Vlasov 1D2V Solver

Example decks live in configs/vlasov-1d2v/. To run one:

uv run run.py --cfg configs/vlasov-1d2v/epw

solver: vlasov-1d2v selects this module.

Deck	What it shows
`epw.yaml`	Driven EPW with the marginal-coefficient `dougherty` operator
`epw-cylindrical-landau.yaml`	The same wave with the full-geometry `cylindrical_landau` operator and per-channel weights

This solver exists to run the same electrostatic problems as Vlasov-1D but with a velocity space that can support a genuinely 3V-geometry collision operator — pitch-angle scattering has no meaning in 1V, and the anisotropy it produces is exactly what distinguishes a real Coulomb operator from a 1D drift-diffusion caricature.

Things You Might Care About

Infinite length (single mode) plasma waves — Landau damping, trapping
Finite length plasma waves — everything in 1. plus wavepackets
Wave dynamics on density gradients — 2. plus density gradients
How pitch-angle scattering versus speed diffusion modifies a trapped-particle wave, using the cylindrical_landau channel weights

Equations and Quantities

The distribution is \(f = f(x, v_\parallel, v_\perp)\), with \(v_\parallel\) along the single spatial direction and \(v_\perp\) the magnitude of the two-dimensional perpendicular velocity, so the phase-space measure is

\[ d^3v = 2\pi v_\perp \, dv_\perp \, dv_\parallel, \qquad w_\perp = 2\pi v_\perp \, \Delta v_\perp \]

Only \(v_\parallel\) couples to the electric field:

\[ \frac{\partial f}{\partial t} + v_\parallel \frac{\partial f}{\partial x} + \frac{q}{m} (E + E_D) \frac{\partial f}{\partial v_\parallel} = C[f] \]

\[ \partial_x^2 \phi = 1 - \int f \, d^3v, \qquad E = -\partial_x \phi \]

The ions are static.

Reuse of the 1D field machinery

The field solver only ever consumes velocity moments, and the moments of \(f\) under the cylindrical weight are exactly the 1D moments of the marginal

\[ F(x, v_\parallel) = \int f \, w_\perp \, dv_\perp \]

So the electrostatic and EM field machinery from Vlasov-1D is reused verbatim, fed the marginal. The marginal is the bridge to the 1D solver in every respect: it is initialized to exactly the vlasov-1d initialization, saved in exactly the vlasov-1d layout, and — for the marginal-coefficient collision operators — evolves with exactly the vlasov-1d dynamics. That is what the test_1d_limit.py suite checks.

Collisions

Two families of collision operator are available through terms.fokker_planck.type.

Marginal-coefficient operators

dougherty, dougherty_nodrag, and lenard_bernstein act along \(v_\parallel\) only, with drift and diffusion coefficients computed from the marginal \(F\):

\[ C[f] = \nu \frac{\partial}{\partial v_\parallel} \left[ (v_\parallel - \bar v) f + \frac{1}{2\beta} \frac{\partial f}{\partial v_\parallel} \right], \qquad (\bar v, \beta) \ \text{from} \ F \]

This is deliberate rather than approximate bookkeeping: the discrete energy-flux condition is linear in \(f\) and the marginal is a \(w_\perp\)-weighted sum of slices, so computing the coefficients from the marginal and applying the same tridiagonal operator to every \(v_\perp\) slice conserves \(n\), \(P_\parallel\), and \(E_\parallel\) to the same standard as the 1D operator. For a separable \(f = F(v_\parallel) M(v_\perp)\) the marginal dynamics are exactly the 1D operator’s.

Cylindrical Landau

cylindrical_landau is the full-velocity-geometry linearized electron-electron operator:

\[ C[f] = \nu \, \nabla_v \cdot \left[ \mathbf{D}(v) \cdot M \nabla_v (f/M) \right], \qquad M = e^{-|\mathbf{v} - \mathbf{u}|^2 / 2T} \]

with the anisotropic test-particle tensor built about the bulk frame,

\[ \mathbf{D} = \alpha_{\text{speed}} \hat{D}_\parallel(s) \, \hat{s}\hat{s} + \alpha_{\text{lorentz}} \hat{D}_\perp(s) \, (\mathbf{I} - \hat{s}\hat{s}), \qquad s = |\mathbf{v} - \mathbf{u}| / \sqrt{T} \]

where \(\hat{D}_\parallel = \psi(x)/s^3\) and \(\hat{D}_\perp = [(1 - 1/s^2)\psi + \psi']/(2s)\) (with \(x = s^2/2\)) are the erf-exact Rosenbluth-potential coefficients of a Maxwellian field species. The two channel weights are independently configurable, and that is the main reason to reach for this operator: it lets you attribute an effect on the wave to pitch-angle scattering or to speed diffusion separately.

Writing the flux in the Einstein-relation form \(M \nabla_v (f/M)\) means each channel annihilates the bulk Maxwellian separately. Discretizing as \(D \, M_{\text{edge}} (g_{i+1} - g_i)/dv\) with \(g = f/M\) and \(M_{\text{edge}}\) the geometric mean makes the sampled Maxwellian an exact discrete fixed point of any channel mix, at any timestep.

Moment restoration. Momentum and energy exchanged with the implicit field particles are restored each step by projecting onto shifted and heated bulk-Maxwellian modes — the discrete analogue of the field-particle back-reaction. This is required for long-run stability, not just bookkeeping: the operator re-linearizes about the live \((n, u, T)\) every call, and without restoration that loop chases the \(O(dv_\perp^2)\) midpoint-rule bias in the measured \(T\) and drifts secularly. Leave moment_restoration: true unless you are deliberately measuring the drift.

Note

The Krook operator is not implemented for this solver and raises NotImplementedError.

Boundary Conditions

Axis / quantity	Condition	Notes
\(x\)	Periodic	Inherited from the Vlasov-1D field solvers, which are spectral.
\(v_\parallel\) (advection)	Periodic	`edfdv: exponential` is the only supported push, so the same caveat as Vlasov-1D applies — keep \(f \approx 0\) at both parallel edges.
\(v_\parallel\), \(v_\perp\) (`cylindrical_landau`)	Zero-flux	Finite-volume on the cylindrical grid with zero-flux outer boundaries, so density is conserved to solver precision.
\(v_\perp\) (lower edge)	Cylindrical axis	\(v_\perp \in (0, v_{\perp,\text{max}})\) with cell-centered points; there is no lower half to the axis.
Legendre/Dirichlet	—	Not applicable to this solver.

Forcing and Drivers

Forcing is inherited from Vlasov-1D and enters only the \(v_\parallel\) force term:

drivers.ex — a prescribed longitudinal field with tanh envelopes in space and time. The wavenumber, frequency, and amplitude set the wave; see the configuration reference for the schema.
terms.fokker_planck.time.baseline sets \(\nu\), and the time/space envelopes shape it exactly as the driver envelopes shape the driver. Collisions will modify the dynamics substantially depending on how far the distribution is driven from Maxwellian.

What Gets Saved

binary/:

File	Contents
`scalars-t=<t>.nc`	Scalar time series. Saved moments include `pperp` \(= \int f v_\perp^2 w_\perp \, dv_\perp dv_\parallel\) alongside the parallel `p`; with two perpendicular degrees of freedom, \(T_\perp = p_\perp / 2n\).
`dist-<save_key>.nc`	One file per species save block. A `{t, x, v}` save writes the rank-3 marginal \(F(x, v_\parallel)\) in the `vlasov-1d` layout; a `{t}`-only save writes the full rank-4 \(f(x, v_\parallel, v_\perp)\).

plots/:

Path	Contents
`plots/fields/spacetime_<field>.png`	Space-time plots of the shared fields, with a `logplots/` variant
`plots/fields/<species>/spacetime_<field>.png`	Per-species moments
`plots/scalars/<name>.png`	Scalar time series
`plots/dists/<species>/phase_space.png`	Phase-space snapshots

postprocess_time_min is logged to MLflow as a metric.

Cumulative diagnostics

diagnostics.diag-vlasov-cumulative and diag-fp-cumulative accumulate each term’s contribution to \(\partial F / \partial t\) as marginal (nx, nv) arrays holding a running time integral, not a sampled rate. Difference them between save points to recover exact interval-averaged rates — sampling a rate instead would alias the \(2\omega\) wave-particle energy exchange. Enable them under diagnostics, then add matching entries under save to write them out.

Solver Options

Time Integration (`terms.time`)

leapfrog — 2nd-order.
sixth — 6th-order Hamiltonian splitting (Crouseilles et al.). Worth the extra cost when the collision time is long compared to the wave period and time-integration error would otherwise dominate the measurement.

Velocity Advection (`terms.edfdv`)

Only exponential is supported; anything else raises NotImplementedError.

Practical Notes

Density profile. Uniform is easy. For a non-uniform profile the density can be parameterized as a sinusoidal perturbation or a tanh flat top; see initialization for the tanh flat-top parameters. The \(v_\perp\) dependence is always a Maxwellian at the component’s T0 — super-Gaussian shapes apply to \(v_\parallel\) only.

Velocity grid. Beyond the 1D nv/vmax you must set nvperp and vperp_max; they have no defaults. How fine the perpendicular axis needs to be depends on the collision operator — see choosing nvperp.

Configuration Reference

See the Configuration Reference for complete YAML schema documentation.