[rogeriojorge]

Target: main. Head: 30e67c4 (rebased on main ab87385; old head ff6a0a0). No dependencies. Supersedes #24.

Problem

• compute_C_nmp assumed every Maxwellian has exactly its basis width.
• Its factorial formula overflowed to NaN at high order (Nn = 112 and 256 on main).
• A beam expanded in its own narrow basis diverges once it is trapped and spreads.

Change

• Independent widths. compute_C_nmp(..., vth_s=None) takes physical thermal speeds separately from the basis widths alpha_s. The default alpha_s/√2 keeps the previous result.
• Stable recurrence. Coefficients cⁿHₙ(d/c)/√(2ⁿn!), with c² = 1 − 2vₜ²/α², come from a three-term recurrence run as a lax.scan. There are no factorials, and compile time does not depend on Nn.
• Corrected convergence statement.
  • The basis weight is exp(-(v-u)²/α²). A Maxwellian of standard deviation σ has finite weighted norm ∫ f²/F_b iff σ < α, and its normalised coefficients decay like |1 − 2σ²/α²|^(n/2).
  • The matched Maxwellian is σ = α/√2. So Maxwellians broader than matched (α/√2 ≤ σ < α) are admissible. The previous text wrongly said a Maxwellian wider than α/√2 does not converge.
  • This is a statement about this fixed weighted norm and the initial expansion only. It says nothing about other formulations, or about whether a truncated nonlinear run stays resolved.
• Examples/1D_bump_on_tail.py (Nx 33, Nn 128, ν = 3, t = 100; about 9 s on a laptop CPU).
  • The beam is expanded in the bulk basis.
  • The dashed γ = 0.198 line is labelled as linear theory, for orientation only.
  • The docstring says that neither the line nor this single resolution certifies the nonlinear saturation phase.
  • The example prints max|E_k| and min f.

Tests (local, macOS CPU, Python 3.12, jax 0.11.2, at 30e67c4)

python -m pytest -q -n 2 tests: 19 passed.

Tests in tests/test_initialize_maxwellian.py:

• Closed form: the default and explicit alpha_s/√2 reproduce the old formula (rtol 1e-12).
• Narrow beam: a narrow Maxwellian is reconstructed to 1e-12.
• High order: Nn = 256 stays finite.
• New: σ/α = 0.5, 1/√2, 0.85, 0.95 against independent quadrature.
  • Setup: two species, anisotropic α per axis, shifted centres, σ < α on every axis, Nn = 96 × Nm 6 × Np 4.
  • Every C_nmp equals the product of 1D coefficients projected by independent quadrature, to 1e-11 of the maximum.
  • The tail decay rate matches √|1−2σ²/α²| within 10%.
• New: at and beyond the bound.
  • At σ = α, the even coefficients stop converging: Σ|g|² keeps growing.
  • At σ = 1.1α they grow.
  • The recurrence stays finite in both cases.

Example run: python Examples/1D_bump_on_tail.py gives max|E_k| = 0.5298 and min f = −0.019, in 8.5 s wall time. This is a single resolution and not a convergence study.

GitHub CI: see the checks on this head. The local results above are not CI results.

Owner decision

Please personally check the weighted-norm statement and the test fixtures before merging.

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.

Files with missing lines	Coverage Δ
spectrax/_initialize_maxwellian.py	100.00% <100.00%> (+62.96%)	⬆️

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