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

Gennaro Coppola, Alessandro Aiello, Carlo De Michele

arXiv:2605.03617v2physics.flu-dyncs.NAmath.NA

TL;DR

Real-gas Euler discretizations can produce nonphysical oscillations across contact discontinuities because pressure depends nonlinearly on density and internal energy. This paper develops equation-of-state-dependent fluxes that preserve conservation and pressure equilibrium; its exact formulation demonstrates that these properties can coexist, though it shows strong instabilities in a tested transcritical case.

  • Problem

    Real-gas Euler discretizations can generate nonphysical oscillations across contact discontinuities because pressure depends nonlinearly on density and internal energy.

    Source: §1 · pp. 1–2

  • Method

    The paper develops equation-of-state-dependent mass and internal-energy fluxes to discretely preserve linear invariants, kinetic energy by convection, and pressure equilibrium for arbitrary equations of state.

    Source: §2.3 · p. 5, §3.1 · p. 6, §7 · p. 17

  • Results

    The exact formulation demonstrates that discrete conservation of mass, momentum, and total energy is compatible with exact pressure-equilibrium preservation.

    Source: §1 · p. 2, §7 · p. 17

  • Takeaways & Limitations

    The resulting two-point fluxes can also be applied in structured or unstructured finite-volume and discontinuous-Galerkin formulations.

    Source: §7 · p. 17

    In a tested Peng–Robinson transcritical case, EPEP-RG showed strong instabilities, probably due to near-singular thermodynamic derivatives near the pseudo-critical line.

    Source: §6.1 · p. 12

Abstract

from arXiv · show

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.

1 Introduction

The paper addresses spurious oscillations and the tension between exact total-energy conservation and pressure-equilibrium preservation in compressible real- and thermally perfect-gas flows. It proposes a fully conservative, kinetic-energy-preserving, exactly pressure-equilibrium-preserving discretization for arbitrary equations of state, plus an approximate formulation for transcritical conditions.

1 Introduction

  • Nonlinear real-gas equations of state can generate spurious pressure and velocity oscillations across contact discontinuities, potentially causing non-physical states and simulation failure.The pressure dependence on density and internal energy is highly nonlinear in real gases.Source: §1 · pp. 1–2
  • Fully conservative real-gas schemes can lose pressure equilibrium after nonlinear EoS inversion, while pressure-equilibrium-preserving modifications or primitive-variable formulations sacrifice exact discrete total-energy conservation and may fail to recover the correct weak solution at strong shocks.Primitive-variable formulations avoid the energy-to-pressure EoS inversion; auxiliary thermodynamic-advection schemes pair standard conservative fluid equations with non-conservative advection equations and sacrifice exact discrete total-energy conservation.Source: §1 · p. 2
  • Double-flux methods enforce pressure equilibrium locally but remain non-conservative and depend on empirical sensors that are sensitive to tuning for nonlinear real gases.Other fully conservative specialized fluxes achieve only approximate pressure-equilibrium preservation, leaving residual oscillations under severe gradients or phase transitions.Source: §1 · p. 2
  • The proposed framework is fully conservative, kinetic-energy-preserving, and exactly pressure-equilibrium-preserving for thermally perfect and real gases with arbitrary equations of state.It retains exact discrete evolution of total energy without triggering spurious interfacial oscillations.Source: §1 · pp. 2–3
  • The proposed approximate flux formulation slightly modifies the exact fluxes and retains excellent pressure-equilibrium preservation with robust behavior in extreme transcritical conditions.The paper describes its performance as unrivaled among existing formulations.Source: §1 · p. 3

2 Problem formulation

The section frames the compressible Euler equations and their pressure-equilibrium property, then sets out a conservative central-flux discretization constrained to preserve kinetic energy by convection.

2.1 Governing equations

  • The Euler system conserves mass, momentum, and total energy, with total energy per unit mass equal to internal plus kinetic energy and closure supplied by an equation of state.The equation of state relates pressure, density, and temperature, while internal energy depends on temperature and density.Source: §2.1 · p. 3
  • A spatially constant pressure and velocity remain constant in time, defining pressure equilibrium and permitting density waves to travel at constant velocity in uniform pressure.The pressure and velocity evolution equations show that their time derivatives vanish when those fields are spatially constant.Source: §2.1 · pp. 3–4

2.2 Discretization setting

  • The analysis focuses on internal spatial points and assumes continuous-style handling of time derivatives to isolate spatial errors.The theory is developed for one dimension and finite differences on a uniform mesh; the authors state that most results can extend to finite-volume or finite-element methods and nonuniform or curvilinear meshes.Source: §2.2 · p. 4
  • The numerical flux is a consistent approximation of the convective flux, and its divergence structure yields global conservation of the transported invariant.Pressure contributions to momentum and total energy are discretized separately using central derivative formulas.Source: §2.2 · p. 4
  • The second-order discretization is determined by two-point convective fluxes and central pressure-term derivatives at neighboring nodes.The analysis uses a one-dimensional finite-difference treatment on a uniform mesh, with extension to higher orders and other frameworks described as possible.Source: §2.2 · p. 4

2.3 Kinetic Energy Preserving formulation

  • The KEP formulation links mass and momentum fluxes so the discrete kinetic-energy convective term is locally and globally conservative.Total-energy flux combines the kinetic-energy flux with an internal-energy flux; the resulting scheme conserves mass, momentum, and total energy locally while preserving kinetic-energy conservation by convection.Source: §2.3 · pp. 4–5
  • KEEP, zeroth-order AEC, and KEEPPE are examples of KEP flux choices, with the latter two stated to be pressure-equilibrium-preserving for ideal gases.The KEEP scheme uses Fρ = ρu and Fρe = ρue; AEC uses a harmonic-mean internal-energy flux, while KEEPPE uses Fρe = ρeu.Source: §2.3 · p. 5

3 PEP formulation

The section derives a pressure-equilibrium-preserving discretization for compressible Euler equations without assuming a particular equation of state, with specific gas models treated in later sections.

3.1 Theoretical derivation of exact PEP fluxes

  • With the adopted kinetic-energy-preserving discretization, uniform pressure and velocity yield zero velocity time derivative, so the velocity field remains uniform.This velocity-equilibrium result is independent of the equation of state.Source: §3.1 · pp. 5–6
  • Exact pressure preservation requires the mass and internal-energy fluxes to satisfy a constraint that makes dp_i/dt = 0 for uniform pressure and velocity.The resulting fluxes in Eqs. (20) and (22) satisfy this constraint and form the basis for the general PEP fluxes.Source: §3.1 · pp. 6–7
  • The exact flux construction uses α(ρ) as a locally invertible variable and a primitive function ψ to express the mass and internal-energy fluxes as exact differences.The resulting fluxes satisfy the pressure-equilibrium constraint.Source: §3.1 · p. 7

3.2 Treatment of the singularity and approximate PEP formulation

  • The exact formulation satisfies both KEP and PEP for an arbitrary equation of state, with its details represented by α(ρ,p) and λ(ρ,p).Source: §3.2 · p. 8
  • The exact mass flux can become singular for uniform α; because its full series expansion is cumbersome for arbitrary equations of state, the authors replace ρλ with arithmetic-mean ρ.The formulation also describes fixes that locally revert to nonsingular fluxes when the singular-average denominator falls below a tolerance.Source: §3.2 · p. 8
  • The arithmetic-mean flux defines an approximate PEP formulation that can be used throughout a simulation and showed excellent performance on challenging tests; harmonic and geometric density means also performed well.The authors note that the interpolation choice appears to have only marginal effects on PEP enforcement.Source: §3.2 · p. 8
  • For a calorically perfect gas, the approximate formulation reduces to KEEPPE, while the exact mass flux remains undefined and its internal-energy flux reduces to ρe u.Source: §3.2 · p. 8

4 Thermally perfect gases

The section specializes the pressure-equilibrium-preserving fluxes to thermally perfect gases, using temperature-dependent heat capacity and deriving an efficient mass-flux formulation that avoids singularities.

4 Thermally perfect gases

  • The model assumes p = ρRT and temperature-dependent isochoric specific heat cv(T), represented by polynomial fits with typically 5, 7, or 9 coefficients.The derivation is given for general polynomial order N.Source: §4 · pp. 8–9
  • The polynomial heat-capacity model yields a mass-flux formulation that is singularity-free in all conditions without a fix, derived from the PEP condition using uniform pressure and arithmetic interpolation.The derivation starts from the ideal-gas relation for density at uniform pressure and assumes arithmetic interpolation for pressure and velocity.Source: §4 · pp. 8–9
  • The adopted efficient PEP formulation combines the mass flux in Eq. (32) with the internal-energy flux in Eq. (31), and extends to heat-capacity fits with negative exponents [36].This is the formulation used in the numerical tests in Section 6.Source: §4 · p. 9

5 Real gases

The section represents real-gas thermodynamics using departure functions and develops specializations for van der Waals and Peng–Robinson equations of state.

5 Real gases

  • Real-gas internal energy is expressed using thermally perfect contributions and departure functions, with sound speed evaluated using the standard expression for each equation of state.The departure function is defined as the difference between a thermodynamic quantity and its thermally perfect contribution; the internal-energy departure’s temperature derivative at fixed density is the heat-capacity departure.Source: §5 · pp. 9–10

5.1 Van der Waals model

  • The van der Waals model provides a simple real-gas correction to the ideal-gas equation of state.Source: §5.1 · p. 10

5.2 Peng–Robinson model

  • The Peng–Robinson model is widely applied to high-pressure, low-temperature flows and overcomes van der Waals instability near the critical region.Source: §5.2 · p. 10

6 Numerical results

The numerical tests assess pressure-equilibrium preservation and total-energy conservation in density-wave and double-jet flows across thermodynamic regimes. They show that scheme robustness and conservation vary with the gas model and flow conditions.

6.1 One-dimensional density wave

  • Across thermally perfect, van der Waals, and Peng–Robinson density-wave tests, EPEP-RG kept kinetic-energy error at machine-zero, while all shown non-PEP schemes blew up.APEP-RG, though not shown in the figure, performed almost indistinguishably from EPEP-RG over this time interval.Source: §6.1 · p. 12, Figure 1 · p. 12
  • At the density-wave snapshot, EPEP-RG kept pressure at its initial value and density close to the exact solution, whereas APEC showed spurious pressure and density oscillations.The snapshot was taken after KEEP and KEEPPE had become unstable, while APEC was still running.Source: §6.1 · p. 12, Figure 2 · p. 13
  • In the supercritical Peng–Robinson density wave run to t = 100, EPEP-RG and APEP-RG conserved total energy to machine precision, while KGPPt accumulated an error of order 10^-3.All three schemes ran without recorded instability; APEP-RG and KGPPt density profiles had similar dispersion features.Source: §6.1 · p. 12, Figure 3 · p. 14
  • In the transcritical density wave, EPEP-RG also became unstable, and long runs blew up for APEP-RG at t ≈ 96 and KGPPt at t ≈ 107.At t = 50, KGPPt’s total-energy error had accumulated to approximately 6 × 10^-4; APEP-RG showed slightly better agreement with the exact density solution.Source: §6.1 · pp. 12, 14, Figure 4 · p. 14

6.2 Two-dimensional inviscid double-jet flow at high-enthalpy and supercritical conditions

  • In the double-jet tests, KEEPPE and APEC developed pressure oscillations; these were much more visible for the supercritical van der Waals case, where APEP-RG and KGPPt remained essentially free of oscillation.KGPPt steadily deviated from exact total-energy conservation in the high-enthalpy case, and all considered schemes violated exact entropy preservation.Source: §6.2 · p. 15, Figure 5 · p. 15, Figure 6 · p. 16

6.3 Two-dimensional inviscid double-jet flow at transcritical conditions

  • For the transcritical Peng–Robinson double jet, KEEPPE and APEC showed more evident pressure oscillations, while entropy oscillations were exacerbated for KGPPt.The pressure-field snapshot also showed contamination from growing oscillations in KEEPPE and APEC.Source: §6.3 · p. 16, Figure 7 · p. 17

7 Conclusions

The conclusions present EPEP-RG as reconciling exact pressure-equilibrium preservation with full discrete conservation, and APEP-RG as a practical option for difficult transcritical regimes.

7 Conclusions

  • The resulting two-point fluxes extend beyond finite differences to structured and unstructured finite-volume and discontinuous-Galerkin formulations.Source: §7 · p. 17
  • EPEP-RG combines exact pressure-equilibrium preservation with discrete mass, momentum, and total-energy conservation, addressing a challenge for general equations of state.The framework provides a generic formula for arbitrary equations of state and computationally viable fluxes for thermally perfect, van der Waals, and Peng–Robinson gases.Source: §7 · p. 17
  • APEP-RG approximates pressure-equilibrium preservation while maintaining full primary conservation, including kinetic-energy preservation by convection.It is proposed as a practical alternative for thermodynamic regimes where the exact formulation can encounter numerical singularities.Source: §7 · p. 18
  • Benchmarks show EPEP-RG preserves pressure equilibrium across equations of state, while mixing-layer simulations conserve total energy and eliminate pressure oscillations; APEP-RG avoids transcritical singularities.The paper reports favorable robustness, stability, and accuracy for the proposed schemes, with APEP-RG handling severe thermodynamic nonlinearities without sacrificing primary conservation or numerical stability.Source: §7 · p. 18
  • EPEP-RG needs a specialized fix for thermodynamic-derivative singularities; future work includes studying density averages and extending the framework to multicomponent, multiphase, and wall-bounded turbulent flows.Exploring density-averaging operators could help identify formulations with additional properties such as discrete entropy conservation or stability.Source: §7 · p. 18
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