Vlasov-Fokker-Planck 1D Solver

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

uv run run.py --cfg configs/vfp-1d/epp-short

This solver models electron transport over hydrodynamic / collisional time-scales using a spherical harmonic expansion of the distribution function. It is based on the OSHUN algorithm described in Tzoufras et al. (2011, 2013).

Equations and Quantities

The electron distribution function is expanded in spherical harmonics:

\[ f(t, x, v, \theta) = \sum_{l=0}^{l_{\max}} f_{l0}(t, x, v) P_l(\cos\theta) \]

where \(P_l\) are Legendre polynomials, \(v = |\mathbf{v}|\) is the speed (positive-only grid), and \(\theta\) is the angle between \(\mathbf{v}\) and \(\hat{x}\).

The isotropic component \(f_0\) and the first anisotropic component \(f_{10}\) satisfy:

\[ \frac{\partial f_0}{\partial t} = -\frac{v}{3} \frac{\partial f_{10}}{\partial x} + \frac{E}{3}\left(\frac{2}{v} f_{10} + \frac{\partial f_{10}}{\partial v}\right) + C_{00}[f_0] \]

\[ \frac{\partial f_{10}}{\partial t} = -v \frac{\partial f_0}{\partial x} + E \frac{\partial f_0}{\partial v} + C_{10}[f_{10}, f_0] \]

where \(E\) is the electric field, \(C_{00}\) is the isotropic collision operator, and \(C_{10}\) is the anisotropic (FLM) collision operator.

Staggered Spatial Grid

The solver uses a staggered (Yee-like) spatial grid:

Cell centers: \(f_0\) (scalar, isotropic distribution) and ion quantities (\(Z\), \(n_i\))
Cell edges: \(f_{10}\) (vector, anisotropic distribution), \(E\), \(B\), \(j\)

edges:   x_0       x_1       x_2              x_{nx}
          |         |         |                  |
centers:     x_1/2     x_3/2     ...   x_{nx-1/2}

f0:        [0]       [1]       ...     [nx-1]         (nx cells)
f10:     [0]      [1]      [2]    ... [nx-1]   [nx]   (nx+1 edges)
E:       [0]      [1]      [2]    ... [nx-1]   [nx]   (nx+1 edges)

Spatial derivatives naturally map between grids: \(\partial f_0 / \partial x\) (centers \(\to\) edges) and \(\partial f_{10} / \partial x\) (edges \(\to\) centers).

Boundary Conditions

Configurable via grid.boundary:

periodic (default): Wrapping ghost cells. \(f_{10}[0] = f_{10}[n_x]\).
reflective: \(f_{10} = 0\) and \(E = 0\) at boundary edges. \(f_0\) ghost cells use mirror reflection.

Collision Operators

Isotropic (\(f_0\)): Lenard-Bernstein / Dougherty

The \(f_0\) collision operator is a drift-diffusion operator in velocity space, solved implicitly via a tridiagonal system. Configurable model and differencing scheme.

Anisotropic (\(f_{10}\)): FLM

The FLM collision operator (Tzoufras 2013) includes:

Electron-ion: Pitch-angle scattering diagonal term \(\propto Z^2 n_i / v^3\)
Electron-electron (optional): Rosenbluth potential integrals for diagonal and off-diagonal contributions. When disabled, the electron-ion collision frequency is boosted by the “collision fix”: \((Z+4.2)/(Z+0.24)\).

Electric Field Solver

oshun: Implicit E-field via a Taylor expansion of \(J(E)\) (Tzoufras 2013). Computes \(\Delta J / \Delta E\) by perturbation and solves \(E = -J_0 \cdot \Delta E / (J(E + \Delta E) - J_0)\).
ampere: Explicit Ampere update \(E^{n+1} = E^n + \Delta t \cdot J\).

Solver Algorithm

Each timestep:

Explicit spatial streaming: \(v \partial f / \partial x\) (Tsit5 integrator)
Implicit \(f_0\) collision solve (Lenard-Bernstein)
Implicit E-field solve (OSHUN method)
Explicit \(E \partial f / \partial v\) push (Tsit5 integrator)
Implicit \(f_{10}\) collision solve (FLM)

Boundary Conditions

Set with grid.boundary and grid.geometry:

Axis / quantity	Condition	Notes
\(x\)	`periodic` or `reflective`	The default is `periodic` for `geometry: cartesian`. The spatial grid is staggered — cell centers for \(f_0\), cell edges for \(f_1\) and \(E\).
\(r\) (`geometry: spherical`)	Reflective, required	Spherical geometry requires `boundary: reflective` and `xmin >= 0`. \(f_{10}\) and \(E\) vanish at the inner boundary — by symmetry when `xmin = 0` — and at the outer wall. `xmin > 0` gives an annular domain with a reflecting inner wall.
\(v\)	Positive-only with a reflecting axis	The velocity grid runs from 0 to `vmax`, since the spherical-harmonic expansion carries the angular dependence.
\(v\) (collisions)	Zero-flux	At both ends of the speed grid.

Forcing and Drivers

The physics here is transport over collisional timescales, so the forcing is heating rather than a coherent wave driver:

Block	What it does
`drivers.ib`	Inverse-bremsstrahlung heating from a laser of wavelength `units.laser_wavelength`.
`drivers.maxwellian_heating`	A heating operator that drives the distribution toward a hotter Maxwellian, used to set up a temperature profile without resolving the absorption physics.
`drivers.ex`, `ey`	Present in the schema and normally left empty (`{}`) — the example decks drive the system through heating and the initial temperature profile, not through a field.

Initial conditions carry most of the setup: a hotspot or a temperature ramp relaxes and you measure the resulting heat flow.

What Gets Saved

binary/:

File	Contents
`scalars-t=<t>.nc`	Scalar time series
`<prefix>-t=<t>.nc`	Field and moment quantities on the requested grids
`dist.nc`	The distribution function, i.e. the Legendre harmonics \(f_{lm}(x, v)\)
`heat-flux-comparison.nc`	The kinetic heat flux alongside the Spitzer-Härm and SNB predictions

plots/: plots/fields/, plots/scalars/, plots/dist/, plus heat-flux-comparison.png.

That heat-flux comparison is the headline diagnostic — it is the direct read on nonlocal transport, which is what the solver exists to compute.

Configuration Reference

See the Configuration Reference for complete YAML schema documentation.