{"id":"a04deaf3-24f4-4331-b458-0a94d3cc972e","arxiv_id":"2505.13374","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"New entropy-conserving and kinetic-energy-preserving fluxes for the Euler equations modify only the energy flux, and are combined with a Rankine-Hugoniot diffusion and an entropy-distance shock sensor.","lead":"The authors construct finite-volume flux formulas for the Euler equations that conserve entropy and kinetic energy in a semi-discrete sense, and they add a shock sensor to keep the method stable at discontinuities. The practical promise is a cheaper, eigen-structure-free shock-capturing scheme for compressible flow, reported to avoid carbuncle and expansion-shock artifacts.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"EC2/ECKEP are not entropy-conservative when ΔV3=0: Eqs (27)/(31) divide by (ΔV3)^2, and the δ=1e-16 fix leaves the Tadmor residual uncanceled, so the energy-only construction is only conditional.","rationale":"The reader's weakest-assumption analysis identifies the most load-bearing weakness of the paper: the energy-only entropy-conservation construction requires ΔV3≠0, and the δ-regularization masks rather than fixes the degeneracy. I confirmed this with an explicit pair of states that have ΔV3=0 but nonzero Tadmor residual, so the claim that EC2/ECKEP satisfy Eq (18) is false as stated. The numerical experiments are not contradicted because the smooth test problems chosen have generically nonzero ΔV3, but the theoretical claim needs a stated condition and a discussion of behavior near ΔV3=0. Other concerns, such as the absence of a derived 2D extension or of released code, are secondary: the 2D results are consistent with a dimension-by-dimension application of the 1D fluxes, and code availability does not affect the mathematical validity. The CONDITIONAL verdict remains appropriate because the issue is concrete and addressable, not a fundamental collapse of the method.","tokens_in":25506,"tokens_out":13664,"duration_ms":130778,"concrete_test":"Implement Eqs (27)-(31) without the δ-regularization and evaluate the Tadmor residual ΔV·F^c−Δψ at the single interface L=(ρ=1,u=0.1,p=0.4), R=(ρ=2,u=0.1,p=0.8), γ=1.4. Since ΔV3=0, the energy-diffusion term is identically zero; confirm that the residual is approximately 4×10^-3 instead of 0. Repeat with δ=10^-16 and verify that the residual is unchanged. This directly tests whether EC2 and ECKEP satisfy Eq (18) for all admissible states.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of Section 3 is that entropy conservation can be enforced by adding diffusion only to the energy flux. For EC2 and ECKEP, α2/2 and α3/2 are defined by dividing by (ΔV3)^2 in Eqs (27) and (31), so the derivation requires ΔV3≠0 at every interface. This condition is never stated, and the δ=10^-16 regularization added in Section 3.2 does not restore the entropy identity: when ΔV3=0, the energy-diffusion term vanishes regardless of α, so the flux reduces to the arithmetic-average flux and the Tadmor residual R=F̄·ΔV−Δψ is left uncanceled. The degenerate set is not exotic. Take γ=1.4, L=(ρ=1,u=0.1,p=0.4) and R=(ρ=2,u=0.1,p=0.8). Both states have internal energy e=p/((γ−1)ρ)=1, hence ΔV3=0, while ΔV1≈0.694 and Δψ=0.1; the residual is about 4×10^-3. Near this hypersurface the coefficient α blows up as R/((ΔV3)^2+δ), which can also introduce severe stiffness or spurious energy diffusion. Thus the flagship fluxes EC2/ECKEP satisfy Tadmor's condition (18) only away from ΔV3=0, contradicting the unqualified claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes structure-preserving finite-volume schemes for the one- and two-dimensional Euler equations. The central construction is a family of entropy-conservative fluxes: EC1, which applies scalar diffusion to all equations; EC2, which applies diffusion only to the energy equation; and ECKEP, which combines a kinetic-energy-preserving momentum flux with energy-equation diffusion. A Rankine-Hugoniot-based diffusion term is then added to obtain an entropy-stable scheme, and a hybrid scheme uses an entropy-distance shock sensor to blend low and high diffusion. The authors report second-order convergence, exact steady-contact preservation, entropy and kinetic-energy conservation on smooth problems, and a broad set of one- and two-dimensional benchmark results.","tokens_in":25822,"tokens_out":10684,"duration_ms":99747,"significance":"If the central claims hold, the energy-only diffusion construction is an elegant and computationally attractive idea: it avoids logarithmic averages and leaves the mass and momentum fluxes free for other structure-preserving constraints. The paper's strengths include explicit algebraic flux formulas, clean EOC tables, an explicit entropy-stability argument for the ES scheme, exact steady-contact preservation, and an unusually wide benchmark suite covering carbuncle, odd-even decoupling, and shock-vortex interactions. However, the unqualified entropy-conservation claim for EC2/ECKEP is conditional on a non-degeneracy condition that is never stated, the results section uses an undefined 'EC3' flux, and the hybrid scheme's entropy-stability proof does not cover the fourth-order diffusion term. These issues bear directly on the paper's central claims and on the reproducibility of the numerical validation.","major_comments":[{"comment":"The EC2 and ECKEP fluxes satisfy Tadmor's condition (18) only when ΔV3 ≠ 0. Since V3 = -ρ/p = -1/((γ-1)e), ΔV3 = 0 whenever the left and right states have equal internal energy. For example, with γ = 1.4, left state (ρ=1, u=0.1, p=0.4) and right state (ρ=2, u=0.1, p=0.8), one has ΔV3 = 0, ΔV1 ≈ 0.69, and Δψ = 0.1, so the residual in (18) is about 4×10^-3. In this situation the energy-diffusion term in (28)/(30) vanishes regardless of α, and adding δ = 10^-16 to the denominator does not restore the identity; it merely changes the flux when ΔV3 is small but nonzero, producing O(1/δ) coefficients and potential stiffness or spurious energy diffusion. The unqualified statement that entropy conservation 'can theoretically be achieved by adding diffusion only to the energy equation' therefore requires a non-degeneracy assumption that is nowhere stated. The reported divergence of EC2 in Sec. 4.3.2 is consistent with this mechanism and should be analyzed quantitatively rather than attributed only to temporal error accumulation.","section":"Sec. 3.1, Eqs. (27)-(28); Sec. 3.2, Eqs. (30)-(31); note after Eq. (31)"},{"comment":"The sentence 'Note that EC3 flux, kinetic energy preserving, was taken as the entropy conservative flux in ES (41) and HES (52) schemes' refers to a flux called EC3 that is never defined in the paper. The fluxes introduced in Section 3 are EC1 (26), EC2 (28), and ECKEP (30), and Appendix A derives ECKEP; there is no EC3 anywhere in the manuscript. Since the one-dimensional results in Figures 7-14 are obtained with this flux, the experiments are not reproducible and the numerical validation cannot be assessed until either EC3 is defined or the reference is corrected to the intended flux.","section":"Sec. 7.1"},{"comment":"The entropy-stability proof in Sec. 5, especially Eq. (42), applies to the ES scheme F = F_EC + F_RH, not to the hybrid scheme F = F_EC + (1-φ)F_R + φF_RH. The JST-type term F_R in Eq. (49) is a fourth-order difference, and no bound or sign statement is derived for ΔV·F_R. Therefore the paper's claim that the HES scheme is 'entropy stable' is not established by the analysis presented. Either a proof covering the F_R contribution should be added, or the claim should be softened to something like 'entropy stable at shocks and entropy-conservative with added numerical filtering in smooth regions.'","section":"Sec. 6, Eq. (52)"}],"minor_comments":[{"comment":"The final paragraph says the fact that the total energy equation contains both entropy and kinetic energy structures 'is used in section 2 to construct entropy conservative flux'; this should refer to Section 3.","section":"Sec. 2.2"},{"comment":"There are several typographical errors, including 'Reimann' (Sec. 4.1), 'disctontinuities' (Sec. 4.1), 'apprproiate' (Abstract), 'affilication' (author footnote), 'tangecy' (Secs. 7.2.2, 7.2.5, 7.2.6), 'Gudunov' (Sec. 7.2.8), 'Rusnaov' (Sec. 5), 'csse' (Sec. 7.2.9), 'seens' (Sec. 7.2.4), and 'catpures' (Sec. 5).","section":"Throughout"},{"comment":"The caption labels the HES grid as '240x160' while the text states the simulations use 240x80 and 480x160; please reconcile the inconsistency.","section":"Fig. 17 caption"},{"comment":"The quadratic approximation of the exponential sensor is not a Taylor truncation of exp(-q·SED); its accuracy and its effect on the sensor's monotonicity are not discussed and should be clarified.","section":"Sec. 6, Eq. (47)"}],"recommendation":"major_revision","confidential_remarks":"The paper has a substantial and generally well-executed experimental component, and the core idea is worth pursuing, but the undefined EC3 flux and the unqualified ΔV3 = 0 degeneracy are blocking issues for the correctness and reproducibility of the main claims. I would not recommend acceptance before these are resolved, and the authors should also clarify whether the entropy-stability proof can be extended to the JST contribution in the hybrid scheme."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper constructs explicit entropy-conservative fluxes for the Euler equations by adding diffusion only to the energy equation, plus a kinetic-energy-preserving variant, and wraps them in a hybrid shock-capturing scheme using Rankine-Hugoniot-based diffusion and an entropy-distance sensor. The engineering is mostly solid: the 1D semi-discrete algebra is coherent, the EOC tables are clean, and the 2D test suite (odd-even decoupling, carbuncle, double Mach reflection, forward step) shows the scheme to be robust and largely free of the usual shock instabilities. The entropy-distance sensor and RH diffusion combination is a genuinely useful practical idea, and the authors are honest that EC1 reduces to Abgrall's optimization.\n\nThe soft spots are not minor. The central claim that entropy conservation can be enforced by modifying only the energy flux is only true away from ΔV3 = 0. Equations (27) and (31) divide by (ΔV3)^2, and when ΔV3 = 0 the energy-diffusion term vanishes regardless of the coefficient, so the Tadmor residual is not canceled. The δ = 10^-16 regularization does not fix the identity; it just masks the singularity and can produce large spurious diffusion nearby. The condition ΔV3 ≠ 0 is never stated. This is a load-bearing gap: EC2 and ECKEP are not unconditionally entropy conservative, and the paper says they are. A referee can ask for a clear statement of the condition, or a modified construction using a different entropy-variable component when ΔV3 vanishes.\n\nThere are smaller issues. Section 7.1 refers to an \"EC3 flux\" that is never defined, and the 2D extension is asserted rather than derived—no 2D entropy or kinetic-energy analysis is given. No code is provided, which limits reproducibility, though the test case descriptions are fairly detailed.\n\nOn the other hand, the hybrid scheme does not collapse even if EC2/ECKEP lose exact entropy conservation in special states: the added RH diffusion provides entropy dissipation, and the numerical tests demonstrate stability in practice. So the paper's practical contribution may survive the theoretical gap, but the gap needs to be acknowledged and fixed.\n\nMy recommendation: send it to peer review, but with a referee who will check the ΔV3-degenerate case. In its current form I would not cite the entropy-conservation claim; I might cite the sensor/RH-diffusion hybrid after it matures. If the authors add the caveat and define EC3, this could be a useful paper for CFD practitioners.","headline":"Useful hybrid shock-capturing scheme, but the new entropy-conservative fluxes EC2/ECKEP have an unstated degeneracy at ΔV3=0 that undermines their flagship claim.","tokens_in":26381,"tokens_out":3824,"would_cite":false,"duration_ms":37161,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65M08","65M12","35L65"],"pacs":[],"model":"deepseek-v4-flash","headline":"By adding numerical diffusion only to the energy flux of the Euler equations, the paper constructs entropy-conservative and kinetic-energy-preserving finite-volume fluxes, then stabilizes them with a Rankine-Hugoniot-based shock sensor.","keywords":["entropy conservation","kinetic energy preservation","Euler equations","finite volume methods","entropy stability","shock capturing","structure-preserving schemes","Rankine-Hugoniot diffusion"],"falsifier":"At an interface where $\\rho_L/p_L=\\rho_R/p_R$ but velocity differs (so $\\Delta V_3=0$ while other entropy-variable jumps are nonzero), evaluate $\\Delta V\\cdot F_{ECKEP}-\\Delta\\psi$: the ECKEP flux cannot enforce the identity there, and the residual will equal the un-damped contribution from the first two entropy-variable jumps. Computing that residual numerically would settle whether the energy-only construction is valid in full generality.","tokens_in":25291,"feed_emoji":"⚡","tokens_out":8509,"duration_ms":83003,"temperature":0.7,"pith_summary":"Numerical solutions of the compressible Euler equations need to respect two structures hidden inside the total-energy equation: entropy conservation in smooth flow and kinetic-energy balance. The paper tries to turn that observation into explicit finite-volume fluxes. Its key move is to make a numerical flux entropy-conservative by adding diffusion only to the energy flux, so that mass and momentum fluxes remain free to be designed for kinetic-energy preservation. The resulting EC2 and ECKEP fluxes satisfy the discrete entropy-conservation identity and stay second-order accurate. The paper then adds a Rankine-Hugoniot-based diffusion selected by an entropy-distance shock sensor, producing a hybrid scheme that is entropy-stable and kinetic-energy-stable and preserves steady contact discontinuities exactly. A reader should care because the construction is explicit, avoids logarithmic averages, and survives demanding shock tests without expansion shocks or carbuncle artifacts.","feed_headline":"Adding diffusion only to energy conserves entropy in Euler solvers","feed_subtitle":"New finite-volume fluxes preserve entropy and kinetic energy, keep steady contacts exact, and pass tough shock tests.","key_machinery":"The load-bearing identity is the discrete entropy-conservation condition $\\Delta V\\cdot F^c=\\Delta\\psi$ between jumps of the entropy variables $V=\\partial\\eta/\\partial U$ and jumps of the entropy potential $\\psi$. The fluxes are built as average flux minus half a diffusion term $\\frac{1}{2}\\alpha\\Delta V$; the novel choice is $\\alpha=(0,0,\\alpha_3)$, so diffusion appears only in the energy flux, with $\\alpha_3$ solved from the identity. The shock-capturing part uses a Rankine-Hugoniot-based diffusion $F_{RH}=-\\frac{1}{2}\\min(s_1,s_2,s_3)\\tilde{I}\\Delta U$, whose wave speeds come from $\\Delta F=\\tilde{D}\\Delta U$, and an entropy-distance sensor $\\phi=1-|\\exp(-q\\,\\mathrm{SED})|$ that switches between this diffusion and fourth-order JST-type background diffusion. The entropy distance $\\Delta V\\cdot\\Delta U$ is non-negative because the entropy function is convex, which makes the dissipation term sign-definite.","core_discovery":"The paper's central claim is that numerical entropy conservation for the Euler equations can be achieved by adding diffusion only to the energy equation. Starting from an average flux and an entropy-variable jump $\\Delta V$, the authors select a diffusion vector with only its third component nonzero, obtaining the EC2 flux, and impose the standard kinetic-energy-preserving momentum-flux condition to obtain the ECKEP flux. Both satisfy the semi-discrete entropy-conservation identity $\\Delta V\\cdot F^c = \\Delta\\psi$, and the paper verifies second-order spatial accuracy, entropy and kinetic-energy conservation in vortex tests, and exact preservation of steady contact discontinuities. For shock flows, the flux is augmented with a Rankine-Hugoniot-based diffusion; the resulting scheme is shown to dissipate entropy and kinetic energy, and the hybrid version switches this diffusion on only near shocks through an entropy-distance sensor. On the reported one- and two-dimensional benchmarks the schemes produce oscillation-free shocks and avoid common numerical instabilities.","pith_inferences":["The energy-only diffusion construction suggests a general recipe: whenever an additional conservation law is embedded in one equation of a system, the numerical constraint can be enforced by modifying only that equation's flux; the degeneracy when the corresponding entropy-variable jump vanishes is the price of that economy.","Because the Rankine-Hugoniot diffusion does not depend on the eigenstructure, the same hybrid sensor plus RH diffusion could be applied to other hyperbolic systems such as shallow water or MHD, provided a consistent entropy pair and a positivity condition analogous to $\\Delta V\\cdot\\Delta U\\ge 0$ hold.","The entropy distance $\\Delta V\\cdot\\Delta U$ itself could serve as an adaptive smoothness indicator; one testable extension is to make $q$ and $\\epsilon$ functions of the local entropy distance rather than fixed ranges, which might reduce tuning in multidimensional problems.","The reported long-time drift of EC2 in vortex tests suggests that semi-discrete entropy conservation is not enough for fully discrete simulations; pairing these fluxes with a fully discrete entropy-stable time integrator is a natural next step the paper explicitly leaves open."],"forward_implications":["EC2 shows that entropy conservation does not force diffusion in every conserved variable; only the energy flux needs a correction, so the other fluxes can be designed independently.","ECKEP inherits both discrete entropy conservation and discrete kinetic-energy preservation in the semi-discrete setting, while remaining second-order accurate and explicit.","The Rankine-Hugoniot diffusion term makes the scheme entropy-stable and kinetic-energy-stable, because the entropy production is non-positive and the kinetic-energy source is always non-positive when the diffusion coefficient is non-negative.","The entropy-distance sensor lets the hybrid scheme keep low diffusion in smooth regions, so it resolves contact discontinuities and slipstreams while still suppressing expansion shocks, carbuncles, and odd-even decoupling in the reported tests.","The new fluxes are computationally cheaper than logarithmic-average entropy-conservative fluxes; in the paper's timing test EC2 is a few percent faster than the other entropy-conservative fluxes tested."],"supporting_citations":[{"why":"Defines the discrete entropy-conservation condition that the new fluxes are engineered to satisfy.","marker":"[1; 2]"},{"why":"Provides the first explicit affordable entropy-conservative flux family that the paper extends.","marker":"[4]"},{"why":"Gives the numerical momentum-flux condition used to make the new flux kinetic-energy preserving.","marker":"[9]"},{"why":"Supplies the pressure-averaging and kinetic-energy-preserving flux analysis the ECKEP construction relies on.","marker":"[10]"},{"why":"Is the comparison entropy-conserving kinetic-energy-preserving flux used in the numerical tests.","marker":"[11]"},{"why":"Is the entropy-consistent diffusion baseline and the comparison entropy-conservative flux used in efficiency measurements.","marker":"[15]"},{"why":"Furnishes the fourth-order JST diffusion used in the hybrid smooth-region stabilization.","marker":"[17]"},{"why":"Is the MOVERS Rankine-Hugoniot central solver whose diffusion structure the RH flux adopts.","marker":"[29]"},{"why":"Supplies the RICCA diffusion coefficient used in the fourth-order background dissipation.","marker":"[33]"}],"fun_headline_variants":["Energy-only diffusion yields entropy-conserving Euler fluxes","Entropy and kinetic energy preserved by energy-only diffusion","Add diffusion only to energy: entropy-conserving Euler scheme","Energy-targeted diffusion conserves entropy in shock tests","Kinetic energy and entropy conserved by one Euler flux"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction assumes the jump in the third entropy variable $\\Delta V_3$ never vanishes, because the EC2 and ECKEP diffusion coefficients divide by $(\\Delta V_3)^2$; adding $\\delta=10^{-16}$ hides the degeneracy without restoring the entropy-conservation identity at such an interface.","fun_headline_variants_meta":{"raw":{"variants":["Energy-only diffusion yields entropy-conserving Euler fluxes","Entropy and kinetic energy preserved by energy-only diffusion","Add diffusion only to energy: entropy-conserving Euler scheme","Energy-targeted diffusion conserves entropy in shock tests","Kinetic energy and entropy conserved by one Euler flux"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000817,"raw_usage":{"total_tokens":3544,"prompt_tokens":878,"completion_tokens":2666,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":494,"completion_tokens_details":{"reasoning_tokens":2588}},"tokens_in":494,"tokens_out":2666,"duration_ms":17909,"temperature":1.0,"reasoning_tokens":2588,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:15:50.177600+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"At an interface where $\\rho_L/p_L=\\rho_R/p_R$ but velocity differs (so $\\Delta V_3=0$ while other entropy-variable jumps are nonzero), evaluate $\\Delta V\\cdot F_{ECKEP}-\\Delta\\psi$: the ECKEP flux cannot enforce the identity there, and the residual will equal the un-damped contribution from the first two entropy-variable jumps. Computing that residual numerically would settle whether the energy-only construction is valid in full generality.","supporting_citations":[{"cited_title":"URL https://scholar.google.co.in/citations?view_op=view_citation&hl=en&user= 4fNzp4oAAAAJ&cstart=20&pagesize=80&citation_for_view=4fNzp4oAAAAJ:Tiz5es2fbqcC 30","cited_arxiv_id":null,"evidence_quote":"Provides the first explicit affordable entropy-conservative flux family that the paper extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the numerical momentum-flux condition used to make the new flux kinetic-energy preserving."},{"cited_title":"Ranocha, Entropy Conserving and Kinetic Energy Preserving Numerical Methods for the Euler Equations Using Summation-by-Parts Operators, in: S","cited_arxiv_id":null,"evidence_quote":"Supplies the pressure-averaging and kinetic-energy-preserving flux analysis the ECKEP construction relies on."},{"cited_title":"Kinetic energy preserving and entropy stable finite volume schemes for compressible Euler and Navier-Stokes equations","cited_arxiv_id":"1209.4994","evidence_quote":"Is the comparison entropy-conserving kinetic-energy-preserving flux used in the numerical tests."},{"cited_title":"Jaisankar, S.V","cited_arxiv_id":null,"evidence_quote":"Is the MOVERS Rankine-Hugoniot central solver whose diffusion structure the RH flux adopts."},{"cited_title":"Novel, simple and robust contact-discontinuity capturing schemes for high speed compressible flows","cited_arxiv_id":"2003.10695","evidence_quote":"Supplies the RICCA diffusion coefficient used in the fourth-order background dissipation."}],"review_version":1}