Reinterpretation and Extension of Entropy Correction Terms for Residual Distribution and Discontinuous Galerkin Schemes: Application to Structure Preserving Discretization

Hendrik Ranocha; Philipp \"Offner; R\'emi Abgrall

REVIEW 2 major objections 5 minor 58 references

Reinterpretation and Extension of Entropy Correction Terms for Residual Distribution and Discontinuous Galerkin Schemes: Application to Structure Preserving Discretization

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Pith's one-line read This paper shows that entropy conservation for residual distribution and discontinuous Galerkin schemes can be enforced by adding one explicit, optimally small correction term to each degree of freedom.

desk verdict Sound semidiscrete core, honest limitations, but the fully discrete claim is over-sold and needs either proof or careful rescoping. read the letter →

arxiv 1908.04556 v4 pith:NHBSTELX submitted 2019-08-13 math.NA cs.NA

Rémi Abgrall , Philipp Öffner , Hendrik Ranocha This is my paper · ORCID

classification math.NAcs.NA MSC 65M1265M6065M7065M06

keywords entropystabilitykineticenergypreservationconservationlawsresidualdistributionschemesdiscontinuousGalerkinEulerequationsdeferredcorrectionsummation-by-parts

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

This paper tries to establish that any numerical scheme in the residual-distribution class—which includes many continuous/discontinuous Galerkin and flux-reconstruction methods—can be made entropy-conservative or entropy-dissipative by adding one explicitly computable correction term to each degree of freedom. The correction is the unique minimizer of a quadratic program that enforces local conservation and a discrete entropy condition, and the paper gives closed-form formulas for it in both the residual-distribution and the SBP/SAT/DG settings. It then extends the same idea to entropy inequalities, to multiple simultaneous constraints such as entropy plus kinetic energy for the Euler equations, and to fully discrete schemes via deferred correction time integration. If correct, this provides a systematic universal tool that removes the need to derive special entropy-conservative numerical fluxes case by case.

What carries the argument

The load-bearing object is the explicit correction term $r_i = \lambda (F_i - \bar F)$ (with $\bar F$ the mass-matrix-weighted element average of the entropy variables), chosen as the unique minimizer of the quadratic program $\min \frac{1}{2}\|r\|_M^2$ subject to local conservation and the discrete entropy equality. The coefficient $\lambda$ is given in closed form as $\mathcal{E}/\|F - \bar F\|_M^2$, so no optimization solver is needed; the same quadratic program with linear constraints is the template for every extension in the paper, including the inequality, multi-constraint, kinetic-energy, and fully discrete variants.

What would settle it

Compute the discrete entropy budget $F^T M \,\partial_t U$ plus the numerical entropy flux boundary term on one element of a corrected nodal DG scheme for a smooth Euler vortex, using the paper's own quadrature; Theorem 2.8 predicts the budget is zero up to rounding whenever $F$ is not constant, so any significantly nonzero value would contradict the claimed identity.

Watch

Extended reading notes

Core claim

The central claim is that entropy conservation can be imposed on any semidiscretization of the form (25) by adding $r_i = \lambda\bigl(F_i - \frac{1^T M F}{1^T M 1}\mathbf{1}\bigr)$, where $F$ are the entropy variables, $M$ is the mass matrix, and $\lambda = \mathcal{E}/\|F - \bar F\|_M^2$; Theorem 2.8 shows this is the unique solution of $\min \frac{1}{2}\|r\|_M^2$ subject to the conservation constraint $1^T M r_i = 0$ and the entropy constraint $F^T M r = \mathcal{E}$. The paper also claims the same quadratic-program characterization yields closed-form corrections for entropy inequalities (Theorem 3.1), for two simultaneous constraints such as entropy and kinetic energy (Theorem 3.2), and for a fully discrete DeC update, so the framework is a unified recipe rather than a single formula for a single scheme.

Load-bearing premise

The whole construction assumes the entropy variables $F(u)$ can be evaluated from the numerical solution and are not constant inside an element, so the denominator in formulas (26) and (42) is nonzero; the paper's own experiments in Section 4.2.2 show that once density or pressure turns negative this assumption fails and the corrected scheme blows up.

Editorial extensions

If this is right

Any locally conservative semidiscretization in the RD/DG/SBP class can be made discretely entropy-conservative without changing quadrature rules or grid structure, as demonstrated for continuous Galerkin, psi-scheme, and nodal DG discretizations.
The inequality variant yields entropy-dissipative schemes that reduce to the baseline scheme when it is already dissipative, so no spurious dissipation is added to a good method.
Multiple linear constraints can be imposed simultaneously, e.g., entropy conservation and kinetic energy preservation for the Euler equations, with a closed-form two-by-two solve.
Applying the correction to the full space-time residual in the deferred correction framework gives fully discrete entropy-conservative schemes, removing the need for special time integrators.
The framework sidesteps the case-by-case derivation of entropy-conservative numerical fluxes, which the paper argues works cleanly for calorically perfect gases but becomes unclear for general or tabulated equations of state.

Reading between the lines

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

The optimization formulation is constraint-agnostic: any additional linear conservation law (e.g., angular momentum) should be enforceable by adding its generalized variables to the constraint set, a route the paper does not explore.
Because the correction is the minimal-norm repair, one can view the method as a stabilization-by-projection that preserves the baseline scheme's behavior where it is already good; this suggests using it as a general filter for any unstable discretization, not only for entropy.
The vanishing-denominator case limits the approach at strong shocks: an element whose entropy variables are nearly constant acts like a finite-volume element, so a robust implementation would need to switch to the flux-difference correction there; the paper leaves this hybrid as future work.
A testable extension is to use the same quadratic-program correction for grid refinement and coarsening with explicit closed-form weights, since the paper only solves the inequality-constrained version with a generic optimizer.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Worth a look. The semidiscrete correction formulas are solid, and the optimization reinterpretation is a genuinely useful way to think about Abgrall's construction. The paper delivers what it promises on the semidiscrete side: Theorem 2.8 and its extensions to inequality constraints, multiple constraints, and kinetic energy preservation are clearly derived, explicitly solvable, and do not need an optimizer. Section 5 also makes a fair case that this framework is attractive for non-calorically-perfect equations of state, where classical entropy-conservative flux algebra gets ugly. Credit where due: the algebra is correct, the explicit formulas are reproducible, and the paper is unusually candid about its own limitations.

Now the soft spots, in proportion. The abstract and Section 2.1 float a fully discrete entropy-conservative/dissipative (FEC/D) claim via DeC, but Remark 2.2 admits that the paper focuses on semidiscrete corrections and that relaxation combinations are deferred. No theorem proves a discrete entropy balance for the full multi-stage DeC update, and the numerical section reports fully discrete results without quantitative entropy residuals. That is a real gap between headline and content, though not a fatal one: the semidiscrete core stands, and the FEC claim could likely be repaired by proving a discrete identity or by rescoping the abstract. The positivity issue is also real but honestly acknowledged. The correction formulas require admissible states with nonconstant entropy variables; the Taylor-Green experiments show crashes when density or pressure goes negative. This makes the "universal tool" framing too strong, but the paper says "no free lunch" itself and points to limiters as future work. Minor, but note: no code or data is provided, which slows reproduction.

Who is this for? Numerical analysts working on entropy-stable schemes for hyperbolic conservation laws, especially those tired of equation-specific entropy flux derivations. They will get a clean, useful toolkit for semidiscrete corrections and a fair comparison against flux-differencing approaches. I would bring it to a reading group and would cite it.

Recommendation: send it to peer review. It deserves a serious referee. The referee should push on the fully discrete claim and ask for either a proof, a counterexample, or a title/abstract that sticks to semidiscrete results. The core contribution is solid enough that this is fixable rather than fatal.

Referee Report

2 major / 5 minor

Summary. The paper proposes a unified optimization-based interpretation of entropy correction terms for residual distribution (RD) and summation-by-parts/Simultaneous-Approximation-Term (SBP-SAT) semidiscretizations, including discontinuous Galerkin and finite difference methods. It characterizes the correction as the unique minimizer of a quadratic program subject to conservation and entropy constraints (Theorem 2.8), extends the setting to entropy inequalities (Theorem 3.1), multiple linear constraints (Theorem 3.2), and kinetic-energy preservation for the Euler equations (Prop. 3.5), and supplies explicit formulas in each case. The authors also describe a fully discrete variant obtained by correcting the whole space-time residual in a deferred-correction (DeC) framework, present numerical comparisons for CG, psi, and DG schemes, and give entropy-stable grid transfer operators in an appendix.

Significance. If the semidiscrete results are taken as the contribution, the paper is a valuable systematization: the explicit formulas remove the need for optimization solvers, the framework is independent of special quadrature rules, and the extension to multiple constraints and kinetic-energy preservation is a genuine generalization of Abgrall's earlier construction. The algebraic derivations in Theorems 2.8, 3.1, and 3.2 are sound, and the paper is commendably explicit about the positivity limitation in Section 4.2.2. The numerical tests confirm that the corrections enforce the imposed constraints, as expected by construction rather than as empirical discovery. The main weakness is that the fully discrete entropy-conservation claim is asserted rather than proved, and the abstract presents this fully discrete extension as a headline contribution.

major comments (2)

[2.1] The fully discrete FEC/D claim is not supported by a proof. The paper shows that the correction is optimal in the semidiscrete setting (Theorem 2.8), but for the DeC update (18) the final value U^{n+1} is obtained after p correction sweeps over several substages, and no discrete entropy identity for the complete step is derived. The algorithm described after Eq. (19) is also underspecified: it gives the steps conceptually but no explicit correction formula for the full space-time residual analogous to (26). The numerical section reports 'fully discrete EC' results but does not provide a quantitative fully discrete entropy residual (for example, entropy change as a function of dt or number of correction sweeps). Without such an identity or error analysis, the abstract's statement that the approach yields 'for the first time a fully discrete entropy conservative/dissipative RD scheme' is a claim rather than a demonstrated result. I recommend either proving the discrete entropy balance for the DeC correction or reframing the contribution as semidiscrete, with the fully discrete variant presented as a numerical demonstration.
[4.2.2] The 'universal tool' claim should be qualified in the abstract and introduction. The correction formulas (26) and (42) require the entropy variables F to be well-defined, which for the Euler equations requires positive density and pressure; the denominator in (26) also vanishes for constant states. The Taylor-Green experiments in Section 4.2.2 show that when negative density or pressure appears, the correction cannot be applied and the simulation crashes. Since the correction itself does not preserve admissibility, the paper should state clearly that the framework applies to any scheme only for admissible states, and that positivity preservation is an additional requirement outside the scope of the correction. This is not an internal inconsistency, but it limits the claimed universality and should be stated prominently.

minor comments (5)

[Eq. (13)] The denominator in (13) should use a norm squared, e.g., sum_i ||F_i - Fbar||^2, to avoid confusion about vector-valued entropy variables.
[2.1] Remark 2.2 emphasizes that the paper focuses on semidiscrete corrections, yet the abstract presents the fully discrete extension as a major novelty. The abstract should reflect the actual emphasis of the paper.
[4.2.2] There is a typo 'formulation formulation' in the sentence about motivating examples; it should read 'formulation'.
[A.2] The observation that the entropy change decreases as dt^4 in Figure 10 is explained by the order of the time integrator, but the text should state this explicitly rather than leaving the reader to infer it.
[3.2] The nondegeneracy condition for the multiple-constraint theorem is only expressed verbally as 'the constraints do not contradict each other'. It would help to state explicitly that the coefficient matrix in the theorem must be invertible, and to comment on when this can fail for the combined entropy and kinetic-energy constraints.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the correction terms are explicit solutions of the stated constrained optimization problems, and the numerical checks verify the imposed constraints rather than predict them.

full rationale

The paper's central construction is to add correction terms that enforce discrete entropy and/or kinetic-energy identities. These identities are imposed as equality constraints in quadratic programs (Theorems 2.8, 3.1, 3.2 and Proposition 3.5), and the correction terms are derived as explicit solutions, e.g. r_i = lambda (F_i - mean) in (26). The verification that these terms satisfy (23) and (24) is a direct algebraic substitution, not a prediction of an independent quantity. The numerical experiments therefore confirm the design constraints rather than test a theory that is independent of the construction. The baseline correction formula is cited from Abgrall (2018), a work by one of the authors, but the paper re-derives it as the unique minimizer of a well-posed QP and proves the required optimality in the text using standard convex-optimization arguments; it does not rely on the citation for the load-bearing step. The fully discrete DeC claim is underdeveloped: Remark 2.2 explicitly states that the paper focuses on semidiscrete corrections and that fully discrete combinations with relaxation are deferred, and no discrete entropy identity is proven for the full DeC update. That is a missing justification or correctness gap, not circularity, because the proposed fully discrete correction, if implemented, would again enforce the entropy balance by construction rather than as a fitted result. No fitted parameters are used, no empirical input is renamed as a prediction, and no author-imposed uniqueness theorem is used to forbid alternatives. Hence the derivation chain is self-contained and the appropriate finding is no significant circularity.

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

The central construction assumes the standard hyperbolic conservation law setup: a convex entropy with one-to-one entropy variables, SBP operators with a diagonal mass matrix for DG, and a nondegenerate denominator in the correction formula. No free parameters are fitted to data. The fully discrete FEC claim is the least supported assumption, since it is asserted algorithmically and tested numerically without a convergence or entropy proof.

assumptions (4)

domain assumption The entropy function U is convex and the entropic variable map F = grad_U U is one-to-one on the state space.
Invoked throughout Sections 1 and 2. The correction is defined in terms of entropy variables, and switching between conservative and entropy variables requires uniqueness.
domain assumption The discrete operators satisfy the SBP property (20) and the mass matrix M is diagonal.
Assumed in Section 2.2 for the DG/SBP-SAT setting. The correction formulas and the discrete entropy identity rely on this structure.
domain assumption Inside each element, the entropy variables F are not constant, so the denominator in (26) is nonzero.
Theorems 2.8 and 3.1 require nondegenerate constraints. The paper handles the constant case only by referring to EC numerical fluxes in Remarks 2.5-2.7.
ad hoc to paper Correcting the full space-time residual in the DeC update (19) preserves both the order of accuracy and the entropy balance.
The FEC algorithm in Section 2.1 is described but no theorem or proof is provided. The paper relies on numerical experiments to support this claim.

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Cite this review

Pith. "Pith review of Reinterpretation and Extension of Entropy Correction Terms for Residual Distribution and Discontinuous Galerkin Schemes: Application to Structure Preserving Discretization." pith.science (2026). https://pith.science/paper/NHBSTELX

@misc{pith2026190804556,
  author       = {Pith},
  title        = {Pith review of: Reinterpretation and Extension of Entropy Correction Terms for Residual Distribution and Discontinuous Galerkin Schemes: Application to Structure Preserving Discretization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NHBSTELX}},
  note         = {Machine review of arXiv:1908.04556}
}

read the original abstract

For the general class of residual distribution (RD) schemes, including many finite element (such as continuous/discontinuous Galerkin) and flux reconstruction methods, an approach to construct entropy conservative/ dissipative semidiscretizations by adding suitable correction terms has been proposed by Abgrall (J.~Comp.~Phys. 372: pp. 640--666, 2018). In this work, the correction terms are characterized as solutions of certain optimization problems and are adapted to the SBP-SAT framework, focusing on discontinuous Galerkin methods. Novel generalizations to entropy inequalities, multiple constraints, and kinetic energy preservation for the Euler equations are developed and tested in numerical experiments. For all of these optimization problems, explicit solutions are provided. Additionally, the correction approach is applied for the first time to obtain a fully discrete entropy conservative/dissipative RD scheme. Here, the application of the deferred correction (DeC) method for the time integration is essential. This paper can be seen as describing a systematic method to construct structure preserving discretization, at least for the considered example.

Figures

Figures reproduced from arXiv: 1908.04556 by the authors.

**Figure 2.** = 0.25, DeC(2,2), B1, both correction terms, -plot, 3576 elements. Up to this point, only SEC/D schemes have been presented. In [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗

**Figure 3.** = 0.25, DeC(2,2), B1, residual correction, -plot, 8576 elements. The semidiscrete (dotted) and the fully discrete (solid) Galerkin schemes are compared in [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: = 0.25, DeC(2,2), B1, residual correction, -plot, 8576 elements. guarantee that both the pressure as well as the density remain positive. This is important for the application of the entropy correction term, not only because of physical reasons but also because of the …

**Figure 5.** Figure 5: = 0.25, DeC(4,4), B3, residual correction, -plot, 13548 elements Here, the classical 1D Shu-Osher test is extended to two dimension with radial speeds and initial conditions (0, ,0, ,0, 0) =    (3.857143, p 2.6290 [PITH_FULL_IMAGE:figures/full_fig_p023_5.png]

**Figure 6.** Figure 6: shows the initial conditions, an intermediate result after 150 steps, and the final result at = 1.8 (373 steps) using the FEC Galerkin method. The SEC Galerkin method is nearly indistinguishable from this result due to the properties of the DeC approach [3, 8] and the …

**Figure 7.** Figure 7: Relative kinetic energy kin and entropy of numerical solutions of the compressible Euler equations with a Taylor-Green for(, ) ∈ [0, 2] 2 and periodic boundary conditions. Then, a nodal DG scheme on Lobatto Legendre nodes and the associated Lobatto Legendre quadrature…

**Figure 8.** Figure 8: Relative kinetic energy kin and entropy of numerical solutions of the compressible Euler equations with a Taylor-Green vortex initial condition. The nodal discontinuous Galerkin schemes use polynomials of degree = 5 on Lobatto Legendre nodes [11, Eq. 25.4.14], the flux…

**Figure 9.** Figure 9: refinement and coarsening using polynomials of degree ≤ 6. The 2 entropy/energy () = 1 2 [PITH_FULL_IMAGE:figures/full_fig_p032_9.png]

**Figure 10.** Figure 10: Numerical solutions of the linear advection equation and their energy/entropy, computed using a nodal DG scheme based on closed Newton Cotes quadrature. Central Scheme w/o Correction Central Scheme w/ Correction (Ekin) Central Scheme w/ Correction (U) Central Scheme …

**Figure 11.** Figure 11: Relative kinetic energy kin and entropy of numerical solutions of the compressible Euler equations with a Taylor-Green vortex initial condition. The finite difference schemes use sixth order classical central stencils and either the classical central scheme with or …

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