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arxiv: 1906.11648 · v1 · pith:ATROYEINnew · submitted 2019-06-26 · 🧮 math.NA · cs.NA

Consistent Internal Energy Based Schemes for the Compressible Euler Equations

R. Herbin (LATP) , T. Gallou\"et (I2M) , J.-C Latch\'e (IRSN) , N Therme This is my paper

Pith reviewed 2026-05-25 15:26 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords compressible Euler equationsinternal energy discretizationpositivity preservationdiscrete entropy inequalityupwind schemespressure correctionMUSCL reconstructionLax-Wendroff consistency
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The pith

Schemes discretizing internal energy with material-velocity upwinding satisfy positivity and a local discrete entropy inequality 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 constructs numerical schemes for the compressible Euler equations that discretize the internal energy equation together with corrective terms to achieve Lax-Wendroff consistency. Convection operators for mass and internal energy are built to upwind exclusively according to the material velocity, eliminating any need for Riemann solvers. Fractional-step algorithms handle time discretization, either through semi-implicit pressure correction or explicit steps. The resulting methods preserve the integral of total energy, keep density, internal energy and pressure positive, and remain stable either unconditionally or under a CFL restriction. The central claim is that the semi-implicit first-order upwind variant satisfies a local discrete entropy inequality, while MUSCL-like extensions satisfy the inequality up to a remainder that vanishes with mesh and time-step refinement under L-infinity and BV control of the solution.

Core claim

By discretizing the internal energy equation with corrective terms for consistency and designing discrete convection operators that upwind mass and internal energy fluxes solely with respect to the material velocity, the schemes achieve positivity of density, internal energy and pressure, preservation of the integral of total energy, and, for the semi-implicit first-order upwind scheme, a local discrete entropy inequality. When a MUSCL-like reconstruction is introduced, the entropy inequality holds up to a remainder term that tends to zero with the space and time steps provided the discrete solution remains bounded in L-infinity and BV norms. Explicit variants satisfy analogous weaker forms,

What carries the argument

Discrete convection operators for the mass and internal energy equations, constructed with upwinding only with respect to the material velocity, inside a fractional-step time discretization that may be pressure-corrected or fully explicit.

If this is right

  • Preservation of the integral of total energy over the computational domain.
  • Positivity of density, internal energy and pressure.
  • Stability without time-step restriction for the pressure-correction variant and under a CFL condition for explicit variants.
  • Local discrete entropy inequality satisfied by the semi-implicit first-order upwind scheme.
  • Entropy inequality up to a remainder that tends to zero with discretization parameters under L-infinity and BV control for MUSCL-like schemes.

Where Pith is reading between the lines

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

  • The same upwinding construction might be applied to other hyperbolic conservation laws that require strict positivity, such as certain multi-species or shallow-water models.
  • The BV-control hypothesis for the remainder term to vanish indicates that practical high-order implementations may need additional limiting or artificial viscosity to retain the entropy property on coarse meshes.
  • Explicit variants could be made fully robust by adding a stabilization term to the momentum equation that supplies the required velocity estimate.
  • Verification on multi-dimensional unstructured meshes would test whether the entropy remainder still vanishes without extra geometric assumptions.

Load-bearing premise

Upwinding the discrete convection operators for mass and internal energy only with respect to the material velocity is sufficient to guarantee positivity of density and internal energy together with the stated stability and entropy properties.

What would settle it

A numerical run of the first-order semi-implicit scheme on the Sod shock tube in which either density or internal energy becomes negative or the computed discrete entropy production is negative at some cell.

Figures

Figures reproduced from arXiv: 1906.11648 by J.-C Latch\'e (IRSN), N Therme, R. Herbin (LATP), T. Gallou\"et (I2M).

Figure 1
Figure 1. Figure 1: Meshes and unknowns – Left: unstructured discretizations (the present sketch illustrates the possibility, implemented in our software CALIF3 S [4], of mixing simpli￾cial and quadrangular cells); scalars variables are associated to the primal cells (here K, L and M) while velocity vectors are associated to the faces (here, σ and σ ′ ) or, equiva￾lently, to dual cells (here, Dσ and Dσ′ ). – Right: MAC discre… view at source ↗
Figure 2
Figure 2. Figure 2: We observe that both schemes seem to converge, but th [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: Test 5 of [39, chapter 4] - Density obtained with n = 2000 cells, with and without corrective source terms in pressure correction scheme, and analytical solution. When solving the Euler equations numerically, it is thus natural to design numerical schemes such that some entropy inequalities are satisfied by the ap￾proximate solutions; these inequalities should enable to prove that, as the mesh and time ste… view at source ↗
read the original abstract

Numerical schemes for the solution of the Euler equations have recently been developed, which involve the discretisation of the internal energy equation, with corrective terms to ensure the correct capture of shocks, and, more generally, the consistency in the Lax-Wendroff sense. These schemes may be staggered or colocated, using either struc-tured meshes or general simplicial or tetrahedral/hexahedral meshes. The time discretization is performed by fractional-step algorithms; these may be either based on semi-implicit pressure correction techniques or segregated in such a way that only explicit steps are involved (referred to hereafter as "explicit" variants). In order to ensure the positivity of the density, the internal energy and the pressure, the discrete convection operators for the mass and internal energy balance equations are carefully designed; they use an upwind technique with respect to the material velocity only. The construction of the fluxes thus does not need any Rie-mann or approximate Riemann solver, and yields easily implementable algorithms. The stability is obtained without restriction on the time step for the pressure correction scheme and under a CFL-like condition for explicit variants: preservation of the integral of the total energy over the computational domain, and positivity of the density and the internal energy. The semi-implicit first-order upwind scheme satisfies a local discrete entropy inequality. If a MUSCL-like scheme is used in order to limit the scheme diffusion, then a weaker property holds: the entropy inequality is satisfied up to a remainder term which is shown to tend to zero with the space and time steps, if the discrete solution is controlled in L $\infty$ and BV norms. The explicit upwind variant also satisfies such a weaker property, at the price of an estimate for the velocity which could be derived from the introduction of a new stabilization term in the momentum balance. Still for the explicit scheme, with the above-mentioned MUSCL-like scheme, the same result only holds if the ratio of the time to the space step tends to zero.

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.

Referee Report

2 major / 2 minor

Summary. The paper constructs finite-volume schemes for the compressible Euler equations by discretizing the internal energy balance with corrective terms for shock consistency and Lax-Wendroff sense. Convection operators for mass and internal energy are upwinded solely with respect to material velocity (no Riemann solvers). Fractional-step time marching is used, either semi-implicit pressure correction or fully explicit. The schemes are claimed to preserve positivity of density, internal energy and pressure, global total-energy conservation, and to satisfy a local discrete entropy inequality (first-order upwind, semi-implicit case) or the same inequality up to a remainder that vanishes with mesh and time step under L∞/BV control of the discrete solution (MUSCL-like reconstructions). Explicit variants require an additional velocity estimate or a vanishing time-to-space-step ratio.

Significance. If the positivity, conservation and entropy statements are fully rigorous, the work supplies a family of implementable, parameter-free schemes on general meshes that avoid approximate Riemann solvers while guaranteeing thermodynamic consistency and favorable stability (unconditional for the pressure-correction variant). Such properties are practically relevant for compressible-flow codes.

major comments (2)
  1. [Abstract / entropy statements] Abstract (and the corresponding theorem statements): the entropy inequality for MUSCL-like reconstructions is asserted to hold up to a remainder that tends to zero under L∞ and BV control of the discrete solution, yet the manuscript supplies no a priori L∞ or BV estimates, discrete maximum principles, or maximum-principle-preserving limiters that would justify the required control. Without these estimates the remainder term cannot be shown to vanish, so the consistency claim for the practically relevant MUSCL extension remains conditional on an unproven regularity assumption.
  2. [Abstract / explicit variant] Abstract (explicit variant): the weaker entropy property for the explicit upwind scheme is stated to require an estimate for the velocity that “could be derived from the introduction of a new stabilization term in the momentum balance,” but no such term is constructed or analyzed in the provided text, leaving the explicit-scheme entropy result incomplete.
minor comments (2)
  1. Notation for the corrective terms that restore consistency should be introduced with an explicit equation number rather than described only in prose.
  2. The precise definition of the “MUSCL-like scheme” (slope limiter, reconstruction stencil) is not given; a short algorithmic box or pseudocode would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract / entropy statements] Abstract (and the corresponding theorem statements): the entropy inequality for MUSCL-like reconstructions is asserted to hold up to a remainder that tends to zero under L∞ and BV control of the discrete solution, yet the manuscript supplies no a priori L∞ or BV estimates, discrete maximum principles, or maximum-principle-preserving limiters that would justify the required control. Without these estimates the remainder term cannot be shown to vanish, so the consistency claim for the practically relevant MUSCL extension remains conditional on an unproven regularity assumption.

    Authors: We agree that the result for MUSCL-like reconstructions is conditional on L∞ and BV control of the discrete solution, and that the manuscript does not supply a priori estimates or maximum-principle-preserving limiters. Deriving such bounds for the compressible Euler equations remains an open and technically difficult question that lies outside the scope of the present work. The theorem is therefore stated under this hypothesis, which we view as a transparent way to present the consistency result. We will revise the abstract (and the corresponding theorem statement) to make the conditional nature of the claim fully explicit. revision: yes

  2. Referee: [Abstract / explicit variant] Abstract (explicit variant): the weaker entropy property for the explicit upwind scheme is stated to require an estimate for the velocity that “could be derived from the introduction of a new stabilization term in the momentum balance,” but no such term is constructed or analyzed in the provided text, leaving the explicit-scheme entropy result incomplete.

    Authors: The manuscript indeed only remarks that an estimate for the velocity could potentially be obtained via a new stabilization term in the momentum equation, without constructing or analyzing any such term. This remark was intended to indicate a possible route rather than to claim a completed result. We will revise the abstract to clarify that the entropy property for the explicit scheme is stated under the additional assumption of such a velocity estimate, and that the construction of the stabilization term is left for future work. revision: yes

Circularity Check

0 steps flagged

No circularity: properties derived directly from discrete operator definitions via standard finite-volume arguments

full rationale

The paper defines convection operators with material-velocity upwinding, then proves positivity, integral energy preservation, and (conditional) entropy inequalities by direct manipulation of the resulting discrete balance equations. No parameter is fitted on a data subset and renamed a prediction; no self-citation supplies a uniqueness theorem or ansatz that the present derivation relies upon; the L∞/BV control hypothesis for the MUSCL remainder is stated explicitly as an external assumption rather than smuggled in by definition or prior self-work. The derivation chain is therefore self-contained against the scheme's own equations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction rests on standard finite-volume assumptions for hyperbolic systems and on the design choice that material-velocity upwinding alone suffices for positivity; no free parameters or new entities are introduced.

axioms (2)
  • domain assumption Finite-volume discretization on general simplicial or hexahedral meshes is appropriate for the compressible Euler equations.
    Invoked when the paper states that schemes may be staggered or colocated on structured or unstructured meshes.
  • domain assumption Upwind convection operators defined with respect to material velocity alone preserve positivity of density, internal energy and pressure.
    Central premise used to justify the flux construction and the absence of Riemann solvers.

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Reference graph

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