A structure-preserving multiscale solver for particle-wave interaction in non-uniform magnetized plasmas

Chi-Wang Shu; Irene M. Gamba; Kun Huang

REVIEW 3 major objections 8 minor 12 references

A structure-preserving multiscale solver for particle-wave interaction in non-uniform magnetized plasmas

T0 review · 3 major / 8 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A structure-preserving LDG scheme conserves mass and energy exactly in electron-plasmon kinetics.

desk verdict Genuinely new trajectory-bundle discretization and clean mass/energy conservation, but momentum is only exponentially-small under an unproved boundary assumption, and the cost claim lacks a benchmark. read the letter →

arxiv 2505.20210 v1 pith:K7KJ2NIB submitted 2025-05-26 math.NA cs.NAphysics.plasm-ph

Kun Huang , Irene M. Gamba , Chi-Wang Shu This is my paper · ORCID

classification math.NAcs.NAphysics.plasm-ph MSC 65M6082D10

keywords electron-plasmoninteractionquasilineardiffusionrunawayelectronslocaldiscontinuousGalerkintrajectoryaveragingstructure-preservingschemeplasmonskineticplasmasimulation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper addresses the numerical simulation of electron-plasmon interactions in a magnetized, radially inhomogeneous plasma, the kind of system relevant to runaway electrons in fusion devices. It proposes a solver that pairs a local discontinuous Galerkin discretization of the quasilinear collision operator with a trajectory-averaging treatment of the fast plasmon Hamiltonian flow. The authors establish that the semi-discrete scheme conserves total mass and energy exactly and total momentum up to an exponentially small boundary error $O(e^{-L^2})$, for any regularized wave-particle interaction kernel. The trajectory-bundle construction reduces the per-step cost of the plasmon dynamics to $O(n^5)$ operations while preserving the Hamiltonian structure. If correct, the method makes spatially non-uniform electron-plasmon kinetic simulations feasible without sacrificing the conservation laws that a physically reliable runaway-electron simulation must obey.

What carries the argument

At the center is a pair of structured discretizations. On the particle side, an LDG method defines a discrete gradient $\nabla_{p,h}$ through alternating fluxes, leading to a bilinear weak form, and the paper forms the projected directional operator $L_h$ from $L_h g_h := k_{\parallel,h}(\partial_{\perp,h}E_h)\frac{p}{p_\perp}(\partial_{\parallel,h}g_h) + (\omega_h-k_{\parallel,h}(\partial_{\parallel,h}E_h))\frac{p}{p_\perp}(\partial_{\perp,h}g_h)$, with all coefficient functions projected into the test spaces. On the plasmon side, the advection operator $Tg = \nabla_k H\cdot\nabla_x g - \nabla_x H\cdot\nabla_k g$ is skew-symmetric, and trajectory averaging is identified with the orthogonal projection onto $\ker T$; the paper discretizes $\ker T$ by trajectory bundles, which are connected level-set components of the piecewise-linear Hamiltonian constructed by the connection-proportion algorithm. Their indicator functions form an orthogonal test space $\mathcal{N}_h$, and the interaction tensor collapses to a sparse object because $\eta_q\eta_s = \delta_{qs}\eta_q$, which is what makes the $O(n^5)$ cost possible.

What would settle it

Run the semi-discrete scheme (5.12) with a deliberately small momentum cut-off so the initial electron bump has visible support on $\partial\Omega_p^L$, integrate for several diffusion times, and compare the numerical $\partial_t P^h_{z,\mathrm{tot}}$ with $e^{-L^2}$. If the drift scales with the boundary gradient of $f_h$ rather than with $e^{-L^2}$, the momentum result is controlling the boundary assumption instead of the interior scheme.

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Extended reading notes

Core claim

The central claim is Theorem 5: for the semi-discrete scheme (5.12) posed in cut-off domains and using the projected operators $L_h$, $E_h$, $\omega_h$, and $k_{\parallel,h}$, any solution $f_h \in G_h$, $N_h \in \mathcal{N}_h$ satisfies $\partial_t M^h_{\mathrm{tot}}=0$, $\partial_t E^h_{\mathrm{tot}}=0$, and $\partial_t P^h_{z,\mathrm{tot}}=O(e^{-L^2})$, where $L$ is the radius of the momentum cut-off. In other words, replacing the exact directional operator $L$ by its discrete, projected counterpart makes the discrete collision term an exact conservative exchange between electrons and plasmons for mass and energy, leaving only a boundary-size momentum defect. The same structure holds for any regularized kernel $B_\varepsilon$. The paper further claims that the trajectory-averaging operator for the plasmon Hamiltonian flow can be discretized by first constructing the null space of the advection operator via trajectory bundles, and that this reduces the per-step complexity of the plasmon dynamics to $O(n^5)$.

Load-bearing premise

The momentum conservation result rests on the assumption that the discrete electron distribution is effectively zero, gradient included, at the boundary of the chosen finite momentum box.

Editorial extensions

If this is right

Mass and energy conservation are exact at the semi-discrete level for any regularized kernel $B_\varepsilon$, so the choice of kernel cannot break the scheme's conservation balance.
With a sufficiently large momentum cut-off, momentum conservation is restored up to $O(e^{-L^2})$, so a practical run can choose $L$ to drive this error below machine precision.
The trajectory-bundle projection converts the expensive fast plasmon Hamiltonian flow into a static geometric data structure, giving a per-step cost of $O(n^5)$ and a natural parallelization along the radial direction.
The LDG formulation is compatible with additional convection terms, extending the earlier continuous-Galerkin approach to more complete runaway-electron models.
For positive plasmon densities and small enough time steps, the fully discrete explicit scheme preserves positivity of the plasmon density and gives $L^2$-stability of the electron distribution.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the conservation algebra in Theorem 5 is essentially combinatorial, so the same weak form should apply to any quasilinear wave-particle system with an operator $L$ satisfying the algebraic identity used in Theorem 1; testing it on a synthetic kernel would isolate that structure.
Editorial inference: the connection-proportion algorithm is stated to extend to higher-dimensional simplex covers, so a natural next experiment is to apply the trajectory-bundle projection to a Hamiltonian flow in more than one spatial dimension, where level sets are no longer simple strips.
Editorial inference: because the momentum estimate rests on boundary decay of $f_h$, a production code should monitor $f_h$ and $\nabla_{p,h}f_h$ on $\partial\Omega_p^L$ and adaptively enlarge the box; that check is not in the paper and would make the $O(e^{-L^2})$ statement machine-verifiable.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

The short version: this is a real numerical-methods paper with a genuinely new algorithmic idea—the trajectory-bundle basis built by the connection-proportion algorithm—and clean algebraic conservation proofs for mass and energy. It is not, however, the "rigorously preserves all" paper the abstract advertises. Momentum conservation is only proved up to O(e^{-L^2}) under an unproved boundary-decay assumption, and the computational-cost reduction is claimed without a head-to-head comparison.

The connection-proportion algorithm is the best part. It turns the problem of discretizing the null space of the advection operator into a graph problem on the minimal triangle cover, and the orthogonality and partition-of-unity properties of the resulting indicator basis are nice. The LDG weak form (5.12) is a natural extension of [7], and the algebraic identities for mass and energy are exactly right—substituting the right test functions gives conservation regardless of the kernel B_epsilon. The numerical experiments do show machine-precision conservation for the test case.

Now the soft spots. The stress-test note is right: in Theorem 5, the momentum balance reduces to a boundary integral, and the proof simply asserts that the discrete gradient of f_h on the boundary can be made arbitrarily small by taking the cut-off domain large enough. That is a reasonable heuristic but not a theorem; the O(e^{-L^2}) rate is never derived. The abstract's "rigorously preserves" is an overstatement. Second, the interpolated Hamiltonian in §6.1 is introduced to make the trajectory bundles finite, but the approximation error it introduces is never quantified. Third, the complexity analysis gives an O(n^5) flop count, but there is no comparison with a direct solve of the advection term, so "significantly reduces computational cost" is plausible but unverified. Finally, no code or data are provided, though the method is described in enough detail to reimplement.

None of this sinks the core method. The mass/energy conservation is solid, the trajectory-bundle construction is a real contribution, and the flaws are mainly omissions and an overclaimed abstract. A revision that states the boundary assumption honestly, proves or at least bounds the interpolation error, and adds a benchmark against a direct solver would make this a strong paper.

Send it to peer review. It deserves a serious referee; the referee should push on Theorem 5 and the cost comparison.

Referee Report

3 major / 8 minor

Summary. The paper develops a numerical method for a coupled kinetic system describing electrons and plasmons (wave packets) in a non-uniform magnetized cylindrical plasma. The model consists of a quasilinear diffusion equation for the electron distribution and a Liouville-reaction equation for the plasmon distribution, with a fast Hamiltonian advection term. The authors eliminate the fast advection by projecting onto the null space of the Poisson-bracket operator, realized discretely through 'trajectory bundles' constructed by a connection-proportion algorithm on a triangulated (r,k_r) domain. The electron equation is discretized with a local discontinuous Galerkin method in momentum space. Theorem 5 claims that the semi-discrete scheme conserves mass and energy exactly and momentum to O(e^{-L^2}), for any regularized interaction kernel; a complexity analysis estimates O(n^5) work per time step. Numerical tests in a non-uniform density cylinder verify conservation to machine precision and illustrate spatial inhomogeneity.

Significance. The central idea of this paper is attractive and, to the best of my knowledge, original: instead of numerically averaging along sampled trajectories, the authors discretize the kernel of the Hamiltonian advection operator and use the orthogonal projection onto it as the averaging operator. This preserves the Hamiltonian structure by construction and is compatible with conservative finite-element discretizations. The conservation statements are derived algebraically from the discrete weak form and involve no fitted parameters; the numerical conservation check is a legitimate self-consistency test of the implementation. The paper also gives an explicit complexity estimate and a positivity-preserving time-stepping analysis. If the momentum-conservation gap identified below is closed, and the approximation error of the interpolated Hamiltonian is quantified, this would be a solid contribution to structure-preserving kinetic plasma simulation.

major comments (3)

[§5.3, Theorem 5] The proof of momentum conservation reduces the momentum error to the boundary integral ∫_{pkx} N_h B_ε ω_h k_{∥,h}(∂_{⊥,h}E_h)[1-(∂_{∥,h}p_{z,h})] L_h f_h and then states that this integral vanishes because ∇_{p,h} f_h on the boundary can be arbitrarily small for large enough cut-off. This is an unproved assumption about the discrete solution, not a consequence of the scheme; no decay of f_h or of its discrete gradient is established, no propagation of such decay under the nonlinear evolution is shown, and the O(e^{-L^2}) rate is asserted without derivation. In the piecewise-constant implementation the boundary contribution cancels only because of the ghost-cell zero-gradient rule (6.3), so the numerical experiment in §8.4 (relative momentum error 1.8×10^{-14}) does not validate the general theorem. The claim in the abstract that the scheme 'rigorously preserves all the conservation laws' therefore overstates what is proved; the theorem should either prove exponential decay of the boundary term or be restated with the precise conditions under which it holds.
[§6.1, Theorem 6] The trajectory-bundle construction is performed for the interpolated Hamiltonian ω ∈ H (piecewise constant in (q_ϕ,k_z) and piecewise linear in (r,k_r)). Theorem 6 shows that S×R is a trajectory bundle only when ω ∈ H; the physical dispersion relation ω is not in H. The manuscript gives no error estimate relating the trajectory bundles of ω to those of ω, nor any bound on the resulting perturbation of the averaged collision operator. Since the entire multiscale reduction depends on this interpolation, the lack of quantification is a load-bearing gap in the claim that the method is a reliable structure-preserving solver.
[§6.4] The complexity analysis states that each time step costs O(n^5) after an O(n^6) precomputation and that this 'significantly reduces the computational cost' relative to direct Hamiltonian-flow simulation, but no baseline solver is defined and no comparison is made. As written, the O(n^5) estimate is a cost count for the proposed method, not a demonstration of reduction; the abstract's 'significantly reduces' claim needs either a comparative analysis or a more modest wording.

minor comments (8)

[Abstract and §4.2] There are typos: 'struture-preserving' in the abstract and 'connnection' in the Section 4.2 title; these should be corrected.
[§2] The notation R3p, R3k and the integrals 'RR px' is nonstandard; using $\mathbb{R}^3_p$ and $\int_p\int_x$ would improve readability.
[§3.1] The phrase 'with H ≡ Hb and ∇H ≠ 0 on ∂Ωb' should state explicitly that H is required to be constant on the boundary; as written it could be misread as H being identically equal to a single number on the whole domain.
[§3.2, Theorem 1] The proof of the continuous conservation theorem drops boundary terms at infinity; please state the decay assumptions on f and N under which the theorem is valid.
[§5.3, Equation (5.11)] The operator L_h uses the discrete gradient ∇_{p,h} from Definition 4, but the expression for ∂_{⊥,h}g_h contains a norm of a vector; please clarify the notation so that the product (∂_{⊥,h}E_h) · p/p_⊥ is unambiguous.
[§6.3] In the definition of D_{ij}[η_q,λ_ξ], the bracket β^ij_h appears to involve a row vector times a column vector; make the tensor structure explicit.
[§7] Lemma 1 and Theorem 7 establish L2-stability and positivity through mutual hypotheses; please state the induction argument explicitly and define the constants C1 and C2, which otherwise appear without specification.
[§8.4] Figure 4 reports only final-time relative errors; plotting the errors as a function of time would more convincingly demonstrate that the conservation is maintained throughout the evolution.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the conservation theorems are algebraic identities from the discrete weak form, and the self-citations are contextual rather than load-bearing.

full rationale

Walking the derivation chain, the continuous conservation theorem (Theorem 1) is proved in the paper itself by substituting test functions and using the cancellation k_z L E - omega L p_z = 0. The discrete analogue (Theorem 5) repeats the same substitutions with the projected operators L_h, E_h, omega_h, k_{z,h}; mass and energy conservation are exact algebraic identities of the semi-discrete weak form (5.12), with no fitted parameter or empirical input. The numerical conservation check in Section 8.4 merely verifies that the implementation reproduces these identities, so it is a self-consistency test, not a prediction from tuned data. The trajectory-averaging construction is also self-contained: the operator P is defined as the orthogonal projection onto ker T, and the trajectory-bundle basis functions are indicators of connected level-set components, with the connection-proportion algorithm constructing them from the discrete Hamiltonian. This is a test-space construction, not a renamed empirical pattern. The self-citations to the authors' prior works [6,7] supply the dispersion model, the cut-off domain discussion, and the continuous-Galerkin predecessor, but the load-bearing L-operator cancellation is re-derived in Theorem 1 and used again in Section 5.3, so those citations are not load-bearing. The only substantive weakness is a rigor gap: Theorem 5's momentum statement O(e^{-L^2}) relies on the unproved assumption that grad_{p,h} f_h is arbitrarily small on the boundary of the cut-off domain. That is a hypothesis about the discrete solution rather than a consequence of the scheme, so it is a correctness/rigor concern, not a circularity. No equation is equivalent to its input by construction, and no fitted quantity is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central numerical claims rest on the standard weak-turbulence model, the validity of the two-scale Hilbert expansion, the approximation of the dispersion relation by piecewise polynomials, and unproved boundary decay of the electron distribution. None of these are fitted to data, but several are introduced by the paper without error estimates.

assumptions (5)

domain assumption The electron-plasmon kinetic system (2.1), with quasilinear diffusion and reaction operators, is an accurate reduced model.
The model is taken from the weak turbulence and WKB literature [4,11]; the paper does not derive or benchmark it.
domain assumption The Hilbert expansion N = N0 + epsilon N1 + ... and the projection onto ker T describe the fast-advection limit correctly.
Section 3.1 substitutes the expansion and ignores higher order terms; convergence for this nonlinear problem is not justified.
domain assumption The trajectory-bundle indicator basis can approximate any element of ker T as the mesh is refined.
Section 5.3 assumes the projection Pi_kx,h converges for every xi in ker T, but no approximation rate for the bundle basis is proven.
ad hoc to paper The interpolated Hamiltonian, piecewise constant in (q_phi,k_z) and piecewise linear in (r,k_r), is a sufficient approximation of the true dispersion relation.
Section 6.1 introduces this interpolation to make only finitely many trajectory bundles necessary (Theorem 6); no error estimate between the interpolated and physical Hamiltonian is given.
domain assumption The electron distribution has negligible mass and negligible discrete gradient at the boundary of the cut-off momentum domain.
Used in Theorem 5 to drop boundary terms and obtain momentum conservation to O(e^{-L^2}); it is asserted, not proven.

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Cite this review

Pith. "Pith review of A structure-preserving multiscale solver for particle-wave interaction in non-uniform magnetized plasmas." pith.science (2026). https://pith.science/paper/K7KJ2NIB

@misc{pith2026250520210,
  author       = {Pith},
  title        = {Pith review of: A structure-preserving multiscale solver for particle-wave interaction in non-uniform magnetized plasmas},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K7KJ2NIB}},
  note         = {Machine review of arXiv:2505.20210}
}

read the original abstract

Particle-wave interaction is of fundamental interest in plasma physics, especially in the study of runaway electrons in magnetic confinement fusion. Analogous to the concept of photons and phonons, wave packets in plasma can also be treated as quasi-particles, called plasmons. To model the ``mixture" of electrons and plasmons in plasma, a set of ``collisional" kinetic equations has been derived, based on weak turbulence limit and the Wentzel-Kramers-Brillouin (WKB) approximation. There are two main challenges in solving the electron-plasmon kinetic system numerically. Firstly, non-uniform plasma density and magnetic field results in high dimensionality and the presence of multiple time scales. Secondly, a physically reliable numerical solution requires a structure-preserving scheme that enforces the conservation of mass, momentum, and energy. In this paper, we propose a struture-preserving multiscale solver for particle-wave interaction in non-uniform magnetized plasmas. The solver combines a conservative local discontinuous Galerkin (LDG) scheme for the interaction part with a trajectory averaging method for the plasmon Hamiltonian flow part. Numerical examples for a non-uniform magnetized plasma in an infinitely long symmetric cylinder are presented. It is verified that the LDG scheme rigorously preserves all the conservation laws, and the trajectory averaging method significantly reduces the computational cost.

Figures

Figures reproduced from arXiv: 2505.20210 by the authors.

**Figure 2.** Spatial inhomogeneity 19 [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗

**Figure 3.** Temporal evolution. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Conservation laws 9 Summary In this paper, we introduced a structure-preserving solver for particle-wave interaction in a cylinder with radially non-uniform plasma density. We preserve all the conservation laws during “collision” by adopting the unconditionally conserv…

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[7]

A conservative galerkin solver for the quasilinear diffusion model in magnetized plasmas

Kun Huang, Michael Abdelmalik, Boris Breizman, and Irene M Gamba. A conservative galerkin solver for the quasilinear diffusion model in magnetized plasmas. Journal of Computational Physics , 488:112220, 2023. 21

work page 2023
[1]

Stability analysis of runaway-driven waves in a tokamak

Pavel Aleynikov and Boris Breizman. Stability analysis of runaway-driven waves in a tokamak. Nuclear Fusion, 55(4):043014, 2015

work page 2015
[2]

Unified analysis of discontinuous galerkin methods for elliptic problems

Douglas N Arnold, Franco Brezzi, Bernardo Cockburn, and L Donatella Marini. Unified analysis of discontinuous galerkin methods for elliptic problems. SIAM journal on numerical analysis , 39(5):1749– 1779, 2002

work page 2002
[3]

Transport equations with disparate advection fields

Mihai Bostan. Transport equations with disparate advection fields. application to the gyrokinetic models in plasma physics. Journal of Differential Equations , 249(7):1620–1663, 2010

work page 2010
[4]

Breizman, Pavel Aleynikov, Eric M

Boris N. Breizman, Pavel Aleynikov, Eric M. Hollmann, and Michael Lehnen. Physics of runaway electrons in tokamaks. Nuclear Fusion, 59(8):083001, August 2019

work page 2019
[5]

Non-linear stability of plasma oscillations

WE Drummond and D Pines. Non-linear stability of plasma oscillations. 1962

work page 1962
[6]

A numerical and analytical study of kinetic models for particle-wave interaction in plasmas

Kun Huang. A numerical and analytical study of kinetic models for particle-wave interaction in plasmas . PhD thesis, 2023

work page 2023
[8]

Reduced quasilinear treatment of energetic electron instabilities in nonuniform plasmas

Dmitrii Kiramov and Boris Breizman. Reduced quasilinear treatment of energetic electron instabilities in nonuniform plasmas. In APS Division of Plasma Physics Meeting Abstracts , volume 2021, pages JP11–068, 2021

work page 2021

Show all 12 references

[9]

On the vibrations of the electronic plasma

L Landau. On the vibrations of the electronic plasma. Zhurnal eksperimentalnoi i teoreticheskoi fiziki , 16(7):574–586, 1946

1946
[10]

Structure-preserving strategy for conservative simulation of the relativistic nonlinear landau-fokker-planck equation

Takashi Shiroto and Yasuhiko Sentoku. Structure-preserving strategy for conservative simulation of the relativistic nonlinear landau-fokker-planck equation. Physical Review E , 99(5):053309, 2019

2019
[11]

Modern classical physics: optics, fluids, plasmas, elasticity, relativity, and statistical physics

Kip S Thorne and Roger D Blandford. Modern classical physics: optics, fluids, plasmas, elasticity, relativity, and statistical physics . Princeton University Press, 2017

2017
[12]

Nonlinear oscillations of rarified plasma

AA Vedenov, EP Velikhov, and RZ Sagdeev. Nonlinear oscillations of rarified plasma. Nuclear Fusion, 1(2):82, 1961. 22

1961

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [7]

A conservative galerkin solver for the quasilinear diffusion model in magnetized plasmas

Kun Huang, Michael Abdelmalik, Boris Breizman, and Irene M Gamba. A conservative galerkin solver for the quasilinear diffusion model in magnetized plasmas. Journal of Computational Physics , 488:112220, 2023. 21

work page 2023

[2] [1]

Stability analysis of runaway-driven waves in a tokamak

Pavel Aleynikov and Boris Breizman. Stability analysis of runaway-driven waves in a tokamak. Nuclear Fusion, 55(4):043014, 2015

work page 2015

[3] [2]

Unified analysis of discontinuous galerkin methods for elliptic problems

Douglas N Arnold, Franco Brezzi, Bernardo Cockburn, and L Donatella Marini. Unified analysis of discontinuous galerkin methods for elliptic problems. SIAM journal on numerical analysis , 39(5):1749– 1779, 2002

work page 2002

[4] [3]

Transport equations with disparate advection fields

Mihai Bostan. Transport equations with disparate advection fields. application to the gyrokinetic models in plasma physics. Journal of Differential Equations , 249(7):1620–1663, 2010

work page 2010

[5] [4]

Breizman, Pavel Aleynikov, Eric M

Boris N. Breizman, Pavel Aleynikov, Eric M. Hollmann, and Michael Lehnen. Physics of runaway electrons in tokamaks. Nuclear Fusion, 59(8):083001, August 2019

work page 2019

[6] [5]

Non-linear stability of plasma oscillations

WE Drummond and D Pines. Non-linear stability of plasma oscillations. 1962

work page 1962

[7] [6]

A numerical and analytical study of kinetic models for particle-wave interaction in plasmas

Kun Huang. A numerical and analytical study of kinetic models for particle-wave interaction in plasmas . PhD thesis, 2023

work page 2023

[8] [8]

Reduced quasilinear treatment of energetic electron instabilities in nonuniform plasmas

Dmitrii Kiramov and Boris Breizman. Reduced quasilinear treatment of energetic electron instabilities in nonuniform plasmas. In APS Division of Plasma Physics Meeting Abstracts , volume 2021, pages JP11–068, 2021

work page 2021