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, filamentation) → 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

• Explicit time integration by choice. Both free-streaming operators are symmetric-tridiagonal in mode index (Hermite eqn 7, Legendre eqn 11), so each is integrated exactly via a prediagonalized matrix exponential; the E-field force, the Legendre Dirichlet penalty, and the Hermite→Legendre coupling go in an explicit Lawson-RK4 stage. The paper uses an implicit-midpoint/JFNK integrator for machine-precision energy conservation, but that carries a large memory footprint — so this module uses an explicit integrator instead. Consequence: mass and momentum are conserved to machine precision; energy to the time-integrator order (dt-convergent, ~4e-8 at dt=1e-2).
• Space is spectral (Fourier, periodic) rather than the paper's 2nd-order central FD — required for the exact free-streaming exponential and consistent with the other ADEPT spectral solvers.
• Conservation constraint J_{Nh,0}=J_{Nh,1}=J_{Nh,2}=0 (paper §3.4/§4) and Lenard-Bernstein artificial collisions (cubic spectrum, zero on the first three moments) are both 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,bump-on-tail}.yaml (paper parameters)
• Docs: docs/source/solvers/hermite_legendre_1d/config.md + quick links

Validation

• Linear advection (§4.2): reconstructed f = f0 + δf matches the analytic f(x−vt, v, 0) to <2% L2 before recurrence.
• Conservation (§3, two-stream, J=0 enforced): mass & momentum to ~1e-13; energy to ~1e-5 (time-integration-limited, dt-convergent).
• Unit: streaming-matrix eigenvalues match Gauss nodes (Golub–Welsch); J_{Nh,m} parity matches eqns 27/29/34.
• End-to-end: uv run run.py --cfg configs/hermite-legendre-1d/linear-advection runs through ergoExo/MLflow to completion.
• uv run pytest tests/test_hermite_legendre_1d → 12 passed.

Test plan

• uv run pytest tests/test_hermite_legendre_1d -q
• 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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