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REVIEW 4 major objections 6 minor 23 references

Long-term behaviour of symmetric partitioned linear multistep methods I. Global error and conservation of invariants

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Pith's one-line read This paper proves an asymptotic expansion of the global error of partitioned linear multistep methods and uses it to show that symmetric PLMMs with no common roots except 1 keep parasitic error and Hamiltonian drift under control over…

desk verdict Solid extension of PLMM error analysis to nonseparable Hamiltonians, but the double-pendulum boundedness argument is hand-wavy and the main proof leans on an unpublished thesis. read the letter →

arxiv 2411.13487 v1 pith:Q4SQ4IZ3 submitted 2024-11-20 math.NA cs.NA

B. Cano , A. Durán , M. Rodríguez This is my paper · ORCID
classification math.NAcs.NA MSC 65L0665P1037M15
keywords partitionedlinearmultistepmethodssymmetricglobalerrorasymptoticexpansionHamiltoniansystemsnonseparableinvariantpreservationdoublependulumlong-timeintegration
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 attempts to establish that a partitioned linear multistep method, meaning a pair of linear multistep formulas that advance the two halves of a first-order system separately, can be explicit and still behave well over very long integration intervals, even for nonseparable Hamiltonian systems. The key restriction is that the method be symmetric and that the two first characteristic polynomials share no root of modulus one except the consistency root 1. For such methods the paper proves an asymptotic expansion of the global error in powers of the stepsize, shows that the parasitic error components attached to non-common roots stay of order $O(h^r)$ uniformly for times of order $h^{-2}$, and derives an explicit expression for the drift of a Hamiltonian along the smooth part of the numerical solution. Applied to small oscillations of a double pendulum, the analysis explains why this method class keeps the Hamiltonian error bounded, while symmetric non-partitioned LMMs drift exponentially and non-symmetric methods drift linearly with time.

What carries the argument

The engine is the asymptotic expansion (9): the global error at time $t_n$ is written as a sum over the roots of $\rho_p$ and $\rho_q$ of terms $h^j x_i^n e_{j,i}(t_n)$, where each coefficient $e_{j,i}$ solves a linear differential equation along the exact solution. Roots common to both polynomials give the smooth error; roots belonging to only one polynomial give parasitic components whose growth is governed by the scalar systems (13)-(14); symmetry makes the odd expansion coefficients vanish and makes certain ratios purely imaginary, which is what lets Theorem 3.1 and Theorem 4.1 turn the expansion into a long-time statement. The double-pendulum analysis works by identifying the transition matrices of (13)-(14) as solutions of $2\times 2$ systems whose eigenvalues are $0$ and $\pm(b-a)$, and arguing that these remain bounded because the modulating function $\alpha(t)$ oscillates erratically relative to the sine frequencies when $\omega_1/\omega_2$ is irrational.

What would settle it

Repeat the double-pendulum experiment with a mass ratio chosen so that the two normal frequencies have a rational ratio; if the Hamiltonian error then grows without bound instead of staying flat, the irrational-ratio boundedness assumption is essential to the case-study conclusion and the uniform $O(h^r)$ parasitic bound would fail for that problem.

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Extended reading notes

Core claim

The central discovery is that the long-time behaviour of a PLMM decouples into contributions attached to the roots of the two first characteristic polynomials, and for symmetric PLMMs the dangerous contributions vanish or stay bounded. Theorem 3.1 states that, when the transition matrices of the scalar auxiliary problems (13)-(14) are bounded, the error terms attached to non-common unitary roots are bounded by a constant times $h^r$ uniformly for $t-t_0$ up to order $h^{-2}$, provided the starting values are accurate enough. Theorem 4.1 gives an explicit formula for the Hamiltonian drift of the smooth numerical solution, showing that odd-order error coefficients vanish and that the remaining drift is controlled by boundary terms and integrals of the form $p^{(k+1)}(s)^T q^{(k+1)}(s)$. The double-pendulum case study then shows that for small oscillations the relevant transition matrices are bounded because the two normal frequencies have irrational ratio, so the symmetric PLMM (42) keeps the Hamiltonian error bounded, unlike the symmetric non-partitioned LMM (43), the Adams method (44), and the mixed PLMM (45).

Load-bearing premise

The load-bearing premise for the double-pendulum conclusion is that the oscillatory integrals in (37) and the determinant factors in (41) stay bounded with time because the ratio of the two normal frequencies is irrational; the paper does not supply a quantitative Diophantine condition or a proof, so if that boundedness fails the parasitic coefficients could grow and destroy the long-term energy conservation the experiments show.

Editorial extensions

If this is right

  • If the central claim is right, symmetric explicit PLMMs with no common roots except 1 provide a practical way to integrate nonseparable Hamiltonian systems over long intervals with bounded energy error, without implicit solves.
  • The same expansion gives a general tool: for any invariant of (1), formula (30) splits the drift into smooth, parasitic, and starting-value contributions, so one can predict whether an invariant will drift linearly, stay bounded, or grow exponentially.
  • For Hamiltonian problems, Theorem 4.1 shows that a symmetric PLMM has no odd-order terms in the drift and that the even-order drift is expressed through boundary terms plus integrals of $p^{(k+1)T} q^{(k+1)}$; when the two methods coincide this becomes a total differential, linking to a modified Hamiltonian.
  • The double-pendulum analysis predicts which method classes are safe: symmetric non-partitioned LMMs will show exponential error growth, non-symmetric methods linear growth, and symmetric PLMMs bounded error, matching the numerical figures.

Reading between the lines

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

  • One could test the same boundedness argument on other nonseparable two-degree-of-freedom Hamiltonians whose linearization has two frequencies with rational ratio; the theory suggests energy drift will appear unless another mechanism bounds the transition matrices.
  • A quantitative version of the double-pendulum conclusion would need explicit Diophantine conditions on the frequency ratio; the paper's heuristic 'erratic oscillation' argument could be replaced by small-divisor bounds, giving explicit ranges of stepsize and time for the $O(h^{-2})$ validity.
  • The same expansion-based reasoning could be carried over to semidiscretizations of dispersive wave equations, where a similarly good long-time behaviour is announced in the paper's companion work.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper develops an asymptotic expansion, in powers of the stepsize, of the global error of partitioned linear multistep methods (PLMMs) for general first-order ODE systems. Theorem 2.1 gives the expansion, Theorem 3.1 states conditions under which the parasitic components associated with non-common unit-modulus roots remain small over long times, and Theorem 4.1 analyses the drift of invariants, in particular the Hamiltonian, along the smooth part of the numerical solution. The authors then apply this framework to small oscillations of the double pendulum, arguing that symmetric PLMMs whose first characteristic polynomials have no common roots except unity are efficient for nonseparable Hamiltonian systems. Numerical experiments compare the symmetric PLMM2 with a symmetric non-partitioned LMM, an Adams method, and a mixed PLMM.

Significance. If the main theorems are correct, this is a valuable contribution: it provides explicit, parameter-free asymptotic formulas for PLMM errors, separates smooth from parasitic components, and identifies a conditional mechanism by which explicit symmetric PLMMs could be competitive for nonseparable Hamiltonian problems. The derivation of the expansion is detailed and the theorems are stated in falsifiable form; the numerical experiments cover four methods and illustrate the claimed qualitative differences. However, the broad claim about nonseparable Hamiltonians is supported only by the double-pendulum case study, and the boundedness arguments in that case study are not rigorous as written. The self-containedness of the main theorem is also compromised by reliance on the unpublished thesis [20].

major comments (4)
  1. [Section 5.2, after Eq. (41)] The verification that the transition matrices associated with (13)-(14) are bounded is incomplete. From the determinant formula (41) the authors obtain information only about the area spanned by two solutions. Together with the existence of one bounded solution along span{(1,-1)^T} or span{(1,1)^T}, this gives |v(t)| sinθ(t) = |det(u(t),v(t))|/|u(t)|, which bounds |v(t)| sinθ(t), not |v(t)| itself. The angle θ(t) can tend to zero while |v(t)| grows, so the determinant being bounded and bounded away from zero does not imply that the second linearly independent solution is bounded. Thus the hypothesis of Theorem 3.1 is not established for the double-pendulum problem, and the paper's central nonseparable claim rests on this case study.
  2. [Section 5.1.1, Eq. (37); Section 5.2, after Eq. (41)] The assertion that the integrals in (37) and the exponents in (41) are bounded because ω1/ω2 is irrational is not a valid general principle: a quasiperiodic function with two irrationally related frequencies can have a nonzero mean and hence a linearly growing integral. The manuscript does not supply a zero-mean computation, a Diophantine condition, or a parity argument for the specific integrands. I do not regard the bare matching-Fourier-components objection as decisive, because the phases of cos(α) and of the sine factors in (37) may cancel, but as written the boundedness statement is unproved. Since this is the only nonseparable verification of the hypothesis of Theorem 3.1, the gap is load-bearing.
  3. [Appendix A, Theorem 2.1] Theorem 2.1 is stated under assumptions (i)-(v), but the proof in Appendix A begins by assuming that all roots of ρp and ρq are single and non-zero and refers to the unpublished Master's thesis [20] for the more general case. In addition, the step from (A3) to εn,ηn = O(h^{2r}) invokes a variant of Lemma 5.6 of [13] whose proof is also deferred to [20]. This means the main expansion theorem is not self-contained as stated. The theorem should either be proved in full generality in the paper, with the stability lemma included, or its statement should be restricted to the case actually proved.
  4. [Abstract and Section 6, Eq. (28)] The closing conclusions state that symmetric PLMMs with no common roots except x1=1 are very efficient for nonseparable Hamiltonian systems. The general theorem for the smooth part, Eq. (28), only gives a Hamiltonian drift of O(t h^{2r}); for t = O(h^{-2}) this is O(h^{2r-2}), which is O(1) for the second-order method PLMM2 used in Section 5. Moreover, Section 5 replaces the exact solution by the linearized approximation (33) without controlling the difference between the two. Consequently the paper's general nonseparable claim goes beyond what is proved; the only support is the double-pendulum case study, whose boundedness arguments are incomplete as noted above.
minor comments (6)
  1. [Section 1] There are typos: 'Hamitonian systems' should be 'Hamiltonian systems', and 'very advantageos results' should be 'very advantageous results'.
  2. [Remark 4.1] In the sentence 'common roots of ρp snd ρq', 'snd' should be 'and'.
  3. [Section 5.2] The word 'paralelogram' is misspelled; it should be 'parallelogram'.
  4. [Reference [19]] The title contains 'Symmetric muultistep methods'; 'muultistep' should be 'multistep'.
  5. [Theorem 3.1] The statement 'Equivalently, they behave as O(h^r) uniformly in time for t−t0 = O(h^{-2})' is not literally equivalent to the preceding O(h^{r+1}) bound for t−t0 = O(h^{-1}); the intended time scales and the nature of the equivalence should be clarified.
  6. [Section 5.3] The 'exact' starting values are computed with ode45 to tolerance 10^{-13}; this is not a precise O(h^{r+1}) starting procedure, so the numerics illustrate but do not exactly test the asymptotic statement for accurate starting values.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the expansion and error-growth results are derived from method coefficients and exact-solution quantities, not from the conclusions; the only self-references are background citations and proof deferrals to a coauthor's thesis.

full rationale

The derivation chain is not circular. Theorem 2.1 builds the asymptotic expansion by substituting an ansatz for the global error into the PLMM residual and choosing the coefficient functions to annihilate terms at each order; the resulting ODEs (10)-(22) are solved from starting-value data, and the error remainder is controlled by a discrete stability lemma attributed to Henrici ([13], Lemma 5.6), with the PLMM variant deferred to coauthor Rodríguez's Master's thesis [20]. This is a self-citation in the proof, but it is not a re-insertion of the theorem's conclusion: the lemma is a standard external stability fact and does not assume the parasitic-root behaviour being proved. The other self-citations ([2], [3], [4], [20]) are background or proofs of standard lemmas. No parameter is fitted to the energy-error data, and no predicted quantity is constructed from the invariant being predicted. The weak spot highlighted by the skeptical analysis is real but is not circularity: Section 5.2's assertion that the integrals and determinants in (41) are bounded because the modulating functions oscillate 'in an erratic way' with ω_1/ω_2 irrational is an unproved analytic claim; if false, the bounded-transition-matrix hypothesis of Theorem 3.1 would fail for the double pendulum, but that would be an incorrect bound rather than a circular reduction. Likewise, the Appendix A restriction to single nonzero roots, with the general case in [20], is a proof-deferral issue, not a by-construction equivalence. The central claims therefore have independent mathematical content; the score reflects only the minor, non-reduction self-citations and deferred proofs.

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

The central theorems rest on standard smoothness and zero-stability assumptions, plus a deferred stability lemma and a problem-specific boundedness condition. No numerical constants are fitted to data, and no new entities are introduced.

assumptions (5)
  • domain assumption Existence, uniqueness and C^∞ regularity of the solution of the IVP (1).
    Section 2 first paragraph assumes smooth enough f and g for all theorems.
  • domain assumption Both PLMM methods are zero-stable, have the same order r, satisfy the local truncation expansion (3), and the starting values are O(h^r) away from the exact values.
    Assumptions (i)-(iii) in Section 2; standard for LMM analysis but they restrict the scope of the theorem.
  • standard math A discrete stability lemma, close to Lemma 5.6 in [13], ensures that the residuals ε_n and η_n are O(h^{2r}) from the recurrence (A3).
    Invoked at the end of Appendix A; the proof is said to be in the unpublished Master's thesis [20].
  • ad hoc to paper All roots of ρp and ρq are single and non-zero for the proof of Theorem 2.1; the more general case is deferred to [20].
    First sentence of Appendix A restricts the proof, while the theorem's stated hypotheses only force simple roots on the unit circle.
  • domain assumption For the double-pendulum study, the linearized small-oscillation solution (33)-(35) is a good enough approximation of the exact solution, with bounded derivatives and rationally independent normal frequencies ω1 and ω2.
    Section 5 uses (33) as a good enough approximation and relies on the irrationality of ω1/ω2 to claim boundedness of the oscillatory integrals.

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Pith. "Pith review of Long-term behaviour of symmetric partitioned linear multistep methods I. Global error and conservation of invariants." pith.science (2026). https://pith.science/paper/Q4SQ4IZ3

@misc{pith2026241113487,
  author       = {Pith},
  title        = {Pith review of: Long-term behaviour of symmetric partitioned linear multistep methods I. Global error and conservation of invariants},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q4SQ4IZ3}},
  note         = {Machine review of arXiv:2411.13487}
}
read the original abstract

In this paper an asymptotic expansion of the global error on the stepsize for partitioned linear multistep methods is proved. This provides a tool to analyse the behaviour of these integrators with respect to error growth with time and conservation of invariants. In particular, symmetric partitioned linear multistep methods with no common roots in their first characteristic polynomials, except unity, appear as efficient methods to approximate non-separable Hamiltonian systems since they can be explicit and show good long term behaviour at the same time. As a case study, a thorough analysis is given for small oscillations of the double pendulum problem, which is illustrated by numerical experiments.

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