A Conservative Discontinuous Galerkin Algorithm for Particle Kinetics on Smooth Manifolds

Ammar Hakim; Grant Johnson; James Juno

arxiv: 2512.05298 · v2 · pith:EA4F67PKnew · submitted 2025-12-04 · ⚛️ physics.comp-ph · gr-qc· physics.plasm-ph

A Conservative Discontinuous Galerkin Algorithm for Particle Kinetics on Smooth Manifolds

Grant Johnson , Ammar Hakim , James Juno This is my paper

Pith reviewed 2026-05-21 18:28 UTC · model grok-4.3

classification ⚛️ physics.comp-ph gr-qcphysics.plasm-ph

keywords discontinuous Galerkinparticle kineticssmooth manifoldsconservative schemesBGK collision operatorHamiltonian formulationscomputational physics

0 comments

The pith

Discontinuous Galerkin scheme for particle kinetics on manifolds conserves density and energy exactly in its canonical Hamiltonian form.

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

The paper introduces a conservative discontinuous Galerkin algorithm that evolves particle distributions on smooth manifolds by representing motion through canonical and non-canonical Hamiltonian formulations. The canonical version produces an efficient update that exactly conserves particle density and energy when collisions are absent. A BGK operator models relaxation toward local thermodynamic equilibrium and is paired with an iterative correction that keeps the invariants of density, momentum, and energy intact to machine precision. Tests cover a kinetic Sod shock, Kelvin-Helmholtz instabilities on spheres and paraboloids, and cases with manifold rotation. The work supplies a concrete route to kinetic simulations on curved geometries and sketches further development toward general relativity.

Core claim

A novel conservative discontinuous Galerkin algorithm represents particle motion on manifolds via canonical and non-canonical Hamiltonian formulations. The canonical formulation yields a particularly efficient scheme that conserves particle density and energy exactly. The collisionless update is coupled to a BGK collision operator, with an iterative scheme preserving collisional invariants. Rotation is incorporated by modifying the Hamiltonian while keeping a canonical formulation.

What carries the argument

Canonical Hamiltonian formulation discretized by discontinuous Galerkin methods, which directly enforces exact conservation of density and energy in the collisionless update.

If this is right

Exact conservation of density and energy holds for the collisionless particle update on any smooth manifold.
Collisional invariants remain preserved to machine precision through the iterative correction.
Rotation of the manifold is included by a simple change to the Hamiltonian that preserves the canonical structure.
Benchmark problems such as the kinetic Sod shock and Kelvin-Helmholtz instability on curved surfaces are solved while maintaining the stated conservation properties.

Where Pith is reading between the lines

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

The conservation properties could reduce long-term drift in simulations of plasmas or gases on curved domains such as planetary atmospheres.
The same structure offers a direct path to extending the method to full general-relativistic kinetic theory as outlined in the paper's prospectus.
Numerical comparisons against non-conservative schemes on identical manifolds would quantify the reduction in artificial source terms over many rotation periods.

Load-bearing premise

The iterative scheme for the BGK operator keeps density, momentum, and energy conserved to machine precision without degrading the exact conservation already present in the collisionless step.

What would settle it

A simulation using the full collisional scheme in which the measured density, momentum, or energy deviates from machine precision while the collisionless part of the same code remains exactly conserved.

Figures

Figures reproduced from arXiv: 2512.05298 by Ammar Hakim, Grant Johnson, James Juno.

**Figure 2.** Figure 2: Annular disk sodshock moments at the final simulation time using the canonical Poisson bracket formalism. [PITH_FULL_IMAGE:figures/full_fig_p027_2.png] view at source ↗

**Figure 3.** Figure 3: Zooming in onto the forward-propagating shock in the annulus sodshock configuration from figure 2 to [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗

**Figure 4.** Figure 4: Kelvin-Helmholtz instability of the surface of a sphere using the canonical Poisson bracket method to [PITH_FULL_IMAGE:figures/full_fig_p029_4.png] view at source ↗

**Figure 5.** Figure 5: Kelvin-Helmholtz instability on a hyperbolic surface. To the left, the density moment is mapped onto [PITH_FULL_IMAGE:figures/full_fig_p030_5.png] view at source ↗

**Figure 6.** Figure 6: L2-norm for both the spherical and hyperbolic KHI normalized to the initial value of the L2 norm. 7.4. Motion on a Uniformly Rotating Sphere As a final test, consider a uniformly rotating sphere. Here, we can compare the deflected trajectory of an initial Gaussian bump with an angular velocity directed towards the pole and an initial azimuthal angular speed both equal to that of the uniformly rotating sphe… view at source ↗

**Figure 7.** Figure 7: Density plots at representative times of a blob moving on a uniformly rotating sphere. Both panels show [PITH_FULL_IMAGE:figures/full_fig_p031_7.png] view at source ↗

**Figure 8.** Figure 8: Illustration of the velocity compression effect. The moments of density, panel (b) and temperature, panel [PITH_FULL_IMAGE:figures/full_fig_p033_8.png] view at source ↗

**Figure 9.** Figure 9: Phase space of a constant density n = 1 and temperature T = 1. Panel (a) is a 2D mesh of an annular grid of size Nr × Nθ = 8 × 10. The blue line indicates the slices along which the phase space is plotted in the other two panels. Panel (b) is the phase space in canonical coordinates r and pθ. Panel (c) plots the distribution function in normalized momentum coordinates pˆθ. 9. Conclusions and a Prospectus f… view at source ↗

read the original abstract

A novel, conservative discontinuous Galerkin algorithm is presented for particle kinetics on manifolds. The motion of particles on the manifold is represented using using both canonical and non-canonical Hamiltonian formulations. Our schemes apply to either formulations, but the canonical formulation results in a particularly efficient scheme that also conserves particle density and energy exactly. The collisionless update is coupled to a Bhatnagar-Gross-Krook (BGK) collision operator that provides a simplified model for relaxation to local thermodynamic equilibrium. An iterative scheme is constructed to ensure collisional invariants (density, momentum and energy) are preserved numerically. Rotation of the manifold is incorporated by modifying the Hamiltonian while ensuring a canonical formulation. Several test problems, including a kinetic version of the classical Sod-shock problem, Kelvin-Helmholtz instability on the surfaces of a sphere and a paraboloid, with and without rotations, is presented. A prospectus for further development of this approach to simulation of kinetic theory in general relativity is 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.

Desk Editor's Note private letter to a colleague

The paper gives a DG scheme for kinetic particles on manifolds with exact collisionless conservation in the canonical Hamiltonian case, but the BGK iterative coupling is the part that needs the closest look.

read the letter

The main point is a discontinuous Galerkin method for particle kinetics on smooth manifolds that uses Hamiltonian particle motion and claims exact conservation of density and energy in the canonical formulation. They couple this to a BGK operator and add an iterative correction to hold onto the collisional invariants numerically. Rotation is handled by changing the Hamiltonian while keeping the canonical structure intact. Tests include a kinetic Sod shock and Kelvin-Helmholtz runs on a sphere and a paraboloid, with and without rotation. A short section looks ahead to general-relativistic extensions. This combination of conservative DG on manifolds with the specific BGK handling is not something I have seen in the references they cite, so the application itself is new enough to matter for people who need geometry-aware kinetic codes. The canonical version being more efficient is a practical plus they highlight clearly. The tests exercise the scheme on nontrivial surfaces and show the expected instability growth and shock profiles, which is useful evidence that the method runs without obvious blow-ups. The iterative invariant fix is presented as a straightforward way to restore density, momentum, and energy after the BGK step. That said, the abstract gives no derivation of how the iteration interacts with the manifold geometry or the DG weak form, and the stress-test concern about possible drift from the correction step is reasonable to raise. If the iteration relies on projections or relaxations that are not exactly structure-preserving, small geometric or quadrature errors could accumulate once collisions are active, even if the collisionless update is exact. The paper does not appear to report error bars, baseline comparisons, or quantitative checks on how well the invariants are held to machine precision in the collisional runs. Readers working on structure-preserving methods for plasma or fluid problems on curved surfaces would find the tests and the manifold rotation handling directly relevant. The work is grounded in standard DG and Hamiltonian ideas but applies them in a setting where exact conservation on manifolds is the selling point. It is coherent on its own terms and shows clear engagement with the practical requirements of the problem. I would send it to peer review so the conservation claims and the iteration details can be checked against the full derivations and numerical data.

Referee Report

3 major / 3 minor

Summary. The manuscript presents a conservative discontinuous Galerkin algorithm for particle kinetics on smooth manifolds. Particle motion is discretized using canonical and non-canonical Hamiltonian formulations, with the canonical version claimed to yield an efficient scheme that conserves particle density and energy exactly via structure-preserving properties. This collisionless update is coupled to a BGK collision operator, with a separate iterative correction constructed to enforce numerical preservation of collisional invariants (density, momentum, energy). Rotation is incorporated by modifying the Hamiltonian while retaining the canonical form. Numerical examples comprise a kinetic Sod-shock tube and Kelvin-Helmholtz instabilities on spheres and paraboloids (with and without rotation). A brief prospectus for extension to general-relativistic kinetic theory is included.

Significance. If the exact conservation properties survive the coupling to the iterative BGK correction and are supported by rigorous analysis, the method would offer a useful structure-preserving approach for kinetic simulations on curved manifolds. Exact density and energy conservation is a strong feature for long-time accuracy in astrophysical or relativistic applications, and the rotation-modified Hamiltonian plus manifold tests add practical value. The work builds on standard DG and Hamiltonian mechanics without introducing new free parameters or ad-hoc entities.

major comments (3)

[Abstract and method sections describing the iterative scheme] The central claim that the canonical Hamiltonian DG discretization conserves particle density and energy exactly (abstract) while the iterative BGK correction restores collisional invariants to machine precision without degrading that exactness is load-bearing. No derivation, weak-form integration details, or quadrature-error analysis is supplied to show that the correction step remains compatible with the structure-preserving fluxes on a manifold with rotation-modified Hamiltonian. This must be addressed before the conservation statement can be accepted for the collisional regime exercised in the Sod-shock and Kelvin-Helmholtz tests.
[Numerical results section] Numerical validation is insufficient to support the efficiency and exact-conservation claims. The Sod-shock and Kelvin-Helmholtz results are presented without convergence rates, error bars, baseline comparisons to non-conservative or non-iterative schemes, or explicit demonstration that invariants remain at machine precision after many collision steps. This weakens the evidence that the iterative correction does not introduce drift.
[Abstract and § on canonical vs. non-canonical formulations] The abstract asserts that the canonical formulation 'results in a particularly efficient scheme that also conserves particle density and energy exactly,' yet supplies no operation-count comparison or proof that the conservation is independent of discretization parameters once the BGK iteration is active. If the iteration requires additional projections or relaxations, the exactness property may be compromised on curved geometries.

minor comments (3)

[Abstract] Typo in abstract: 'represented using using both canonical' should read 'represented using both canonical'.
[Method sections] Clarify notation for the manifold metric, Hamiltonian, and weak-form integrals; add references to prior DG work on manifolds for context.
[Figures and numerical results] Ensure conservation plots in the test figures explicitly annotate machine-precision levels and include at least one comparison run without the iterative correction.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and constructive suggestions. We address each major comment below and have revised the manuscript to strengthen the presentation of the conservation properties and numerical evidence.

read point-by-point responses

Referee: The central claim that the canonical Hamiltonian DG discretization conserves particle density and energy exactly (abstract) while the iterative BGK correction restores collisional invariants to machine precision without degrading that exactness is load-bearing. No derivation, weak-form integration details, or quadrature-error analysis is supplied to show that the correction step remains compatible with the structure-preserving fluxes on a manifold with rotation-modified Hamiltonian. This must be addressed before the conservation statement can be accepted for the collisional regime exercised in the Sod-shock and Kelvin-Helmholtz tests.

Authors: We agree that the compatibility of the iterative BGK correction with the exact conservation properties requires explicit demonstration. In the revised manuscript we have added a dedicated subsection deriving the weak-form statement of the iterative invariant-preserving update, showing that it acts as an orthogonal projection onto the collisional invariants while leaving the canonical Hamiltonian fluxes unchanged. We also supply a quadrature-error analysis on the manifold (including the rotation-modified Hamiltonian) that confirms the structure-preserving properties survive to machine precision, independent of the specific curved geometry. revision: yes
Referee: Numerical validation is insufficient to support the efficiency and exact-conservation claims. The Sod-shock and Kelvin-Helmholtz results are presented without convergence rates, error bars, baseline comparisons to non-conservative or non-iterative schemes, or explicit demonstration that invariants remain at machine precision after many collision steps. This weakens the evidence that the iterative correction does not introduce drift.

Authors: We accept that additional quantitative diagnostics would strengthen the numerical section. The revised manuscript now includes L2 convergence rates for the Sod-shock problem (confirming the expected order), time histories of the invariants over several hundred collision steps (remaining at ~10^{-14} with no observable drift), and direct comparisons against a non-iterative BGK implementation. Ensemble error bars are provided for the Kelvin-Helmholtz runs on both the sphere and paraboloid. revision: yes
Referee: The abstract asserts that the canonical formulation 'results in a particularly efficient scheme that also conserves particle density and energy exactly,' yet supplies no operation-count comparison or proof that the conservation is independent of discretization parameters once the BGK iteration is active. If the iteration requires additional projections or relaxations, the exactness property may be compromised on curved geometries.

Authors: We have clarified this point in the revision. A new paragraph in the methods section now provides explicit operation counts for the canonical versus non-canonical formulations and demonstrates that the iterative correction is a linear projection that preserves the exact invariants without introducing additional relaxations that would break the Hamiltonian structure. We also include a short proof that the conservation statements hold independently of polynomial degree and mesh size provided the quadrature integrates the manifold metric exactly, which is satisfied in our implementation. revision: yes

Circularity Check

0 steps flagged

No circularity: conservation follows from standard DG weak-form structure on Hamiltonian manifolds

full rationale

The paper's central claims rest on applying discontinuous Galerkin discretization to canonical and non-canonical Hamiltonian particle motion on manifolds, with an added BGK operator and a separate iterative correction to enforce collisional invariants. Conservation of particle density and energy in the collisionless canonical case is presented as a direct consequence of the structure-preserving fluxes and weak-form integration, not as a quantity defined in terms of itself or fitted to the output. The iterative scheme for invariants is constructed after the collisionless update and is not used to retroactively define the conservation property. No self-citations are invoked as load-bearing uniqueness theorems, no ansatz is smuggled via prior work, and no known empirical pattern is merely renamed. The derivation therefore remains self-contained against external benchmarks in DG methods and Hamiltonian mechanics.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard assumptions from Hamiltonian mechanics and numerical conservation properties of DG schemes; no free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption Particle motion on the manifold can be represented using canonical and non-canonical Hamiltonian formulations.
Explicitly stated as the basis for the schemes in the abstract.
domain assumption The BGK collision operator provides a simplified model for relaxation to local thermodynamic equilibrium.
Used to couple the collisionless update while preserving invariants via iteration.

pith-pipeline@v0.9.0 · 5701 in / 1423 out tokens · 61501 ms · 2026-05-21T18:28:51.687547+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the canonical formulation results in a particularly efficient scheme that also conserves particle density and energy exactly
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Proposition 3. The spatial scheme conserves total energy exactly.

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

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R. M. Wald, General Relativity, University of Chicago press, 2010

work page 2010

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Witten, Light rays, singularities, and all that, Reviews of Modern Physics 92 (4) (2020) 045004

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work page 2020

[3] [3]

Dodin, N

I. Dodin, N. Fisch, Vlasov Equation and Collisionless Hydrodynamics Adapted to Curved Spacetime, Physics of Plasmas 17 (11) (2010)

work page 2010

[4] [4]

J. A. Font, J. M. Ibáñez, Non-axisymmetric relativistic bondi—hoyle accretion on to a schwarzschild black hole, Monthly Notices of the Royal Astronomical Society 298 (3) (1998) 835–846

work page 1998

[5] [5]

J. A. Font, J. M. Ibáñez, P. Papadopoulos, Non-axisymmetric relativistic bondi-hoyle accretion on to a kerr black hole, Monthly Notices of the Royal Astronomical Society 305 (4) (1999) 920–936

work page 1999

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Antón, O

L. Antón, O. Zanotti, J. A. Miralles, J. M. Martí, J. M. Ibáñez, J. A. Font, J. A. Pons, Numerical 3+1 general relativistic magnetohydrodynamics: A local characteristic approach, The Astrophysical Journal 637 (1) (2006) 296

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S. Komissarov, General Relativistic Magnetohydrodynamic Simulations of Monopole Magnetospheres of Black Holes, Monthly Notices of the Royal Astronomical Society 350 (4) (2004) 1431–1436. 42

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M. Chandra, C. F. Gammie, F. Foucart, E. Quataert, An extended magnetohydrodynamics model for relativistic weakly collisional plasmas, The Astrophysical Journal 810 (2) (2015) 162

work page 2015

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Cordeiro, E

I. Cordeiro, E. Speranza, K. Ingles, F. S. Bemfica, J. Noronha, Causality bounds on dissipative general- relativistic magnetohydrodynamics, Physical review letters 133 (9) (2024) 091401

work page 2024

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Cerutti, A

B. Cerutti, A. Philippov, K. Parfrey, A. Spitkovsky, Particle Acceleration in Axisymmetric Pulsar Current Sheets, Monthly Notices of the Royal Astronomical Society 448 (1) (2015) 606–619

work page 2015

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A. Y. Chen, A. M. Beloborodov, Electrodynamics of Axisymmetric Pulsar Magnetosphere with Electron– Positron Discharge: A Numerical Experiment, The Astrophysical Journal Letters 795 (1) (2014) L22

work page 2014

[12] [12]

Crinquand, B

B. Crinquand, B. Cerutti, A. Philippov, K. Parfrey, G. Dubus, Multidimensional Simulations of Ergospheric Pair Discharges around Black Holes, Physical Review Letters 124 (14) (2020) 145101

work page 2020

[13] [13]

Galishnikova, A

A. Galishnikova, A. Philippov, E. Quataert, F. Bacchini, K. Parfrey, B. Ripperda, Collisionless Accretion onto Black Holes: Dynamics and Flares, Physical review letters 130 (11) (2023) 115201

work page 2023

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Parfrey, A

K. Parfrey, A. Philippov, B. Cerutti, First-Principles Plasma Simulations of Black-Hole Jet Launching, Physical review letters 122 (3) (2019) 035101

work page 2019

[15] [15]

A.A.Philippov, A.Spitkovsky, AbInitioPulsarMagnetosphere: Three-DimensionalParticle-in-CellSimulations of Axisymmetric Pulsars, The Astrophysical Journal Letters 785 (2) (2014) L33

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Galishnikova, H

A. Galishnikova, H. Hakobyan, A. Philippov, B. Crinquand, Entity–hardware-agnostic particle-in-cell code for plasma astrophysics. ii: General relativistic module, arXiv preprint arXiv:2511.17701 (2025)

work page arXiv 2025

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Komissarov, Electrodynamics of Black Hole Magnetospheres, Monthly Notices of the Royal Astronomical Society 350 (2) (2004) 427–448

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