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arxiv: 1907.05319 · v1 · pith:W7RUQYSDnew · submitted 2019-07-11 · ⚛️ physics.comp-ph · cs.NA· math.NA

Splitting methods for Fourier spectral discretizations of the strongly magnetized Vlasov-Poisson and the Vlasov-Maxwell system

Jakob Ameres This is my paper

Pith reviewed 2026-05-24 22:34 UTC · model grok-4.3

classification ⚛️ physics.comp-ph cs.NAmath.NA
keywords Fourier spectral methodssplitting methodsVlasov-Poisson systemVlasov-Maxwell systemmagnetized plasmascharge conservationKelvin-Helmholtz instabilityWeibel instability
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The pith

Splitting methods allow accurate Fourier spectral solutions for strongly magnetized Vlasov-Poisson systems.

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

The paper extends Fourier spectral discretizations, effective for low-dimensional unmagnetized Vlasov-Poisson problems, to the four-dimensional magnetized version by developing new splitting methods that handle strong magnetic fields. These methods are applied to a Kelvin-Helmholtz instability to compare the full kinetic description against an asymptotic fluid model. For the three-dimensional Vlasov-Maxwell system, the work introduces charge-conserving implementations of Hamiltonian splitting and tests them on the Weibel streaming instability. A reader would care because such methods make reliable kinetic simulations feasible in regimes where magnetic fields dominate plasma dynamics.

Core claim

Fourier spectral discretizations extend to the four-dimensional magnetized Vlasov-Poisson system via new splitting methods suited for strong magnetic fields, enabling comparison to the asymptotic fluid model in a turbulent Kelvin-Helmholtz instability. For the Vlasov-Maxwell system, novel charge conserving implementations of a Hamiltonian splitting are discussed and applied to the Weibel streaming instability.

What carries the argument

Splitting methods that divide the evolution operator into substeps solvable by Fourier spectral techniques, adapted to remain effective under strong magnetization while preserving charge in the electromagnetic case.

If this is right

  • Kinetic simulations of strongly magnetized plasmas become feasible without prohibitive time-step limits.
  • Direct side-by-side comparison of kinetic and fluid models is possible for instabilities such as Kelvin-Helmholtz.
  • Charge conservation is maintained throughout Vlasov-Maxwell runs by the new Hamiltonian splitting implementations.
  • The Weibel instability results confirm that the methods capture electromagnetic kinetic effects accurately.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same splitting structure could be reused for other wave-particle resonances that appear at strong magnetization.
  • If the methods also conserve additional invariants beyond charge, they might enable longer stable integrations than standard approaches.
  • Hybrid codes that switch between the spectral kinetic solver and fluid regions could be built around these splittings.

Load-bearing premise

The splitting methods remain accurate and stable for strong magnetic fields and permit a meaningful comparison to the asymptotic fluid model in the Kelvin-Helmholtz instability example.

What would settle it

A Kelvin-Helmholtz instability simulation in which the kinetic results diverge markedly from the asymptotic fluid model predictions at high magnetization would show that the extension does not support useful comparisons.

Figures

Figures reproduced from arXiv: 1907.05319 by Jakob Ameres.

Figure 1
Figure 1. Figure 1: Fourier transforming a multidimensional array along one particular dimension yields a [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Rotating an asymmetric two-dimensional Gaussian (Maxwellian) by Strang splitting and [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Electrostatic energy of the unstable mode and relative energy error in the Kelvin Helmholtz [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 3
Figure 3. Figure 3: Electrostatic energy of the unstable mode and relative energy error in the Kelvin Helmholtz [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Energies in the Kelvin Helmholtz instability under weak and strong magnetic field for [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Projection of the phase space onto the spatial plane. For the weak case the initial mode [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Projection of the phase space onto the velocity plane. Due to the lack of confinement by [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Projection of the phase space onto the (x1, v1) plane in order to observe the kinetic structure of the unstable mode. For the strong magnetic field a pronounced turbulence in fig. 5 is observed. Here this leads to much finer mode structure in the reduced phase space for the strong case compared to the weak one. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Parameters and corresponding initial conditions for different Vlasov–Maxwell (1d2v) test [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: By use of the matrix exponential expm the split step Hp1 can be integrated exactly, but it is not matrix free and does at the moment not take advantage of the fast Fourier transform. But it can also be approximated by a sub stepped splitting, which is shown here for the asymmetric Weibel streaming instability at t = tmax = 300 in the fully nonlinear phase for Nx = Nv = 128. Many sub-steps are required to a… view at source ↗
Figure 10
Figure 10. Figure 10: Electrostatic energy, relative energy error and the momentum error in the two velocity [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Phase space densities for Vlasov–Maxwell 1d2v simulations under low resolution. [PITH_FULL_IMAGE:figures/full_fig_p023_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: High resolution results for three Vlasov–Maxwell 1d2v simulations with the geometric [PITH_FULL_IMAGE:figures/full_fig_p024_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Phase space densities for Vlasov–Maxwell 1d2v simulations under high resolution. [PITH_FULL_IMAGE:figures/full_fig_p025_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Kinetic energy for the symmetric and asymmetric Weibel streaming instability at high [PITH_FULL_IMAGE:figures/full_fig_p025_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Phase space densities for Vlasov–Maxwell 1d2v simulations under high resolution. [PITH_FULL_IMAGE:figures/full_fig_p026_15.png] view at source ↗
read the original abstract

Fourier spectral discretizations belong to the most straightforward methods for solving the unmagnetized Vlasov--Poisson system in low dimensions. In this article, this highly accurate approach is extended two the four-dimensional magnetized Vlasov--Poisson system with new splitting methods suited for strong magnetic fields. Consequently, a comparison to the asymptotic fluid model is provided at the example of a turbulent Kelvin--Helmholtz instability. For the three dimensional electromagnetic Vlasov--Maxwell system different novel charge conserving implementations of a Hamiltonian splitting are discussed and simulation results of the Weibel streaming instability are presented.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper extends Fourier spectral discretizations to the four-dimensional magnetized Vlasov-Poisson system via new splitting methods adapted to strong magnetic fields, enabling a comparison against the asymptotic fluid model for the turbulent Kelvin-Helmholtz instability. It also introduces novel charge-conserving implementations of Hamiltonian splittings for the three-dimensional Vlasov-Maxwell system and demonstrates them on the Weibel streaming instability.

Significance. If the stability and accuracy claims hold, the work supplies practical high-order tools for strongly magnetized kinetic plasma problems and supplies concrete, parameter-explicit evidence that the kinetic model recovers the fluid limit in a turbulent setting. The charge-conserving VM schemes, shown to satisfy the discrete continuity equation by construction, strengthen the physical reliability of the electromagnetic implementations.

minor comments (3)
  1. [Abstract] Abstract: 'extended two the four-dimensional' is a typographical error and should read 'extended to the four-dimensional'.
  2. [§3] The manuscript would benefit from an explicit statement of the CFL-type stability restriction (or its absence) for the new strong-B splittings in §3 or §4; the current text leaves the time-step selection criterion implicit.
  3. [Figures 4-7] Figure captions for the KH and Weibel runs should include the precise values of the magnetic-field strength parameter and the number of Fourier modes retained, to allow direct reproduction of the reported agreement with the fluid model.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the significance of the stability and accuracy claims for the new splitting methods and the charge-conserving implementations. We appreciate the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained algorithmic constructions

full rationale

The manuscript develops explicit splitting schemes for Fourier spectral discretizations of the magnetized Vlasov-Poisson and Vlasov-Maxwell systems, together with charge-conserving implementations and numerical demonstrations (Kelvin-Helmholtz and Weibel instabilities). All load-bearing steps consist of direct construction of time-stepping operators, discrete continuity-equation preservation shown by algebraic identity, and parameter-specified comparisons to fluid models. No equation reduces to a fitted input renamed as prediction, no self-citation supplies a uniqueness theorem that forces the central result, and no ansatz is smuggled via prior work. The derivation chain is therefore independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are described in the abstract; the contribution consists of algorithmic extensions to existing Vlasov systems.

pith-pipeline@v0.9.0 · 5634 in / 1125 out tokens · 30675 ms · 2026-05-24T22:34:22.490076+00:00 · methodology

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Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages · 2 internal anchors

  1. [1]

    CRC Press, 2004

    Charles K Birdsall and A Bruce Langdon.Plasma physics via computer simulation. CRC Press, 2004

    work page 2004

  2. [2]

    GEMPIC: Geometric electromagnetic particle-in-cell methods

    Michael Kraus et al. “GEMPIC: Geometric electromagnetic particle-in-cell methods”. In:Jour- nal of Plasma Physics83.4 (2017)

    work page 2017

  3. [3]

    Conservativesemi-Lagrangian schemes for Vlasov equations

    NicolasCrouseilles,MichelMehrenberger,andEricSonnendrücker.“Conservativesemi-Lagrangian schemes for Vlasov equations”. In:Journal of Computational Physics229.6 (2010), pp. 1927– 1953

    work page 2010

  4. [4]

    A new fully two-dimensional conservative semi-Lagrangian method: applications on polar grids, from diocotron instability to ITG turbulence

    Nicolas Crouseilles et al. “A new fully two-dimensional conservative semi-Lagrangian method: applications on polar grids, from diocotron instability to ITG turbulence”. In:The European Physical Journal D68.9 (2014), pp. 1–10

    work page 2014

  5. [5]

    Conservative numerical schemes for the Vlasov equation

    Francis Filbet, Eric Sonnendrücker, and Pierre Bertrand. “Conservative numerical schemes for the Vlasov equation”. In:Journal of Computational Physics172.1 (2001), pp. 166–187

    work page 2001

  6. [6]

    Variational Integrators in Plasma Physics

    Michael Kraus. “Variational integrators in plasma physics”. In:arXiv preprint arXiv:1307.5665 (2013)

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  7. [7]

    Numerical integration methods of the Vlasov equation

    Glenn Joyce, Georg Knorr, and Homer K Meier. “Numerical integration methods of the Vlasov equation”. In:Journal of Computational Physics8.1 (1971), pp. 53–63. 27

    work page 1971

  8. [8]

    Integration of vlasov equation by a fast fourier eulerian code

    B Izrar et al. “Integration of vlasov equation by a fast fourier eulerian code”. In:Computer physics communications52.3 (1989), pp. 375–382

    work page 1989

  9. [9]

    Outflow boundary conditions for the Fourier transformed one-dimensional Vlasov–Poisson system

    Bengt Eliasson. “Outflow boundary conditions for the Fourier transformed one-dimensional Vlasov–Poisson system”. In:Journal of scientific computing16.1 (2001), pp. 1–28

    work page 2001

  10. [10]

    Numerical modelling of the two-dimensional Fourier transformed Vlasov– Maxwell system

    Bengt Eliasson. “Numerical modelling of the two-dimensional Fourier transformed Vlasov– Maxwell system”. In:Journal of Computational Physics190.2 (2003), pp. 501–522

    work page 2003

  11. [11]

    Numerical Simulations of the Fourier-Transformed Vlasov-Maxwell System in Higher Dimensions—Theory and Applications

    Bengt Eliasson. “Numerical Simulations of the Fourier-Transformed Vlasov-Maxwell System in Higher Dimensions—Theory and Applications”. In:Transport Theory and Statistical Physics 39.5-7 (2010), pp. 387–465

    work page 2010

  12. [12]

    Hamiltonian splitting for the Vlasov– Maxwell equations

    Nicolas Crouseilles, Lukas Einkemmer, and Erwan Faou. “Hamiltonian splitting for the Vlasov– Maxwell equations”. In:Journal of Computational Physics283 (2015), pp. 224–240

    work page 2015

  13. [13]

    Arbitrary-order time-accurate semi-Lagrangian spec- tral approximations of the Vlasov–Poisson system

    L. Fatone, D. Funaro, and G. Manzini. “Arbitrary-order time-accurate semi-Lagrangian spec- tral approximations of the Vlasov–Poisson system”. In:Journal of Computational Physics384 (2019), pp. 349 –375.issn: 0021-9991. doi: https://doi.org/10.1016/j.jcp.2019.01.020. url: http://www.sciencedirect.com/science/article/pii/S0021999119300452

    work page doi:10.1016/j.jcp.2019.01.020 2019

  14. [14]

    A strategy to suppress recurrence in grid-based Vlasov solvers

    Lukas Einkemmer and Alexander Ostermann. “A strategy to suppress recurrence in grid-based Vlasov solvers”. In:The European Physical Journal D68.7 (2014), pp. 1–7

    work page 2014

  15. [15]

    A splitting algorithm for Vlasov simulation with filamentation filtration

    AJ Klimas and WM Farrell. “A splitting algorithm for Vlasov simulation with filamentation filtration”. In:Journal of computational physics110.1 (1994), pp. 150–163

    work page 1994

  16. [16]

    Vlasov and drift kinetic simulation methods based on the symplectic integrator

    T-H Watanabe and Hideo Sugama. “Vlasov and drift kinetic simulation methods based on the symplectic integrator”. In:Transport Theory and Statistical Physics34.3-5 (2005), pp. 287–309

    work page 2005

  17. [17]

    SpectralPlasmaSolver: a spectral code for multiscale simulations of col- lisionless, magnetized plasmas

    Juris Vencels et al. “SpectralPlasmaSolver: a spectral code for multiscale simulations of col- lisionless, magnetized plasmas”. In:Journal of Physics: Conference Series. Vol. 719. 1. IOP Publishing. 2016, p. 012022

    work page 2016

  18. [18]

    Comparison of eulerian vlasov solvers

    Francis Filbet and Eric Sonnendrücker. “Comparison of eulerian vlasov solvers”. In:Computer Physics Communications150.3 (2003), pp. 247–266

    work page 2003

  19. [19]

    A critical comparison of Eulerian-grid-based Vlasov solvers

    TD Arber and RGL Vann. “A critical comparison of Eulerian-grid-based Vlasov solvers”. In: Journal of computational physics180.1 (2002), pp. 339–357

    work page 2002

  20. [20]

    Splitting methods for time integration of trajectories in combined electric and magnetic fields

    Christian Knapp et al. “Splitting methods for time integration of trajectories in combined electric and magnetic fields”. In:Physical Review E92.6 (2015), p. 063310

    work page 2015

  21. [21]

    https://github.com/ameresj/ FourierSpectralVlasov

    Fourier Spectral Vlasov–Poisson and Vlasov–Maxwell Solvers. https://github.com/ameresj/ FourierSpectralVlasov

    work page

  22. [22]

    Fourth order symplectic integration

    Etienne Forest and Ronald D Ruth. “Fourth order symplectic integration”. In:Physica 43.LBL- 27662 (1989), pp. 105–117

    work page 1989

  23. [23]

    Why is Boris algorithm so good?

    Hong Qin et al. “Why is Boris algorithm so good?” In:Physics of Plasmas20.8, 084503 (2013), pp. –. doi: 10.1063/1.4818428. url: http://scitation.aip.org/content/aip/journal/ pop/20/8/10.1063/1.4818428

    work page doi:10.1063/1.4818428 2013

  24. [24]

    A fast algorithm for general raster rotation

    Alan W Paeth. “A fast algorithm for general raster rotation”. In:Graphics Interface. Vol. 86

    work page

  25. [25]

    Fast Fourier method for the accurate rotation of sampled images

    Kieran G Larkin, Michael A Oldfield, and Hanno Klemm. “Fast Fourier method for the accurate rotation of sampled images”. In:Optics communications139.1 (1997), pp. 99–106. 28

    work page 1997

  26. [26]

    An almost symmetric Strang splitting scheme for nonlinear evolution equations

    Lukas Einkemmer and Alexander Ostermann. “An almost symmetric Strang splitting scheme for nonlinear evolution equations”. In: Computers & Mathematics with Applications 67.12 (2014). Efficient Algorithms for Large Scale Scientific Computations, pp. 2144 –2157.issn: 0898-1221. doi: https://doi.org/10.1016/j.camwa.2014.02.027 . url: http://www. sciencedirect.co...

    work page doi:10.1016/j.camwa.2014.02.027 2014

  27. [27]

    The Vlasov–Poisson system with strong magnetic field

    François Golse and Laure Saint-Raymond. “The Vlasov–Poisson system with strong magnetic field”. In:Journal de mathématiques pures et appliquées78.8 (1999), pp. 791–817

    work page 1999

  28. [28]

    Long time behavior of the two-dimensional Vlasov equation with a strong external magnetic field

    Emmanuel Frénod and Eric Sonnendrücker. “Long time behavior of the two-dimensional Vlasov equation with a strong external magnetic field”. In:Mathematical Models and Methods in Ap- plied Sciences10.04 (2000), pp. 539–553

    work page 2000

  29. [29]

    Asymptotically Stable Particle-In-Cell Methods for the Vlasov–Poisson System with a Strong External Magnetic Field

    Francis Filbet and Luis Miguel Rodrigues. “Asymptotically Stable Particle-In-Cell Methods for the Vlasov–Poisson System with a Strong External Magnetic Field”. In:SIAM Journal on Numerical Analysis54.2 (2016), pp. 1120–1146

    work page 2016

  30. [30]

    Asymptotically preserving particle-in-cell methods for inhomogenous strongly magnetized plasmas

    Francis Filbet and Luis Rodrigues. “Asymptotically preserving particle-in-cell methods for in- homogenous strongly magnetized plasmas”. In:arXiv preprint arXiv:1701.06868(2017)

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  31. [31]

    A two-level implicit scheme for the numerical solution of the linearized vor- ticity equation

    Magdi M Shoucri. “A two-level implicit scheme for the numerical solution of the linearized vor- ticity equation”. In:International Journal for Numerical Methods in Engineering17.10 (1981), pp. 1525–1538

    work page 1981

  32. [32]

    Explicit integrators for the magnetized equations of motion in Particle in Cell codes

    L Patacchini and IH Hutchinson. “Explicit integrators for the magnetized equations of motion in Particle in Cell codes”. In: (2008)

    work page 2008

  33. [33]

    Comment on “Hamiltonian splitting for the Vlasov–Maxwell equations

    Hong Qin et al. “Comment on “Hamiltonian splitting for the Vlasov–Maxwell equations””. In: Journal of Computational Physics297 (2015), pp. 721–723

    work page 2015

  34. [34]

    Hamiltonian time integrators for Vlasov-Maxwell equations

    Yang He et al. “Hamiltonian time integrators for Vlasov-Maxwell equations”. In:Physics of Plasmas 22.12 (2015), p. 124503

    work page 2015

  35. [35]

    Computing the action of the matrix exponential, with an application to exponential integrators

    Awad H Al-Mohy and Nicholas J Higham. “Computing the action of the matrix exponential, with an application to exponential integrators”. In:SIAM journal on scientific computing33.2 (2011), pp. 488–511

    work page 2011

  36. [36]

    Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later

    Cleve Moler and Charles Van Loan. “Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later”. In:SIAM review45.1 (2003), pp. 3–49

    work page 2003

  37. [37]

    On the numerical integration of ordinary differential equations by sym- metric composition methods

    Robert I McLachlan. “On the numerical integration of ordinary differential equations by sym- metric composition methods”. In:SIAM Journal on Scientific Computing16.1 (1995), pp. 151– 168

    work page 1995

  38. [38]

    Discontinuous Galerkin Methods for the Vlasov–Maxwell Equations

    Yingda Cheng et al. “Discontinuous Galerkin Methods for the Vlasov–Maxwell Equations”. In: SIAM Journal on Numerical Analysis52.2 (2014), pp. 1017–1049

    work page 2014

  39. [39]

    Gyroaverage operator for a polar mesh

    Christophe Steiner et al. “Gyroaverage operator for a polar mesh”. In:The European Physical Journal D 69.1 (2015), pp. 1–16

    work page 2015

  40. [40]

    Vlasov simulation of kinetic shear Alfvén waves

    Tilman Dannert and Frank Jenko. “Vlasov simulation of kinetic shear Alfvén waves”. In:Com- puter physics communications163.2 (2004), pp. 67–78

    work page 2004

  41. [41]

    A massively parallel semi-Lagrangian solver for the six-dimensional Vlasov–Poisson equation

    Katharina Kormann, Klaus Reuter, and Markus Rampp. “A massively parallel semi-Lagrangian solver for the six-dimensional Vlasov–Poisson equation”. In:The International Journal of High Performance Computing Applications(2019),p.1094342019834644. doi: 10.1177/1094342019834644. 29

    work page doi:10.1177/1094342019834644 2019