feat: mixed Hermite-Legendre 1D Vlasov-Poisson solver (arXiv:2606.12322)

[joglekara]

Summary

New ADEPTModule (hermite-legendre-1d) implementing the mixed Hermite–Legendre spectral method of Issan, Delzanno & Roytershteyn (arXiv:2606.12322) for the 1D-1V electrostatic Vlasov–Poisson system.

The electron distribution is split f = f0 + δf:

• f0 (near-Maxwellian bulk) → asymmetrically-weighted (AW) Hermite expansion in velocity, coeffs C_n(x,t)
• δf (strongly non-Maxwellian features: beams, plateaus, trapping structure) → Legendre expansion on a bounded velocity window [v_a, v_b], coeffs B_m(x,t)

The highest Hermite coefficient C_{Nh-1} feeds the Legendre modes (one-way), and both feed the field through Poisson. At fixed total velocity DOFs the mixed method is more accurate than pure Hermite or pure Legendre when non-Maxwellian features are localized in velocity.

Structurally modeled on BaseVlasov1D; reuses the Lawson-RK4 design of the sibling _hermite_poisson_1d.

Design notes

• Three integrators, config-selected: lawson (explicit Lawson-RK4; both free-streaming operators are symmetric-tridiagonal so streaming+collisions are integrated exactly via prediagonalized matrix exponentials), imex (Lawson + frozen-E backward-Euler Lorentz substep; lifts the ~Nl²/width·|E| force CFL), and implicit (implicit midpoint via AD-JFNK: matrix-free GMRES with exact autodiff JVPs; A-stable, conserves mass exactly and energy to solve tolerance — required for saturated/long-time regimes, e.g. bump-on-tail, where the explicit and IMEX paths blow up).
• JFNK preconditioning: streaming+collision factor only (per-k tridiagonal, exact for the stiff skew streaming spectrum). Preconditioning the Lorentz-force block stalls GMRES at saturation-scale fields — the force factors are strongly non-normal nilpotent ladders; measured and documented in config.md. At dt ≳ 0.05 with large |E|, unpreconditioned is the robust choice.
• Choosing Nh (documented in config.md): keep the Hermite basis bulk-only. At large Nh a spurious k=0 mid-n velocity-space cascade (nonlinear force-ladder pump vs the weak mid-n window of the cubic collision spectrum) destroys post-saturation runs at any practical ν_H; small Nh closes the window structurally (bump-on-tail: Nh=32 matches Nh=128 field observables to 3 digits).
• External longitudinal driver (drivers.ex, vlasov1d conventions) for driven-wave studies; never enters the Poisson solve.
• Space is spectral (Fourier, periodic); conservation constraint J_{Nh,0..2}=0 (paper §3.4/§4) and Lenard-Bernstein artificial collisions (cubic spectrum, zero on the first three moments) implemented and config-exposed.

What's included

• adept/_hermite_legendre_1d/{vector_field,modules,storage}.py + public entry; registered in _base_ dispatch and adept.__init__
• Configs: configs/hermite-legendre-1d/{linear-advection,two-stream,two-stream-imex,bump-on-tail}.yaml
• Docs: docs/source/solvers/hermite_legendre_1d/config.md + quick links
• CI: test-hermite-legendre-1d job (plus the previously-missing test-hermite-poisson-1d job from feat: native 1D Hermite-Poisson solver (_hermite_poisson_1d) #287)

Physics validation

• Linear advection (§4.2): reconstructed f = f0 + δf matches analytic f(x−vt, v, 0) to <2% L2 before recurrence; streaming-matrix eigenvalues match Gauss nodes; J_{Nh,m} parity matches eqns 27/29/34.
• Driven Landau damping (kλ_D = 0.30, 0.35): free-ringdown frequency to <1% and damping to ~10% of the kinetic dispersion root; with the window on the resonance, Nh=128 reproduces the root to 4 digits (ω 1.1597 vs 1.1598, γ −0.01261 vs −0.01262).
• Two-stream: conservation gate (mass/momentum ~1e-13); IMEX stable at dt where explicit blows up, saturating at the converged field energy; implicit midpoint mass-exact and energy-conserving at dt=0.05.
• Bump-on-tail (n_b=0.01, v_d=10, Chapurin-style params; the hardest case): runs through saturation to tmax=120 with integrator: implicit. Measured on E(k=1): γ=0.0895 vs kinetic theory 0.0898, ω=0.9058 vs 0.9052; field energy saturates at 2.7 (t≈94) relaxing to 1.55; energy drift 2e-16; Nh=32 vs Nh=128 field observables agree to 3 digits.
• Driven nonlinear EPW (exploratory, on-branch scripts): trapping physics captured — O'Neil damping shutoff (γ_eff ≈ γ_L/4), bounce-period envelope oscillation matching τ_b = 2π/√(kE), Morales–O'Neil-like frequency downshift.
• uv run pytest tests/test_hermite_legendre_1d → 17 passed.

GPU validation (Perlmutter A100, 1 GPU)

Canonical run of the shipped configs/hermite-legendre-1d/bump-on-tail.yaml through ergoExo on a Perlmutter A100 — MLflow run (experiment hermite-legendre-1d, run c92312ac).

	Mac CPU (M-series)	A100
per implicit step (Nh=32, Nx=Nl=128)	~100–220 ms	30 ms
full tmax=120 through saturation	~35–60 min	3.0 min
per step (Nh=128)	~224 ms	37–41 ms

• Physics identical to CPU to 4 digits: field energy saturates at 2.726 (t≈94) → 1.548 (t=120) vs 2.727 → 1.548 on CPU; energy drift 3.4e-16; mass exact.
• The complex-banded tridiagonal preconditioner solve (jax.lax.linalg.tridiagonal_solve) works on the GPU/cuSPARSE path.
• The Nh=128 mid-n cascade reproduces identically on GPU — platform-independent semi-discrete dynamics, consistent with the config.md guidance.

Test plan

• uv run pytest tests/test_hermite_legendre_1d -q (17 passed locally; now in CI)
• Bump-on-tail through saturation via ergoExo/MLflow (config as shipped)
• uv run run.py --cfg configs/hermite-legendre-1d/two-stream → phase-space vortex (Fig 8), flat conservation diagnostics (Fig 7)

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