Vlasov-Fokker-Planck 2D Solver

VFP-2D is a 2D3P electron Vlasov-Maxwell-Fokker-Planck solver. Configuration space is a periodic Cartesian \((x,y)\) mesh; momentum space is represented by its radial coordinate and an arbitrary spherical-harmonic expansion,

\[ f(\mathbf r,p,t)=\sum_{\ell=0}^{\ell_{\max}}\sum_{m=-\ell}^{\ell} f_\ell^m(\mathbf r,p,t)P_\ell^{|m|}(\cos\theta)e^{im\phi}, \qquad f_\ell^{-m}=(f_\ell^m)^*. \]

The implementation follows equations (5)-(24) of Tzoufras et al., Journal of Computational Physics 230 (2011), 6475-6494, and equations (28)-(39) of Bell et al., Plasma Physics and Controlled Fusion 48 (2006), R37-R57 (the KALOS review).

Explicit preparation and execution

vfp-2d is registered with the logging-free core API. Existing YAML configurations can be prepared without MLflow or filesystem writes:

import jax

jax.config.update("jax_enable_x64", True)

from adept import SimulationSpec, run_prepared, solver_registry

prepared = solver_registry.prepare(SimulationSpec.from_legacy_config(config), key=42)
completed = run_prepared(prepared, key=jax.random.key(42))
raw = completed.raw_result
final_state = raw.final_state
history = raw.observations["state"]
times = raw.times["state"]
dataset = completed.report.result["vfp2d"]

The builder shares initialization and numerical operators with BaseVFP2D. It supports all four field modes, collisions and heating, permitted moving-ion and reservoir configurations, and existing x-axis sharding. Existing physics restrictions still apply. Capabilities declare float64, differentiation, and single/multiple-device placement; batching and multi-host execution are not advertised. The default LocalExecutor currently admits only single-device placement.

The numerical program is a ScanProgram over the complete split-step map. Its arrays and nested operators are explicit PyTree children. Pass program, params, state, inputs, and key individually to JAX transformations, as described in the explicit-program guide. params is initially empty; inputs contains normalized runtime arrays such as ni, Z, D0_heating, and enabled IB/hidden-gradient controls. Use partition_parameters to select these controls for differentiation while keeping the remaining inputs fixed. Preparation itself is host-side and is not differentiated. The program records executed inputs in raw.stats["runtime_inputs"] so the host analyzer uses updated controls when reconstructing Ohm-law diagnostics.

Saved histories retain the existing save.t schedule, including linear interpolation between timesteps. final_state always contains the last complete step, even when saving stops earlier. observation_plan declares the complete-state sample schema, retained byte budget, and any sharding collectives. The analyzer returns the existing xarray diagnostics and metrics in a Report; explicit runs do not write plots or NetCDF files automatically. To export a report, call adept.vfp2d.plotting.save_artifacts(dataset, output_directory) on the host.

Ordinary ergoExo forward calls use this prepared path and preserve legacy output and artifact behavior. Custom modules, replaced states, and explicit legacy arguments retain the compatibility fallback.

Harmonic storage

Only \(m\geq0\) is stored. The retained modes satisfy

\[0\leq\ell\leq\ell_{\max},\qquad 0\leq m\leq\min(\ell,m_{\max}).\]

They occupy one packed harmonic axis rather than a nested Python dictionary. Output includes ell(harmonic) and m(harmonic) coordinates. Both execution APIs use a real-valued final axis of length two; the operator restores complex values for the angular algebra. This avoids relying on experimental complex-state support while remaining compatible with JIT, vmap, autodiff, and array sharding.

Physics

The solver includes:

  • all \(x\) and \(y\) streaming couplings between \((\ell,m)\) modes;

  • all three electric-force components, with the radial \(G_\ell^m\) and \(H_\ell^m\) operators;

  • all three magnetic-field components, algebraic and tridiagonal in \(m\) at fixed \(\ell\);

  • all three components of Maxwell’s equations with \(\partial_z=0\);

  • a spectral initial Poisson solve;

  • density-conserving implicit isotropic electron-electron collisions;

  • the linearized Tzoufras anisotropic electron-electron and electron-ion operator for every retained \((\ell,m)\);

  • spatially shaped inverse-bremsstrahlung or Maxwellian heating;

  • distribution-function diagnostics for the scalar, vector, \(f_2\) tensor, and Nernst moments used in kinetic Ohm’s law.

  • opt-in ideal-fluid ions coupled to ion-frame electrons through pressure feedback, finite-mass moment exchange, and magnetic force and mechanical work.

The default is non-relativistic, matching VFP-1D. With grid.relativistic: true, the radial coordinate is momentum in \(m_ec\) units, streaming uses \(v=p/\sqrt{1+p^2}\), current moments use \(p^2v\), and initialization uses a Maxwell-Juttner distribution. The current collision operator is non-relativistic, so relativistic mode presently requires terms.fokker_planck.active: false.

Time integration

Stationary-ion steps use collision half-steps around the kinetic/field update. The field-solver hierarchy is:

  • maxwell: physical explicit Vlasov–Maxwell with midpoint stepping.

  • ampere: the same explicit update with the complete Ampere residual divided by a required relative_permittivity >= 1; Faraday remains unchanged.

  • oshun-implicit: explicit Faraday and midpoint non-electric transport, followed by a local discrete kinetic-current response solve for the electric field. It updates the distribution through the electric-force operator without an f1 projection.

  • kinetic-ohm: an RK4 kinetic/Faraday step with the inertia-free generalized Ohm law and current-moment projection onto quasistatic Ampere’s law.

The OSHUN-style response is not a fully implicit Maxwell integrator. Coupled-ion runs use the symmetric hydro/source/kinetic/source/hydro composition described in the configuration reference. The origin derivative enforces the KALOS regularity condition \(f_\ell^m\sim p^\ell\). Unsharded spatial derivatives and field curls are spectral on a periodic box; the sharded x derivative uses finite differences.

Current limitations

  • Coupled spatial boundaries are periodic; standalone IonEuler2D also supports outflow.

  • Moving ions are opt in and require non-relativistic, unsharded, quasineutral kinetic-ohm evolution, no active hidden density gradient, and lmax >= 1, mmax >= 1.

  • oshun-implicit is stationary-ion only and rejects spatial sharding.

  • kinetic-ohm is inertia-free and uses a current-moment projection; OSHUN’s implicit current response still leaves Faraday and transport explicit.

  • Local electron–ion exchange is a weak-drift moment model, not a full finite-mass Landau operator.

  • Atomic kinetics, ionization, and relativistic collisions are not implemented.

  • Positivity of the full distribution is not guaranteed by a truncated harmonic expansion.

  • Long-duration and production-scale moving-ion validation remain outstanding.

Ion-fluid verification gates

IonEuler2D advances cell averages of \((\rho_i,\rho_i u_x,\rho_i u_y,\rho_i u_z,\mathcal E_i)\) with MUSCL reconstruction, HLLC fluxes, periodic or outflow boundaries, and SSP-RK2 time stepping. The following components and acceptance tests are implemented:

  1. Gate 0a: conservative Euler core; uniform-flow, smooth-advection, contact, conservation, and coordinate-rotated Sod tests.

  2. Gate 0b: quantitative strong-shock, Sedov, translating isentropic-vortex, and magnetic-divergence benchmarks.

  3. Gate 1a: conservative bulk advection, compression, shear, and frame acceleration for arbitrary spherical harmonics, with sparse angular couplings and a dense verification reference. See ion-frame operators.

  4. Gate 1b: finite-mass electron–ion temperature and momentum relaxation with measured equal-and-opposite ion updates, within the local weak-drift model.

  5. Gates 2a/2b: opt-in coupled split, finite-mass frame remapping, full electron-pressure feedback, magnetic force/work, coupled invariant histories, frozen-ion regression, and quantitative Spitzer–Härm/Epperlein–Haines and Biermann local-limit tests.

  6. Nonlinear energy gate: accounted-energy tolerance and timestep, spatial, and radial refinement tests for the periodic coupled benchmark. Projection work is saved separately.

  7. Magnetic flow milestone: \(\mathbf J\times\mathbf B\) force and \(\mathbf u_i\cdot(\mathbf J\times\mathbf B)\) ion work, with source half-kicks around induction. Tests cover pressure and tension, discrete ideal-work balance, one-period Alfvén propagation of the ideal magnetic subsystem, the full coupled initial tension response, and finite-field radial energy convergence.

The default vfp-2d time loop uses stationary ions. Moving ions must be enabled explicitly. Passing these local and nonlinear tests does not complete production-scale validation; moving-ion parameter scans remain a future gate. The nonrelativistic electric operator corrects its known interior radial work defect while retaining the upper-tail flux and separate current-projection accounting. A finite-field coupled check requires less than 0.1% accounted energy defect relative to the transverse magnetic perturbation at its specified end time. The raw energy and projection work remain separately visible. See the electric-work derivation and limits and coupling details before choosing a resolution. These tests do not close the long-time energy gate. Sustained driven/open boundaries, cooling/ionization, and convergence over experimental flow times remain separate development gates.

See the configuration reference, the Joglekar 2014 reconstruction design, and configs/vfp-2d/landau-damping.yaml.