Physics Validation Tests¶

Running the Tests¶

Tests are run with bash run_physics_validation.slm (or sbatch on HPC); use --particle-pusher, --field-solver, --particle-mesh-coupling, or --all to choose groups. On first run the script creates a local .venv in this directory and installs dependencies from requirements.txt (numpy, openpmd-api); later runs reuse that venv. The HEXAPIC executable needs libopenPMD.so at runtime — the script sets LD_LIBRARY_PATH from OPENPMD_ROOT (default ../openpmd relative to the repo); override with export OPENPMD_ROOT=/path/to/openpmd if your openPMD install is elsewhere.

I. Particle Pusher Verification¶

These tests verify the Boris integrator and 3V vector logic. Maxwell solver and Deposition are disabled (FieldSolverFlag = 0).

Threshold framework: All thresholds are adaptive, computed as

\[ \text{threshold} = K \times (\text{physics\_err} + \text{float\_err}) \]

where \(K=10\) is a safety factor. Physics error captures Boris integrator \(O(\Delta t^2)\) phase drift, and float error captures \(O(\epsilon_m \sqrt{N})\) accumulation over \(N\) output steps (\(\epsilon_m \approx 2.22 \times 10^{-16}\)).

1) Multi-Species Larmor Orbit (In-Plane)¶

Description: Verifies basic cyclotron rotation in the simulation plane (x, y) and charge sign handling using an electron–positron pair.

Field E: Disabled
Field B: \(B=(0,\,0,\,0.1)\) T
Particles:
- Species 1 (e−): \(q=-e,\; m=m_e,\; v_0=(10^6,\; -8794,\; 0)\) m/s
- Species 2 (e+): \(q=+e,\; m=m_e,\; v_0=(10^6,\; +8794,\; 0)\) m/s
(The non-zero \(v_y\) is the Boris half-step back correction: \(v_y \approx \mp v_\perp \omega_c \Delta t / 2\).)
Simulation Time: \(\Delta t = 10^{-12}\) s, \(T_{\text{total}}= 3 \times T_{\text{Larmor}}\). Only the last 1/3 of output steps is analyzed (initial transient skipped).
Metrics & Pass Criteria:
1. Energy Conservation (Absolute):
  
  \[ \epsilon_E = \max|E_k(t)-E_k(0)|, \quad E_k = \tfrac{1}{2}m(v_x^2+v_y^2+v_z^2) \]
  
  Pass if: \(\epsilon_E < K \cdot \epsilon_m \cdot \sqrt{N}\)
2. 4-Radius Separation:
  
  \[ \Delta_{\text{sep}} = \max_t \left|\mathbf{x}_{e^-}(t) - \mathbf{x}_{e^+}(t)\right| \]
  
  Expected \(4 R_L\) where \(R_L = \dfrac{m\, v_\perp}{|q|\, B_0}\).
  
  Pass if:
  
  \[ |\Delta_{\text{sep}} - 4 R_L| < K \cdot \!\left(4 R_L \cdot \frac{20(\omega_c \Delta t)^2}{12} + 4 R_L \cdot \epsilon_m \sqrt{N}\right) \]
3. Y-Projection Trajectory:
  
  \[ \text{Extent} = \bigl(\max(y) - \min(y)\bigr) \cdot \Delta y \]
  
  Expected \(2 R_L\).
  
  Pass if:
  
  \[ |\text{extent} - 2 R_L| < K \cdot \!\left(2 R_L \cdot \frac{10(\omega_c \Delta t)^2}{12} + 2 R_L \cdot \epsilon_m \sqrt{N}\right) \]
4. Z-Axis Stability:
  
  \[ \Delta z = \max|z(t)-z(0)| \]
  
  Pass if: \(\Delta z < K \cdot R_L \cdot \epsilon_m \cdot \sqrt{N}\)

2) Multi-Species Larmor Orbit (Out-of-Plane / 3V)¶

Description: Verifies the coupling between the spatial grid (x) and the virtual velocity component (\(v_z\)) using a \(B_y\) field. Only the electron trajectory is analyzed.

Field E: Disabled
Field B: \(B=(0,\,0.1,\,0)\) T
Particles:
- Species 1 (e−) & Species 2 (e+): \(v_0=(0,\;0,\;10^6)\) m/s
Simulation Time: \(\Delta t = 10^{-12}\) s, \(T_{\text{total}}= 3 \times T_{\text{Larmor}}\). Only the last 1/3 of output steps is analyzed.
Metrics & Pass Criteria:
1. Energy Conservation (Absolute):
  
  \[ \epsilon_E = \max|E_k(t)-E_k(0)| \]
  
  Pass if: \(\epsilon_E < K \cdot \epsilon_m \cdot \sqrt{N}\)
2. X-Projection Trajectory:
  
  \[ \text{Extent} = \bigl(\max(x) - \min(x)\bigr) \cdot \Delta x \]
  
  Expected \(2 R_L\).
  
  Pass if:
  
  \[ |\text{extent} - 2 R_L| < K \cdot \!\left(2 R_L \cdot \frac{10(\omega_c \Delta t)^2}{12} + 2 R_L \cdot \epsilon_m \sqrt{N}\right) \]
3. Y-Axis Stability:
  
  \[ \Delta y = \max|y(t)-y(0)| \]
  
  Pass if: \(\Delta y < K \cdot R_L \cdot \epsilon_m \cdot \sqrt{N}\)

3) E×B Drift¶

Description: Verifies the Lorentz force combination by creating a drift along the Y-axis via crossed E and B fields.

Field E: From boundary potentials: \(V=0\) at \(x=0\), \(V=100\) V at \(x=L_x=0.01\) m. Gives \(E_x = -10^4\) V/m.
Field B: \(B=(0,\,0,\,0.1)\) T
Particles:
- Species 1 (e−): \(v_0=(0,\; 10^5,\; 0)\) m/s. Placed at domain center.
Simulation Time: \(\Delta t = 10^{-11}\) s, 400 steps, output every 20 steps.
Metrics & Pass Criteria:
1. Hamiltonian Conservation:
  
  \[ H = \tfrac{1}{2}m|\mathbf{v}|^2 + q\phi \]
  
  where \(\phi = -E_x \cdot x\).
  
  Pass if:
  
  \[ \max|H(t) - H(0)| < K \cdot \!\left(|H_0| \cdot \frac{(\omega_c \Delta t)^2}{8} + |H_0| \cdot \epsilon_m \sqrt{N}\right) \]
2. Drift Velocity (Y): Mean \(v_y\) over the second half of the simulation, expected \(v_{\text{drift}} = -E_x / B_z\).
  
  Pass if:
  
  \[ |\bar{v}_y - v_{\text{drift}}| < K \cdot \!\left(|v_{\text{drift}}| \cdot \frac{(\omega_c \Delta t)^2}{8} + |v_{\text{drift}}| \cdot \epsilon_m \sqrt{N}\right) \]

II. Field Solver Verification¶

These tests verify the Poisson solver logic

\[ \nabla^2\phi = -\frac{\rho}{\epsilon_0} \]

and boundary condition implementation. The solver uses an iterative method with target relative residual \(\text{rtol} \sim 10^{-6}\).

Threshold framework: Thresholds combine solver residual error (\(O(\text{rtol})\)), floating point accumulation (\(O(\epsilon_m \sqrt{N_{\text{cells}}})\)), and grid truncation error (\(O(1/N)\)), scaled by safety factor \(K=10\).

4) Mixed Boundaries (Capacitor / Dirichlet-Neumann)¶

Description: Verifies the transition between fixed potential (electrodes) and open space (dielectric). Simulates a finite parallel-plate capacitor.

Domain: \(32 \times 32\) grid, \(L = 0.001\) m
Boundary Conditions:
- Y-Axis (Electrodes): \(\phi(x, 0) = -10\) V, \(\phi(x, L_y) = +10\) V (Dirichlet)
- X-Axis (Open): \(\dfrac{\partial \phi}{\partial x} = 0\) at \(x=0,\, x=L_x\) (Neumann / Dielectric)
Source: No particles (vacuum solve)
Metrics & Pass Criteria:
1. Boundary Value Integrity:
  
  \[ \epsilon_{\text{BC}} = \max\!\bigl(|\phi_{y=0} - (-10)|,\; |\phi_{y=L_y} - 10|\bigr) \]
  
  Pass if: \(\epsilon_{\text{BC}} < K \cdot \epsilon_m \cdot V_0\)
2. Field Uniformity (Interior): Standard deviation of \(E_y\) in the interior (excluding 2-cell border near Neumann/Dirichlet corners).
  
  Pass if:
  
  \[ \sigma(E_y) < K \cdot \!\left(|\bar{E}_y| \cdot \text{rtol} + |\bar{E}_y| \cdot \epsilon_m \sqrt{N_{\text{cells}}}\right) \]

5) Fully Open Boundaries (Isolated System)¶

Description: Verifies solver stability for a floating potential system using all-Neumann boundaries. Uses a neutral dipole to ensure \(\int \rho \, dV = 0\).

Domain: \(32 \times 32\) grid, \(L = 0.001\) m
Boundary Conditions:
- All boundaries: \(\dfrac{\partial \phi}{\partial n} = 0\) (Neumann everywhere)
Source: Neutral Dipole — electron cloud at \(x \approx 0.35\, L\), positron cloud at \(x \approx 0.65\, L\).
Metrics & Pass Criteria:
1. Boundary Flux Leakage: Maximum of mean \(|E_n|\) across all four boundaries.
  
  Pass if:
  
  \[ \max\!\bigl(\overline{|E_x|}_{\text{walls}},\; \overline{|E_y|}_{\text{walls}}\bigr) < K \cdot \!\left(E_{\text{scale}} \cdot \text{rtol} + E_{\text{scale}} \cdot \epsilon_m \sqrt{N} + \frac{E_{\text{scale}}}{N_{\min}}\right) \]
  
  where \(E_{\text{scale}} = \max|\mathbf{E}|\) over the domain, \(N_{\min} = \min(N_x, N_y)\).

6) Closed Metal Box (All-Dirichlet)¶

Description: Verifies the basic "grounded box" scenario. Tests solver matrix construction and solution symmetry.

Domain: \(32 \times 32\) grid, \(L = 0.001\) m
Boundary Conditions:
- All boundaries: \(\phi = 0\) (Ground)
Source: Electron charge cloud centered at \((L_x/2,\, L_y/2)\).
Metrics & Pass Criteria:
1. Boundary Zero Check:
  
  \[ \epsilon_{\text{wall}} = \max_{\text{all boundaries}} |\phi| \]
  
  Pass if: \(\epsilon_{\text{wall}} < K \cdot \epsilon_m \cdot \max(|\phi_{\text{boundary}}|, 1)\)
2. Symmetry:
  
  \[ \epsilon_{\text{sym}} = P_{99.9}\!\bigl(|\phi(x,y) - \phi(L_x-x,\, L_y-y)|\bigr) \]
  
  over interior points (excluding 2-cell border and 3-cell radius around center singularity).
  
  Pass if:
  
  \[ \epsilon_{\text{sym}} < K \cdot \!\left(V_{\text{scale}} \cdot \text{rtol} + V_{\text{scale}} \cdot \epsilon_m \sqrt{N} + \frac{V_{\text{scale}}}{N^{1.5}}\right) \]

7) Periodic Boundaries¶

Description: Verifies matrix topology connectivity. Signals leaving the right boundary must wrap to the left boundary. Periodic in X direction with Neumann in Y.

Domain: \(32 \times 32\) grid, \(L = 0.001\) m
Boundary Conditions:
- X-Axis: \(\phi(0, y) = \phi(L_x, y)\) (Periodic, PeriodicFlagX1 = 1)
- Y-Axis: \(\dfrac{\partial \phi}{\partial y} = 0\) (Neumann)
Source: Neutral dipole — electron and positron clouds at symmetric x-positions.
Metrics & Pass Criteria:
1. Periodicity (Derivative Continuity): Checks that the first derivative of \(\phi\) is continuous across the periodic boundary (left–right wrap-around). Automatically detects whether the solver duplicates the endpoint (\(\phi[0] = \phi[N_x]\)) or not.
  
  Pass if:
  
  \[ \text{derivative jump} < K \cdot \!\left(\text{max internal curvature} + V_{\text{range}} \cdot \text{rtol} + V_{\text{range}} \cdot \epsilon_m \sqrt{N}\right) \]

8) Analytical Accuracy (Green's Function)¶

Description: Quantifies numerical error by comparing the grid solution to the analytical \(\ln(r)\) profile for a 2D point charge.

Domain: \(64 \times 64\) grid, \(L = 0.001\) m
Boundary Conditions: All Dirichlet \(\phi = 0\).
Source: Electron charge cloud at center \((L_x/2,\, L_y/2)\).
Comparison: Fit \(\phi_{\text{num}}(r)\) to \(A \ln(r) + B\) via linear regression in the annular region \(3\Delta x < r < 0.35 \min(L_x, L_y)\).
Metrics & Pass Criteria:
1. Analytical Fit (\(1 - R^2\)): Coefficient of determination for \(\phi\) vs \(\ln(r)\) regression. Measures how well the numerical potential follows the expected logarithmic profile.
  
  Pass if:
  
  \[ 1 - R^2 < K \cdot \!\left((\Delta x / r_{\min})^2 + 100 \cdot \epsilon_m\right) \]

III. Particle-Mesh Coupling Verification¶

These tests verify the projection of particle quantities to the grid (Deposition) and the sampling of grid fields to the particle (Interpolation/Gathering).

Threshold framework: All thresholds are adaptive, computed as

\[ \text{threshold} = K \times (\text{physics\_err} + \text{float\_err}) \]

where \(K=10\) is a safety factor. Physics error captures solver residual (\(O(\text{rtol})\)), discretization (\(O(\Delta x^2)\)), and CIC shape-function effects. Float error captures \(O(\epsilon_m \sqrt{N})\) accumulation (\(\epsilon_m \approx 2.22 \times 10^{-16}\)). Grid charge is inferred via Gauss's law:

\[ Q_{\text{grid}} = -\epsilon_0 \sum_{\text{interior}} (\nabla^2 \phi)_{i,j}\, \Delta x \, \Delta y \]

9) Charge Deposition & Potential Topology¶

Description: Verifies that deposited charge creates a quantitatively correct and symmetric potential well. Ensures coupling between CIC particle weighting and the Poisson solver, and that physical constants (\(\epsilon_0\)) are correctly applied.

Domain: \(32 \times 32\) grid, \(L = 0.001\) m, All-Dirichlet (\(\phi = 0\)).
Source: Charge cloud at center \((L_x/2,\, L_y/2)\). Total charge \(Q_{\text{total}} = -q_e \cdot n \cdot \Delta x \cdot \Delta y\) where \(n = 2.1 \times 10^{15}\) m\(^{-3}\) (from static_deposit.inp).
Simulation: 5 steps, \(\Delta t = 10^{-12}\) s. Field read from final iteration.
Metrics & Pass Criteria:
1. Integrated Charge Conservation:
  
  \[ \epsilon_Q = |Q_{\text{grid}} - Q_{\text{total}}| \]
  
  Pass if:
  
  \[ \epsilon_Q < K \cdot |Q_{\text{total}}| \cdot \!\left(\text{rtol} + \frac{1}{N} + \epsilon_m \sqrt{N_{\text{cells}}}\right) \]
  
  (Combines solver residual, \(O(1/N)\) Laplacian discretization error, and float accumulation.)
2. Symmetry:
  
  \[ \epsilon_{\text{sym}} = P_{99.9}\!\bigl(|\phi(x,y) - \phi(L_x-x,\, L_y-y)|\bigr) \]
  
  over interior points (excluding 2-cell border and 3-cell radius around center singularity).
  
  Pass if:
  
  \[ \epsilon_{\text{sym}} < K \cdot V_{\text{scale}} \cdot \!\left(\text{rtol} + \epsilon_m \sqrt{N_{\text{pts}}} + \frac{1}{N_{\text{grid}}}\right) \]
  
  (Combines solver residual, float accumulation, and \(O(1/N)\) grid discretization asymmetry from CIC and Laplacian.)
3. L2 Profile Error: Relative RMS between simulated \(\phi\) and analytic Green's function \(\phi_{\text{ana}}\) (2D Fourier series for point charge in grounded box, 50 modes).
  
  Pass if:
  
  \[ \sqrt{\frac{\sum (\phi - \phi_{\text{ana}})^2}{\sum \phi_{\text{ana}}^2}} < K \cdot \!\left(\text{rtol} + \frac{1}{N_{\text{grid}}} + \epsilon_m \sqrt{N_{\text{pts}}}\right) \]
  
  (Combines solver residual, \(O(1/N)\) Fourier truncation and grid discretization error, and float accumulation.)

10) Field Interpolation (Acceleration)¶

Description: Verifies that a particle in a uniform E-field experiences the correct Lorentz acceleration \(a = qE/m\) by measuring velocity change over multiple timesteps.

Domain: \(32 \times 32\) grid, \(L_y\) from input file (default \(L = 0.001\) m).
Boundary Conditions:
- Y-Axis (Electrodes): \(\phi(x, 0) = -10\) V, \(\phi(x, L_y) = +10\) V (Dirichlet)
- X-Axis (Open): \(\dfrac{\partial \phi}{\partial x} = 0\) (Neumann / Dielectric)
Particle: Single electron (\(q = -e\), \(m = m_e\)) at center, \(v_0 = 0\).
Simulation: 20 steps, \(\Delta t = 10^{-12}\) s, output every step.
Metrics & Pass Criteria:
1. Acceleration Accuracy: Linear fit \(v_y(t) = a_{\text{meas}} \cdot t + v_0\) gives measured acceleration.
  
  Expected: \(a_{\text{expected}} = \dfrac{q\, E_y}{m}\) where \(E_y = -(V_{\text{top}} - V_{\text{bot}}) / L_y\).
  
  Pass if:
  
  \[ |a_{\text{meas}} - a_{\text{expected}}| < K \cdot |a_{\text{expected}}| \cdot \!\left(\text{rtol} + \frac{1}{N_{\text{grid}}} + \epsilon_m \sqrt{N_{\text{pts}}}\right) \]
  
  (Combines solver residual, \(O(1/N)\) grid error from CIC and finite-difference E, and float accumulation in the linear fit.)

11) Self-Force Stability (Stationary Particle)¶

Description: Integration test for the full PIC loop: Deposit \(\rho\) → Solve Poisson → Interpolate \(\mathbf{E}\) → Push. A stationary particle at the center of a symmetric grounded box should feel near-zero net force; any displacement is parasitic (self-force).

Domain: \(32 \times 32\) grid, \(L = 0.001\) m, All-Dirichlet (\(\phi = 0\)).
Particle: Single electron at rest (\(v_0 = 0\)) centered at \((L_x/2,\, L_y/2)\).
Simulation: 20 steps, \(\Delta t = 10^{-12}\) s, output every step.
Metrics & Pass Criteria:
1. Parasitic Displacement:
  
  \[ \Delta r = \sqrt{(x_{\text{final}} - x_0)^2 + (y_{\text{final}} - y_0)^2} \]
  
  Pass if:
  
  \[ \Delta r < K \cdot \!\left(N_{\text{steps}} \cdot \Delta x + \epsilon_m \cdot \Delta x \cdot \sqrt{N_{\text{steps}}}\right) \]
  
  (PIC codes exhibit parasitic motion from CIC shape asymmetry, field gradient noise, and pusher-solver coupling. Threshold is grid-based: displacement bounded by \(O(N \cdot \Delta x)\) over the simulation.)

12) Current Deposition (Charge Conservation)¶

Description: Verifies that moving a particle and re-depositing charge conserves total charge on the grid. This ensures Gauss's law (\(\nabla \cdot \mathbf{E} = \rho / \epsilon_0\)) holds at every timestep, critical for Esirkepov-type current deposition schemes.

Domain: \(32 \times 32\) grid, \(L = 0.001\) m, All-Dirichlet (\(\phi = 0\)).
Particle: Single electron with \(v_x = 10^4\) m/s (traverses \({\sim}\,0.2\) cells over 10 steps).
Simulation: 10 steps, \(\Delta t = 10^{-12}\) s, output every step. Poisson solver active (FieldSolverFlag = 1).
Metrics & Pass Criteria:
1. Charge Conservation (Gauss):
  
  \[ \Delta Q = |Q_{\text{grid}}(t_{\text{final}}) - Q_{\text{grid}}(t_0)| \]
  
  where \(Q_{\text{grid}}\) is computed via discrete Laplacian of \(\phi\).
  
  Pass if:
  
  \[ \Delta Q < K \cdot |Q| \cdot \!\left(\text{rtol} + \epsilon_m \sqrt{N_{\text{cells}}} + \frac{1}{N}\right) \]
  
  (Combines solver residual at two iterations, float accumulation, and \(O(1/N)\) CIC spreading discretization error.)

Output¶

ROOT_DIR: /xxxxx/hexapic
HEXAPIC:  /xxxxx/hexapic/source/HEXAPIC
NPROCS:   1

=======================================================================================================
                                    HEXAPIC PHYSICS VALIDATION SUITE                                   
                 Date: 2026-02-18 14:25 | Host: xxxxxx | Procs: 1 | Commit: 8fc15a8                 
=======================================================================================================

-------------------------------------------------------------------------------------------------------
                                         PARTICLE PUSHER TESTS                                         
-------------------------------------------------------------------------------------------------------

[TEST 01] In-Plane Larmor Orbit
   Context: B=0.1T, v=1e+06 m/s, dt=1ps, Steps=1071
   ----------------------------------------------------------------------------------------------------
   METRIC                              MEASURED        EXPECTED         ERROR       MAX ERR   STATUS
   ----------------------------------------------------------------------------------------------------
   Energy Conservation                 1.16e-33        0.00e+00      1.16e-33      4.20e-14   [PASS]
   4R Separation                     2.2802e-04      2.2743e-04    5.9871e-07    1.1725e-06   [PASS]
   Y-projection trajectory           1.1373e-04      1.1371e-04    1.9311e-08    2.9314e-07   [PASS]
   Z-Axis Drift                        0.00e+00        0.00e+00      0.00e+00      2.39e-18   [PASS]
   ----------------------------------------------------------------------------------------------------
   Result: OK  Time: 7.62s

[TEST 02] 3V Larmor Orbit
   Context: B_y=0.1T, v_z=1e+06 m/s, 3D Rotation Check
   ----------------------------------------------------------------------------------------------------
   METRIC                              MEASURED        EXPECTED         ERROR       MAX ERR   STATUS
   ----------------------------------------------------------------------------------------------------
   Energy Conservation                 6.74e-34        0.00e+00      6.74e-34      1.88e-14   [PASS]
   X-Projection Trajectory           1.1370e-04      1.1371e-04    1.0182e-08    2.9314e-07   [PASS]
   Y-Axis Drift                        0.00e+00        0.00e+00      0.00e+00      1.07e-18   [PASS]
   ----------------------------------------------------------------------------------------------------
   Result: OK  Time: 2.64s

[TEST 03] ExB Drift
   Context: E=-1e+04 V/m, B=0.1T, Expected v_drift=1e+05 m/s
   ----------------------------------------------------------------------------------------------------
   METRIC                              MEASURED        EXPECTED         ERROR       MAX ERR   STATUS
   ----------------------------------------------------------------------------------------------------
   Hamiltonian Conservation            2.10e-22        0.00e+00      2.10e-22      1.87e-19   [PASS]
   Drift Velocity                    1.0023e+05      1.0000e+05    2.3437e+02    3.8668e+03   [PASS]
   ----------------------------------------------------------------------------------------------------
   Result: OK  Time: 2.29s

-------------------------------------------------------------------------------------------------------
                                           FIELD SOLVER TESTS                                          
-------------------------------------------------------------------------------------------------------

[TEST 04] Mixed Boundaries
   Context: Dirichlet Y, Neumann X
   ----------------------------------------------------------------------------------------------------
   METRIC                              MEASURED        EXPECTED         ERROR       MAX ERR   STATUS
   ----------------------------------------------------------------------------------------------------
   Boundary Integrity                  0.00e+00        0.00e+00      0.00e+00      2.22e-14   [PASS]
   Field Uniformity                  3.9426e-02        0.00e+00    3.9426e-02    2.0000e-01   [PASS]
   ----------------------------------------------------------------------------------------------------
   Result: OK  Time: 0.01s

[TEST 05] Fully Open Boundaries
   Context: All Neumann, Neutral Dipole
   ----------------------------------------------------------------------------------------------------
   METRIC                              MEASURED        EXPECTED         ERROR       MAX ERR   STATUS
   ----------------------------------------------------------------------------------------------------
   Boundary Flux Leakage             4.4524e+02        0.00e+00    4.4524e+02    1.5967e+04   [PASS]
   ----------------------------------------------------------------------------------------------------
   Result: OK  Time: 0.01s

[TEST 06] Closed Metal Box
   Context: All phi=0, Centered Charge
   ----------------------------------------------------------------------------------------------------
   METRIC                              MEASURED        EXPECTED         ERROR       MAX ERR   STATUS
   ----------------------------------------------------------------------------------------------------
   Boundary Zero Check                 0.00e+00        0.00e+00      0.00e+00      2.22e-15   [PASS]
   Symmetry Error                    1.8462e-02        0.00e+00    1.8462e-02    6.0076e-01   [PASS]
   ----------------------------------------------------------------------------------------------------
   Result: OK  Time: 0.02s

[TEST 07] Periodic X Boundaries
   Context: Periodic X, Neumann Y
   ----------------------------------------------------------------------------------------------------
   METRIC                              MEASURED        EXPECTED         ERROR       MAX ERR   STATUS
   ----------------------------------------------------------------------------------------------------
   Periodicity                       1.2711e+02        0.00e+00    1.2711e+02    6.3732e+02   [PASS]
   ----------------------------------------------------------------------------------------------------
   Result: OK  Time: 0.01s

[TEST 08] Analytical Accuracy
   Context: Line Charge vs ln
   ----------------------------------------------------------------------------------------------------
   METRIC                              MEASURED        EXPECTED         ERROR       MAX ERR   STATUS
   ----------------------------------------------------------------------------------------------------
   Analytical Fit                    2.0674e-04        0.00e+00    2.0674e-04    1.1111e+00   [PASS]
   ----------------------------------------------------------------------------------------------------
   Result: OK  Time: 0.01s

-------------------------------------------------------------------------------------------------------
                                      PARTICLE MESH COUPLING TESTS                                     
-------------------------------------------------------------------------------------------------------

[TEST 09] Charge Deposition & Potential Topology
   Context: 32x32, L=0.001 m, grounded box, Q_total=-3.29e-13 C
   ----------------------------------------------------------------------------------------------------
   METRIC                              MEASURED        EXPECTED         ERROR       MAX ERR   STATUS
   ----------------------------------------------------------------------------------------------------
   Integrated Charge Conserv...        8.18e-15        0.00e+00      8.18e-15      9.96e-14   [PASS]
   Symmetry                          1.2351e-03        0.00e+00    1.2351e-03    3.2676e-03   [PASS]
   L2 Profile Error                  4.3836e-02        0.00e+00    4.3836e-02    3.0304e-01   [PASS]
   ----------------------------------------------------------------------------------------------------
   Result: OK  Time: 2.11s

[TEST 10] Field Interpolation
   Context: Capacitor dV=20V, L=0.001m
   ----------------------------------------------------------------------------------------------------
   METRIC                              MEASURED        EXPECTED         ERROR       MAX ERR   STATUS
   ----------------------------------------------------------------------------------------------------
   Acceleration Accuracy             4.3808e+13        0.00e+00    4.3808e+13    1.0993e+15   [PASS]
   ----------------------------------------------------------------------------------------------------
   Result: OK  Time: 2.06s

[TEST 11] Self-Force Stability
   Context: Stationary particle, Centered
   ----------------------------------------------------------------------------------------------------
   METRIC                              MEASURED        EXPECTED         ERROR       MAX ERR   STATUS
   ----------------------------------------------------------------------------------------------------
   Parasitic Displacement            1.6419e-04        0.00e+00    1.6419e-04    6.2500e-03   [PASS]
   ----------------------------------------------------------------------------------------------------
   Result: OK  Time: 2.06s

[TEST 12] Current Deposition
   Context: Moving particle, Gauss law charge check
   ----------------------------------------------------------------------------------------------------
   METRIC                              MEASURED        EXPECTED         ERROR       MAX ERR   STATUS
   ----------------------------------------------------------------------------------------------------
   Charge Conservation                 3.02e-17        0.00e+00      3.02e-17      9.71e-14   [PASS]
   ----------------------------------------------------------------------------------------------------
   Result: OK  Time: 2.04s

=======================================================================================================
                                                SUMMARY                                                
=======================================================================================================
PARTICLE PUSHER
  1. In-Plane Larmor Orbit               [PASS]
  2. 3V Larmor Orbit                     [PASS]
  3. ExB Drift                           [PASS]

FIELD SOLVER
  4. Mixed Boundaries                    [PASS]
  5. Fully Open Boundaries               [PASS]
  6. Closed Metal Box                    [PASS]
  7. Periodic X Boundaries               [PASS]
  8. Analytical Accuracy                 [PASS]

PARTICLE MESH COUPLING
  9. Charge Deposition & Potential ...   [PASS]
  10. Field Interpolation                 [PASS]
  11. Self-Force Stability                [PASS]
  12. Current Deposition                  [PASS]

-------------------------------------------------------------------------------------------------------
OVERALL: PASSED (12/12 tests passed, 22/22 metrics)
Total Runtime: 20.89s
=======================================================================================================