps(ilev),omega, ...)

Inside each run_level, the production FDTD kernels perform only Maxwell curl updates. The test then injects MMS source terms so the solution tracks a known analytic field. compute_errors compares numerical and exact fields at native Yee locations, and observed_order converts adjacent grid errors into convergence rates.

call sub_E03_fdtd_3d_cartesian_H(..., Ex,Ey,Ez,Hx,Hy,Hz,dt,dx,dy,dz,mu)
call add_h_source(nx,ny,nz,dx,dy,dz,dt,t_n,omega0,Hx,Hy,Hz)

call sub_E03_fdtd_3d_cartesian_E(..., Ex,Ey,Ez,Hx,Hy,Hz,dt,dx,dy,dz,ep)
call add_e_source(nx,ny,nz,dx,dy,dz,dt,t_n+0.5*dt,omega0,Ex,Ey,Ez)

call compute_errors(..., Ex,Ey,Ez,Hx,Hy,Hz, l2c, linfc, ...)

Core Call Pattern

call sub_*_H(...)
call add_h_source(..., t_n, ...)
call sub_*_E(...)
call add_e_source(..., t_n + 0.5*dt, ...)
call compute_error_against_mms_exact(...)

Result Interpretation

Built-in criteria require L2 observed order at least 1.8 and Linf observed order at least 1.5. Current reference runs pass:

3D Cartesian periodic MMS: L2 orders 2.017, 2.009, result PASS
2D RZ TMz MMS:             L2 orders 2.014, 2.007, result PASS
3D cylindrical m=0 MMS:    L2 orders 2.015, 2.007, result PASS
2D RZ TEz MMS:             L2 orders 2.016, 2.008, result PASS
3D cylindrical m=1 MMS:    L2 orders 2.019, 1.998, result PASS

Common Pitfall

MMS checks operator/source sampling consistency; it is not a free-space wave-packet or CPML test. RZ/cylindrical norms use r weighting and also report axis_band_Linf near the axis.