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arxiv: 1907.03287 · v1 · pith:7MJPDYOGnew · submitted 2019-07-07 · 🧮 math.NA · cs.NA· physics.comp-ph

Entropy stable numerical approximations for the isothermal and polytropic Euler equations

Andrew R. Winters , Christof Czernik , Moritz B. Schily , Gregor J. Gassner This is my paper

Pith reviewed 2026-05-25 01:32 UTC · model grok-4.3

classification 🧮 math.NA cs.NAphysics.comp-ph
keywords entropy stabilitydiscontinuous Galerkinpolytropic Euler equationssummation-by-partsnumerical fluxesisothermal Eulershallow water equations
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The pith

Specially chosen numerical fluxes make high-order discontinuous Galerkin approximations to the polytropic Euler equations entropy stable and consistent with total energy.

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

The paper establishes a way to build entropy-stable high-order schemes for the polytropic Euler equations, a system closed by assuming pressure depends only on density so that total energy functions as a convex mathematical entropy. The authors first replicate the continuous entropy analysis inside a finite-volume discretization to obtain special numerical flux functions. These fluxes are then inserted into a discontinuous Galerkin spectral element method that uses summation-by-parts operators, producing schemes that remain entropy stable while staying consistent with the auxiliary total energy equation. The construction covers the isothermal Euler equations when the adiabatic index equals one and the shallow water equations when it equals two. Such schemes matter because they keep numerical solutions physically plausible for flows governed by these reduced Euler models.

Core claim

When the Euler equations are closed with a polytropic equation of state, pressure is a power law of density alone and total energy serves as a convex entropy; the mass and momentum equations form a closed system. Mimicking the continuous entropy analysis on the discrete level in finite volume produces special numerical flux functions. Transferring those fluxes into a discontinuous Galerkin spectral element framework equipped with summation-by-parts operators yields high-order accurate approximations that are entropy stable and consistent with the auxiliary total energy behavior.

What carries the argument

Special numerical flux functions obtained by discretely mimicking the continuous entropy analysis, transferred into a DG spectral element method that employs summation-by-parts operators.

If this is right

  • The resulting DG scheme is entropy stable for the isothermal Euler equations.
  • The scheme is entropy stable for general polytropic gases including the shallow water equations.
  • The approximation remains consistent with total energy acting as an auxiliary convex entropy.
  • High-order accuracy is retained while the entropic properties hold.

Where Pith is reading between the lines

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

  • The same flux-construction strategy could be tested on other hyperbolic systems whose entropy structure is known only through an auxiliary variable.
  • The approach may help maintain positivity or boundedness in long-time simulations of atmospheric or free-surface flows.
  • Extensions that incorporate source terms or three-dimensional geometry can be checked by repeating the finite-volume entropy analysis step.

Load-bearing premise

The continuous entropy analysis can be directly mimicked on the discrete level through specially chosen numerical flux functions that remain entropy stable when moved from finite volume into the DG framework.

What would settle it

A numerical experiment on a standard test problem in which the new DG scheme produces a decrease in total entropy or violates consistency with the auxiliary total energy equation would falsify the claim.

read the original abstract

In this work we analyze the entropic properties of the Euler equations when the system is closed with the assumption of a polytropic gas. In this case, the pressure solely depends upon the density of the fluid and the energy equation is not necessary anymore as the mass conservation and momentum conservation then form a closed system. Further, the total energy acts as a convex mathematical entropy function for the polytropic Euler equations. The polytropic equation of state gives the pressure as a scaled power law of the density in terms of the adiabatic index $\gamma$. As such, there are important limiting cases contained within the polytropic model like the isothermal Euler equations ($\gamma=1$) and the shallow water equations ($\gamma=2$). We first mimic the continuous entropy analysis on the discrete level in a finite volume context to get special numerical flux functions. Next, these numerical fluxes are incorporated into a particular discontinuous Galerkin (DG) spectral element framework where derivatives are approximated with summation-by-parts operators. This guarantees a high-order accurate DG numerical approximation to the polytropic Euler equations that is also consistent to its auxiliary total energy behavior. Numerical examples are provided to verify the theoretical derivations, i.e., the entropic properties of the high order DG scheme.

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

0 major / 3 minor

Summary. The manuscript constructs entropy-stable high-order discontinuous Galerkin (DG) spectral element approximations for the polytropic Euler equations (including isothermal γ=1 and shallow-water γ=2 limits) by first deriving special numerical interface fluxes in a finite-volume setting that replicate the continuous entropy analysis (with total energy serving as the convex entropy), then transferring those fluxes into an SBP-DG framework. The summation-by-parts property is used to ensure the discrete entropy inequality carries over exactly, yielding schemes that are high-order accurate and consistent with the auxiliary total-energy behavior. Numerical examples are supplied to verify the entropic properties.

Significance. If the central construction holds, the work supplies a systematic route to entropy-stable high-order DG schemes for polytropic closures where the energy equation is auxiliary. The explicit mimicry of the continuous entropy analysis inside the finite-volume flux and its direct transfer via SBP operators is a clear strength; the approach is falsifiable through the supplied numerical tests and extends naturally to other SBP-based discretizations.

minor comments (3)
  1. [Abstract] Abstract: the phrasing 'guarantees a high-order accurate DG numerical approximation ... that is also consistent to its auxiliary total energy behavior' is slightly loose; a one-sentence clarification on whether consistency is proven or observed numerically would improve precision.
  2. [Introduction] The manuscript would benefit from an explicit statement (perhaps in the introduction or conclusions) of how the derived fluxes reduce in the isothermal and shallow-water limits, to make the generality of the construction immediately visible to readers.
  3. [Numerical examples] Numerical examples section: while the abstract states that examples verify the entropic properties, a brief table or plot caption indicating the observed convergence rates and entropy decay for at least one non-trivial test case would strengthen the verification claim without lengthening the text.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of our work on entropy-stable high-order DG schemes for the polytropic Euler equations, as well as the recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper starts from the continuous entropy analysis of the polytropic Euler equations (where total energy is a convex entropy) and mimics it exactly inside a finite-volume discretization to obtain interface fluxes. These fluxes are then inserted into an SBP-DG spectral-element method whose summation-by-parts property supplies a discrete integration-by-parts identity that mirrors the continuous one, allowing the entropy inequality to transfer without additional fitted parameters or self-referential assumptions. No step reduces a claimed prediction or uniqueness result to a prior fit or self-citation; the construction is self-contained against the external continuous entropy property and standard SBP theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that total energy functions as a convex entropy for the polytropic system; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Total energy acts as a convex mathematical entropy function for the polytropic Euler equations.
    Explicitly stated in the abstract as the foundation for the entropy analysis and the auxiliary total energy behavior.

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