{"id":"be100658-985c-404a-b4c1-c13a4329d6e8","arxiv_id":"2501.07934","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"TRT lattice Boltzmann schemes converge to the unique entropy solution of a scalar nonlinear conservation law when the relaxation parameters satisfy two explicit monotonicity inequalities.","lead":"This paper proves that two-relaxation-times (TRT) lattice Boltzmann schemes can be made stable and convergent for nonlinear conservation laws, under two explicit parameter inequalities. The result matters because TRT schemes can be tuned to reduce numerical blurring compared with simpler BGK schemes while still keeping a rigorous convergence guarantee.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of the central convergence claim is not self-contained: the flux-limit step in Theorem 4.10 and the entropy-limit passage in Theorem 4.11 are deferred to supplementary/formal arguments, so the manuscript alone does not yet verify convergence to the entropy solution.","rationale":"The paper's conditional theorem is coherent. I checked the main estimates: Proposition 3.3 reduces Jacobian non-negativity to linear half-space conditions; Proposition 4.4's link-wise \\ell^1 contraction sums to one; Proposition 4.8's recursion is contracting because \\omega_s,\\omega_a\\neq 2; and Theorem 4.11's Kruzhkov-entropy calculation is standard once starred and unstarred quantities coincide in the limit. The affine equilibrium ansatz (3.1)/(3.3) is genuinely load-bearing for these steps, but it is an explicit hypothesis rather than a hidden assumption; the paper even acknowledges alternatives such as flux-decomposition schemes. The genuinely weak spot is verifiability of the central claim: the two limit passages that connect the discrete update to the weak entropy formulation are not written out in the manuscript. Deferring auxiliary lemmas to supplementary material is acceptable, but here the deferred argument covers the main Lax-Wendroff mechanism for the collision term, so the reader's CONDITIONAL verdict is appropriate. I would not reject: the deferred steps are standard and likely correct, and the numerical experiments are consistent with the theory. The proposed check would settle whether the F-limit is actually derivable from the stated estimates.","tokens_in":25746,"tokens_out":29678,"duration_ms":299856,"concrete_test":"Obtain the supplementary material and check the derivation of the limit of the term F in Theorem 4.10. Verify that F converges to the flux integral using only (i) Proposition 4.8, i.e. \\|f_Delta - f_eq(u_Delta)\\|_{L^\\infty_t L^1_x} = O(Delta x); (ii) Lipschitz continuity of R on the compact set K; and (iii) the consistency conditions (2.5). In particular, look for a bound of the form |\\int (R_i(f_Delta)-f_Delta,i) \\psi(\\cdot+\\Delta t c_i) dx dt| \\le C \\|f_Delta - f_eq(u_Delta)\\|_{L^\\infty_t L^1_x} \\to 0. If no such estimate appears, or if an additional assumption is needed beyond Corollary 4.6, the weak-form proof has a gap that should be closed in the main text.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorems 4.10 and 4.11 are the paper's central claim. In Theorem 4.10, after reducing the scheme to the balance D - I = F, the text states: \"We are left to handle F ... but can be treated by analogous arguments—see supplementary material.\" That term F is precisely where the lattice Boltzmann collision/streaming update is converted into the weak form of (1.1); it requires proving that the integral of (R_i(f_Delta) - f_Delta,i) against shifted test functions tends to zero and that the shifted distribution terms produce the flux integral. This relies on Proposition 4.8 and on Lipschitz continuity of R on K, but the derivation is not in the main text. Theorem 4.11's entropy inequality is justified only as \"Formally (rigorous justifications ... can be obtained as for Theorem 4.10)\". Several supporting propositions (3.9, 3.10, 3.11) are also stated with proofs in supplementary material. This is not an internal contradiction, but it means the central convergence result is conditional on external material not reproduced here. The reader's identified weakest assumption, the affine equilibrium ansatz (3.1)/(3.3), is real but explicitly stated; the deferred limit passages are the more immediate obstruction to verifying the claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies two-relaxation-times (TRT) lattice Boltzmann schemes for multidimensional scalar conservation laws with smooth fluxes and BV initial data. Under an affine-in-the-conserved-moment ansatz for the equilibria (Eq. (3.1) with the symmetry assumptions (3.3)), the authors derive monotonicity conditions on the relaxation parameters (Proposition 3.3, conditions (3.5)--(3.6)). They then use monotonicity to prove invariance of a compact set, L1-contractivity, total-variation bounds, and closeness-to-equilibrium estimates (Propositions 4.2--4.8). These estimates are assembled into a Crandall--Majda--Lax--Wendroff type argument (Theorems 4.10 and 4.11) claiming convergence, along a subsequence, to the unique weak entropy solution of the conservation law. Numerical experiments for D1Q3 and D2Q5 schemes illustrate the monotonicity region M and report convergence rates.","tokens_in":26142,"tokens_out":3871,"duration_ms":38291,"significance":"If the deferred proofs are correct, the paper gives the first convergence analysis covering the full TRT parameter range for scalar conservation laws, improving on the BGK-specific results of reference [1]. The explicit monotonicity conditions (3.5)--(3.6) are concrete and testable, and they depend on the equilibrium coefficients L_i and N_{i,k} in a way that practitioners can check. A notable strength is that the main estimates visible in the main text (Propositions 3.3, 4.4, 4.5, 4.7, 4.8) are algebraically explicit and internally consistent. The paper is also honest about the scope of its assumptions: the affine equilibrium ansatz (3.1)--(3.3) is stated up front and is not hidden. The numerical experiments, including mesh-refinement tables and invariant-compact-set checks, support the qualitative claim that TRT schemes can reduce numerical diffusion while preserving monotonicity.","major_comments":[{"comment":"The convergence to a weak solution is not proven in the main text: after deriving the balance D - I = F, the proof stops at 'We are left to handle F ... but can be treated by analogous arguments—see supplementary material.' The term F contains precisely the lattice Boltzmann collision/streaming contribution that must be converted into the flux integral of the weak form. The omitted step requires showing that finite differences of shifted test functions against R_i(f_Delta) produce the flux term and that the remainder vanishes, using Proposition 4.8 and the Lipschitz continuity of R on K. Without this derivation, Theorem 4.10 is conditional on material not present in the manuscript.","section":"§4.1.5, Theorem 4.10, Eq. (4.8)"},{"comment":"The entropy inequality is justified only formally: the sentence 'Formally (rigorous justifications ... can be obtained as for Theorem 4.10)' covers the limit passage in the discrete Kruzhkov entropy balance. Since Theorem 4.10 itself has an unproved flux limit, the entropy solution claim inherits the same gap. The main text thus does not establish convergence to the entropy solution, which is the paper's headline result.","section":"§4.2, Theorem 4.11"},{"comment":"Three results that are load-bearing for the subsequent estimates are stated with proofs in supplementary material: Proposition 3.9 (monotonicity of the equilibria and the characterization of M_BGK) is used in the proof of Proposition 4.5 to justify the L1-contractivity bound and the TV bound; Proposition 3.10 (ωs ≠ 2 and ωa ≠ 2 for monotone schemes) is used in Proposition 4.8 to ensure that max(|1−ωs|,|1−ωa|) < 1; Proposition 3.11 is used for the geometric description of M. The main text therefore relies on unproved results for its core stability and convergence estimates.","section":"§3.2, Propositions 3.9, 3.10, 3.11"}],"minor_comments":[{"comment":"The title as rendered contains stray spaces ('RELAXA TION-TIMES LA TTICE BOL TZMANN'); these are clearly formatting artifacts and should be fixed.","section":"Title"},{"comment":"The relaxation operator R_i is defined on R^q, but the collision step is written for each velocity index i; it might help to state explicitly that R_i depends on the full vector (f_1,...,f_q) and that the Jacobian entries are evaluated at generic points in K, not only at equilibrium.","section":"§2.2.2, Eq. (2.3)"},{"comment":"The term min(2−ωs−ωa, 0, ωa−ωs) in (3.6) is not immediately transparent; a short explanation of how it arises from taking the minimum of the three right-hand sides in (3.8)--(3.10) would help readability.","section":"§3.2, conditions (3.5)--(3.6)"},{"comment":"For the case ωa = 2 with hat initial data, the reported empirical order is about 1.3, but the row with ∆x = 2.44e-04 is followed by a jump to 1.76e+13 in Table 2 for the BGK case; this is presumably an instability artifact and should be briefly explained in the caption or text.","section":"§5.1.3, Table 1"},{"comment":"References [24] and [23] are cited for monotonicity-preserving linear multistep methods; the connection to the lattice Boltzmann context in Remark 3.5 would be clearer if the specific properties used were indicated.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is not self-contained: the convergence theorems and several supporting propositions depend on proofs relegated to supplementary material. If the journal has a policy of publishing supplementary material, I recommend that the supplementary proofs be made available to reviewers and that the main text clearly state which claims are proven where. The numerical experiments are convincing, but they cannot substitute for the missing flux-limit and entropy-limit arguments."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe short version: this is a genuine extension of the BGK convergence analysis to TRT lattice Boltzmann schemes, with explicit monotonicity conditions and a convergence theorem. If you work on LBM or monotone schemes for conservation laws, it's worth your time. The central proof, however, is not fully self-contained in this preprint.\n\nWhat's actually new: the sufficient conditions (3.5)–(3.6) for the TRT relaxation operator to be monotone, and the resulting Lax–Wendroff-type convergence to the unique entropy solution. The BGK case is recovered, and the TRT conditions give a larger parameter region, confirming the practical claim that TRT reduces numerical diffusion without sacrificing stability. The convexity of the monotonicity region (Lemma 3.8) is a nice structural observation. The numerical experiments are honest and back up the theory; the blow-up at ωa=2 is a good sanity check.\n\nThe soft spots are about completeness. Several load-bearing arguments are deferred to supplementary material: Propositions 3.9–3.11 (monotonicity of equilibria, ω≠2, symmetry of the monotonicity region) and, more importantly, the flux-limit passage in Theorem 4.10 and the entropy-limit passage in Theorem 4.11. The phrase 'can be treated by analogous arguments—see supplementary material' is doing real work: that's where the collision/streaming update becomes a weak form of the conservation law. As the preprint stands, the central claim is conditional on those proofs. The conclusions also contain an unproven claim that the magic-case conditions are optimal; that needs a proof or a softer statement. The affine equilibrium ansatz (3.1)/(3.3) is a real restriction, but it is explicitly assumed and standard in this line of work, so I don't count it as a flaw.\n\nNone of this looks like a fatal defect. The visible algebra is consistent, the strategy is sound, and the missing pieces are likely routine but not trivial. I'd give this a serious referee if it crossed my desk, with the request that the supplementary proofs be included or at least made publicly accessible. It would be a contribution to the LBM convergence literature.\n\nBest.","headline":"A solid, genuinely new TRT extension of the BGK convergence analysis, but the central proof is not fully self-contained in the preprint.","tokens_in":26615,"tokens_out":3072,"would_cite":true,"duration_ms":28751,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65M12","65M08","65M06","35L60"],"pacs":[],"model":"deepseek-v4-flash","headline":"Under two explicit inequalities on the relaxation parameters, two-relaxation-times lattice Boltzmann schemes for scalar conservation laws are monotone and converge to the weak entropy solution as the grid is refined.","keywords":["two-relaxation-times lattice Boltzmann scheme","monotonicity","scalar conservation law","entropy solution","L1-contractivity","convergence","numerical diffusion","kinetic entropy"],"falsifier":"Run the D1Q3 test with flux $\\varphi(u)=u^2/2$, the magic combination $\\omega_s=50/73$, $\\omega_a=96/73$ (inside the monotonicity set $M$), and initial data $u_0(x)=1_{[0,1/2]}(|x|)$ together with a small shift of the same data; the paper predicts the discrete $L^1$ distance between the two runs never increases, so a single time step where it grows refutes the $L^1$-contractivity claim that underpins the convergence theorem.","tokens_in":25576,"feed_emoji":"📐","tokens_out":13736,"duration_ms":115613,"temperature":0.7,"pith_summary":"This paper sets out to prove that two-relaxation-times (TRT) lattice Boltzmann schemes—the middle ground between simple BGK and fully general multi-relaxation-times schemes—are convergent for scalar non-linear conservation laws. It gives explicit conditions on the two relaxation parameters $\\omega_s$ and $\\omega_a$ under which the collide-and-stream operator is monotone in the distribution functions, and then follows the monotone-scheme route: invariant compact set, $L^1$-contractivity, total variation bounds, closeness to equilibrium, and a weak-form test-function argument that identifies the limit as the unique weak entropy solution. The point of TRT is that one parameter can be raised toward 2, which lowers numerical diffusion, while the other keeps monotonicity, so the scheme can be sharper than BGK and still provably convergent. Numerical experiments on D1Q3 and D2Q5 lattices confirm the predicted convergence and the reduced diffusion.","feed_headline":"Two-relaxation-time lattice Boltzmann schemes provably converge","feed_subtitle":"Two relaxation parameters let the scheme cut numerical diffusion while still provably reaching the entropy solution.","key_machinery":"The load-bearing object is the relaxation operator written in symmetric/anti-symmetric variables: each pair of opposed velocities $c_{2\\ell}=-c_{2\\ell+1}$ is split into $s_{2\\ell}=(f_{2\\ell}+f_{2\\ell+1})/2$ and $a_{2\\ell}=(f_{2\\ell}-f_{2\\ell+1})/2$, relaxed at rates $\\omega_s$ and $\\omega_a$. With equilibria of the form $f_i^{eq}=L_i u + \\sum_k N_{i,k}\\varphi_k(u)$ and $N_{2\\ell,k}=-N_{2\\ell+1,k}\\ge0$, the Jacobian entries of the relaxation are affine in the flux derivatives, and requiring them non-negative on the invariant box gives exactly conditions (3.5)–(3.6). Monotonicity then yields the full convergence toolbox: invariant set, $\\ell^1$-contractivity of the relaxation (Proposition 4.4), $L^1$-contractivity and total-variation decay, an $O(\\Delta x)$ estimate for the distance to equilibrium (Proposition 4.8), and finally the weak-form limit argument that produces the entropy inequality.","core_discovery":"The central claim is that a TRT lattice Boltzmann scheme for $\\partial_t u + \\sum_{k=1}^d \\partial_{x_k}\\varphi_k(u)=0$ converges to the unique weak entropy solution, provided the equilibria have the structural form $f_i^{eq}(u)=L_i u + \\sum_k N_{i,k}\\varphi_k(u)$ with the linear part symmetric along each link and the flux part anti-symmetric, and provided the relaxation parameters satisfy $\\omega_s L_1 \\ge \\max(0,\\omega_s-1)$ and the per-link bound $\\omega_a \\max_{u\\in[-u_\\infty,u_\\infty]}|\\sum_k N_{2\\ell,k}\\varphi_k'(u)| \\le \\omega_s L_{2\\ell} + \\tfrac12 \\min(2-\\omega_s-\\omega_a,0,\\omega_a-\\omega_s)$. Under these conditions the relaxation operator is monotone non-decreasing on an invariant set of distribution functions, and the scheme inherits $L^\\infty$ bounds, $L^1$ contractivity, total-variation decay, geometric convergence to equilibrium, and ultimately strong $L^1$ convergence to the unique entropy solution.","pith_inferences":["Editorial extension: because the paper notes the monotonicity conditions are sufficient but not necessary, a natural next experiment is to scan pairs $(\\omega_s,\\omega_a)$ outside $M$ and measure the entropy error; finding parameter regions outside $M$ with vanishing error would indicate where the sufficient conditions can be widened.","Editorial extension: the structural assumption (3.1) is the main restriction; the paper remarks that the flux only needs to be locally Lipschitz, so the same proof should extend to fluxes with kinks such as $\\varphi(u)=|u|$ with minor technical changes.","Editorial extension: since the entropy argument is scalar, extending TRT convergence to systems of conservation laws would require a different mechanism, but the same link-wise monotonicity conditions could be checked against known finite-volume entropy stability criteria.","Editorial extension: a quantitative test of the sharpness of (4.5) would be to track $\\|f^n_\\Delta - f^{eq}(u^n_\\Delta)\\|_{L^1}$ for many random initial data and check whether the $O(\\Delta x \\operatorname{TV}(u_0))$ bound and the geometric decay predicted by (4.6) hold with the computed constant."],"forward_implications":["TRT schemes can run with $\\omega_a>1$, and under the magic combination $\\omega_s+\\omega_a=2$ even close to 2, while remaining provably monotone and convergent; the BGK scheme, with one parameter, must stay much closer to 1.","Any scheme satisfying (3.5)–(3.6) keeps all distribution functions inside their equilibrium box and the conserved moment inside $[-u_\\infty,u_\\infty]$ for all times.","Two numerical solutions whose initial data differ by $\\|u_0-v_0\\|_{L^1}$ never grow apart in $L^1$, and the total variation of each solution never exceeds that of the initial datum.","The distribution functions approach their equilibrium at a geometric rate in time and stay within $O(\\Delta x)$ of equilibrium, so the numerical modes become slaves of the physical mode as the grid refines.","The limit $u$ of the conserved moment satisfies both the weak form of the conservation law and the kinetic entropy inequalities for every constant $\\kappa$, which identifies it as the unique entropy solution."],"supporting_citations":[{"why":"The BGK convergence analysis this work extends; supplies the equilibria-monotonicity strategy and the BGK limit to compare against.","marker":"[1]"},{"why":"Provides the monotone-scheme compactness machinery that turns monotonicity, L1 contractivity and TV bounds into convergence to a weak solution.","marker":"[12]"},{"why":"Source of the link-wise Jacobian non-negativity argument used to derive the monotonicity conditions in Proposition 3.3.","marker":"[17]"},{"why":"Provides the kinetic-entropy formulation used to prove the entropy inequality in the limit.","marker":"[25]"},{"why":"Convergence result for under-relaxed one-dimensional schemes whose post-relaxation entropy balance the paper follows.","marker":"[10]"},{"why":"Gives the definition of monotone scheme and the weak-form test-function argument adapted here.","marker":"[21]"},{"why":"Supplies the discrete-kinetic proof skeleton of maximum principle, L1 contractivity and TV estimates.","marker":"[2]"}],"fun_headline_variants":["TRT lattice Boltzmann: monotone, stable, convergent","Two-relaxation-time Boltzmann proves entropy convergence","TRT scheme: less diffusion, same provable convergence","Monotone TRT lattice Boltzmann reaches entropy solution","Two relaxations, one guarantee: TRT convergence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on the structural assumption that every equilibrium distribution is an affine function of the conserved quantity plus a weighted copy of the flux, with the flux term entirely in the anti-symmetric part and the linear term entirely in the symmetric part.","fun_headline_variants_meta":{"raw":{"variants":["TRT lattice Boltzmann: monotone, stable, convergent","Two-relaxation-time Boltzmann proves entropy convergence","TRT scheme: less diffusion, same provable convergence","Monotone TRT lattice Boltzmann reaches entropy solution","Two relaxations, one guarantee: TRT convergence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000713,"raw_usage":{"total_tokens":3192,"prompt_tokens":918,"completion_tokens":2274,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":534,"completion_tokens_details":{"reasoning_tokens":2197}},"tokens_in":534,"tokens_out":2274,"duration_ms":16072,"temperature":1.0,"reasoning_tokens":2197,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:32:00.419435+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the D1Q3 test with flux $\\varphi(u)=u^2/2$, the magic combination $\\omega_s=50/73$, $\\omega_a=96/73$ (inside the monotonicity set $M$), and initial data $u_0(x)=1_{[0,1/2]}(|x|)$ together with a small shift of the same data; the paper predicts the discrete $L^1$ distance between the two runs never increases, so a single time step where it grows refutes the $L^1$-contractivity claim that underpins the convergence theorem.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The BGK convergence analysis this work extends; supplies the equilibria-monotonicity strategy and the BGK limit to compare against."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the monotone-scheme compactness machinery that turns monotonicity, L1 contractivity and TV bounds into convergence to a weak solution."},{"cited_title":"Dubois, B","cited_arxiv_id":null,"evidence_quote":"Source of the link-wise Jacobian non-negativity argument used to derive the monotonicity conditions in Proposition 3.3."},{"cited_title":"Natalini, A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws, Journal of Differential Equations, 148 (1998), pp","cited_arxiv_id":null,"evidence_quote":"Provides the kinetic-entropy formulation used to prove the entropy inequality in the limit."},{"cited_title":"Caetano, F","cited_arxiv_id":null,"evidence_quote":"Convergence result for under-relaxed one-dimensional schemes whose post-relaxation entropy balance the paper follows."},{"cited_title":"Godlewski and P.-A","cited_arxiv_id":null,"evidence_quote":"Gives the definition of monotone scheme and the weak-form test-function argument adapted here."},{"cited_title":"Aregba-Driollet and R","cited_arxiv_id":null,"evidence_quote":"Supplies the discrete-kinetic proof skeleton of maximum principle, L1 contractivity and TV estimates."}],"review_version":1}