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REVIEW 3 major objections 4 minor 41 references

Structure-preserving schemes conserving entropy and kinetic energy

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2505.13374 v1 pith:VT5JCQGU submitted 2025-05-19 math.NA cs.NAphysics.comp-ph

Kunal Bahuguna , Ramesh Kolluru , S.V. Raghurama Rao This is my paper · ORCID
classification math.NAcs.NAphysics.comp-ph MSC 65M0865M1235L65
keywords entropyconservationkineticenergypreservationEulerequationsfinitevolumemethodsstabilityshockcapturingstructure-preservingschemesRankine-Hugoniotdiffusion
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [Sec. 3.1, Eqs. (27)-(28); Sec. 3.2, Eqs. (30)-(31); note after Eq. (31)] 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.
  2. [Sec. 7.1] 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.
  3. [Sec. 6, Eq. (52)] 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.'
minor comments (4)
  1. [Sec. 2.2] 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.
  2. [Throughout] 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).
  3. [Fig. 17 caption] The caption labels the HES grid as '240x160' while the text states the simulations use 240x80 and 480x160; please reconcile the inconsistency.
  4. [Sec. 6, Eq. (47)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the entropy-conservative fluxes are constructed by solving Tadmor's condition algebraically, and the stability proofs are self-contained.

full rationale

The derivation chain is self-contained. In Section 3, the paper inserts the ansatz F^c = F_bar - (1/2) alpha Delta V (Eq. 23) into Tadmor's algebraic entropy-conservation condition Delta V . F^c = Delta psi (Eq. 18) and solves for alpha, giving EC1 in Eq. (25), EC2 in Eq. (27), and ECKEP in Eq. (31). This is a constructive solution of a defining condition, not a prediction of an independent quantity from fitted inputs: the coefficient alpha is determined by the very equation whose satisfaction is claimed. The kinetic-energy-preserving condition of Jameson (Eq. 29) is likewise imposed and then verified by summation identities; no parameter is fitted to observed entropy or kinetic-energy histories and later relabeled as a prediction. The entropy-stability proof for the Rankine-Hugoniot diffusion (Eqs. 41-44) relies only on alpha_S >= 0 and Delta V . Delta U >= 0, the latter established independently in Appendix B from convexity of the entropy function, so it does not smuggle in the target result. Self-citations to MOVERS [29] and RICCA [33] provide building blocks for the dissipation and the sensor coefficient, but the central conservation identities do not reduce to those citations. The only caveat found is a correctness risk rather than a circular step: EC2 and ECKEP divide by (Delta V3)^2 in Eqs. (27) and (31), and the paper's note that 'a small parameter delta is added to the denominators (delta = 10^-16 in our computations)' (Sec. 3.2) masks but does not cure the failure of Tadmor's identity when Delta V3 = 0. That degeneracy affects the validity of the unqualified entropy-conservation claim on that hypersurface, but it is not an input-output equivalence and therefore does not raise the circularity score.

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

The central flux construction rests on standard entropy-variable theory and the ideal-gas closure; no new physical entities are introduced. The only adjustable inputs are regularization and sensor constants, listed above. The 2D assumption is outsourced to prior work.

free parameters (5)
  • denominator regularization delta = 10^-16
    Added to denominators in Eqs (25), (27), and (31) to avoid division by zero; chosen by hand and can alter the entropy-conservation identity near vanishing DeltaV3.
  • sonic-point fix Theta = 0.1
    Used in Eq (40) to smooth the RH diffusion coefficient at sonic points; fixed by the authors without a sensitivity study.
  • shock-sensor scaling q = 8-16 (q=10 in 2D)
    Controls the exponential entropy-distance sensor in Eqs (46)-(47); values are chosen from numerical experiments rather than derived.
  • shock-sensor clipping threshold epsilon = 0.01-0.5
    Clipping threshold in Eq (48); ranges differ between 1D (0.01-0.1) and 2D (0.1-0.5), indicating per-problem tuning.
  • fourth-order diffusion coefficient = 1/32
    Coefficient alpha_R in Eq (50) for the JST-style smoothing; fixed by hand.
assumptions (5)
  • domain assumption Ideal gas closure p=(gamma-1)rho e with constant gamma
    Used throughout Sec 2 to derive the entropy pair and internal-energy equation.
  • standard math Entropy pair eta=-rho s/(gamma-1), zeta=-rho u s/(gamma-1) with s=ln(p/rho^gamma) satisfies zeta'=eta'F'
    Invoked in Sec 2.1 to obtain entropy variables and the potential identity (18).
  • standard math The entropy Hessian is positive definite on admissible states, so DeltaU*DeltaV >= 0
    Used in Sec 5 and Appendix B to sign the entropy-stability and kinetic-energy-stability inequalities.
  • domain assumption Semi-discrete finite-volume framework is the relevant target; fully discrete conservation is not required
    The authors explicitly restrict claims to semi-discrete form in Sec 4.3 and do not prove properties under SSPRK time stepping.
  • domain assumption The 2D scheme is obtained by applying the 1D fluxes dimension-by-dimension with the unstructured diffusion of [34]
    Assumed in Sec 7.2 without a derivation of the 2D entropy or kinetic-energy identities.

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Cite this review

Pith. "Pith review of Structure-preserving schemes conserving entropy and kinetic energy." pith.science (2026). https://pith.science/paper/VT5JCQGU

@misc{pith2026250513374,
  author       = {Pith},
  title        = {Pith review of: Structure-preserving schemes conserving entropy and kinetic energy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VT5JCQGU}},
  note         = {Machine review of arXiv:2505.13374}
}
read the original abstract

This paper presents a novel structure-preserving scheme for Euler equations, focusing on the numerical conservation of entropy and kinetic energy. Explicit flux functions engineered to conserve entropy are introduced within the finite-volume framework. Further, discrete kinetic energy conservation too is introduced. A systematic inquiry is presented, commencing with an overview of numerical entropy conservation and formulation of entropy-conserving and kinetic energy-preserving fluxes, followed by the study of their properties and efficacy. A novelty introduced is to associate numerical entropy conservation to the discretization of the energy conservation equation. Furthermore, an entropy-stable shock-capturing diffusion method and a hybrid approach utilizing the entropy distance to manage smooth regions effectively are also introduced. The addition of artificial viscosity in appropriate regions ensures entropy generation sufficient to prevent numerical instabilities. Various test cases, showcasing the efficacy and stability of the proposed methodology, are presented.

Figures

Figures reproduced from arXiv: 2505.13374 by the authors.

Figure 1
Figure 1. Density and velocity plots for a stationary contact discontinuity at T=2 s by the three entropy conservative fluxes [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Log-log plot of errors with grid size for EC1, EC2 and ECKEP fluxes respectively [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 4
Figure 4. Total Kinetic Energy for Taylor-Greens vortex [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figures from the paper (24 more)
Figure 5
Figure 5. Figure 5: Total entropy and total kinetic energy v/s Time for Isentropic vortex test case for different entropy conservative fluxes [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: (a): Shock sensor value for SOD tube test case. (b): Shock sensor value and contour of density variation for oblique [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: Plots of density, velocity and internal energy for various schemes for the first test case [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: Plots of density, velocity and internal energy using HES scheme for the first test case [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: Plots of density, velocity and internal energy using ES (top) and HES (bottom) schemes for the third test case [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: Plots of density, velocity and internal energy using ES (top) and HES (bottom) schemes for the fourth test case [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]
Figure 11
Figure 11. Figure 11: Plots of density, velocity and internal energy using ES (top) and HES (bottom) schemes for the stationary shock [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]
Figure 12
Figure 12. Figure 12: Plots of density, velocity and internal energy using ES scheme for stationary contact wave. [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]
Figure 13
Figure 13. Figure 13: Plots of density, velocity and internal energy using ES (top) and HES (bottom) schemes slowly moving shock wave. [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]
Figure 14
Figure 14. Figure 14: Plots of density, velocity and internal energy using ES (top) and HES (bottom) schemes slowly moving contact wave. [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]
Figure 15
Figure 15. Figure 15: Oblique shock reflection test case 0 1 2 3 4 5 Time 10 -14 10 -12 10 -10 10 -8 10 -6 10 -4 Residuals Mass Residual X-Momentum Residual Y-Momentum Residual Energy Residual 0 1 2 3 4 5 Time 10 -14 10 -12 10 -10 10 -8 10 -6 10 -4 Residuals Mass Residual X-Momentum Residu…
Figure 16
Figure 16. Figure 16: Semi-log plot of residuals for ES scheme for 240x80 grid (left) and 480x160 grid (right) [PITH_FULL_IMAGE:figures/full_fig_p024_16.png]
Figure 17
Figure 17. Figure 17: Semi-log plot of residuals for HES scheme for 240x160 grid (left) and 480x160 grid (right) [PITH_FULL_IMAGE:figures/full_fig_p024_17.png]
Figure 18
Figure 18. Figure 18: Pressure contours (50 in 0.6-2.4) for 15 [PITH_FULL_IMAGE:figures/full_fig_p025_18.png]
Figure 19
Figure 19. Figure 19: Density contours (30 in 1.3-9.0) for hypersonic flow over cylinder on 320x40 grid with (a) Roe scheme, (b) ES scheme [PITH_FULL_IMAGE:figures/full_fig_p025_19.png]
Figure 20
Figure 20. Figure 20: Density contours (30 in 0.3-6.8) for forward step on 480x160 grid using ES (left) and HES (right) scheme [PITH_FULL_IMAGE:figures/full_fig_p025_20.png]
Figure 21
Figure 21. Figure 21: Density contours (30 in 0.05-7.1) for backwards-facing step. (a) and (b) using the ES scheme on 400x400 and [PITH_FULL_IMAGE:figures/full_fig_p025_21.png]
Figure 22
Figure 22. Figure 22: (a): Part of the grid (x=0 to 35) used for odd-even decoupling with centerline perturbations. (b): Contours of density [PITH_FULL_IMAGE:figures/full_fig_p026_22.png]
Figure 23
Figure 23. Figure 23: Density contours (30 in 1.3-22) for double mach reflection test case on a grid of 960x240 using ES (left) and HES [PITH_FULL_IMAGE:figures/full_fig_p026_23.png]
Figure 24
Figure 24. Figure 24: Color plot of density for the double mach reflection test case on a grid of 960x240 using the HES scheme. [PITH_FULL_IMAGE:figures/full_fig_p026_24.png]
Figure 25
Figure 25. Figure 25: entropy (s = ln(p/ργ)) contours (15 in -0.6-4.2) for shock vortex filament interaction with (a) LLF (b) ES (c) HES schemes [PITH_FULL_IMAGE:figures/full_fig_p027_25.png]
Figure 26
Figure 26. Figure 26: Structured mesh for NACA0012 airfoil 27 [PITH_FULL_IMAGE:figures/full_fig_p027_26.png]
Figure 27
Figure 27. Figure 27: NACA 0012 Airfoil test case 0 0.5 1 x/c -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Cp Top Surface Bottom Surface (a) 0 0.5 1 x/c 0.4 0.6 0.8 1 1.2 1.4 1.6 Pressure Top Surface Bottom Surface (b) 0 0.5 1 x/c -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Cp Top Surface Bottom Surface (c) 0 0.5 1…
Figure 28
Figure 28. Figure 28: Pressure and Cp plots along top and bottom surfaces of airfoil using HES scheme : (a) and (b) Transonic case. (c) and (d) Supersonic case. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_28.png]

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