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Nodal discontinuous Galerkin methods for non-ideal equations of state: pressure equilibrium preservation and entropy correction

Jesse CHan, Hendrik Ranocha, Raymond Park, Joshua Lampert, Eric Ching, Ayaboe Edoh

arXiv:2608.14506v1math.NAcs.CEcs.NA

TL;DR

Spurious pressure oscillations and under-resolved features challenge robust high-order DG simulations of non-ideal fluids. This paper combines EPEC/APEC flux-differencing DG with minimally dissipative entropy correction. The authors report reduced pressure-equilibrium errors and improved robustness in long-time and under-resolved simulations.

  • Problem

    High-order simulations of real fluids can be unstable near shocks and under-resolved features, while non-ideal EOS flows can develop pressure oscillations that standard shock-capturing methods do not effectively suppress.

    Source: §1 · pp. 1–2

  • Method

    The paper develops conservative EPEC and APEC flux-differencing DG formulations and adds minimally dissipative entropy correction to enforce a semi-discrete cell entropy inequality.

    Source: §1 · p. 2, §5 · p. 11

  • Results

    Pressure-equilibrium-conserving formulations reduce spurious oscillations and distortions near pressure equilibria, while entropy correction is especially helpful for robustness in long-time simulations, higher polynomial degrees, and under-resolved features.

    Source: §7 · p. 27

  • Takeaways & Limitations

    At higher approximation orders, interface-dissipation pressure-equilibrium violations are small, allowing DG interface penalties to stabilize under-resolved solutions while retaining advantages of APEC formulations.

    Source: §7 · p. 27

    The entropy correction introduces small pressure-equilibrium errors, and the FCT-based variant assumes nodal collocation and SBP operators.

    Source: §5 · p. 11, §7 · p. 27

Abstract

from arXiv · show

Structure-preserving discontinuous Galerkin (DG) methods typically improve the robustness of high order simulations of real fluids. In addition to conservation, key structures include the preservation of pressure equilibrium and satisfaction of at least one entropy inequality. In this work, we investigate conservative discretizations using exactly pressure equilibrium conserving (EPEC) and approximately pressure equilibrium conserving (APEC) flux differencing DG formulations, as well as entropy stable formulations through the use of minimally dissipative corrections for non-ideal equations of state (EOS). We introduce an analysis of EPEC schemes and a new procedure for designing such fluxes based on a generalization of Tadmor's shuffle condition. We also analyze APEC DG schemes and show that the incorporation of dissipative interface penalization terms does not significantly increase pressure equilibrium errors, especially at higher orders of approximation. Finally, we observe that when combined with APEC flux differencing formulations, entropy correction improves robustness for under-resolved solutions and long-time simulations.

1. Introduction

The introduction motivates structure-preserving DG methods as a way to improve robustness in real-fluid simulations while retaining conservation. It extends pressure-equilibrium-conserving fluxes to nodal DG and proposes entropy correction for non-ideal EOS.

1. Introduction

  • High-order methods offer low dissipation and dispersion errors but can become unstable around shocks and under-resolved features; stabilization often requires problem-dependent heuristic tuning.The introduction identifies robustness and parameter generalizability as concerns.Source: §1 · p. 1
  • Spatially varying specific-heat ratios can produce pressure oscillations that grow and crash simulations, and common artificial-viscosity or shock-capturing methods do not effectively suppress them.Earlier responses included non-conservative formulations, while recent conservative schemes have EPEC or APEC properties.Source: §1 · pp. 1–2
  • The work extends conservative EPEC and APEC techniques to nodal DG formulations for non-ideal EOS flows.The paper also adapts conditions from [6] to characterize pressure-equilibrium conservation and derive a new EPEC flux.Source: §1 · p. 2
  • Entropy-stable DG methods enforce a semi-discrete cell entropy inequality using entropy-conservative flux differencing and added interface dissipation.The paper instead develops minimally dissipative entropy correction, including a flux-corrected-transport-like variant, for a single-component non-ideal EOS fluid.Source: §1 · p. 2
  • Entropy correction enforces a cell entropy inequality without requiring derivation of entropy-conservative two-point fluxes for the non-ideal EOS.The introduction presents this as an alternative with additional flexibility and portability.Source: §1 · p. 2

2. Governing equations

The paper studies the compressible Euler equations closed by a general pressure EOS and a temperature-dependent internal-energy model. Its structural targets are a cell entropy inequality and preservation of constant pressure and velocity.

2. Governing equations

  • The governing system is the single-component compressible Euler equations in one or two dimensions, closed by a general pressure EOS and a temperature-dependent internal-energy relation.For an ideal gas, the model reduces to e = cvT.Source: §2 · pp. 2–3

2.1. Entropy inequality

  • The entropy framework assumes a strictly convex scalar entropy, whose conservative-to-entropy-variable mapping is invertible, and uses an integrated cell entropy inequality.For a general EOS, the paper gives the entropy-flux pair and defines entropy variables through the thermodynamic relation Tds = de + pdV.Source: §2.1 · pp. 3–4

2.2. Pressure equilibrium

  • For uniform pressure and velocity, the Euler equations keep both quantities constant in time; discrete schemes aim to preserve this equilibrium exactly or approximately.Failure to preserve it can cause spurious pressure oscillations that pollute the solution and may cause divergence; velocity equilibrium preservation is a prerequisite.Source: §2.2 · p. 4

3. Flux differencing nodal discontinuous Galerkin formulation

The section introduces a conservative flux-differencing nodal DG formulation built from SBP operators, nodal approximations, volume fluxes, and interface fluxes.

3.1. A conservative flux differencing nodal DG formulation

  • The section formulates flux-differencing nodal DG, first in one dimension and with multidimensional extensions noted for later use.The method is intended for nodal collocation DG and DG-like discretizations based on multidimensional SBP operators.Source: §3.1 · p. 4, §3 · p. 4
  • The solution uses a degree-N nodal Lagrange basis on (N+1)-point Legendre-Gauss-Lobatto nodes with positive quadrature weights.These are defined on the one-dimensional reference element.Source: §3.1 · p. 5
  • The element formulation uses an SBP differentiation matrix, boundary interpolation matrix, and diagonal reference mass matrix.The operators are defined on the reference element and satisfy a summation-by-parts property.Source: §3.1 · p. 5
  • The flux-differencing scheme combines a volume flux with an interface numerical flux; the volume flux is consistent and skew-symmetric.The interface flux is taken along the boundary normal, while the volume flux is a two-point flux.Source: §3.1 · p. 5
  • The formulation is conservative because summing over element nodes leaves the scaled solution average balanced by interface fluxes, with the volume contribution vanishing.This cancellation follows from skew-symmetry of the algebraic normal term and symmetry of the volume flux.Source: §3.1 · p. 5
  • A symmetric, consistent volume flux gives a conservative, high-order-accurate formulation.The formulation extends to multiple dimensions through tensor-product constructions or multidimensional SBP operators.Source: §3.1 · p. 6

4. Conservative pressure-equilibrium-conserving fluxes

The section develops conditions for exact pressure-equilibrium conservation in flux differencing and uses them to construct EPEC and APEC fluxes. The analysis generalizes Tadmor-style reasoning and motivates a corrected density average.

4. Conservative pressure-equilibrium-conserving fluxes

  • Flux differencing can use pressure-equilibrium-conserving two-point fluxes, and the section derives a new EPEC flux by analyzing conditions for exact preservation.The EPEC condition is related to Tadmor-style entropy-conservative flux conditions, with pressure derivatives playing an analogous role to entropy variables.Source: §4 · p. 6
  • The single-fluid exact-PEC condition is constrained by an asymmetric partial derivative, so locally defined half-point values cannot generally satisfy it.The paper presents the condition as necessary and sufficient for exact pressure-equilibrium conservation.Source: §4 · pp. 6–7

4.1. A new EPEC flux based on density average correction

  • A jump-based correction to the density average, paired with the corresponding internal-energy density average, preserves pressure equilibria exactly up to regularization.The paper reports similar numerical behavior for a sharp switch and a regularized ratio.Source: §4.1 · pp. 8–9

4.2. Connection to the EPEC flux of Coppola, De Michele, and Aiello [25]

  • The correction-based EPEC flux is closely related to Coppola, De Michele, and Aiello’s EPEC flux [25], differing in the definition of the density average.Numerical experiments suggest that including the additional consistency term improves the maximum stable time-step.Source: §4.2 · pp. 9–10

4.3. Approximately pressure-equilibrium conserving (APEC) fluxes

  • Choosing the arithmetic mean for the density average decouples it from other thermodynamic quantities but gives up exact PEC, a trade-off reported to improve robustness for highly non-ideal EOS.This choice yields the APEC flux; the cited discussion notes only the internal-energy-average modification is necessary for its APEC property.Source: §4.3 · pp. 10–11

5. Enforcing a cell entropy inequality via entropy correction

The section adds minimally dissipative entropy correction to pressure-equilibrium-focused flux differencing schemes to enforce a semi-discrete cell entropy inequality. Its FCT formulation blends high- and low-order fluxes, with interface dissipation creating pressure-equilibrium errors for some non-ideal EOS.

5. Enforcing a cell entropy inequality via entropy correction

  • The contact-preserving entropy correction is orders of magnitude smaller than approximation error, targeting aliasing and high-wavenumber instabilities without suppressing Gibbs oscillations near stronger shocks.The correction residuals are described as products of two L2 best approximation errors.Source: §5 · p. 11
  • For inviscid experiments, FCT is more efficient than artificial-viscosity correction because it avoids a viscous DG discretization, while their one-dimensional results are virtually identical.The paper omits the artificial-viscosity results for brevity.Source: §5 · p. 11
  • FCT correction assumes nodal collocation and summation-by-parts operators, whereas the artificial-viscosity approach extends to arbitrary bases and quadrature.The authors defer a careful comparison of the approaches to future work.Source: §5 · p. 11

5.1. A semi-discrete entropy inequality and volume entropy residual

  • The volume entropy residual can have either sign and violate the target inequality, so the correction adds entropy dissipation to make the residual nonnegative.The section also interprets the residual as a quadrature approximation.Source: §5.1 · p. 13

5.2. Entropy correction via flux corrected transport

  • FCT blends high- and low-order fluxes with an element-wise coefficient to enforce a cell entropy inequality, using an APEC low-order flux with local Lax-Friedrichs jump penalization.The high-order flux may be EPEC or APEC; the low-order flux is chosen to limit pressure-equilibrium errors from the central contribution.Source: §5.2 · p. 13

5.2.1. Dissipation and pressure equilibrium errors

  • For van der Waals and Peng-Robinson EOS, local Lax-Friedrichs dissipation violates pressure equilibrium; for ideal and stiffened gases, that flux is EPEC.The pressure-equilibrium condition for the dissipative term is not satisfied for the former EOS cases.Source: §5.2.1 · p. 15

6. Numerical experiments

The numerical experiments assess EPEC and APEC flux-differencing schemes and entropy correction for non-ideal equations of state, examining pressure errors, stability, interface dissipation, and shock-tube and transcritical-flow behavior. In the two-dimensional inviscid transcritical jet, solutions differ due to Euler-equation sensitivity, while similar behavior is observed at both N = 3 and N = 7.

6.1. Comparison of EPEC fluxes

  • For N = 3, Coppola, de Michele, and Aiello’s EPEC flux (23) needs CFL 0.05 on 4 elements and 0.002 on 8, while correction-based EPEC (15) and APEC (24) remain stable at CFL 1.5.The flux (23) has intermittent, refinement-amplified Jacobian eigenvalue growth; at N = 7 it remains unstable on 8 elements across tested CFLs, while the other fluxes are stable at CFL 0.5 on both meshes.Source: §6.1 · pp. 16–17, Figure 1 · p. 17
  • With CFL 0.001, the EPEC flux of Coppola, de Michele, and Aiello (23) reduces pressure-equilibrium error to order 1 × 10^-13 on both meshes; the correction-based EPEC (15) maintains about 1 × 10^-8 to 1 × 10^-9.Flux (23) has the lowest pressure-equilibrium error overall, but its error grows on the coarser mesh; reducing the CFL lowers the correction-based flux error further.Source: Figure 2 · p. 17, §6.1 · p. 17

6.2. APEC fluxes with interface dissipation

  • For the transcritical Peng-Robinson density wave, both EPEC fluxes diverge regardless of mesh size or CFL, whereas the APEC flux of remains stable through one period.At longer final time 0.1, interface dissipation improves pressure errors except for N = 0; dissipative schemes run to final time for N = 0 and N = 1, unlike pure APEC, with no significant difference between the two dissipative surface-flux choices.Source: §6.2 · p. 18

6.3. APEC fluxes with interface dissipation and entropy correction

  • For the sharp transcritical wave at N = 3 on 4 elements, entropy correction stabilizes the run to final time after uncorrected APEC diverges before t = 0.025; at 3 elements, correction also diverges.With and without correction, errors are virtually identical for N = 3 and N = 7, while θ relative to the error decreases as polynomial degree increases for both smooth and steep-gradient waves.Source: §6.3 · pp. 20–21

6.3.1. Transcritical shock tube

  • In the transcritical shock tube, APEC DG with APEC volume flux and APEC plus LxF interface penalization runs stably with or without entropy correction, which slightly reduces oscillations but makes no significant overall difference.With central volume flux, spurious oscillations arise between the rarefaction and contact, and the uncorrected simulation diverges; APEC and APEC-C reduce those oscillations, while the correction’s activity varies with volume-flux choice.Source: §6.3.1 · pp. 21–22

6.4. Two-dimensional transcritical mixing layer

  • At 4tc, APEC without entropy correction diverges around t ≈ 0.11, while entropy correction remains stable despite visible density artifacts from under-resolution.For the shorter 2tc mixing-layer runs, corrected and uncorrected solutions are virtually identical, with maximum θ values of 1.194×10^-4 for N = 3 and 5.843 × 10^-5 for N = 7.Source: §6.4 · p. 23

6.5. Two-dimensional inviscid transcritical jet injection

  • In the transcritical jet, entropy correction alone diverges with negative density and temperature, while smoothing the blending factors remedies this failure; scaling θ by 1.5 yields a 5 −10× larger time-step size.At N = 3 on a 320 × 160 mesh, θ is relatively small and sparsely activated, while the specific-heat ratio strongly deviates from the ideal-gas value of 1.4.Source: §6.5 · p. 24, Figure 9 · p. 25
  • For the transcritical jet, solutions differ due to Euler-equation sensitivity [35], but behavior is similar at N = 3 and N = 7.Figure 10 shows density, pressure, ratio of specific heats γ, and entropy-correction blending coefficient at final time 10^-3 on a 160 × 80 mesh with N = 7.Source: Figure 10 · p. 26, §6.5 · p. 27

7. Conclusion

The conclusion presents pressure-equilibrium-preserving DG and entropy correction as complementary tools for real-gas simulations. It reports small high-order interface-dissipation errors and improved robustness from entropy enforcement, while noting remaining limitations.

7. Conclusion

  • Pressure-equilibrium-conserving formulations reduce spurious oscillations near equilibria, while cell entropy enforcement helps robustness in long-time simulations, at higher polynomial degrees, and for under-resolved features.The entropy correction introduces small pressure-equilibrium errors, motivating future corrections designed to preserve equilibria.Source: §7 · p. 27
  • Combining APEC fluxes with flux differencing nodal DG extends pressure-equilibrium preservation to DG, and interface-dissipation violations are small at higher approximation orders.This permits interface penalties to stabilize under-resolved solutions while retaining advantages of approximate pressure-equilibrium conservation.Source: §7 · p. 27
  • For non-ideal EOS, positivity-preserving limiters that respect pressure equilibria remain an open problem, to the authors’ knowledge.The minimally dissipative entropy correction still introduces small pressure-equilibrium errors.Source: §7 · p. 27

Appendix A. Thermodynamic identities

The appendix lists the thermodynamic quantities required by the implementations and describes derived quantities and a temperature-evaluation procedure.

Appendix A. Thermodynamic identities

  • The implementations require pressure and its partial derivatives, internal energy, specific entropy, and sound speed for a Lax-Friedrichs interface flux.Source: Appendix A · p. 28
  • These inputs are used to calculate Gibbs free energy and enthalpy, and to obtain mappings between conservative and entropy variables and the entropy Hessian inverse.Source: Appendix A · p. 28
  • Temperature as a function of specific volume and internal energy is computed by Newton inversion of the internal-energy relation at fixed volume for admissible states with positive specific volume.Source: Appendix A · p. 28

Appendix B. Entropy correction via artificial viscosity

The artificial-viscosity entropy correction (ECAV) adds a Laplacian viscosity term whose coefficient is based on δk(uh). Its formulation uses the entropy Hessian, which creates a limitation when that Hessian is singular.

Appendix B. Entropy correction via artificial viscosity

  • ECAV adds a Laplacian artificial-viscosity term with a coefficient calculated from δk(uh).It is closely related to local entropy corrections and entropy-viscosity regularization tied to chain-rule violation.Source: Appendix B · p. 28
  • The artificial-viscosity correction uses the inverse entropy Hessian to discretize the Laplacian of the solution.Source: Appendix B · p. 28
  • For a general EOS, the Hessian can be calculated analytically by factoring it into Jacobian matrices with respect to primitive variables q = [V, v1, v2, T].The chosen variables make the stated Jacobians triangular, and the inverse is calculated using thermodynamic relations and pressure derivatives with respect to V and T.Source: Appendix B · p. 28
  • Entropy-correction AV can fail when the entropy is non-convex and its Hessian is singular.The passage contrasts this with discrete graph viscosity, which is entropy dissipative for any convex entropy; FCT-based correction or graph-viscosity AV are cited as alternatives.Source: Appendix B · p. 29

Appendix B.1. Stiffened gas

The stiffened-gas example compares liquid water and steam in a quasi-1D nozzle flow. Water remains smooth, while steam develops a shock and its uncorrected simulations crash at both tested approximation degrees.

Appendix B.1. Stiffened gas

  • Under the stiffened-gas EOS, PEC conditions are automatically satisfied for standard DG with central fluxes and a local Lax-Friedrichs surface flux, so pressure-equilibrium errors are not an issue here.Source: Appendix B.1 · p. 29
  • For the quasi-1D system, the entropy is the standard 1D Euler entropy scaled by nozzle width a(x), while the entropy variables remain identical to those of the original system.This entropy choice is used to construct the Hessian matrix for ECAV.Source: Appendix B.1 · p. 29
  • AV-based entropy correction applies to non-conservative systems, whereas constructing an FCT subcell finite-volume scheme is more involved for the quasi-1D non-conservative equations.Source: Appendix B.1 · p. 30
  • The nozzle experiments use liquid water and steam with otherwise shared settings and are intended to illustrate ECAV performance, not replace available efficient exact solvers.Source: Appendix B.1 · p. 30
  • The nozzle-flow behaviors differ with the stiffened-gas stiffness parameter p∞: water remains subsonic, while steam develops a shock as it accelerates through the nozzle.Source: Appendix B.1 · p. 30
  • Without entropy-correction stabilization, steam simulations crash at both N = 3 and N = 7, whereas liquid-water solutions are smooth and behave similarly with or without ECAV.Source: Appendix B.1 · p. 30
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