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arxiv: 2604.19272 · v1 · submitted 2026-04-21 · 🧮 math.NA · cs.NA

Symplectic Error of Implicit Symplectic Integrators: A Qualitative Structural Analysis

Mat\v{e}j Gajdo\v{s} , Ond\v{r}ej Brichta , V\'aclav Ku\v{c}era This is my paper

Pith reviewed 2026-05-10 02:12 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords symplectic integratorspseudo-symplecticityfixed-point iterationSymplectic EulerStormer-VerletHamiltonian systemsnumerical analysis
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The pith

Finite fixed-point iterations in implicit symplectic integrators yield a skew-symmetric perturbed structure matrix with one vanishing block and O(h^{M+1}) errors in the rest.

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

The paper shows that Symplectic Euler and Stormer-Verlet schemes applied to nonseparable Hamiltonians lose exact symplecticity when the implicit equations are solved with only a fixed number M of fixed-point iterations. It gives a block-wise description of the resulting pseudo-symplecticity by examining how the Jacobian's symplectic matrix J is perturbed to a nearby matrix tilde J. The analysis proves that tilde J stays skew-symmetric, that one diagonal block is identically zero depending on the variant chosen, and that the other blocks differ from J by terms no larger than order h to the power M+1. These structural facts matter because they directly control the size of the volume-preservation error and the growth of energy drift along computed trajectories.

Core claim

For a fixed finite number M of fixed-point iterations, the numerical flow of the Symplectic Euler method produces a Jacobian whose associated structure matrix tilde J remains skew-symmetric. One of its two diagonal blocks vanishes exactly, while the off-diagonal blocks and the remaining diagonal block each differ from the exact J by an O(h^{M+1}) term. The same block-wise decay pattern, with distinct orders across blocks, carries over to compositions that realize the Stormer-Verlet method. The orders are shown to be sharp by an explicit quadratic Hamiltonian example, and the volume-preservation defect is traced solely to the off-diagonal blocks.

What carries the argument

The perturbed symplectic structure matrix tilde J extracted from the Jacobian of the numerical flow, examined block-wise according to the position-momentum splitting.

If this is right

  • The deviation from exact volume preservation is produced only by the off-diagonal blocks of tilde J.
  • Energy error along trajectories can be bounded directly from the size of the O(h^{M+1}) blocks.
  • p-implicit and q-implicit variants of Symplectic Euler place the identically zero block in opposite diagonal positions.
  • Stormer-Verlet inherits distinct decay rates for the symplectic defect in its different blocks.

Where Pith is reading between the lines

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

  • Raising M by one order should improve the symplectic defect by one extra power of h, suggesting a simple way to trade iteration cost for geometric fidelity.
  • In problems where one variable is linear, the linearly implicit variant already achieves the better block structure at lower cost.
  • The block-wise picture could guide the design of hybrid solvers that iterate only on the blocks that contribute to the defect.

Load-bearing premise

The nonlinear equations arising from the implicit integrator are solved with exactly M fixed-point iterations for some fixed finite M.

What would settle it

Compute the full symplectic defect matrix for the quadratic Hamiltonian test case at several step sizes h and several iteration counts M; check whether one diagonal block is exactly zero while the other blocks scale as h^{M+1} and the matrix remains skew-symmetric.

Figures

Figures reproduced from arXiv: 2604.19272 by Mat\v{e}j Gajdo\v{s}, Ond\v{r}ej Brichta, V\'aclav Ku\v{c}era.

Figure 1
Figure 1. Figure 1: Various coordinate systems used to describe the ve [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Trajectory of a charged particle in a tokamak magne [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The diagonal perturbation of the symplectic struc [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The antidiagonal perturbation of the symplectic s [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Energy (Hamiltonian) drift of both the q-implicit (pseudo-symplectic) and the p-implicit (symplectic) schemes over a long time interval (h = 0.25T0, number of steps 3 · 106 ). The almost-explicit scheme (29) exhibit a stable Hamiltonian development with a bounded time-independent error, just as is expected from a symplectic integrator. The implicit schemes (28) get eventually disturbed during calculation, … view at source ↗
read the original abstract

We study how inexact nonlinear solvers lead to a loss of exact symplecticity in the Symplectic Euler (SE) and Stormer-Verlet (SV) schemes when applied to general nonseparable Hamiltonian systems. These schemes are implicit and require nonlinear solvers in practice. Here, we consider a fixed number $M$ of fixed-point iterations (FPI). While SE is exactly symplectic under exact solves, a finite $M$ gives only pseudo-symplecticity. Compared to previous results, we provide a more qualitative, block-wise characterization of the induced pseudo-symplecticity by analyzing the resulting perturbations to the matrix of symplectic structure $J$. We prove that the perturbed matrix $\tilde{J}$ is skew-symmetric, that one diagonal block vanishes identically (depending on the SE variant), and that the remaining blocks are $O(h^{M+1})$ perturbations of their counterparts in $J$, with time step $h$. A quadratic Hamiltonian example shows these bounds are sharp. Extending to compositions, we quantify how SV inherits distinct decay orders across different blocks of the symplectic defect. As a corollary, we show that the perturbation of volume preservation in phase space arises solely from the off-diagonal blocks of $\tilde{J}$, and we bound the induced energy error along trajectories. Numerical experiments on a tokamak magnetic-field Hamiltonian, where q-implicit SE is fully nonlinear (requiring FPI) but p-implicit SE is linearly implicit, confirm the sharpness of the theory and highlight the gap to the exactly symplectic counterpart.

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 / 3 minor

Summary. The manuscript examines the symplectic error introduced by using a finite number M of fixed-point iterations to solve the implicit equations in Symplectic Euler (SE) and Stormer-Verlet (SV) integrators for nonseparable Hamiltonian systems. Through perturbation analysis of the effective symplectic matrix tilde J, it proves that tilde J remains skew-symmetric, that one diagonal block vanishes identically (depending on the SE variant), and that the other blocks deviate from those in J by O(h^{M+1}). Sharpness is demonstrated with a quadratic Hamiltonian example, and the analysis is extended to compositions and SV. Corollaries bound the perturbation to volume preservation (arising only from off-diagonal blocks) and the induced energy error. Numerical tests on a tokamak magnetic-field Hamiltonian confirm the predictions and contrast with the exactly symplectic case.

Significance. If the structural claims hold, the block-wise characterization of pseudo-symplecticity supplies a useful qualitative tool for analyzing inexact implicit symplectic integrators, complementing prior quantitative bounds. The exact properties (skew-symmetry of tilde J, vanishing of a diagonal block) and the sharpness example with quadratic Hamiltonians, together with the tokamak numerics, constitute concrete strengths. The corollaries on volume preservation and energy error are directly applicable to long-term integration in applications such as plasma physics.

major comments (2)
  1. [§3] §3 (perturbation analysis of tilde J): the proof that the remaining blocks are O(h^{M+1}) perturbations appears to rest on the contraction mapping property of the fixed-point iteration; a concrete expansion of the iteration map to order M+1 would make the order claim fully explicit and allow direct verification of the block structure.
  2. [§5] §5 (extension to SV compositions): the distinct decay orders inherited by different blocks of the symplectic defect are stated as a quantification, but the precise relation between the composition weights and the per-block orders is not derived in the provided outline; this step is load-bearing for the SV claim.
minor comments (3)
  1. [§2] The notation distinguishing the two SE variants (q-implicit vs. p-implicit) and their corresponding vanishing blocks should be introduced with a small table or diagram early in §2.
  2. [Numerical experiments] In the tokamak numerical section, the reported energy-error curves would benefit from explicit mention of the number of independent runs or error bars to quantify variability.
  3. [Introduction] A brief comparison paragraph placing the new block-wise bounds against the earlier pseudo-symplecticity literature cited in the introduction would help readers gauge the advance.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major comment below and will incorporate the suggested clarifications in the revised version.

read point-by-point responses

  1. Referee: [§3] §3 (perturbation analysis of tilde J): the proof that the remaining blocks are O(h^{M+1}) perturbations appears to rest on the contraction mapping property of the fixed-point iteration; a concrete expansion of the iteration map to order M+1 would make the order claim fully explicit and allow direct verification of the block structure.

    Authors: We agree that an explicit expansion of the fixed-point iteration would make the O(h^{M+1}) claim and the block structure more transparent. In the revised manuscript we will insert a direct Taylor expansion of the iteration map through order M+1, showing term-by-term how the perturbations enter each block of tilde J while preserving skew-symmetry and the vanishing diagonal block. revision: yes

  2. Referee: [§5] §5 (extension to SV compositions): the distinct decay orders inherited by different blocks of the symplectic defect are stated as a quantification, but the precise relation between the composition weights and the per-block orders is not derived in the provided outline; this step is load-bearing for the SV claim.

    Authors: We acknowledge that the link between the composition weights and the per-block orders requires an explicit derivation. In the revision we will add a step-by-step calculation that traces how each weight in the Stormer-Verlet composition propagates into the orders of the individual blocks of the symplectic defect, thereby making the quantification self-contained. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained perturbation analysis

full rationale

The paper derives its central claims (skew-symmetry of the perturbed matrix tilde J, exact vanishing of one diagonal block depending on SE variant, and O(h^{M+1}) bounds on remaining blocks) directly from algebraic expansion of the fixed-point iteration map applied to the implicit symplectic Euler and Stormer-Verlet update equations for nonseparable Hamiltonians. These steps rely only on the contraction properties of the iteration and the block structure of the exact symplectic matrix J; no parameters are fitted to data, no results are renamed, and no load-bearing premise reduces to a self-citation or self-definition. The finite-M FPI setting is the explicit object of study, and the quadratic example and tokamak numerics serve only to illustrate sharpness rather than to establish the bounds. The reference to 'previous results' is comparative and does not substitute for the new block-wise proofs.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard properties of symplectic integrators and convergence of fixed-point iteration; no new entities are introduced.

free parameters (1)
  • M
    Fixed finite number of fixed-point iterations chosen for the nonlinear solver.
axioms (2)
  • domain assumption Fixed-point iteration with finite M produces a well-defined map for the implicit schemes.
    Required for the pseudo-symplecticity analysis to apply.
  • domain assumption The Hamiltonian is general and nonseparable.
    Ensures the schemes are fully implicit and require nonlinear solves.

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Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

  1. [1]

    Geometric Numerical Integration

    Ernst Hairer, Christian Lubich, and Gerhard Wanner. Geometric Numerical Integration. Springer, 2002

    work page 2002

  2. [2]

    Numerical Hamiltonian problems

    Jesus-Maria Sanz-Serna and Mar´ ıa Paz Calvo. Numerical Hamiltonian problems . V ol. 7. Courier Dover Publications, 2018

    work page 2018

  3. [3]

    Ernst Hairer, Gerhard Wanner, and Syvert P . Nørsett. Solving ordinary differential equations I: Nonstiff problems. Springer, 1993

    work page 1993

  4. [4]

    Pseudo-symplectic Ru nge-Kutta methods

    Anne Aubry and Philippe Chartie. “Pseudo-symplectic Ru nge-Kutta methods”. In: BIT Numerical Math- ematics 38.3 (1998), pp. 439–461. DOI : https : / / doi . org / 10 . 1007 / BF02510253. URL : https : //api.semanticscholar.org/CorpusID:17088488

    work page 1998

  5. [5]

    Almost symplectic Runge–Kutta schemes for Hamiltonian systems

    Xiaobo Tan. “Almost symplectic Runge–Kutta schemes for Hamiltonian systems”. In: Journal of Com- putational Physics 203.1 (2005), pp. 250–273. ISSN : 0021-9991. DOI : https : / / doi . org / 10 . 1016 / j . jcp . 2004 . 08 . 012. URL : https : / / www . sciencedirect . com / science / article / pii / S0021999104003262

    work page 2005

  6. [6]

    Initializers for RK-Gauss methods based on pseudo-symplecticity

    Manuel Calvo, Mar´ ıa Pilar Laburta, and Juan Ignacio Mon tijano. “Initializers for RK-Gauss methods based on pseudo-symplecticity”. In: Journal of computational and applied mathematics 189.1-2 (2006), pp. 228–

    work page 2006

  7. [7]

    URL : https://www.sciencedirect

    DOI : https://doi.org/10.1016/j.cam.2005.04.029 . URL : https://www.sciencedirect. com/science/article/pii/S0377042705002803

    work page doi:10.1016/j.cam.2005.04.029 2005

  8. [8]

    M´ ethodes pseudo-symplectiques

    Anne Aubry. “M´ ethodes pseudo-symplectiques”. PhD the sis. INRIA, 1996

    work page 1996

  9. [9]

    An introduction to plasma physics

    William Bell Thompson. An introduction to plasma physics . Elsevier, 2013

    work page 2013

  10. [10]

    Classical electrodynamics

    John David Jackson. Classical electrodynamics. John Wiley & Sons, 2021

    work page 2021

  11. [11]

    Chaotic motion of charged parti cles in toroidal magnetic configurations

    Benjamin Cambon et al. “Chaotic motion of charged parti cles in toroidal magnetic configurations”. In: Chaos: An Interdisciplinary Journal of Nonlinear Science 24.3 (2014). DOI : 10.1063/1.4885103

    work page doi:10.1063/1.4885103 2014

  12. [12]

    Amann and G

    H. Amann and G. Metzen. Ordinary Differential Equations: An Introduction to Nonli near Analysis . De Gruyter Studies in Mathematics. De Gruyter, 1990

    work page 1990

  13. [13]

    Petersen

    Karl E. Petersen. Ergodic Theory . Cambridge Studies in Advanced Mathematics. Cambridge Uni versity Press, 1983

    work page 1983

  14. [14]

    Meyer and Glen R

    Kenneth R. Meyer and Glen R. Hall. Introduction to Hamiltonian Dynamical Systems and the N-Bo dy Problem. Springer, 1992

    work page 1992

  15. [15]

    Safko, Herbert Goldstein, and Charles P

    John L. Safko, Herbert Goldstein, and Charles P . Poole. Classical Mechanics. Pearson, 2011. ISBN : 978-81- 317-5891-5. 15

    work page 2011

  16. [16]

    Ergodicity and the Numerical Simulati on of Hamiltonian Systems

    Paul F. Tupper. “Ergodicity and the Numerical Simulati on of Hamiltonian Systems”. In: SIAM Journal on Applied Dynamical Systems 4.3 (2005), pp. 563–587. DOI : 10.1137/040603802 . URL : https://doi. org/10.1137/040603802

    work page doi:10.1137/040603802 2005

  17. [17]

    Julia: A fresh approach to numeric al computing

    Jeff Bezanson et al. “Julia: A fresh approach to numeric al computing”. In: SIAM Review 59.1 (2017), pp. 65–

    work page 2017

  18. [18]

    Bezanson , author A

    DOI : 10.1137/141000671. URL : https://epubs.siam.org/doi/10.1137/141000671

    work page doi:10.1137/141000671

  19. [19]

    Determinants of block matrices

    John R. Silvester. “Determinants of block matrices”. I n: The Mathematical Gazette 84.501 (2000), pp. 460–

    work page 2000

  20. [20]

    URL : http://www.jstor.org/stable/3620776

    DOI : 10.2307/3620776. URL : http://www.jstor.org/stable/3620776. 7 Appendix 7.1 Appendix A. Proof of Theorem 1 and Theorem 2 In this section, denote H[n] := H(q,pn), where the values pn are given by the scheme (10). Lemma 9. F or the scheme(10), it holds that pn − pn− 1 = O(hn), h → 0, and H [n] pq − H[n− 1] pq = O(hn), h → 0. Proof. For the difference o...

    work page doi:10.2307/3620776