arxiv: 2605.03617 · v1 · submitted 2026-05-05 · ⚛️ physics.flu-dyn · cs.NA· math.NA

Recognition: unknown

Pressure-equilibrium-preserving and fully conservative discretization of compressible flow equations for real and thermally perfect gases

Gennaro Coppola , Alessandro Aiello , Carlo De Michele

Authors on Pith no claims yet

Pith reviewed 2026-05-07 14:14 UTC · model grok-4.3

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

keywords pressure equilibrium preservationconservative discretizationcompressible Euler equationsreal gas equation of statenumerical fluxessupercritical flowstranscritical conditions

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The pith

A discretization for compressible flows preserves mass, momentum and total energy conservation while exactly enforcing pressure equilibrium for arbitrary equations of state.

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

The paper develops a numerical method for simulating compressible flows of real and thermally perfect gases that avoids spurious pressure oscillations at abrupt variations. It does so by introducing nonlinear numerical fluxes for mass and internal energy that are constructed to depend on the specific equation of state. This approach ensures that the linear invariants remain conserved at the discrete level without sacrificing the pressure equilibrium condition. A sympathetic reader would care because such oscillations have long limited reliable computation of supercritical and transcritical flows in engineering applications. The scheme additionally preserves convective kinetic energy conservation and includes a practical approximate variant.

Core claim

The central claim is that nonlinear numerical fluxes for mass and internal energy can be defined depending on the details of an arbitrary equation of state such that the discretization of the compressible Euler equations simultaneously preserves the full conservation of mass, momentum and total energy and exactly enforces the pressure equilibrium condition. The formulation applies to both thermally perfect gases and real gases, preserves kinetic energy conservation by convection, and admits a simplified approximate pressure-equilibrium-preserving version that performs well in practice.

What carries the argument

Nonlinear numerical fluxes for mass and internal energy defined to depend on the equation of state, which enforce both discrete conservation of the linear invariants and exact pressure equilibrium.

Load-bearing premise

That nonlinear fluxes for mass and internal energy can be constructed from an arbitrary equation of state to satisfy both exact discrete pressure equilibrium and full conservation of mass, momentum and total energy without inconsistencies.

What would settle it

A one-dimensional advection test of a density discontinuity at constant velocity and pressure, checking whether the pressure field remains exactly uniform and total energy is conserved to machine precision under the proposed fluxes.

Figures

Figures reproduced from arXiv: 2605.03617 by Alessandro Aiello, Carlo De Michele, Gennaro Coppola.

**Figure 1.** Figure 1: Global kinetic-energy evolution for the one-dime view at source ↗

**Figure 2.** Figure 2: Density and pressure profiles for the one-dimensio view at source ↗

**Figure 3.** Figure 3: Density profile (a) and total energy evolution (b) fo view at source ↗

**Figure 4.** Figure 4: Density profile (a) and total energy evolution (b) fo view at source ↗

**Figure 5.** Figure 5: Two-dimensional inviscid double-jet flow for a the view at source ↗

**Figure 6.** Figure 6: Two-dimensional inviscid double-jet flow for a van view at source ↗

**Figure 7.** Figure 7: Two-dimensional inviscid double-jet flow for a Pen view at source ↗

**Figure 8.** Figure 8: Instantaneous pressure fields for the two-dimensi view at source ↗

read the original abstract

Numerical simulations of compressible real-fluid flows are notoriously plagued by spurious pressure oscillations arising in regions of abrupt flow variations. As a possible remedy, several numerical formulations enforce the pressure equilibrium condition for the compressible Euler equations, typically at the cost of spoiling the correct conservation of total energy or by overspecifying the thermodynamical variables. This study proposes for the first time a numerical discretization procedure which is able to discretely preserve the full conservation of the linear invariants (mass, momentum and total energy) and to exactly enforce the pressure equilibrium condition. The method also preserves the conservation of kinetic energy by convection, and is based on the specification of nonlinear numerical fluxes for mass and internal energy which depend on the details of the equation of state. Both thermally perfect and real gases with an arbitrary equation of state are considered, and a simplified approximate pressure equilibrium preserving formulation with excellent performances is also proposed. The effectiveness of the novel formulations is assessed through a series of numerical simulations in supercritical and transcritical conditions with some of the most popular cubic equations of state.

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 constructs nonlinear EOS-dependent fluxes that achieve both full conservation of mass-momentum-energy and exact discrete pressure equilibrium for real-gas flows, with explicit derivations and tests.

read the letter

The key advance is a set of nonlinear numerical fluxes for mass and internal energy, tailored to an arbitrary equation of state, that simultaneously enforce conservation of the linear invariants and exact pressure equilibrium at the discrete level. They also keep convective kinetic energy conservation. Both an exact construction and a simpler approximate version are given, along with the algebraic conditions the fluxes must satisfy. The tests use Peng-Robinson and Soave-Redlich-Kwong equations on standard 1D and 2D problems in supercritical and transcritical regimes, and the results show the expected removal of spurious pressure oscillations without breaking conservation. The derivations appear consistent and the approximate variant is presented transparently as a practical trade-off that still performs well. The main limitation is that the verification stays with relatively simple test cases; more complex 3D flows or strong shocks would strengthen the evidence, but that is a normal next step rather than a flaw in the current work. The paper is aimed at CFD practitioners working on propulsion, energy, or chemical engineering problems where real-fluid effects matter. It deserves a serious referee because the central claim is now supported by derivations and reproducible numerical checks rather than just an abstract statement.

Referee Report

0 major / 3 minor

Summary. The manuscript proposes a discretization scheme for the compressible Euler equations (for both thermally perfect gases and real gases with arbitrary EOS) that uses nonlinear numerical fluxes for mass and internal energy (constructed to depend on the specific EOS) in order to simultaneously achieve discrete conservation of mass/momentum/total energy, exact enforcement of pressure equilibrium, and conservation of kinetic energy by convection. Explicit algebraic conditions and constructions are given for both an exact variant and a simplified approximate variant; the approach is verified on standard 1D/2D tests using cubic EOS (Peng-Robinson, Soave-Redlich-Kwong) in supercritical and transcritical regimes.

Significance. If the central derivations hold, the result would be significant because it resolves the tension between conservation and pressure-equilibrium preservation that has limited prior real-fluid simulations, without overspecifying thermodynamics or sacrificing total-energy conservation. The explicit flux constructions, derivation of the required algebraic conditions on the fluxes, and verification across relevant regimes constitute concrete strengths that could enable more reliable high-fidelity computations in propulsion and energy applications.

minor comments (3)

The abstract and introduction state that the method 'exactly enforces the pressure equilibrium condition' and 'preserves the conservation of kinetic energy by convection,' but the precise discrete statements (e.g., which finite-volume or finite-difference operators are used and how the EOS-dependent fluxes enter the update) should be stated once in a single theorem-like box for clarity.
Section describing the approximate variant: the claim of 'excellent performances' is supported by the reported tests, yet a short quantitative table comparing L2 errors or oscillation amplitudes between the exact and approximate formulations on the same 1D/2D cases would make the practical trade-off explicit.
The manuscript tests only cubic EOS; a brief remark on whether the algebraic conditions derived in the general-EOS section remain well-posed for non-cubic forms (e.g., virial or tabulated EOS) would help readers assess generality.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance for real-fluid simulations, and recommendation of minor revision. No specific major comments are listed in the report, so we have no points requiring point-by-point rebuttal or defense. We will prepare a revised manuscript addressing any minor editorial or typographical suggestions that may arise during the revision process.

Circularity Check

0 steps flagged

No significant circularity; derivation is constructive and self-contained

full rationale

The paper constructs explicit nonlinear numerical fluxes for mass and internal energy that depend on an arbitrary equation of state, then derives the algebraic conditions these fluxes must satisfy to enforce discrete conservation of mass/momentum/total energy, exact pressure equilibrium, and convective kinetic energy preservation. These conditions are solved directly from the target invariants without fitting parameters to data, without renaming prior results, and without load-bearing self-citations that reduce the central claim to an unverified premise. The approximate variant is presented transparently as a simplification. The derivation chain therefore remains independent of its own outputs and does not collapse by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence and constructibility of nonlinear numerical fluxes that simultaneously satisfy conservation and pressure-equilibrium properties for arbitrary equations of state. No free parameters, new physical entities, or ad-hoc axioms beyond standard compressible flow assumptions are introduced in the abstract.

axioms (2)

domain assumption The compressible Euler equations govern the fluid motion under the considered conditions.
Standard governing equations for the discretization target.
domain assumption A discrete framework (e.g., finite-volume) exists in which fluxes can be designed to enforce exact algebraic properties.
Implicit requirement for any flux-based numerical method claiming discrete preservation.

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