Mixed Hermite-Legendre 1D Solver

Example decks live in configs/hermite-legendre-1d/. To run one:

uv run run.py --cfg configs/hermite-legendre-1d/bump-on-tail

solver: hermite-legendre-1d selects this module.

A 1D-1V electrostatic Vlasov-Poisson solver that uses two velocity-space bases at once, following Issan, Delzanno & Roytershteyn (arXiv:2606.12322).

The motivation is that a pure Hermite expansion is very efficient for a near-Maxwellian plasma and very inefficient for anything else: resolving a beam, a plateau, or a sharp velocity-space feature takes a large number of Hermite modes, because those features are poorly represented by Maxwellian-weighted polynomials. Splitting the distribution lets each basis do what it is good at.

Equations and Quantities

The electron distribution is split into a near-Maxwellian bulk and a correction:

\[ f(x, v, t) = f_0(x, v, t) + \delta f(x, v, t) \]

The bulk is expanded in an asymmetrically-weighted (AW) Hermite basis, and the correction in a Legendre basis on a bounded velocity window \([v_a, v_b]\):

\[ f_0 = \sum_{n=0}^{N_h - 1} C_n(x, t) \, \psi_n(v; \alpha, u), \qquad \delta f = \sum_{m=0}^{N_l - 1} B_m(x, t) \, \xi_m(v; v_a, v_b) \]

These evolve under the electrostatic Vlasov-Poisson system

\[ \frac{\partial f}{\partial t} + v \frac{\partial f}{\partial x} - E \frac{\partial f}{\partial v} = 0, \qquad \partial_x E = 1 - \int f \, dv \]

with a single electron species against an immobile neutralizing ion background of density 1. Normalization is time by \(1/\omega_{pe}\), space by the Debye length \(\lambda_D\), and velocity by the electron thermal velocity \(v_{th,e}\).

The coupling between the two bases is one-way: the highest Hermite coefficient \(C_{N_h - 1}\) feeds the Legendre modes, and both sets of coefficients feed the self-consistent field through Poisson. The method pays off when the non-Maxwellian features are localized in velocity — a bump-on-tail beam is the canonical case — since then a modest Legendre window resolves what would otherwise cost many Hermite modes.

Numerics

Space is treated spectrally on a periodic domain (Fourier, \(\partial_x \to i k_x\)). Both free-streaming operators are symmetric-tridiagonal in mode index and are integrated exactly via prediagonalized matrix exponentials. The recommended split integrator uses exact half linear steps around a local velocity-force update. The Hermite force is solved by a bidiagonal recurrence; the Legendre implicit-midpoint Cayley transform uses the derivative matrix’s lower-triangular plus rank-2 structure. This removes the global Newton/GMRES solve and its Krylov-vector memory cost.

Note

With gamma=0.5 on every Legendre mode, the force generator is skew-symmetric to roundoff and the Cayley update preserves its coefficient norm. A minimum-norm correction of the six low k=0 Hermite/Legendre coefficients restores total mass, momentum, and energy after each split step.

Artificial collision rates nu_H and nu_L damp the highest modes of each basis to control filamentation, in the same spirit as the hypercollisions in Spectrax-1D.

For collisional physics, physics.collisions offers two total-distribution models: bgk is a linear-cost relaxation to the same-moment Maxwellian, while dougherty is the higher-fidelity Fokker–Planck option with quadratic modal cost. Both use native Hermite/Legendre updates with no conversion or least-squares projection and restore density, momentum, and kinetic energy exactly. These remain 1D model operators; full Coulomb pitch-angle and perpendicular-energy scattering require the vlasov-1d2v solver’s cylindrical Landau operator.

Initialization

The initialization.type key selects the initial condition:

linear-advection — free-streaming with no field, for verifying the streaming operators
two-stream — counter-propagating beams, for the two-stream instability
bump-on-tail — a Maxwellian bulk plus a fast beam, the case the mixed basis is built for
custom — user-specified

An external longitudinal driver drivers.ex can be applied. It enters only the \(E \cdot \partial_v f\) force term and never the Poisson solve, so the self-consistent field-energy diagnostic excludes the driver — useful for a clean Landau-damping measurement.

Boundary Conditions

Axis / quantity	Condition	Notes
\(x\)	Periodic	Spatial dependence is carried in Fourier space, \(\partial_x \to i k_x\).
\(v\) (Hermite part \(f_0\))	None — spectral	The AW-Hermite basis is defined on the whole line; the truncation at \(N_h\) is the effective closure.
\(v\) (Legendre part \(\delta f\))	Weak Dirichlet	\(\delta f(v_a) = \delta f(v_b) = 0\), enforced by a rank-2 penalty term \(P[m,j] = (\gamma_m/\text{width})(\xi_b[m]\xi_b[j] - \xi_a[m]\xi_a[j])\) with strength `gamma`. This is what confines the Legendre correction to its velocity window instead of letting it leak.

In the split integrator the penalty is part of the structured local Cayley solve; both free-streaming operators are integrated exactly.

Forcing and Drivers

Block	What it does
`initialization.type`	The dominant “forcing” here is the initial condition: `linear-advection`, `two-stream`, `bump-on-tail`, or `custom`, each with its own parameters.
`drivers.ex`	A prescribed longitudinal field \(E_\text{drive}(x,t) = \sum \text{env}(x,t)(\omega_0 + \delta\omega_0) a_0 \sin(k_0 x - (\omega_0+\delta\omega_0)t)\). It enters only the \(E \cdot \partial_v f\) force term and never the Poisson solve, so the self-consistent field-energy diagnostic excludes the driver — which is what makes a clean Landau-damping measurement possible.
`physics.nu_H`, `nu_L`	Artificial collision rates damping the highest modes of each basis, to control filamentation.
`physics.collisions`	Optional conservative `bgk` or `dougherty` total-distribution collision model.

What Gets Saved

binary/:

File	Contents
`fields-t=<t>.nc`	Field quantities on the requested time grid
`<name>-t=<t>.nc`	One file per configured save stream
`distribution-f_xv.nc`	The reconstructed \(f(x, v)\) — both bases evaluated back onto a velocity grid

plots/:

File	Contents
`spacetime-<field>.png`	Space-time plots per field
`scalar-<name>.png`	Scalar time series, including the conservation diagnostics
`coefficients-facets.png`	Hermite and Legendre coefficient spectra
`phase-space-f_xv.png`	Reconstructed phase space

The scalar diagnostics directly report the largest Legendre-window boundary value, the high-mode Legendre energy fraction, the step residual, and the low-mode conservation-correction norm. Together with the coefficient-facet plot, these distinguish boundary contamination from spectral recurrence or loss of predictor/corrector convergence.

Configuration Reference

See the Configuration Reference for complete YAML schema documentation.