Develop Legendre-space flux theory and long-time nonlinear EPW/Vlasov validation

[joglekara]

Motivation

Issue #343 replaced the Hermite-Legendre global JFNK step with a structured split/Cayley integrator and produced an approximately 150x long-horizon time-to-solution improvement. The widened [2, 18] Legendre window also keeps the current warm bump-on-tail case bounded through t=700, whereas [4, 15] becomes boundary-contaminated and fails near t=659.

The next result should exploit Legendre structure rather than stop at a longer solver benchmark. The target is a JPP-style physics paper on long-time nonlinear electrostatic wave-particle interactions, with driven nonlinear EPWs as the controlled theory problem and bump-on-tail as the autonomous long-time test. Nonlinear SRS should remain a separate PoP application paper.

Research question

How does velocity-space free energy flow through a localized Legendre hierarchy as an EPW crosses from linear Landau damping into nonlinear trapping, and can that mode-space description predict long-time fidelity and failure relative to grid Vlasov?

Legendre theory and diagnostics

For

$$\delta f(x,v,t) = \sum_m B_m(x,t)\,\xi_m(v),$$

the basis normalization gives a Legendre free-energy-like norm proportional to

$$W_L = (v_b-v_a)\int dx\sum_m |B_m|^2/2.$$

With gamma=0.5 applied to all Legendre modes, the local force/penalty generator is skew-symmetric. Use that structure to derive and implement an exact modal transfer budget, including:

• W_m(t): Legendre spectrum, optionally resolved by spatial Fourier mode;
• Pi_M(t): cumulative conservative flux through Legendre cutoff M;
• Hermite-to-Legendre injection from the coupling term;
• Legendre hypercollision removal;
• contribution from the post-step conservation correction;
• budget residual and sign/normalization unit tests.

The useful physical test is whether forward Legendre flux during phase mixing arrests or reverses when trapping begins, and whether the reversal/oscillation timescale follows the bounce frequency.

Add a projection diagnostic for a grid-Vlasov reference:

$$B_m^{Vlasov}(x,t) = \frac{1}{v_b-v_a}\int_{v_a}^{v_b} [f_{Vlasov}(x,v,t)-f_0(x,v,t)]\,\xi_m(v)\,dv.$$

This should make spectra and fluxes comparable in the same representation rather than relying only on pointwise phase-space error.

Campaign A: driven nonlinear EPW

Use the existing longitudinal Ex driver and a near-Maxwellian initial state.

• Scan a small set such as k*lambda_D = {0.26, 0.30, 0.34, 0.38}.
• Scan driver amplitude across omega_B << |gamma_L|, omega_B ~ |gamma_L|, and omega_B >> |gamma_L|.
• Turn the driver off cleanly and evolve for many bounce periods.
• Center the Legendre window on v_phi; relate its minimum safe width to the nonlinear trapping/separatrix width.
• Run matched grid-Vlasov references.

Compare:

• EPW envelope and linear damping rate;
• nonlinear frequency shift and bounce period;
• plateau/trapping width and trapped-particle fraction;
• W_m, Pi_M, Hermite-to-Legendre injection, and hypercollision removal;
• coarse-grained distribution and field harmonics after drive-off.

Look for collapse of the trapping transition and Legendre-flux reversal when time is normalized by omega_B^-1.

Campaign B: long-time bump-on-tail

• Extend the validated [2, 18], Nh=32, Nl=128 run directly to at least t=1200 (preferably t=2000).
• Run a matched, converged grid-Vlasov reference.
• Use at least three beam densities or perturbation amplitudes so the result is a physics scan rather than a single benchmark.
• Compare long-time field spectrum, bounce dynamics, plateau/island geometry, trapped/passing populations, Legendre flux, and coarse-grained VDF.
• Determine whether boundary occupancy and modal flux predict loss of validity before field observables fail.

Required convergence and ablations

• dt = 0.1, 0.05;
• Nl = 128, 256 and at least one Nh refinement;
• multiple Legendre windows, including [4, 15] and [2, 18] for the current discriminator;
• sensitivity to nu_H and nu_L, treating them explicitly as artificial cutoff controls rather than physical collision frequencies;
• conservation-correction magnitude and an ablation where practical;
• converged Vlasov velocity resolution and declared coarse-graining/filter scale.

At long times, do not require pointwise agreement of arbitrarily fine Vlasov filaments. Use field observables, projected modal budgets, trapped-particle observables, Casimir/entropy-like diagnostics, and distributions compared at a controlled coarse-graining scale.

Completion criteria

• Legendre modal budget derived and documented.
• Spectrum, flux, source, sink, correction, and residual diagnostics implemented with tests.
• Grid-Vlasov-to-Hermite/Legendre projection implemented and tested.
• Driven-EPW amplitude scan spans linear damping through strong trapping.
• Flux reversal/arrest is either demonstrated and related to bounce dynamics, or cleanly ruled out.
• Bump-on-tail reaches t>=1200 in both solvers with convergence evidence.
• Artificial-collision, timestep, resolution, and window sensitivity are quantified.
• Cost-to-accuracy is reported against grid Vlasov; the Replace Hermite-Legendre full JFNK stepping with a fast split structured integrator #343 speedup is reported separately against the old JFNK implementation.
• Paper-ready long-time phase-space, modal-flux, convergence, and cost figures are produced.

Publication boundary

JPP paper: Legendre-space transport theory; driven nonlinear EPW; long-time bump-on-tail; Vlasov validation; structured-integrator performance.

Separate PoP paper: apply the validated representation and diagnostics to inhomogeneous nonlinear SRS, reflectivity/saturation, nonlinear detuning, and hot-electron production.

Related work

• Replace Hermite-Legendre full JFNK stepping with a fast split structured integrator #343 -- structured split/Cayley Hermite-Legendre integrator
• Issan, Delzanno & Roytershteyn, Mixed Hermite-Legendre spectral method for kinetic plasma simulations, arXiv:2606.12322
• Manzini et al., A Legendre-Fourier spectral method with exact conservation laws for the Vlasov-Poisson system, JCP 317, 82-107 (2016)