{"id":"8b3012b7-05e0-4612-99eb-d97dd44afaca","arxiv_id":"2601.20494","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Monotone finite-volume schemes with approximated nonlocal terms converge to the unique weak entropy solution of 2D nonlocal conservation-law systems, at worst-case rate O(√Δt).","lead":"This paper proves that standard monotone numerical fluxes — Lax-Friedrichs, Upwind, Godunov — can be used for two-dimensional systems of nonlocal conservation laws (e.g., pedestrian flow) after approximating the nonlocal terms, with a proven worst-case error of order √Δt, plus uniqueness of the entropy solution. Generalists read it because it extends trustworthy one-dimensional solver theory to a practically important 2D setting with nonlinear, space- and time-dependent fluxe","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Discrete entropy condition (Prop. 3.12) is asserted without proof; convergence and the O(√Δt) rate both rest on it.","rationale":"The reader’s weakest_assumption was the scope of Assumption 4 in Def. 3.2: it is verified only for Lax–Friedrichs and multiplicative/Godunov-type fluxes, not for arbitrary monotone fluxes. That is a legitimate scope limitation, but it does not threaten the theorem for fluxes that do satisfy Assumption 4. In contrast, Prop. 3.12 is used directly in the proof of Theorem 3.5 for every flux in the declared class, and it is also used in Lemma 4.3 for the error rate. Because the proposition is omitted rather than proved, the central convergence claim is not fully established even for the fluxes whose scope is admitted. This is therefore the most load-bearing concern: if Prop. 3.12 cannot be proved (or is false), the main theorem collapses; if it can be proved, the theorem stands independent of how wide the class of admissible fluxes is. The reader did mention the delegated proofs in their rationale, so we partially agree, but their chosen weakest assumption was the other issue. Our verdict remains CONDITIONAL, same as the reader’s, because the concern calls for an explicit proof of Prop. 3.12 before full acceptance, but does not amount to evidence of incorrectness. There is no ad hominem and no suggestion of dishonesty; the paper openly flags the omission. The proposed test—an independent derivation or a counterexample—directly settles whether the gap is real or merely cosmetic.","tokens_in":39701,"tokens_out":5353,"duration_ms":60327,"concrete_test":"Independently derive Prop. 3.12 from Def. 3.2 and the CFL condition (6) for the two-dimensional scheme (5). Write out the discrete entropy inequality using the entropy flux F^{κ} and verify that the extra terms involving sgn(ρ^{n+1}_{i,j} − κ)(f^k_1(t_n,x_{i+1/2,j},κ,R^n_{i+1/2,j}) − f^k_1(t_n,x_{i−1/2,j},κ,R^n_{i−1/2,j})) and the analogous y-term are controlled by the CFL condition and Lipschitz constants. If the inequality cannot be established for every flux satisfying Def. 3.2, construct a counterexample flux in that class for which the residual is positive, which would refute Thm. 3.5 as stated. A successful complete derivation would resolve the concern.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3.5’s convergence proof relies on a Lax-Wendroff-type argument that passes to the limit in the entropy inequality. For that limit to be a weak entropy solution, the numerical scheme must satisfy the discrete entropy condition of Prop. 3.12. The proposition is stated without proof: the text says “the proof is analogue to [5, 18] and we do not go into detail here.” This is a self-declared omission in a load-bearing step. The 2D system setting is not a trivial translation of the 1D scalar arguments: the inequality contains extra terms involving sgn(ρ^{n+1}_{i,j} − κ) times differences of f^k across interfaces in both coordinate directions, and the nonlocal approximations R^n enter the numerical fluxes. A missing proof here does not merely narrow the scope; it leaves the central convergence theorem unverified even for fluxes that fully satisfy Def. 3.2. The same proposition is later “exploited for Λ1” in Lemma 4.3, so the O(√Δt) error estimate inherits the same gap. The concern is not that Prop. 3.12 is false; it is that the main theorem and the error estimate are not independently established without a complete proof of this key estimate.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a class of first-order finite-volume schemes for two-dimensional systems of nonlocal conservation laws (1), based on approximating the nonlocal convolutions at cell interfaces via (4a)-(4b) and then applying well-known monotone numerical fluxes satisfying Definition 3.2. The main convergence theorem (Theorem 3.5) states that, under Assumption 2.2 and the CFL condition (6), any scheme of the form (5) with a flux from Definition 3.2 converges in L1_loc to the unique weak entropy solution of Definition 2.1. The proof is built from a maximum principle (Theorem 3.7), a BV estimate (Theorem 3.9), time-continuity (Proposition 3.10), an L∞ bound (Lemma 3.11), and a discrete entropy condition (Proposition 3.12) that is stated without proof. The paper also proves a Kuznetsov-type lemma (Lemma 4.1), uniqueness of the weak entropy solution (Theorem 4.2), and an O(√Δt) error estimate (Lemma 4.4) via a relative entropy bound (Lemma 4.3). Numerical experiments are reported for a reversible encryption-decryption model and a two-population pedestrian flow model, comparing Lax-Friedrichs-type, Upwind-type, and Godunov-type schemes.","tokens_in":39865,"tokens_out":5931,"duration_ms":65387,"significance":"If the missing proofs are supplied, this is a meaningful advance: it extends the one-dimensional general monotone-scheme framework of [18] to two-dimensional systems and provides a worst-case O(√Δt) error estimate without dimensional splitting. The paper's strengths are the detailed flux-difference estimates (e.g., (15b)-(19), (20a)-(22), and the five-term bound (25)), the verification of the structural condition for Lax-Friedrichs and multiplicative fluxes, and the numerical validation—in particular the clear separation between the worst-case rate 0.5 and the observed smooth-data rate approximately 1, which indicates the bound is not reverse-engineered. However, the central convergence theorem and the error estimate currently rest on an unproved discrete entropy inequality and on several estimates delegated to [3], so the contribution is not yet fully established as written.","major_comments":[{"comment":"The discrete entropy condition is asserted with the sentence \"the proof is analogue to [5, 18] and we do not go into detail here.\" This proposition is load-bearing: Theorem 3.5 invokes it in the Lax-Wendroff limit, and Lemma 4.3 uses it to control the term Λ1. The inequality is not a direct translation of the 1D scalar argument: it contains extra sgn(ρ^{n+1}_{i,j}−κ) terms coupled to differences of f^k across interfaces in both coordinate directions, and the nonlocal approximations R^n enter the numerical fluxes. The manuscript should provide a complete proof, or a precise reduction that accounts for all 2D and nonlocal terms, rather than leaving this to references.","section":"Section 3.1, Proposition 3.12"},{"comment":"The O(√Δt) error estimate depends on the Kuznetsov-type lemma and the relative entropy bound, but their proofs are only sketches. Lemma 4.1 refers to [3, Lem. 4.1] and says \"the remaining terms can be estimated as in [3]\"; Lemma 4.3 repeatedly says \"similar to [3, pp.…]\" and \"we can proceed as in [3]\" for key bounds such as (28), (29), and the eε terms. Because the flux in (1) is more general than the one in [3]—it depends explicitly on t, x, and R[ρ], and the system is genuinely two-dimensional—these references do not by themselves establish the needed estimates. Please supply full proofs or a detailed, step-by-step translation of every referenced estimate, especially for the terms involving the nonlocal flux differences and the test-function estimates.","section":"Section 4, Lemma 4.1 and Lemma 4.3"},{"comment":"Assumption 4 of Definition 3.2 is the extra structural condition that makes the BV estimate (Theorem 3.9) work. It is verified only for the Lax-Friedrichs flux (7) in Appendix A and for multiplicative fluxes (8) in Proposition A.1. The introduction and abstract claim that \"any monotone numerical flux\" can be used, but for a generic monotone flux outside these two classes condition 4 is not established. In particular, the advertised Godunov-type scheme (10) is defined only for multiplicative fluxes. The statements should be sharpened to say that Theorem 3.5 applies to fluxes satisfying all conditions of Definition 3.2, and the paper should either prove condition 4 for a wider class or explicitly acknowledge this limitation in the main claims.","section":"Section 3, Definition 3.2 and Theorem 3.5"}],"minor_comments":[{"comment":"The space BV(R; R^K) should be BV(R^2; R^K); the symbol R appears without dimension.","section":"Definition 2.1"},{"comment":"The text says the smooth initial-data example uses encryption time T=0.3, while the caption of Figure 3 and the table say t=0.75. Please harmonize these values.","section":"Section 5.1"},{"comment":"The phrase \"a uniformly convergent subsequence in L1_loc(R^2) on every bounded interval [0,T]\" is imprecise. The compactness via [29, Lem. 1] yields convergence in a space such as C([0,T];L1_loc(R^2)) or L1_loc([0,T]×R^2); please state the exact topology.","section":"Proof of Theorem 3.5"},{"comment":"There are typos: \"Montone-based\" in Definition 3.2, \"Kruž zkov\" instead of \"Kružkov\", \"Encypted\" and \"inital\" in Section 5.1 figures. These do not affect the mathematics but should be corrected.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is promising and the overall strategy appears sound, but the unproved discrete entropy condition (Proposition 3.12) and the extensive delegation to [3] in the error-analysis section are too large to remain as \"we do not go into detail here.\" I would be willing to accept after a revision that supplies these proofs or clearly delineates exactly which parts are imported and verifies their hypotheses in the present 2D system setting. The numerical experiments and the appendix verifications are positive indicators, and the claimed rate separation (0.5 worst case vs. observed ~1 for smooth data) supports the credibility of the analysis."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper genuinely extends [18] and [3] to 2D systems with t,x-dependent nonlinear fluxes, and the O(√Δt) error estimate is a real result. The BV estimate and the five-term flux-difference bound are written out in enough detail to be convincing. The numerical section is honest: the discontinuous test case tracks the 0.5 rate, the smooth case does better, and the authors explicitly note the worst-case bound is not sharp. No red flags on data or fabrication.\n\nThe soft spots are real. The discrete entropy condition (Prop. 3.12) is load-bearing for the convergence theorem and for Lemma 4.3, and the proof is delegated: 'the proof is analogue to [5, 18] and we do not go into detail here.' In 2D with nonlocal terms and sgn(ρ^{n+1}−κ) differences across both interfaces, this is not a trivial copy; a referee needs to see the proof or a precise statement-to-reference mapping. Lemma 4.1 and parts of Lemma 4.3 are similarly sketched with references to [3]. I think the reader's soundness score of 6 is about right: the architecture is credible, but the central estimate is currently unverified.\n\nSecond issue: Assumption 4 in Def. 3.2 is verified only for Lax-Friedrichs and multiplicative/Godunov-type fluxes. Theorem 3.5 states convergence for any flux from Def. 3.2, but the generic class is wider than the verified examples. That mismatch should be fixed, either by proving Assumption 4 for all monotone fluxes or by narrowing the theorem's scope. This is a proportionate concern, not a fatal one: the verified examples cover most practical models, including pedestrian and material flow.\n\nBottom line: this is accept-shaped with revision, not a desk reject. The 2D framework and the error estimate are worth having. I would send it to a referee and ask for a complete proof of Prop. 3.12, a check on Assumption 4's scope, and ideally code for the numerical examples. The paper shows clear thinking and honest engagement with the literature, and the self-critical remarks count in its favor.","headline":"A serious 2D convergence framework for monotone-based nonlocal schemes, with a real but fixable gap: the discrete entropy condition is stated without proof.","tokens_in":40533,"tokens_out":1789,"would_cite":true,"duration_ms":21184,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35L65","35L03","65M12","65M15","76A30"],"pacs":[],"model":"deepseek-v4-flash","headline":"A class of monotone finite-volume schemes converges for 2D systems of nonlocal conservation laws.","keywords":["nonlocal conservation laws","monotone schemes","finite volume methods","Godunov flux","Lax-Friedrichs flux","weak entropy solutions","convergence rates","pedestrian flow models"],"falsifier":"A decisive test: implement the Godunov-type scheme (10) on the two-population crowd model (33) on a sequence of refined grids and check whether the discrete total variation stays uniformly bounded as Δx→0. If TV grows without bound while the flux satisfies all of Definition 3.2, the BV estimate (Theorem 3.9) and hence the convergence proof would fail; if it stays bounded but the L1 error at a fixed time decays slower than O(√Δt), the error estimate Lemma 4.4 would be contradicted.","tokens_in":39388,"feed_emoji":"🧮","tokens_out":7810,"duration_ms":78840,"temperature":0.7,"pith_summary":"The paper proves that any finite-volume scheme built from a classical monotone numerical flux — such as Lax-Friedrichs or Godunov — converges to the unique weak entropy solution of a two-dimensional system of nonlocal conservation laws, provided the nonlocal convolutions are approximated consistently at cell interfaces and the flux satisfies one additional structural bound. This matters because multidimensional nonlocal models of pedestrian flow, crowd movement, and conveyor-belt traffic had previously been handled almost exclusively with more diffusive Lax-Friedrichs-type schemes; the new result makes sharper Godunov-type fluxes available for nonlinear, nonlocal fluxes. The paper also derives an existence–uniqueness theorem for the system and a worst-case L1 error estimate of order O(√Δt).","feed_headline":"Monotone schemes converge for 2D nonlocal conservation laws","feed_subtitle":"General class includes Godunov-type fluxes; worst-case L1 error is O(√Δt) to a unique entropy solution.","key_machinery":"The central object is the monotone-based numerical flux function from Definition 3.2 — a numerical flux that is consistent with the approximated nonlocal flux at each cell interface, nondecreasing in its own cell value and nonincreasing in the neighbour's, Lipschitz, and constrained by a fourth condition: shifting the flux evaluation one cell in the transverse direction must change it by at most a local variation Δx|ρ_{i,j}−ρ_{i−1,j}| plus a quadratic term Δx²M|ρ|. This fourth condition, assumption 4, is the load-bearing piece: it makes the spatial total-variation estimate (Theorem 3.9) close, from which compactness and convergence follow. The nonlocal convolution is approximated by a compos","core_discovery":"The central result, Theorem 3.5, states that a numerical scheme of the form (5), using a numerical flux satisfying the four conditions of Definition 3.2 (consistency with the approximated interface flux, one-sided monotonicity, Lipschitz continuity, and a transverse-variation bound) and run under the CFL condition (6), converges in L1_loc to the unique weak entropy solution of the nonlocal system (1). The scheme is not monotone in every argument because the nonlocal terms enter through the approximated flux, but it is 'monotone-based' in the local sense. Combined with Lemma 4.4, the theorem gives the worst-case convergence rate ∥ϱ(T,·) − ρΔ(T,·)∥_{L1} = O(√Δt). The results cover, in particul","pith_inferences":["Assumption 4 of Definition 3.2 is verified in the paper only for the Lax-Friedrichs flux and for multiplicative fluxes (which include Godunov-type); the theorem's practical coverage is therefore narrower than its statement 'any flux from Definition 3.2' suggests. Checking this transverse-variation bound for other monotone fluxes (e.g., Engquist-Osher) would extend the result.","The O(√Δt) error bound is a worst-case estimate; the numerical experiments on smooth data in Section 5 show roughly first-order convergence, so the bound may not be tight for smooth solutions.","The decoupling at each time step via interface approximation of the nonlocal terms suggests that asynchronous or parallel-in-time treatments of multi-population models could inherit the same convergence guarantees.","The same framework could be carried to three spatial dimensions, provided the analogue of assumption 4 is formulated and verified for the chosen flux and the quadrature error of the convolution at face centers is controlled."],"forward_implications":["Godunov-type fluxes are now rigorously justified for two-dimensional nonlocal conservation laws with multiplicative or Lax-Friedrichs-type structure, not just for local or one-dimensional problems.","A worst-case L1 error of O(√Δt) is guaranteed for the entire scheme class, matching the optimal rate known for monotone schemes on local nonlinear problems.","The existence and uniqueness proof applies to the general nonlocal system (1), strengthening earlier results that required linear or special fluxes.","Since the scheme is derived from a semi-discrete form without dimensional splitting, it can serve as a starting point for higher-order extensions.","The framework also accommodates dimensional splitting, doubling the allowable CFL number."],"fun_headline_variants":["2D nonlocal conservation laws: monotone-based schemes converge","Monotone-based flux schemes solve 2D nonlocal systems","Convergent monotone-based schemes for 2D nonlocal models","O(√Δt) error for monotone schemes on nonlocal 2D systems","Monotone-based numerics for nonlocal conservation in 2D"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The convergence theorem leans on the fourth condition of Definition 3.2 — a bound on how the numerical flux changes when shifted one cell in the transverse direction — which the paper verifies only for the Lax-Friedrichs and multiplicative flux classes, so the result's reach is narrower than the definition's generic wording suggests.","fun_headline_variants_meta":{"raw":{"variants":["2D nonlocal conservation laws: monotone-based schemes converge","Monotone-based flux schemes solve 2D nonlocal systems","Convergent monotone-based schemes for 2D nonlocal models","O(√Δt) error for monotone schemes on nonlocal 2D systems","Monotone-based numerics for nonlocal conservation in 2D"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000154,"raw_usage":{"total_tokens":1021,"prompt_tokens":691,"completion_tokens":330,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":435,"completion_tokens_details":{"reasoning_tokens":234}},"tokens_in":435,"tokens_out":330,"duration_ms":3529,"temperature":1.0,"reasoning_tokens":234,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T07:20:17.304425+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A decisive test: implement the Godunov-type scheme (10) on the two-population crowd model (33) on a sequence of refined grids and check whether the discrete total variation stays uniformly bounded as Δx→0. If TV grows without bound while the flux satisfies all of Definition 3.2, the BV estimate (Theorem 3.9) and hence the convergence proof would fail; if it stays bounded but the L1 error at a fixed time decays slower than O(√Δt), the error estimate Lemma 4.4 would be contradicted.","supporting_citations":[],"review_version":1}