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An entropy-stable and kinetic energy-preserving macro-element HDG method for compressible flows

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Pith's one-line read The paper proves an entropy-stable macro-element HDG scheme for compressible Euler flows, preserves kinetic energy, and reports roughly threefold speedups in benchmarks.

desk verdict Solid entropy-stable macro-element HDG method, but the abstract overclaims an order-of-magnitude speedup and the Navier-Stokes extension is asserted, not proven. read the letter →

arxiv 2507.22195 v1 pith:LMJMCPPO submitted 2025-07-29 cs.CE

Vahid Badrkhani , Marco F.P. ten Eikelder , Dominik Schillinger This is my paper · ORCID
classification cs.CE MSC 65M6065M1235L6576N15
keywords hybridizeddiscontinuousGalerkinmacro-elemententropy-stableschemeskineticenergypreservationmatrix-freemethodscompressibleflowTaylor-Greenvortexunder-resolvedDNS
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

High-order discontinuous Galerkin methods for compressible flow are accurate but expensive and tend to crash in under-resolved or turbulent regimes. This paper proposes to fix both problems at once: it groups elements into macro-elements with continuous basis inside and hybridized discontinuous Galerkin coupling across macro-element boundaries, cutting the number of coupled unknowns, and it reformulates the scheme in entropy variables with a kinetic-energy-preserving and entropy-stable (KEPES) numerical flux. The main theoretical result is that the total generalized entropy of the semi-discrete Euler equations on a periodic domain is non-increasing in time, a discrete form of the second law. The same construction keeps the under-resolved Taylor-Green vortex simulations stable where conservative-variable HDG blows up, and the reported benchmarks show about a threefold reduction in computing time relative to standard HDG.

What carries the argument

The machinery has three tiers. First, the macro-element mesh: each macro-element is filled with $C^0$-continuous simplicial elements, and the only globally coupled unknowns are HDG trace variables on macro-element boundaries, which sharply reduces degrees of freedom and makes local solves embarrassingly parallel. Second, the entropy-variable formulation: with entropy variables $v = \partial H/\partial u$ the Euler system symmetrizes, and Proposition 4.2 rewrites the time derivative of total entropy as a sum of face jump terms plus a boundary term; on periodic domains entropy stability is exactly the face condition $[\![v_h]\!]^T \hat{F}_h - [\![\Psi_n]\!] \le 0$. Third, the KEPES numerical flux (49): an entropy-conservative flux (37) built from arithmetic and logarithmic means, plus a matrix dissipation term $-\frac{1}{2} R |\Lambda| T R^T [\![v_h]\!]$ whose sign is fixed by an eigenvector scaling argument, making the entropy production a negative quadratic form. The solver layer is a matrix-free implementation with two-level static condensation and FGMRES.

What would settle it

Run the KEPES scheme on a wall-bounded turbulent flow, such as a channel at friction Reynolds number 180, and monitor the total generalized entropy: the theorem covers only periodic inviscid problems, so an observed entropy increase or a blow-up would falsify the paper's assertion that the entropy-stability result extends to Navier-Stokes with physical boundaries. A cheaper check is to evaluate $[\![v_h]\!]^T \hat{F}_h - [\![\Psi_n]\!]$ on the faces of the viscous Taylor-Green run at $N_{\text{eff}} = 84^3$; a positive face contribution would show the inequality (32) is not satisfied in the viscous case.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that entropy-stability theory, usually developed for finite-volume and DG split-form schemes, survives intact inside a macro-element HDG discretization solved in entropy variables. The proof of Theorem 5.4 reduces entropy stability to the sign of the face quantity $[\![v_h]\!]^T \hat{F}_h - [\![\Psi_n]\!]$; the KEPES flux (49) makes this quantity a negative quadratic form, so the total entropy cannot increase. The same flux is constructed to preserve kinetic energy at the semi-discrete level, which the numerical vortex tests show is necessary for accuracy: the entropy-stable flux without the kinetic-energy correction keeps the simulation stable but smears the vortex. The paper also claims that the entropy-variable formulation removes the need for SUPG stabilization inside macro-elements, and that the result extends from Euler to the compressible Navier-Stokes equations, although the proof is given only for the periodic inviscid case.

Load-bearing premise

The discrete entropy inequality is proven only for the Euler equations on periodic domains; the paper asserts, without proof, that the same result carries over to the viscous Navier-Stokes terms and gives no analysis of wall-boundary conditions.

Editorial extensions

If this is right

  • High-order implicit LES of transitional and turbulent compressible flows could be run without extra stabilization, since the entropy inequality is baked into the numerical flux.
  • The degree-of-freedom reduction cuts both local and global system sizes by roughly two to threefold on the reported meshes, with total wall time reduced by factors of 2.65 and 2.82 in the tabulated benchmarks.
  • Simulations that are stable in conservation variables become stable in entropy variables: the Taylor-Green tests show conservative-variable HDG failing at $M_0 = 0.8$ while the KEPES variant runs to completion.
  • Because the KEPES flux preserves kinetic energy separately from entropy, it should reproduce the correct dissipation physics in under-resolved regimes rather than dissipating through numerical noise.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The tabulated speedups (factors of 2.65 and 2.82 in Tables 5 and 8) are below the 'order of magnitude' mentioned in the abstract; benchmarking at larger problem sizes or on distributed memory might close that gap, since macro-element aggregation reduces the global system size most when meshes are fine.
  • The periodic-domain proof leaves wall boundary treatment as the main open risk: a boundary flux that breaks the discrete inequality would limit the method to flows that can be approximated with periodic or far-field boundary data.
  • The same entropy-conservative and KEPES construction should transfer to other systems with a known entropy pair, such as multi-component or reacting flows, by re-deriving the logarithmic-mean fluxes and the jump identity.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper develops a macro-element hybridized discontinuous Galerkin (HDG) method for the compressible Navier–Stokes equations, using entropy variables and flux differencing to construct two inviscid numerical fluxes: an entropy-stable flux (ES) and a kinetic-energy-preserving, entropy-stable flux (KEPES). The method is analyzed for the compressible Euler equations on periodic domains, where Theorems 5.3 and 5.4 establish entropy conservation and entropy stability under conditions (31) and (32). The authors also present a matrix-free, two-level static condensation solver and evaluate the method on the isentropic vortex, the inviscid Taylor–Green vortex, and the viscous Taylor–Green vortex, reporting improved robustness in under-resolved regimes and computational speedups relative to standard HDG.

Significance. If the claims hold, the paper makes a useful contribution to high-order methods for compressible turbulence: it combines a known entropy-stable flux construction with a macro-element HDG formulation, giving reduced global and local degrees of freedom, a matrix-free implementation, and robust behavior on under-resolved Taylor–Green vortex benchmarks. The entropy-stability proof is based on the standard flux-differencing condition and is self-contained for the periodic Euler case. The main contribution is therefore plausible and potentially valuable. The significance is reduced by the mismatch between what is proven (periodic inviscid Euler, semi-discrete) and what is claimed in the abstract and Section 7.3.1 (Navier–Stokes, wall-bounded DNS), as well as by the speedup claim in the abstract, which is not supported by the reported measurements.

major comments (4)
  1. [Section 7.3.1 / Theorem 5.4 / Remark 5.5] The entropy-stability proof is restricted to the compressible Euler equations on periodic domains, but Section 7.3.1 states that Theorem 5.4 'applies not only to the inviscid Euler equations but also extends to the compressible Navier–Stokes equations.' This extension is not proven. The entropy identity (25) and the proof of (53) control only the inviscid jump term; for the Navier–Stokes system, equation (9) contains additional flux and production terms involving v^T K∇v and ∇v^T K∇v, and the proof says nothing about those contributions. In addition, the boundary operator B_∂Ω in (30) is never shown to be entropy dissipative, and the proof relies on the periodic cancellation (28). Since wall-bounded turbulence is a primary target for DNS, the missing viscous and boundary analysis is load-bearing for the paper's main applicability claim. The authors should either provide the missing analysis or explicitly restrict the theoretical claims to periodic Euler and describe the Navier–Stokes results as numerical evidence only.
  2. [Abstract / Section 7.2.2 / Section 7.3.2] The abstract claims 'up to an order of magnitude speedup over standard HDG methods,' but the paper's own Tables 5 and 8 report total-time ratios of 181.9/68.5 ≈ 2.65 and 707.9/250.9 ≈ 2.82, respectively. These are factors of about three, not an order of magnitude. The introduction cites earlier work [11,12] for an order-of-magnitude advantage, but the present paper's benchmarks do not reproduce that factor. The performance claim in the abstract and conclusions should be reworded to match the measured speedups or supplemented with a configuration in which the order-of-magnitude gain is actually obtained.
  3. [Theorem 5.3 / Equation (52)] The proof of entropy conservation for the flux (37) is not fully verifiable as printed. Equation (52) contains undefined or inconsistent symbols, such as 'f3^es' where the flux components in (37) are denoted f1^ec, f2^ec, f3^ec, and the displayed simplification skips the algebraic steps that connect (50), (51), and the definitions in (37). The stray equation-reference markers ('12', '35', '37', '49', '52') in (27), (29), (51), and (53) further obscure the derivation. Since Theorem 5.3 is the foundation for the entropy-stability result in Theorem 5.4, the authors should rewrite this proof cleanly and verify each cancellation.
  4. [Section 5.2 / Section 5.3] Theorems 5.3 and 5.4 are semi-discrete statements: Proposition 4.2 starts from d/dt of the total entropy, and the proofs use only the spatial discretization. However, the abstract and conclusions say the formulations 'satisfy a discrete entropy inequality,' which a reader may reasonably interpret as a fully discrete statement. The DIRK time integrator in (56) is not shown to preserve or dissipate the discrete entropy, and the numerical results in Figures 5 and 13 integrate in time with DIRK. The authors should state explicitly that the entropy inequality is proven only for the semi-discrete scheme, or extend the analysis to the fully discrete case.
minor comments (5)
  1. [Table 8 / Section 7.3.2] Table 8 is captioned 'Neff = 703,' but Section 7.3.2 and Table 7 describe the same comparison at Neff = 84^3; the caption should be corrected.
  2. [Remark 5.5] Remark 5.5 states that 'the matrix R is positive definite.' This is not needed for the quadratic form in (53), because the positivity follows from positive definiteness of |Λ| and T, and the eigenvector matrix R is not generally symmetric or positive definite. The remark should be corrected or deleted.
  3. [Section 7.1.1] The text refers to 'C 2-continuous elements along a macro-element edge'; the method uses C^0-continuous elements, so this appears to be a typo.
  4. [Throughout] There are several typographical errors, including 'formulatios', 'onbe discontinuous', 'elment', 'intial', and inconsistent notation 'M0' versus 'M∞'. These should be corrected in a careful revision.
  5. [Section 5.3] The time-marching formulation in (56) introduces the matrix d_ij = (a_ij)^{-1} and the update (55) with e_j = Σ_i b_i d_ij, but the connection to the standard DIRK Butcher tableau is not explained. A short derivation or a reference to the exact DIRK variant would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the entropy-stability proof is self-contained; the Navier-Stokes extension assertion is an unsupported proof gap, not a circular step.

full rationale

The entropy-stability derivation is self-contained. Theorem 5.4 reduces the KEPES flux (49) to the entropy-conservative flux (37) minus a dissipation term; Theorem 5.3 verifies the entropy-conservation condition (31) by direct algebraic manipulation of (37), and the remaining quadratic form in (53) is non-positive because T and |Λ| are positive diagonal and the eigenvector matrix is nonsingular. No fitted constants enter the derivation; the pressure-indicator blend in (40)-(43) is algorithmic, not calibrated to the target results. The macro-element construction is taken from the authors' prior work [11,12], but the present paper supplies independent degree-of-freedom counts (Tables 4 and 7) and run-time comparisons (Tables 5 and 8), and the benchmarks are checked against the analytic isentropic-vortex solution and the spectral DNS reference [85], so the self-citations are not load-bearing. The abstract's 'up to an order of magnitude speedup' overstates the factors of 2.65 and 2.82 in Tables 5 and 8, but that is a numerical overclaim, not circularity. The main proof gaps are non-circular: Theorem 5.4 is proven only for the Euler equations on periodic domains where B_∂Ω vanishes (Remark 5.5), while Section 7.3.1 asserts that the result 'extends to the compressible Navier-Stokes equations' without treating viscous contributions or wall-boundary traces; this is missing support, not a reduction to the paper's own inputs.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central derivation rests on standard entropy-stable flux machinery (Tadmor, Chandrashekar, Barth scaling) plus the macro-element HDG framework from the authors' prior work. The only ad hoc element is the periodic-boundary restriction in the stability theorems; the pressure-blending sensor is a design choice rather than a fitted constant.

free parameters (2)
  • pressure sensor blending parameter θ = θ = sqrt(|P - Phat| / (P + Phat)) (Eq. 43), ∈ [0,1]
    Hand-chosen local pressure indicator to blend LF and Roe dissipation in the KEPES flux; it is a function of the solution, not fitted to benchmark data.
  • pseudo-transient continuation parameters = τinit = 1.0, τmax = 1e8 (Sec. 6.1)
    Standard continuation parameters, not fitted to the target results.
assumptions (4)
  • standard math Existence of a convex entropy-entropy flux pair for the compressible Navier-Stokes equations (H = -ρs/(γ-1), F = -ρsV).
    Invoked in Section 2.2 to define entropy variables and the symmetrized system.
  • standard math The entropy Jacobian A0 = ∂u/∂v is symmetric positive definite and satisfies A0 = R T R^t via Barth's eigenvector scaling (Eq. 46).
    Used in the proof of Theorem 5.4 to show the dissipation term in (49) is non-positive definite.
  • ad hoc to paper Entropy stability analysis is restricted to periodic boundary conditions.
    Remark 5.5 and the theorems state periodic domains; general conservative boundary conditions are declared out of scope.
  • domain assumption Simplicial meshes with uniform subdivision of macro-simplices; no hanging nodes within macro-elements.
    Section 3.1 restricts to uniform simplicial subdivision, which is needed for the C0-continuity within macro-elements.

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Pith. "Pith review of An entropy-stable and kinetic energy-preserving macro-element HDG method for compressible flows." pith.science (2026). https://pith.science/paper/LMJMCPPO

@misc{pith2026250722195,
  author       = {Pith},
  title        = {Pith review of: An entropy-stable and kinetic energy-preserving macro-element HDG method for compressible flows},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LMJMCPPO}},
  note         = {Machine review of arXiv:2507.22195}
}
read the original abstract

This paper introduces a high order numerical framework for efficient and robust simulation of compressible flows. To address the inefficiencies of standard hybridized discontinuous Galerkin (HDG) methods in large scale settings, we develop a macro element HDG method that reduces global and local degrees of freedom by embedding continuous Galerkin structure within macro-elements. This formulation supports matrix free implementations and enables highly parallel local solves, leading to substantial performance gains and excellent scalability on modern architectures. To enhance robustness in under resolved or turbulent regimes, we extend the method using entropy variables and a flux differencing approach to construct entropy stable and kinetic energy preserving variants. These formulations satisfy a discrete entropy inequality and improve stability without compromising high order accuracy. We demonstrate the performance of the proposed method on benchmark problems including the inviscid isentropic vortex and the Taylor Green vortex in both inviscid and turbulent regimes. Numerical results confirm optimal accuracy, improved robustness, and up to an order of magnitude speedup over standard HDG methods. These developments mark a significant advancement in high order methods for direct numerical simulation (DNS) of compressible flows.

Figures

Figures reproduced from arXiv: 2507.22195 by the authors.

Figure 1
Figure 1. Illustration of the macro-element HDG discretization of a 3D tetrahedral domain and a triangular [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. 3D isentropic vortex problem. HDG macro-elements are plotted in blue, [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗
Figure 3
Figure 3. 3D isentropic vortex problem: (a) and (b) depict 2D slices of Mach number contours, while (c) [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: 3D isentropic vortex problem (ϵ = 2.5, M∞ = 0.50): Convergence of the L 2 error of the density field in space, obtained the standard conservative-variable formulation (solid lines) and the entropy-variable formulation using the KEPES numerical inviscid flux (dashed lin…
Figure 5
Figure 5. Figure 5: 3D isentropic vortex problem (ϵ = 5.0, M0 = 0.85): conservative-variable formulations become unstable shortly after t = 10.0, while entropy-variable formulations remain stable. 7.1.2. Assessment of stability To analyze the behavior of different types of numerical fluxe…
Figure 6
Figure 6. Figure 6: 3D isentropic vortex (M∞ = 0.85, ϵ = 5.0): comparison of the density and Mach number field solutions obtained with the macro-element HDG method in entropy-variable format (m = 4, p = 2) using the inviscid flux (36) (ES) and the inviscid flux (49) (KEPES) against the an…
Figure 7
Figure 7. Figure 7: Inviscid Taylor–Green vortex: HDG discretization ( [PITH_FULL_IMAGE:figures/full_fig_p031_7.png]
Figure 8
Figure 8. Figure 8: Inviscid Taylor–Green vortex (low-Mach number case with [PITH_FULL_IMAGE:figures/full_fig_p031_8.png]
Figure 9
Figure 9. Figure 9: Inviscid Taylor–Green vortex (moderate-Mach number case with [PITH_FULL_IMAGE:figures/full_fig_p032_9.png]
Figure 10
Figure 10. Figure 10: Inviscid Taylor–Green vortex: Entropy and energy dissipation behavior over time, as reproduced [PITH_FULL_IMAGE:figures/full_fig_p034_10.png]
Figure 11
Figure 11. Figure 11: Viscous Taylor–Green vortex (Re∞ = 1, 600): Macro-element HDG discretizations (m = 2). The discontinuous macro-element edges are plotted in blue, the C0 -continuous element edges in black. 7.3.1. Assessment of accuracy and stability To assess the accuracy of our macro…
Figure 12
Figure 12. Figure 12: Viscous Taylor–Green vortex (Re∞ = 1, 600): Vorticity magnitude plotted on iso-contours of the velocity field, obtained with macro-element HDG (m = 2, p = 3) in entropy-variable format and the KEPES flux formulation (49) on the mesh resolution Neff = 1263 . 36 [PITH_…
Figure 13
Figure 13. Figure 13: Viscous Taylor–Green vortex (Re∞ = 1, 600): Entropy and energy dissipation behavior over time, reproduced by macro-element HDG (m = 2, p = 3) in entropy-variable format with KEPES flux (49). In [PITH_FULL_IMAGE:figures/full_fig_p038_13.png]

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