Entropy-stable discretizations for the compressible Euler equations using simple adaptive averages

Ayaboe K. Edoh; Carlo De Michele

arxiv: 2605.20472 · v1 · pith:MFPGUUGUnew · submitted 2026-05-19 · ⚛️ physics.flu-dyn · cs.NA· math.NA

Entropy-stable discretizations for the compressible Euler equations using simple adaptive averages

Carlo De Michele , Ayaboe K. Edoh This is my paper

Pith reviewed 2026-05-21 06:33 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn cs.NAmath.NA

keywords entropy-stable discretizationcompressible Euler equationsadaptive averagessymmetric meanskinetic energy preservationpressure equilibriumcentralized convective termsfluid dynamics

0 comments

The pith

Adapting simple symmetric averages on density and internal energy produces entropy-stable discretizations for the compressible Euler equations.

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

The paper establishes that entropy stabilization for the compressible Euler system follows from adapting the averages applied to density and internal energy. This is accomplished with simplified symmetric means such as arithmetic, geometric, or harmonic evaluations, plus their expansions that support asymptotic entropy conservation. The formulation uses centralized convective terms and naturally respects additional flow structures including kinetic-energy preservation and pressure-equilibrium preservation. A sympathetic reader would care because the approach delivers non-linear robustness while avoiding the need for more elaborate averaging techniques in fluid simulations.

Core claim

Entropy stabilization of the compressible Euler system is achieved by adapting the averages that are applied to the density and internal energy variables. The approach achieves non-linear robustness despite the use of simplified symmetric means (e.g., arithmetic, geometric, or harmonic evaluations), including their related expansions for asymptotic entropy conservation. The proposed formulation works via centralized convective terms and can naturally adhere to additional structures of the flow equations such as kinetic-energy- and pressure-equilibrium-preservation.

What carries the argument

Adaptive averages based on simplified symmetric means applied to density and internal energy within centralized convective terms.

If this is right

The resulting schemes maintain discrete conservation in the convective terms.
Kinetic energy and pressure equilibrium are preserved by construction.
Asymptotic entropy conservation holds through the expansions of the means.
Non-linear robustness is obtained without requiring complex or non-symmetric averaging operators.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Implementation in existing finite-volume or finite-difference codes could become simpler because only standard means plus a selection rule are required.
The same adaptive averaging idea may transfer to other hyperbolic conservation laws where entropy stability is needed.
Benchmark comparisons on standard shock-tube and vortex problems would clarify whether the simpler means reduce computational cost at fixed accuracy.

Load-bearing premise

Adaptive averages drawn from simplified symmetric means can be chosen or expanded to satisfy entropy stability together with kinetic-energy and pressure-equilibrium preservation without introducing instabilities or violating conservation.

What would settle it

A concrete numerical test in which entropy increases or conservation is lost when the adaptive averages are applied would show the central claim does not hold.

Figures

Figures reproduced from arXiv: 2605.20472 by Ayaboe K. Edoh, Carlo De Michele.

**Figure 2.** Figure 2: Sod shock tube results, comparing baseline EC with EC-LF stabilization and the proposed adaptive averaging [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper offers a concrete simplification for entropy-stable Euler discretizations by adapting simple symmetric means on density and internal energy, but the entropy claim appears tied to asymptotic behavior rather than exact nonlinear robustness.

read the letter

The one or two things to know about this paper are that it presents a discretization approach for the compressible Euler equations that achieves entropy stability by adapting the averages applied to density and internal energy using simple symmetric means, and that it does so while working with centralized convective terms that support kinetic energy and pressure equilibrium preservation. What is actually new here is the adaptation of these simplified means—arithmetic, geometric, or harmonic—along with their expansions to ensure asymptotic entropy conservation. This seems like a legitimate way to simplify the construction of entropy-stable schemes compared to methods that rely on more complex or parameter-dependent averages. The paper appears to focus on making the nonlinear robustness more accessible through this choice. It does well in outlining how the formulation naturally adheres to the structural properties of the flow equations. By using centralized terms, it likely reduces some of the complications that arise in other discretizations, and the emphasis on preservation properties is a positive aspect for applications in engineering simulations of compressible flows. Where the soft spots are is in the details of the entropy stability guarantee. The abstract highlights asymptotic entropy conservation, which suggests that the cancellation of entropy production terms may only hold in the limit of smooth, small perturbations. In regimes with discontinuities or high Mach numbers, there might be remaining positive entropy production that could affect the claimed nonlinear robustness. This aligns with the stress-test note, and it would be important to check the full paper for explicit derivations of the entropy balance or numerical tests that demonstrate behavior in those challenging cases. If those are solid, the concern is minor; otherwise, it could be more significant. This kind of paper is for people working on numerical schemes for hyperbolic conservation laws in fluid dynamics. A reader who is developing or evaluating entropy-stable methods for the Euler equations would find value in the specific technique and the way it integrates preservation properties. It deserves a serious referee because the idea is concrete enough and the potential for simplifying robust discretizations makes it worth a closer look by experts in the field. My recommendation is to send this to peer review rather than desk rejecting it, so the claims can be properly vetted with the full math and results.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes entropy-stable discretizations of the compressible Euler equations obtained by adapting the averages applied to density and internal energy via simple symmetric means (arithmetic, geometric, or harmonic) and their expansions. The resulting centralized convective terms are claimed to satisfy a discrete entropy inequality while also preserving kinetic energy and pressure equilibrium, thereby delivering nonlinear robustness without complex flux corrections.

Significance. If the derivations and numerical evidence hold, the work would supply a comparatively simple route to entropy-stable schemes for compressible flows that still respects additional structural invariants. Such constructions are valuable for practical high-Mach and shock-capturing simulations where standard averaging can produce instabilities.

major comments (2)

[Abstract and §3] Abstract and §3: the central claim of nonlinear robustness rests on the adaptive averages producing non-positive entropy production for the full compressible Euler system. The abstract explicitly invokes 'expansions for asymptotic entropy conservation,' which indicates that exact cancellation may hold only in the smooth, small-perturbation regime. Please supply the complete discrete entropy balance (including the residual terms that remain after the expansion) and demonstrate that the residual is non-positive for discontinuous or high-Mach solutions.
[§4] §4, the kinetic-energy and pressure-equilibrium preservation statements: these additional invariants are asserted to hold 'naturally' via the centralized convective terms. The proof should be written out explicitly, showing that the adaptive averaging operators commute with the required telescoping sums without introducing new source terms that could violate conservation.

minor comments (2)

[§2] Notation for the adaptive averaging operators (e.g., the precise definition of the expansion parameter) should be introduced once in §2 and used consistently thereafter.
Figure captions should state the exact mesh resolution, CFL number, and initial conditions used for each test so that the robustness claims can be reproduced.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will incorporate revisions to strengthen the presentation of the entropy balance and the preservation proofs.

read point-by-point responses

Referee: [Abstract and §3] Abstract and §3: the central claim of nonlinear robustness rests on the adaptive averages producing non-positive entropy production for the full compressible Euler system. The abstract explicitly invokes 'expansions for asymptotic entropy conservation,' which indicates that exact cancellation may hold only in the smooth, small-perturbation regime. Please supply the complete discrete entropy balance (including the residual terms that remain after the expansion) and demonstrate that the residual is non-positive for discontinuous or high-Mach solutions.

Authors: We agree that the distinction between asymptotic and exact entropy behavior merits explicit clarification. The adaptive averages are constructed so that the discrete entropy production remains non-positive for the full compressible Euler system, while the expansions recover asymptotic conservation only in the smooth regime. In the revised manuscript we will derive and display the complete discrete entropy balance, isolating the residual terms after the expansion. We will then show analytically that these residuals are non-positive for discontinuous and high-Mach regimes by exploiting the monotonicity and convexity properties of the chosen symmetric means. revision: yes
Referee: [§4] §4, the kinetic-energy and pressure-equilibrium preservation statements: these additional invariants are asserted to hold 'naturally' via the centralized convective terms. The proof should be written out explicitly, showing that the adaptive averaging operators commute with the required telescoping sums without introducing new source terms that could violate conservation.

Authors: We accept that the preservation statements would benefit from fully expanded proofs. In the revised manuscript we will write out the explicit algebraic steps demonstrating that the adaptive averaging operators commute with the telescoping sums for both kinetic energy and pressure equilibrium, confirming that no extraneous source terms arise and that the invariants are preserved exactly. revision: yes

Circularity Check

0 steps flagged

Derivation is a self-contained construction with no reduction to inputs or self-citations

full rationale

The paper presents a direct mathematical construction for entropy-stable discretizations of the compressible Euler equations by adapting averages applied to density and internal energy using simplified symmetric means (arithmetic, geometric, harmonic) and their expansions. This yields centralized convective terms that satisfy entropy stability, kinetic-energy preservation, and pressure-equilibrium preservation. No load-bearing step reduces by definition or by construction to a fitted parameter renamed as a prediction; no self-citation chain or uniqueness theorem imported from prior author work is invoked to force the result; and the method is not a renaming of a known empirical pattern. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that adaptive averages can enforce entropy stability for the Euler system while using only simple symmetric means. No free parameters or invented entities are mentioned in the abstract.

axioms (1)

domain assumption Adaptive averages applied to density and internal energy can achieve entropy stabilization for the compressible Euler equations even when using simplified symmetric means.
This premise is invoked directly in the abstract as the basis for non-linear robustness.

pith-pipeline@v0.9.0 · 5599 in / 1163 out tokens · 36458 ms · 2026-05-21T06:33:58.987026+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

choosing to have each term in Eq. (4) be positive recovers an ES scheme... ρ⋆ = ρlog and e# = eHlog for EC; otherwise adapt above/below via Table 1 and modifying functions M

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

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G. J. Gassner, A. R. Winters, D. A. Kopriva, Split form nodal discontinuous Galerkin schemes with summation- by-parts property for the compressible Euler equations , J. Comput. Phys. 327 (2016) 39–66

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Rojas, R

D. Rojas, R. Boukharfane, L. Dalcin, D. C. Del Rey Fernández, H. Ranocha, D. E. Keyes, M. Parsani, On the robustness and performance of entropy stable collocated discontinuous Galerkin methods , J. Comput. Phys. 426 (2021) 109891

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Aiello, A

A. Aiello, A. Palumbo, C. De Michele, G. Coppola, Impact of structure-preserving discretizations on com- pressible wall-bounded turbulence of thermally perfect gases , 2026. doi: 10.48550/arXiv.2602.17781. arXiv:2602.17781

work page doi:10.48550/arxiv.2602.17781 2026
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A. K. Edoh, Conservative correction procedures utilizing artiﬁcial dissipation operators , J. Comput. Phys. 504 (2024) 112880

work page 2024
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J. Chan, An artiﬁcial viscosity approach to high order entropy stable discontinuous Galerkin methods, J. Comput. Phys. 543 (2025) 114380

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A. Bercik, D. A. C. Penner, D. W. Zingg, Stable volume dissipation for high-order ﬁnite-difference and spectral- element methods with the summation-by-parts property , J. Sci. Comput. 107 (2026) 28

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F. Ismail, P . L. Roe,Affordable, entropy-consistent Euler ﬂux functions II: Entropy production at shocks, J. Com- put. Phys. 228 (2009) 5410–5436

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H. Ranocha, Comparison of some entropy conservative numerical ﬂuxes for the Euler equations , J. Sci. Comput. 76 (2018) 216–242

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De Michele, A

C. De Michele, A. K. Edoh, G. Coppola, Finite-difference compatible entropy-conserving schemes for the compressible Euler equations, J. Comput. Phys. (2025) 114262

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A. Jameson, Formulation of kinetic energy preserving conservative schemes for gas dynamics and direct numer- ical simulation of one-dimensional viscous compressible ﬂow in a shock tube using entropy and kinetic energy preserving schemes, J. Sci. Comput. 34 (2008) 188–208

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A. E. P . V eldman, A general condition for kinetic-energy preserving discretization of ﬂow transport equations , J. Comput. Phys. 398 (2019) 108894

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Coppola, F

G. Coppola, F. Capuano, S. Pirozzoli, L. de Luca, Numerically stable formulations of convective terms for turbulent compressible ﬂows, J. Comput. Phys. 382 (2019) 86–104

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A. K. Edoh, A new kinetic-energy-preserving method based on the convective rotational form , J. Comput. Phys. 454 (2022) 110971

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Shima, Y

N. Shima, Y . Kuya, Y . Tamaki, S. Kawai, Preventing spurious pressure oscillations in split convective form discretization for compressible ﬂows , J. Comput. Phys. 427 (2021) 110060

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De Michele, G

C. De Michele, G. Coppola, Novel pressure-equilibrium and kinetic-energy preserving ﬂuxes for compressible ﬂows based on the harmonic mean , J. Comput. Phys. 518 (2024) 113338

work page 2024
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De Michele, G

C. De Michele, G. Coppola, Asymptotically entropy-conservative and kinetic-energy preserving numerical ﬂuxes for compressible Euler equations , J. Comput. Phys. 492 (2023) 112439. 6 ES discretizations for the compressible Euler equations using simple adaptive averages A P REPRINT

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Jameson, The construction of discretely conservative ﬁnite volume schemes that also globally conserve energy or entropy, J

A. Jameson, The construction of discretely conservative ﬁnite volume schemes that also globally conserve energy or entropy, J. Sci. Comput. 34 (2008) 152–187

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Y . Kuya, K. Totani, S. Kawai, Kinetic energy and entropy preserving schemes for compressible ﬂows by split convective forms, J. Comput. Phys. 375 (2018) 823–853

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De Michele, G

C. De Michele, G. Coppola, Numerical treatment of the energy equation in compressible ﬂows simulations , Comput. Fluids 250 (2023) 105709

work page 2023
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Ranocha, G

H. Ranocha, G. Gassner, Preventing pressure oscillations does not ﬁx local linear stability issues of entropy-based split-form high-order schemes, Commun. Appl. Math. Comput. 4 (2022) 880–903

work page 2022
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Doehring, J

D. Doehring, J. Chan, H. Ranocha, M. Schlottke-Lakemper, M. Torrilhon, G. Gassner, V olume term adaptiv- ity for discontinuous Galerkin schemes , 2026. URL: https://arxiv.org/abs/2603.24189. doi:10.48550/ arXiv.2603.24189. arXiv:2603.24189

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Artiano, H

M. Artiano, H. Ranocha, On affordable high-order entropy-conservative/stable and well-balanced methods for nonconservative hyperbolic systems, 2026. doi:10.48550/ARXIV.2603.18978

work page doi:10.48550/arxiv.2603.18978 2026
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T. C. Fisher, M. H. Carpenter, High-order entropy stable ﬁnite difference schemes for nonlinear conservation laws: Finite domains , J. Comput. Phys. 252 (2013) 518–557

work page 2013
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Tamaki, Y

Y . Tamaki, Y . Kuya, S. Kawai, Comprehensive analysis of entropy conservation property of non-dissipative schemes for compressible ﬂows: KEEP scheme redeﬁned , J. Comput. Phys. 468 (2022) 111494

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Kawai, S

S. Kawai, S. Kawai, Logarithmic mean approximation in improving entropy conservation in KEEP scheme with pressure equilibrium preservation property for compressible ﬂows , J. Comput. Phys. 530 (2025) 113897

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G. A. Sod, A survey of several ﬁnite difference methods for systems of nonlinear hyperbolic conservation laws , J. Comput. Phys. 27 (1978) 1–31

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Chandrashekar, Kinetic energy preserving and entropy stable ﬁnite volume schemes for compressible Euler and NavierStokes equations, Commun

P . Chandrashekar, Kinetic energy preserving and entropy stable ﬁnite volume schemes for compressible Euler and NavierStokes equations, Commun. Comput. Phys. 14 (2013) 12521286

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Svärd, H

M. Svärd, H. Özcan, Entropy-stable schemes for the Euler equations with far-ﬁeld and wall boundary conditions , J. Sci. Comput. 58 (2014) 61–89

work page 2014
[29]

A. R. Winters, D. A. Kopriva, J. Nordström, Numerical boundary ﬂux functions that give provable bounds for nonlinear initial boundary value problems with open boundaries , J. Comput. Phys. 559 (2026) 114891

work page 2026
[30]

Aiello, C

A. Aiello, C. De Michele, G. Coppola, Entropy conservative discretization of compressible Euler equations with an arbitrary equation of state , J. Comput. Phys. 528 (2025) 113836

work page 2025
[31]

Aiello, C

A. Aiello, C. De Michele, G. Coppola, Formulation of entropy-conservative discretizations for compressible ﬂows of thermally perfect gases , 2026. doi:10.48550/arXiv.2507.08115. arXiv:2507.08115

work page doi:10.48550/arxiv.2507.08115 2026
[32]

Klein, B

R. Klein, B. Sanderse, P . Costa, R. Pecnik, R. Henkes, Generalized Tadmor conditions and structure- preserving numerical ﬂuxes for the compressible ﬂow of real gases , 2026. doi: 10.48550/arXiv.2603.15112. arXiv:2603.15112

work page doi:10.48550/arxiv.2603.15112 2026
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Pirozzoli, Generalized conservative approximations of split convective derivative operators, J

S. Pirozzoli, Generalized conservative approximations of split convective derivative operators, J. Comput. Phys. 229 (2010) 7180–7190. 7

work page 2010

[1] [1]

G. J. Gassner, A. R. Winters, D. A. Kopriva, Split form nodal discontinuous Galerkin schemes with summation- by-parts property for the compressible Euler equations , J. Comput. Phys. 327 (2016) 39–66

work page 2016

[2] [2]

Rojas, R

D. Rojas, R. Boukharfane, L. Dalcin, D. C. Del Rey Fernández, H. Ranocha, D. E. Keyes, M. Parsani, On the robustness and performance of entropy stable collocated discontinuous Galerkin methods , J. Comput. Phys. 426 (2021) 109891

work page 2021

[3] [3]

Aiello, A

A. Aiello, A. Palumbo, C. De Michele, G. Coppola, Impact of structure-preserving discretizations on com- pressible wall-bounded turbulence of thermally perfect gases , 2026. doi: 10.48550/arXiv.2602.17781. arXiv:2602.17781

work page doi:10.48550/arxiv.2602.17781 2026

[4] [4]

A. K. Edoh, Conservative correction procedures utilizing artiﬁcial dissipation operators , J. Comput. Phys. 504 (2024) 112880

work page 2024

[5] [5]

Chan, An artiﬁcial viscosity approach to high order entropy stable discontinuous Galerkin methods, J

J. Chan, An artiﬁcial viscosity approach to high order entropy stable discontinuous Galerkin methods, J. Comput. Phys. 543 (2025) 114380

work page 2025

[6] [6]

Bercik, D

A. Bercik, D. A. C. Penner, D. W. Zingg, Stable volume dissipation for high-order ﬁnite-difference and spectral- element methods with the summation-by-parts property , J. Sci. Comput. 107 (2026) 28

work page 2026

[7] [7]

Ismail, P

F. Ismail, P . L. Roe,Affordable, entropy-consistent Euler ﬂux functions II: Entropy production at shocks, J. Com- put. Phys. 228 (2009) 5410–5436

work page 2009

[8] [8]

Ranocha, Comparison of some entropy conservative numerical ﬂuxes for the Euler equations , J

H. Ranocha, Comparison of some entropy conservative numerical ﬂuxes for the Euler equations , J. Sci. Comput. 76 (2018) 216–242

work page 2018

[9] [9]

De Michele, A

C. De Michele, A. K. Edoh, G. Coppola, Finite-difference compatible entropy-conserving schemes for the compressible Euler equations, J. Comput. Phys. (2025) 114262

work page 2025

[10] [10]

A. Jameson, Formulation of kinetic energy preserving conservative schemes for gas dynamics and direct numer- ical simulation of one-dimensional viscous compressible ﬂow in a shock tube using entropy and kinetic energy preserving schemes, J. Sci. Comput. 34 (2008) 188–208

work page 2008

[11] [11]

A. E. P . V eldman, A general condition for kinetic-energy preserving discretization of ﬂow transport equations , J. Comput. Phys. 398 (2019) 108894

work page 2019

[12] [12]

Coppola, F

G. Coppola, F. Capuano, S. Pirozzoli, L. de Luca, Numerically stable formulations of convective terms for turbulent compressible ﬂows, J. Comput. Phys. 382 (2019) 86–104

work page 2019

[13] [13]

A. K. Edoh, A new kinetic-energy-preserving method based on the convective rotational form , J. Comput. Phys. 454 (2022) 110971

work page 2022

[14] [14]

Shima, Y

N. Shima, Y . Kuya, Y . Tamaki, S. Kawai, Preventing spurious pressure oscillations in split convective form discretization for compressible ﬂows , J. Comput. Phys. 427 (2021) 110060

work page 2021

[15] [15]

De Michele, G

C. De Michele, G. Coppola, Novel pressure-equilibrium and kinetic-energy preserving ﬂuxes for compressible ﬂows based on the harmonic mean , J. Comput. Phys. 518 (2024) 113338

work page 2024

[16] [16]

De Michele, G

C. De Michele, G. Coppola, Asymptotically entropy-conservative and kinetic-energy preserving numerical ﬂuxes for compressible Euler equations , J. Comput. Phys. 492 (2023) 112439. 6 ES discretizations for the compressible Euler equations using simple adaptive averages A P REPRINT

work page 2023

[17] [17]

Jameson, The construction of discretely conservative ﬁnite volume schemes that also globally conserve energy or entropy, J

A. Jameson, The construction of discretely conservative ﬁnite volume schemes that also globally conserve energy or entropy, J. Sci. Comput. 34 (2008) 152–187

work page 2008

[18] [18]

Y . Kuya, K. Totani, S. Kawai, Kinetic energy and entropy preserving schemes for compressible ﬂows by split convective forms, J. Comput. Phys. 375 (2018) 823–853

work page 2018

[19] [19]

De Michele, G

C. De Michele, G. Coppola, Numerical treatment of the energy equation in compressible ﬂows simulations , Comput. Fluids 250 (2023) 105709

work page 2023

[20] [20]

Ranocha, G

H. Ranocha, G. Gassner, Preventing pressure oscillations does not ﬁx local linear stability issues of entropy-based split-form high-order schemes, Commun. Appl. Math. Comput. 4 (2022) 880–903

work page 2022

[21] [21]

Doehring, J

D. Doehring, J. Chan, H. Ranocha, M. Schlottke-Lakemper, M. Torrilhon, G. Gassner, V olume term adaptiv- ity for discontinuous Galerkin schemes , 2026. URL: https://arxiv.org/abs/2603.24189. doi:10.48550/ arXiv.2603.24189. arXiv:2603.24189

work page arXiv 2026

[22] [22]

Artiano, H

M. Artiano, H. Ranocha, On affordable high-order entropy-conservative/stable and well-balanced methods for nonconservative hyperbolic systems, 2026. doi:10.48550/ARXIV.2603.18978

work page doi:10.48550/arxiv.2603.18978 2026

[23] [23]

T. C. Fisher, M. H. Carpenter, High-order entropy stable ﬁnite difference schemes for nonlinear conservation laws: Finite domains , J. Comput. Phys. 252 (2013) 518–557

work page 2013

[24] [24]

Tamaki, Y

Y . Tamaki, Y . Kuya, S. Kawai, Comprehensive analysis of entropy conservation property of non-dissipative schemes for compressible ﬂows: KEEP scheme redeﬁned , J. Comput. Phys. 468 (2022) 111494

work page 2022

[25] [25]

Kawai, S

S. Kawai, S. Kawai, Logarithmic mean approximation in improving entropy conservation in KEEP scheme with pressure equilibrium preservation property for compressible ﬂows , J. Comput. Phys. 530 (2025) 113897

work page 2025

[26] [26]

G. A. Sod, A survey of several ﬁnite difference methods for systems of nonlinear hyperbolic conservation laws , J. Comput. Phys. 27 (1978) 1–31

work page 1978

[27] [27]

Chandrashekar, Kinetic energy preserving and entropy stable ﬁnite volume schemes for compressible Euler and NavierStokes equations, Commun

P . Chandrashekar, Kinetic energy preserving and entropy stable ﬁnite volume schemes for compressible Euler and NavierStokes equations, Commun. Comput. Phys. 14 (2013) 12521286

work page 2013

[28] [28]

Svärd, H

M. Svärd, H. Özcan, Entropy-stable schemes for the Euler equations with far-ﬁeld and wall boundary conditions , J. Sci. Comput. 58 (2014) 61–89

work page 2014

[29] [29]

A. R. Winters, D. A. Kopriva, J. Nordström, Numerical boundary ﬂux functions that give provable bounds for nonlinear initial boundary value problems with open boundaries , J. Comput. Phys. 559 (2026) 114891

work page 2026

[30] [30]

Aiello, C

A. Aiello, C. De Michele, G. Coppola, Entropy conservative discretization of compressible Euler equations with an arbitrary equation of state , J. Comput. Phys. 528 (2025) 113836

work page 2025

[31] [31]

Aiello, C

A. Aiello, C. De Michele, G. Coppola, Formulation of entropy-conservative discretizations for compressible ﬂows of thermally perfect gases , 2026. doi:10.48550/arXiv.2507.08115. arXiv:2507.08115

work page doi:10.48550/arxiv.2507.08115 2026

[32] [32]

Klein, B

R. Klein, B. Sanderse, P . Costa, R. Pecnik, R. Henkes, Generalized Tadmor conditions and structure- preserving numerical ﬂuxes for the compressible ﬂow of real gases , 2026. doi: 10.48550/arXiv.2603.15112. arXiv:2603.15112

work page doi:10.48550/arxiv.2603.15112 2026

[33] [33]

Pirozzoli, Generalized conservative approximations of split convective derivative operators, J

S. Pirozzoli, Generalized conservative approximations of split convective derivative operators, J. Comput. Phys. 229 (2010) 7180–7190. 7

work page 2010