Tests: Derive analytical formula for particle_boundary_interaction

[RemiLehe]

Summary

This PR replaces hard-coded reference values in two test analysis scripts with Python expressions that derive them analytically from the input parameters (read via parse_input_file from warpx_used_inputs).

test_rz_particle_boundary_interaction_picmi

• The analysis script compared the simulated particle position at the end of the run to three hard-coded reference values, which made it unclear what was being tested.
• The new code derives the expected position analytically: relativistic Lorentz factor, straight-line ray-sphere intersection to find the impact point, mirror reflection of the velocity at that point, and free flight until ts.t[-1].
• The relative tolerance had to be loosened from 1e-5 to 2 %, because the simulation result differs from the analytical (purely geometric) result by about 1.8 % on z. This error is due to the discretization of the EB (a sphere approximated by AMReX's cut-cell approach). This was confirmed by running with twice the resolution in r and z and half the timestep: the dominant z error dropped from 1.79 % to 0.61 %, consistent with first-order convergence of the EB scraping.

test_rz_secondary_ion_emission_picmi

• The analysis script compared emitted-electron positions to hard-coded expected values.
• The new code computes analytically the impact point of each ion on the sphere (same ray-sphere intersection formula as above).
• Each emitted secondary electron is back-propagated from the final diagnostic time to the analytically computed ion impact time using its velocity (read from the diagnostic; openpmd_viewer returns proper velocities in units of c). This removes the stochastic thermal-kick displacement from the comparison, so the check is sensitive only to the EB discretization error.
• Each back-propagated electron is then matched to the closest analytical impact point (rather than by index), because the stochastic thermal kick at emission scrambles the correspondence between emitted electrons and parent ions.
• The tolerance is set to 5 % of the sphere radius R, consistent with the relative-error convention used in the companion test, and comfortably above the observed ~3.6 % maximum.

🤖 Generated with Claude Code