{"id":"8674c636-3409-4539-a73d-9d048f95f7c8","arxiv_id":"2411.13487","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Symmetric partitioned linear multistep methods with no common root besides 1 keep the Hamiltonian error bounded over long times for nonseparable systems such as a double pendulum.","lead":"This paper proves an asymptotic expansion of the global error for partitioned linear multistep methods and uses it to explain long-time energy behaviour. It shows that explicit symmetric partitioned methods can keep the Hamiltonian error bounded for nonseparable systems such as the double pendulum, unlike standard multistep alternatives.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The boundedness claim for the double-pendulum case rests on an incorrect 'irrational frequency' argument: cos(α(s)) contains Fourier components at the same frequencies as the sines in (37), so the integrals and the exponents in (41) can grow linearly unless phases cancel; the nonseparable case…","rationale":"The reader's weakest assumption identifies the same load-bearing issue: the double-pendulum analysis relies on an informal irrational-frequency argument instead of a quantitative proof that the integrals in Section 5.1 and the transition matrices of Section 5.2 remain bounded. My stress-test confirms this is the weakest point, and in fact the informal argument is not merely incomplete but mathematically incorrect in general: exact resonances produce secular terms regardless of irrationality. The paper's core theorems (Theorem 2.1, Theorem 3.1, Theorem 4.1) may be correct as conditional statements, and the numerical experiments are consistent with the claims for the special initial conditions used. However, the abstract's broad assertion of efficiency for nonseparable Hamiltonian systems is only as strong as the verification of the boundedness hypothesis, and that verification is flawed. The verdict should remain CONDITIONAL: the paper should be accepted only if the authors supply a rigorous proof of the boundedness of (37) and of the transition matrices in (13)-(14) for the double-pendulum case (or explicitly restrict the claims to cases where the relevant averages vanish). The missing proof from [20] also warrants a self-contained treatment of Lemma 5.6 before the expansion theorem can be fully verified.","tokens_in":19845,"tokens_out":11088,"duration_ms":86395,"concrete_test":"Compute the zero-frequency Fourier coefficient of the integrand in (37) for the paper's parameters but with generic phases, e.g. δ1=0.1, δ2=0.2, over T=10^5: if |(1/T)∫_0^T cos(ω1 s) sin(ω1 s) cos(A' cos(ω1 s−δ1)+B' cos(ω2 s−δ2)) ds| > c > 0, the integral grows linearly and the boundedness claim after (37) is false. Separately evaluate the exponent in (41), λ_p,i,p ∫_0^T (b(τ)−a(τ)) dτ, over the same interval; if it is not bounded, Theorem 3.1's hypothesis fails for the double pendulum.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim that symmetric PLMMs are efficient for nonseparable Hamiltonians is conditional on Theorem 3.1's bounded-transition-matrix hypothesis. The paper's only nonseparable verification is the double pendulum, and the boundedness argument there is not sound. In Section 5.1.1 the authors assert that the second integral in (37), ∫ cos(ω_i s) sin(ω_j s) cos(α(s)) ds, is bounded because ω1/ω2 is irrational and α(s) oscillates 'erratically'. But cos(α(s)) expands as a Fourier series with frequencies mω1+nω2; in particular it has components at 2ω1, 2ω2, ω1+ω2 and ω1−ω2. When these match the sine frequencies in (37), the product has a nonzero constant (zero-frequency) term, producing linear growth in t. For the specific initial conditions δ1=π, δ2=0 these constants vanish, but the paper claims the conclusion holds for generic small oscillations and gives no phase condition. Likewise, Section 5.2's boundedness of the determinants in (41) rests on the same 'erratic' assertion; b(t)−a(t) is a product of trigonometric polynomials, and the integrals in the exponents can drift linearly unless their averages vanish. Thus the case study does not verify the hypothesis of Theorem 3.1, leaving the abstract's efficiency claim for nonseparable systems unsupported. Additionally, the proof of Theorem 2.1 in Appendix A outsources a key stability lemma ('Lemma 5.6 in [13]') to the unpublished Master's thesis [20], so the main expansion theorem is not self-contained.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":20212,"tokens_out":21367,"duration_ms":229965,"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":[{"comment":"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.","section":"Section 5.2, after Eq. (41)"},{"comment":"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.","section":"Section 5.1.1, Eq. (37); Section 5.2, after Eq. (41)"},{"comment":"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.","section":"Appendix A, Theorem 2.1"},{"comment":"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.","section":"Abstract and Section 6, Eq. (28)"}],"minor_comments":[{"comment":"There are typos: 'Hamitonian systems' should be 'Hamiltonian systems', and 'very advantageos results' should be 'very advantageous results'.","section":"Section 1"},{"comment":"In the sentence 'common roots of ρp snd ρq', 'snd' should be 'and'.","section":"Remark 4.1"},{"comment":"The word 'paralelogram' is misspelled; it should be 'parallelogram'.","section":"Section 5.2"},{"comment":"The title contains 'Symmetric muultistep methods'; 'muultistep' should be 'multistep'.","section":"Reference [19]"},{"comment":"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.","section":"Theorem 3.1"},{"comment":"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.","section":"Section 5.3"}],"recommendation":"major_revision","confidential_remarks":"The manuscript fits the scope of a numerical analysis journal and the expansion theorems are likely of interest. The main obstacle is not novelty but verification and self-containedness: the nonseparable case-study argument needs a rigorous boundedness proof, and the reliance on the authors' own unpublished thesis [20] for a key lemma and for the general case of Theorem 2.1 should be addressed before publication. I see this as fixable within the manuscript's scope, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper deserves a serious referee, but the abstract overstates the case. The real contribution is Theorem 2.1: an asymptotic expansion of the global error for partitioned linear multistep methods, extending Console and Hairer's separable analysis to nonseparable systems. The invariant-drift formulas in Theorems 4.1 and 4.2 are also new and useful. That part of the paper is carefully done and, as far as I can see, correct.\n\nThe soft spots are real but not fatal. First, the proof of Theorem 2.1 is not self-contained: a key stability lemma is outsourced to an unpublished Master's thesis [20], and the appendix only treats single non-zero roots. That is a self-containment problem a referee should insist on fixing, perhaps by including a proof of the lemma or a published reference.\n\nSecond, the double-pendulum case study contains a flawed boundedness argument. The paper claims the integrals and transition matrices stay bounded because the frequency ratio is irrational and the modulation is 'erratic'. As the stress-test note correctly points out, cos(alpha(s)) has Fourier components at exactly the frequencies that match the sines in (37), so those integrals can grow linearly unless phases cancel. For the specific initial conditions used in the numerical experiments (which correspond to delta_1=pi, delta_2=0) the dangerous constants do vanish, so the figures are probably fine. But the paper states the conclusion generically and gives no phase condition. That needs to be corrected: either prove the boundedness under a stated Diophantine or phase condition, or explicitly restrict the claim to initial conditions where the cancellation holds.\n\nThe citation pattern is not a problem; self-citation to [5], [7], and [20] is legitimate when the results are actually used. The main theorem is a genuine step forward and the numerical evidence is consistent with the theory. The flaws are in the presentation of the case study and the reliance on an unpublished source. I would send it to peer review and ask for a revision that tightens Section 5 and makes the proof of Theorem 2.1 self-contained or clearly deferred to a citable reference.","headline":"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.","tokens_in":20724,"tokens_out":4740,"would_cite":true,"duration_ms":46627,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65L06","65P10","37M15"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["partitioned linear multistep methods","symmetric methods","global error asymptotic expansion","Hamiltonian systems","nonseparable Hamiltonian","invariant preservation","double pendulum","long-time integration"],"falsifier":"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.","tokens_in":19634,"feed_emoji":"🧮","tokens_out":5698,"duration_ms":58912,"temperature":0.7,"pith_summary":"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.","feed_headline":"Symmetric partitioned multistep methods keep energy error bounded","feed_subtitle":"A new error expansion shows why explicit symmetric PLMMs beat other multistep methods on nonseparable Hamiltonians.","key_machinery":"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.","core_discovery":"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).","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"supplies the earlier long-term stability analysis of symmetric PLMMs for separable Hamiltonians that this paper extends to nonseparable ones.","marker":"[7]"},{"why":"provides the error-growth analysis for multistep methods applied to periodic orbits that motivates the PLMM comparison.","marker":"[5]"},{"why":"supplies the symmetry facts that odd error coefficients vanish and that the growth parameters in (7), (13), (14) are real.","marker":"[22]"},{"why":"provides the linearized double-pendulum solution with normal frequencies $\\omega_1$, $\\omega_2$ used in the case study.","marker":"[15]"},{"why":"provides the discrete stability lemma used to conclude that the remainder in the asymptotic expansion is $O(h^{2r})$.","marker":"[13]"},{"why":"shows that linear multistep methods cannot be symplectic, framing why symmetry plus partitioning is the route taken here.","marker":"[8]"},{"why":"establishes conjugate-symplecticity of symmetric LMMs, the background for their good invariant behaviour.","marker":"[9]"}],"fun_headline_variants":["Explicit symmetric PLMMs tame Hamiltonian drift","Symmetric partitioned multistep methods keep energy bounded","How symmetric PLMMs achieve bounded energy error","New proof: symmetric PLMMs preserve invariants long-term","Bounded Hamiltonian error with explicit symmetric PLMMs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Explicit symmetric PLMMs tame Hamiltonian drift","Symmetric partitioned multistep methods keep energy bounded","How symmetric PLMMs achieve bounded energy error","New proof: symmetric PLMMs preserve invariants long-term","Bounded Hamiltonian error with explicit symmetric PLMMs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000233,"raw_usage":{"total_tokens":1461,"prompt_tokens":884,"completion_tokens":577,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":500,"completion_tokens_details":{"reasoning_tokens":501}},"tokens_in":500,"tokens_out":577,"duration_ms":5998,"temperature":1.0,"reasoning_tokens":501,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:21:47.124096+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Console & E","cited_arxiv_id":null,"evidence_quote":"supplies the earlier long-term stability analysis of symmetric PLMMs for separable Hamiltonians that this paper extends to nonseparable ones."},{"cited_title":"Cano & J.M","cited_arxiv_id":null,"evidence_quote":"provides the error-growth analysis for multistep methods applied to periodic orbits that motivates the PLMM comparison."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the symmetry facts that odd error coefficients vanish and that the growth parameters in (7), (13), (14) are real."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the linearized double-pendulum solution with normal frequencies $\\omega_1$, $\\omega_2$ used in the case study."},{"cited_title":"Henrici, Discrete Variable Methods in Ordinary Differential Equations, Wiley, 1962","cited_arxiv_id":null,"evidence_quote":"provides the discrete stability lemma used to conclude that the remainder in the asymptotic expansion is $O(h^{2r})$."},{"cited_title":"Eirola & J","cited_arxiv_id":null,"evidence_quote":"shows that linear multistep methods cannot be symplectic, framing why symmetry plus partitioning is the route taken here."},{"cited_title":"Hairer, Conjugate-symplecticity of linear multistep methods , J","cited_arxiv_id":null,"evidence_quote":"establishes conjugate-symplecticity of symmetric LMMs, the background for their good invariant behaviour."}],"review_version":1}