pith. sign in

arxiv: 2605.18542 · v1 · pith:5NT6LBZSnew · submitted 2026-05-18 · ⚛️ physics.plasm-ph · cs.NA· math-ph· math.MP· math.NA

An explicit, energy-conserving particle-in-cell scheme for relativistic plasmas

Lee Ricketson , Jingwei Hu This is my paper

Pith reviewed 2026-05-20 08:10 UTC · model grok-4.3

classification ⚛️ physics.plasm-ph cs.NAmath-phmath.MPmath.NA
keywords particle-in-cellrelativistic plasmaenergy conservationVlasov-Maxwellexplicit schemenumerical methodsplasma simulation
0
0 comments X

The pith

Explicit particle-in-cell schemes for relativistic plasmas achieve exact energy conservation by solving a local optimization problem for each particle.

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

This paper extends a prior energy-conserving PIC method from the non-relativistic to the relativistic Vlasov-Maxwell system. It formulates the particle update as an optimization problem that is analytically solvable and depends only on data local to that particle, thereby enforcing exact conservation of total energy. The approach remains explicit and pairs with standard electromagnetic field solvers such as Yee/FDTD and PSATD. Although the optimization can occasionally produce non-real solutions, these cases prove rare enough under typical relativistic simulation parameters to yield large gains in conservation accuracy over conventional explicit schemes. Verification on standard test problems confirms the expected behavior and conservation properties.

Core claim

The scheme enforces exact energy conservation for the relativistic Vlasov-Maxwell system through an analytically solvable, particle-local optimization problem that remains practical for standard simulation parameters and is compatible with Yee/FDTD and PSATD field solvers.

What carries the argument

An analytically solvable optimization problem that is local to each particle and adjusts its momentum to enforce exact total-energy conservation while preserving the explicit character of the update.

Load-bearing premise

Instances where the optimization problem yields non-real solutions remain rare enough under practical relativistic simulation parameters to allow reliable use without additional fixes or rejections.

What would settle it

A relativistic PIC simulation using this scheme in which non-real solutions appear frequently enough to cause instability, rejection rates above a few percent, or measurable deviation from exact energy conservation.

Figures

Figures reproduced from arXiv: 2605.18542 by Jingwei Hu, Lee Ricketson.

Figure 1
Figure 1. Figure 1: Electrostatic potential energy for the two-stream test cases, showing good agree [PITH_FULL_IMAGE:figures/full_fig_p022_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Left: Fractional error in total energy as a function of time for the two-stream test problem. Improved energy accuracy of the new scheme is readily observed. Right: Number of particles at each time-step with imaginary Γ values at each time-step. thermal velocity, with smaller values of c corresponding to more relativistic cases. We use L = 6π, to admit perturbations with wave-number k = 1/3 to mirror the p… view at source ↗
Figure 3
Figure 3. Figure 3: Potential energy as a function of time for relativistic Landau damping test case. [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Energy conservation (left) and number of problematic particles at each time-step [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Total potential energy for Weibel test problem, using Crank-Nicolson and [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Breakdown of sources of potential energy in Weibel test problem. We show only [PITH_FULL_IMAGE:figures/full_fig_p026_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Fractional error in total energy for Weibel instability problem. Improved energy [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Growth of potential energy in the filamentation test case with cold initial beams. [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Fractional energy errors over time for the filamentation test case with cold [PITH_FULL_IMAGE:figures/full_fig_p029_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Temporal snapshots of electron density in a filamentation instability test case [PITH_FULL_IMAGE:figures/full_fig_p030_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Energy error as a function of time for the filamentation instability test case [PITH_FULL_IMAGE:figures/full_fig_p031_11.png] view at source ↗
read the original abstract

We extend the recently-developed explicit, energy-conserving particle-in-cell (PIC) scheme of [1] to the relativistic Vlasov-Maxwell system. As in the non-relativistic case, the method is built on an optimization problem that is analytically solvable, local to each particle, and designed to enforce exact energy conservation. Although the solution to this optimization problem is not guaranteed to be real, we show that such instances are rare enough for practical simulation parameters to permit dramatic improvements in energy conservation over traditional explicit PIC schemes. We show that, as in the non-relativistic case, the scheme is compatible with popular field-solvers for electromagnetic PIC schemes, including the Yee/FDTD and pseudo-spectral analytic time-domain (PSATD) methods. The scheme is verified on standard relativistic test problems, where its conservation properties are confirmed.

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

2 major / 1 minor

Summary. The manuscript extends a prior explicit energy-conserving PIC scheme to the relativistic Vlasov-Maxwell system. The central construction solves a particle-local optimization problem that is analytically solvable and enforces exact energy conservation while remaining explicit and compatible with Yee/FDTD and PSATD field solvers. The authors state that non-real solutions to the optimization occur but are rare for standard simulation parameters, and they verify the scheme on standard relativistic test problems.

Significance. If the claims hold, the work supplies a practical explicit scheme achieving exact energy conservation for relativistic plasmas, which is valuable for long-time simulations where energy drift can dominate errors. The analytical solvability of the local optimization problem is a clear strength, preserving efficiency without iterative solvers per particle.

major comments (2)
  1. [Abstract and optimization section] Abstract and the section describing the relativistic optimization problem: the claim that non-real solutions remain rare enough for reliable practical use rests on empirical observation from tests rather than an a priori bound, measure of parameter space, or documented fallback (rejection, projection, or correction). This is load-bearing for the reliability assertion in long or extreme runs.
  2. [Verification section] Verification section: the confirmation of conservation properties on standard relativistic test problems provides no quantitative error metrics (e.g., relative energy error vs. time or vs. traditional explicit PIC) or handling strategy for complex roots, making the 'dramatic improvements' claim difficult to evaluate.
minor comments (1)
  1. [Introduction] The manuscript should include a brief explicit statement of how the relativistic formulation reduces to the non-relativistic limit of the cited prior work [1] to clarify continuity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We respond to each major comment below and outline the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract and optimization section] Abstract and the section describing the relativistic optimization problem: the claim that non-real solutions remain rare enough for reliable practical use rests on empirical observation from tests rather than an a priori bound, measure of parameter space, or documented fallback (rejection, projection, or correction). This is load-bearing for the reliability assertion in long or extreme runs.

    Authors: We agree that the assertion regarding the rarity of non-real solutions would be strengthened by more than empirical evidence from the presented tests. Deriving a rigorous a priori bound on the measure of parameter space yielding complex roots is nontrivial given the nonlinear character of the local optimization. In the revised manuscript we will expand the optimization section with a more detailed characterization of the conditions under which complex roots appear, include additional numerical scans over a broader range of simulation parameters, and document an explicit fallback procedure (e.g., a minimal projection onto the real line that preserves the energy-conservation property to machine precision). These additions will make the reliability claim more robust for long or extreme runs. revision: yes

  2. Referee: [Verification section] Verification section: the confirmation of conservation properties on standard relativistic test problems provides no quantitative error metrics (e.g., relative energy error vs. time or vs. traditional explicit PIC) or handling strategy for complex roots, making the 'dramatic improvements' claim difficult to evaluate.

    Authors: We concur that quantitative metrics are necessary to substantiate the claimed improvements. The revised verification section will incorporate time histories of relative energy error for both our scheme and conventional explicit PIC, together with direct comparisons of long-time energy drift. We will also report the (rare) occurrences of complex roots encountered in the test suite and describe the concrete handling strategy applied in those instances. These quantitative results and the handling protocol will be added to allow readers to evaluate the performance gains directly. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior non-relativistic scheme; relativistic extension remains independent.

full rationale

The paper extends the explicit energy-conserving PIC scheme of reference [1] to the relativistic Vlasov-Maxwell system via a particle-local, analytically solvable optimization problem that enforces exact energy conservation. This construction is presented as new for the relativistic case and is verified on standard test problems for both conservation properties and compatibility with Yee/FDTD and PSATD solvers. The observation that non-real solutions remain rare is supported by empirical verification rather than by any definitional reduction, fitted parameter, or self-citation chain that would make the central result tautological. No load-bearing step reduces to its own inputs by construction, satisfying the criteria for at most minor self-citation without circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract provides limited detail; the main unexamined premise is analytic solvability of the per-particle optimization.

axioms (1)
  • domain assumption The optimization problem for each particle is analytically solvable and local.
    Stated directly in the abstract as the foundation of the method.

pith-pipeline@v0.9.0 · 5680 in / 1035 out tokens · 32652 ms · 2026-05-20T08:10:08.472638+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages

  1. [1]

    L. F. Ricketson, J. Hu, An explicit, energy-conserving particle-in-cell scheme, Journal of Computational Physics 537 (2025) 114098

    work page 2025

  2. [2]

    D. C. Barnes, L. Chac´ on, Finite spatial-grid effects in energy-conserving particle-in-cell algorithms, Computer Physics Communications 258 (2021) 107560

    work page 2021

  3. [3]

    C. K. Birdsall, A. B. Langdon, Plasma physics via computer simulation, CRC press, 2018

    work page 2018

  4. [4]

    G. Chen, L. Chac´ on, D. C. Barnes, An energy-and charge-conserving, implicit, electrostatic particle-in-cell algorithm, Journal of Computa- tional Physics 230 (18) (2011) 7018–7036

    work page 2011

  5. [5]

    Chac´ on, G

    L. Chac´ on, G. Chen, A curvilinear, fully implicit, conservative electro- magnetic pic algorithm in multiple dimensions, Journal of computa- tional physics 316 (2016) 578–597

    work page 2016

  6. [6]

    G. Chen, L. Chacon, A multi-dimensional, energy-and charge- conserving, nonlinearly implicit, electromagnetic vlasov–darwin particle-in-cell algorithm, Computer Physics Communications 197 (2015) 73–87

    work page 2015

  7. [7]

    G. Chen, L. Chacon, L. Yin, B. J. Albright, D. J. Stark, R. F. Bird, A semi-implicit, energy-and charge-conserving particle-in-cell algorithm for the relativistic vlasov-maxwell equations, Journal of Computational Physics 407 (2020) 109228

    work page 2020

  8. [8]

    Lapenta, Exactly energy conserving semi-implicit particle in cell formulation, Journal of Computational Physics 334 (2017) 349–366

    G. Lapenta, Exactly energy conserving semi-implicit particle in cell formulation, Journal of Computational Physics 334 (2017) 349–366

    work page 2017

  9. [9]

    Markidis, G

    S. Markidis, G. Lapenta, The energy conserving particle-in-cell method, Journal of Computational Physics 230 (18) (2011) 7037–7052

    work page 2011

  10. [10]

    Bacchini, J

    F. Bacchini, J. Amaya, G. Lapenta, The relativistic implicit particle- in-cell method, in: Journal of Physics: Conference Series, Vol. 1225, IOP Publishing, 2019, p. 012011. 35

    work page 2019

  11. [11]

    Gonoskov, Explicit energy-conserving modification of relativistic pic method, Journal of Computational Physics 502 (2024) 112820

    A. Gonoskov, Explicit energy-conserving modification of relativistic pic method, Journal of Computational Physics 502 (2024) 112820

    work page 2024

  12. [12]

    L. Ji, Z. Yang, Z. Li, D. Wu, S. Jin, Z. Xu, An asymptotic-preserving and energy-conserving particle-in-cell method for vlasov–maxwell equa- tions, Journal of Mathematical Physics 64 (6) (2023)

    work page 2023

  13. [13]

    B. N. Breizman, P. Aleynikov, E. M. Hollmann, M. Lehnen, Physics of runaway electrons in tokamaks, Nuclear Fusion 59 (8) (2019) 083001

    work page 2019

  14. [14]

    Atzeni, A

    S. Atzeni, A. Schiavi, F. Califano, F. Cattani, F. Cornolti, D. Del Sarto, T. Liseykina, A. Macchi, F. Pegoraro, Fluid and kinetic simulation of inertial confinement fusion plasmas, Computer physics communications 169 (1-3) (2005) 153–159

    work page 2005

  15. [15]

    Nishikawa, I

    K. Nishikawa, I. Dut ¸an, C. K¨ ohn, Y. Mizuno, Pic methods in astro- physics: simulations of relativistic jets and kinetic physics in astro- physical systems, Living Reviews in Computational Astrophysics 7 (1) (2021) 1

    work page 2021

  16. [16]

    Yee, Numerical solution of initial boundary value problems involving Maxwell’s equations in istropic media, IEEE Trans

    K. Yee, Numerical solution of initial boundary value problems involving Maxwell’s equations in istropic media, IEEE Trans. Antennas Propag. 14 (1966) 302–307

    work page 1966

  17. [17]

    Leh´ e, J.-L

    R. Leh´ e, J.-L. Vay, et al., Review of spectral maxwell solvers for electro- magnetic particle-in-cell: Algorithms and advantages, in: Proceedings of the 13th International Computational Accelerator Physics Confer- ence, Key West, FL, USA, 2018, pp. 20–24

    work page 2018

  18. [18]

    J.-L. Vay, I. Haber, B. B. Godfrey, A domain decomposition method for pseudo-spectral electromagnetic simulations of plasmas, Journal of Computational Physics 243 (2013) 260–268

    work page 2013

  19. [19]

    H. Qin, S. Zhang, J. Xiao, J. Liu, Y. Sun, W. M. Tang, Why is boris algorithm so good?, Physics of Plasmas 20 (8) (2013)

    work page 2013

  20. [20]

    Hairer, C

    E. Hairer, C. Lubich, Energy behaviour of the Boris method for charged- particle dynamics, BIT Numer. Math. 58 (2018) 969–979

    work page 2018

  21. [21]

    J. P. Boris, et al., Relativistic plasma simulation-optimization of a hy- brid code, in: Proc. Fourth Conf. Num. Sim. Plasmas, 1970, pp. 3–67

    work page 1970

  22. [22]

    Vay, Simulation of beams or plasmas crossing at relativistic veloc- ity, Physics of Plasmas 15 (5) (2008)

    J.-L. Vay, Simulation of beams or plasmas crossing at relativistic veloc- ity, Physics of Plasmas 15 (5) (2008). 36

    work page 2008

  23. [23]

    A. V. Higuera, J. R. Cary, Structure-preserving second-order integra- tion of relativistic charged particle trajectories in electromagnetic fields, Physics of Plasmas 24 (5) (2017)

    work page 2017

  24. [24]

    Schmitz, An overview of relativistic particle pushers and their ex- tension to arbitrary order accuracy, arXiv preprint arXiv:2603.06509 (2026)

    H. Schmitz, An overview of relativistic particle pushers and their ex- tension to arbitrary order accuracy, arXiv preprint arXiv:2603.06509 (2026)

    work page arXiv 2026

  25. [25]

    B. M. Cowan, D. L. Bruhwiler, J. R. Cary, E. Cormier-Michel, C. G. Geddes, Generalized algorithm for control of numerical dispersion in ex- plicit time-domain electromagnetic simulations, Physical Review Spe- cial Topics—Accelerators and Beams 16 (4) (2013) 041303

    work page 2013

  26. [26]

    Blaclard, H

    G. Blaclard, H. Vincenti, R. Lehe, J. Vay, Pseudospectral maxwell solvers for an accurate modeling of doppler harmonic generation on plasma mirrors with particle-in-cell codes, Physical Review E 96 (3) (2017) 033305

    work page 2017

  27. [27]

    S. D. Gedney, Yee algorithm for maxwell’s equations, in: Introduction to the Finite-Difference Time-Domain (FDTD) Method for Electromag- netics, Springer, 2011, pp. 39–73

    work page 2011

  28. [28]

    P. G. Petropoulos, Phase error control for fd-td methods of second and fourth order accuracy, IEEE transactions on antennas and propagation 42 (6) (1994) 859–862

    work page 1994

  29. [29]

    J. B. Schneider, R. J. Kruhlak, Dispersion of homogeneous and inho- mogeneous waves in the yee finite-difference time-domain grid, IEEE transactions on microwave theory and techniques 49 (2) (2001) 280– 287

    work page 2001

  30. [30]

    Shapoval, R

    O. Shapoval, R. Lehe, M. Th´ evenet, E. Zoni, Y. Zhao, J.-L. Vay, Over- coming timestep limitations in boosted-frame particle-in-cell simula- tions of plasma-based acceleration, Physical Review E 104 (5) (2021) 055311

    work page 2021

  31. [31]

    L. F. Ricketson, G. Chen, A pseudospectral implicit particle-in-cell method with exact energy and charge conservation, Computer Physics Communications (2023) 108811

    work page 2023

  32. [32]

    W. J. Arrighi, J. W. Banks, R. Berger, T. Chapman, A. G. Odu, J. Gor- man, A new approach to the evaluation and solution of the relativistic kinetic dispersion relation and verification with continuum kinetic sim- ulation, Journal of Computational Physics 508 (2024) 113001. 37

    work page 2024

  33. [33]

    Cheng, A

    Y. Cheng, A. J. Christlieb, X. Zhong, Energy-conserving discontinuous galerkin methods for the vlasov–maxwell system, Journal of Computa- tional Physics 279 (2014) 145–173

    work page 2014

  34. [34]

    A. Bret, L. Gremillet, M. E. Dieckmann, Multidimensional electron beam-plasma instabilities in the relativistic regime, Physics of Plasmas 17 (12) (2010)

    work page 2010

  35. [35]

    J. Yoo, J. Hu, L. F. Ricketson, An explicit energy-conserving par- ticle method for the Vlasov-Fokker-Planck equation, arXiv preprint arXiv:2510.03960 (2025). 38

    work page arXiv 2025