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An artificial viscosity approach to high order entropy stable discontinuous Galerkin methods

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Pith's one-line read A local, parameter-free artificial viscosity recovers the entropy-stability guarantee of flux-differencing DG without entropy-conservative two-point fluxes.

desk verdict Entropy-correction viscosity with a genuinely sharper residual estimate and strong numerics, but the time-step and BR-1 null-mode guarantees are softer than the abstract implies. read the letter →

arxiv 2501.16529 v3 pith:CVFUZ3XW submitted 2025-01-27 math.NA cs.NA

Jesse Chan This is my paper · ORCID
classification math.NAcs.NA MSC 65M6065M1235L65
keywords entropystabilitydiscontinuousGalerkinartificialviscositycorrectionconservationlawshighordermethodsprojectioncompressibleEulerequations
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 proposes making a standard weak-form discontinuous Galerkin (DG) discretization of a nonlinear conservation law entropy stable by adding an artificial viscosity term, with a coefficient chosen cell-by-cell from the local violation of a cell entropy inequality. The coefficient is parameter-free and locally computable, and it requires none of the entropy-conservative two-point flux formulas that flux-differencing entropy stable DG methods depend on. The main theorem states that with this coefficient the semi-discrete scheme satisfies the same global entropy inequality satisfied by entropy-stable flux-differencing DG methods, provided interface fluxes are entropy stable and evaluated using the entropy projection. If correct, any existing weak-form DG code can gain an entropy stability guarantee with a small local modification, while preserving high order accuracy and a hyperbolic explicit time-step bound.

What carries the argument

The load-bearing object is the cell-local ratio defining the viscosity coefficient: the numerator is the volume entropy residual $\delta_k(u_h)$ from (20), a local measure of the failure of the discrete chain-rule identity, and the denominator is the viscous entropy dissipation $\sum_{i,j}(\epsilon_k K_{ij}\Theta_j,\Theta_i)_{D_k}$, where $\Theta_i$ is a BR-1 DG approximation of $\partial \Pi^N v(u_h)/\partial x_i$. The entropy projection $\tilde u = u(\Pi^N v(u_h))$ is the other essential ingredient: evaluating interface fluxes at $\tilde u$ makes inter-element contributions dissipative, and under nodal collocation this projection collapses to evaluation of $u_h$ at quadrature points. The proofs use a Roe-type linearization matrix to show that $\delta_k$ is a commutator of projection with the flux Jacobian, giving the superconvergence estimate that keeps the viscosity small for smooth data.

What would settle it

Initialize a degree N=2 periodic DG simulation with one of the spurious BR-1 null-space modes shown in Figure 1, such as a checkerboard gradient mode, and monitor the viscosity coefficient (24). If the coefficient grows without bound as the denominator approaches zero, or if the explicit step size collapses to an O($h^{2}$) parabolic restriction on a genuinely hyperbolic problem, the load-bearing assumption fails.

Watch

Extended reading notes

Core claim

The central claim is that entropy stability can be enforced by a local viscous term of BR-1 type. Concretely, define the volume entropy residual $\delta_k(u_h)$ as the cell-local failure of the discrete entropy identity, and choose the viscosity coefficient as $\epsilon_k(u_h) = -\min(0,\delta_k(u_h)) / \sum_{i,j}(K_{ij}\Theta_j,\Theta_i)_{D_k}$, using a regularized ratio when the denominator is small. Lemma 2 proves that if the viscous entropy dissipation is at least $-\min(0,\delta_k(u_h))$, the DG scheme satisfies the global semi-discrete entropy inequality; Lemma 3 upgrades this to a global inequality when the interface flux is entropy stable and evaluated at the entropy projection $\tilde u = u(\Pi^N v(u_h))$. The paper also proves that the volume entropy residual is superconvergent, of order $O(h^{2N+2+d})$ for smooth solutions, which explains why the viscosity vanishes for smooth flows and for linear advection. The resulting inequality is the same one satisfied by flux-differencing entropy stable DG methods, achieved without entropy-conservative two-point fluxes.

Load-bearing premise

The scheme assumes that the spurious gradient modes of the BR-1 viscosity operator are damped by the upwind convective flux in practice, so that the denominator in the viscosity coefficient does not become pathologically small; if it did, the viscosity would blow up and the time-step claim would fail.

Editorial extensions

If this is right

  • Any standard weak-form DG code can be made entropy stable by inserting the BR-1 viscous term and the cell-local coefficient, without changing the convective discretization.
  • The approach removes the need for explicit entropy-conservative two-point flux formulas, which may be unavailable or expensive for some systems.
  • High order accuracy is preserved, and numerical experiments show the viscosity coefficient converges to zero for smooth solutions and is sharply localized near shocks.
  • The scheme is less dissipative than flux-differencing entropy stable DG in the accuracy tests reported, and more locally linearly stable, with smaller spurious oscillations on shock problems.
  • Explicit time-stepping does not collapse to a parabolic step restriction in the tested cases, even when large viscosity is generated near shocks.

Reading between the lines

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

  • Because the viscosity coefficient is essentially an entropy-residual sensor, it could be combined with a shock-capturing viscosity by taking the maximum of the two coefficients, a route the paper mentions in an appendix.
  • The superconvergence estimate suggests the coefficient itself could serve as a mesh-adaptive or p-adaptive indicator for smooth versus under-resolved regions, a use the paper does not pursue.
  • If the spurious gradient mode issue is resolved by switching to non-central gradient discretizations such as local DG, the coefficient would be even smaller; the paper notes this as future work.
  • The construction should extend to any system with a strictly convex entropy, including shallow water or magnetohydrodynamics, since only the entropy variables and the viscous Jacobian enter the coefficient.
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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

2 major / 5 minor

Summary. The paper proposes an entropy correction artificial viscosity for standard weak-form discontinuous Galerkin (DG) discretizations of systems of conservation laws. The viscosity coefficient is chosen cell-locally to enforce a discrete cell entropy identity, which yields a global semi-discrete entropy inequality under a sufficient condition (21). The authors prove a superconvergence estimate for the volume entropy residual (Lemma 4), discuss the effect of inexact quadrature, compare the method with existing local entropy corrections and artificial viscosity approaches, and present extensive numerical experiments for the compressible Euler equations, including accuracy tests, shock tubes, the Shu-Osher problem, a 2D Riemann problem, and a long-time Kelvin-Helmholtz instability. The advertised advantages are a parameter-free, locally computable viscosity that recovers the same entropy inequality as flux differencing entropy stable DG methods, reduces oscillations, improves local linear stability, and preserves a hyperbolic explicit time-step restriction.

Significance. If the central claim held unconditionally, the paper would be a useful practical alternative to flux differencing entropy stable DG: it avoids the need for entropy conservative two-point fluxes and can be added to an existing weak-form DG code in a cell-local manner. Lemma 4's superconvergence estimate for the volume entropy residual is a genuine analytical contribution, and the numerical study is thorough, with reproducible code provided. However, the main advertised guarantees are conditional: Lemma 2 requires the sufficient condition (21), and the proposed coefficient (24) does not provably satisfy that condition for all admissible states because of the BR-1 spurious gradient null-space modes discussed in Section 4.5. The paper is transparent about this limitation, but the abstract's unconditional claims overstate what is proved.

major comments (2)
  1. [§4.5, Eq. (24), Lemma 2, Remark 3] The central result is conditional on the sufficient condition (21), but the proposed viscosity coefficient does not guarantee (21) on general meshes. Section 4.5 observes that the BR-1 gradient discretization (16) admits non-constant spurious null-space modes with Θ1=...=Θd=0, for which the denominator in (24) vanishes while the volume entropy residual δk(uh) need not be nonnegative. For such states no finite εk satisfies (21), so the global entropy inequality (22) is not proven. Remark 3's regularized ratio does not cure this: when the denominator is exactly zero it returns εk=0, and for small nonzero denominators it can amplify by O(1/√δ)≈10^7 with δ=10^{-14}, which would destroy the claimed explicit hyperbolic time-step bound. Because the analysis of these modes is explicitly deferred to future work, the abstract's statement that any standard weak-form DG code can be made entropy stable by adding this local parameter-free term is not supported. A concrete numerical test initializing a spurious null mode, or an analytical modification such as an LDG-type gradient discretization or a penalty that removes the null space, is needed before the unconditional claim can stand.
  2. [§7, Abstract, §6.6] The claim that the method preserves a hyperbolic maximum stable time-step size under explicit time-stepping is not established. No theorem or estimate in the paper bounds εk(uh); Section 6.6 reports only adaptive timestep counts and explicitly states that future work will analyze the maximum stable time-step restriction more rigorously. In view of the O(1/√δ) amplification in Remark 3, this advertised property should either be proved under the upwind-dissipation assumption or removed from the abstract and conclusion.
minor comments (5)
  1. [§4.4, Eqs. (25)-(27)] The statement that both factors in the Cauchy-Schwarz estimate are O(h^{N+1+√d}) is inconsistent with the final estimate (27), which is O(h^{2N+2+d}); the intermediate exponent should be O(h^{N+1+d/2}), since the scaling |D_k|=O(h^d) enters through the L2 norm on a d-dimensional simplex.
  2. [Lemma 4 statement] The notation uh ∈ [P^N(D_k)]^d is dimensionally wrong for systems: the number of solution components is n (for example, n=d+2 for the compressible Euler equations), not the spatial dimension d.
  3. [Eq. (30)] The denominator in the derivative-based local entropy correction has a duplicated subscript 'D_k D_k' in the inner product, which should be a single subscript.
  4. [Abstract and Remark 3] The claim that the viscosity coefficients are parameter-free is slightly overstated, since the regularized ratio in Remark 3 uses a fixed tolerance δ=10^{-14}; although the authors report insensitivity to δ, the abstract should acknowledge this safeguard.
  5. [Lemma 1] Lemma 1 is stated with a reference to [13] and a 'straightforward modification' but no proof; a short sketch of the localization argument would make the paper more self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the entropy inequality is enforced by the explicit construction of the viscosity coefficient, and the paper's independent content (residual estimates, accuracy, robustness) does not reduce to its inputs.

full rationale

The paper's central stability claim is constructive: Lemma 2 states that the semi-discrete entropy inequality (22) holds whenever condition (21) is satisfied, and Eq. (24) defines the piecewise-constant coefficient as the smallest value enforcing (21). This is an explicit enforcement mechanism rather than a hidden fit; the coefficient is computed from the local entropy residual and local viscous dissipation, and no data are fitted to obtain the entropy inequality. The independent content is Lemma 4's superconvergent estimate for the volume entropy residual, which is proved using the Roe linearization theorem from Godlewski and Raviart, plus the numerical experiments; neither reduces to the definition of epsilon. Self-citations to the author's prior work (e.g., Lemma 1 adapted from [13] and Lemma 3's proof from [16,11]) are external published results with proofs, not assumptions equivalent to the target conclusion. Section 4.5 explicitly concedes that BR-1 spurious gradient null modes can make the denominator in (24) vanish and that the upwind-flux dissipation mechanism is only numerical intuition deferred to future work. That is an unproven sufficiency or robustness assumption, and Remark 3's regularization is an engineering fix; it does not make the derivation circular. Overall, no circular step was found.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central construction (24) is parameter-free apart from a regularization tolerance delta. The theoretical results rely on standard finite element approximation theory and a cited Roe-linearization theorem, plus two domain assumptions that are only empirically checked: the dissipation of BR-1 spurious null modes (Section 4.5) and the robustness of the entropy projection near low-density states (Section 6.2, Appendix A.2).

free parameters (1)
  • Regularization parameter delta in ratio approximation = 1e-14 (also tested at 1e-13 and 1e-15)
    Remark 3: used to regularize a/b ratios in (24) when the denominator approaches zero; the paper states results are insensitive to nearby values.
assumptions (5)
  • standard math Existence of a Roe-type linearization matrix A_m(u1,u2) for systems with strictly convex entropy (Theorem 1 of Godlewski-Raviart [36]).
    Used in the proof of Lemma 4 to rewrite f(uh tilde) - f(uh) = A(Πv - v), Section 4.4.
  • domain assumption Sufficient regularity of entropy variables and Roe matrices for smooth solutions with density and internal energy bounded away from zero.
    Invoked before (27) in Section 4.4 to apply L2 best-approximation estimates.
  • domain assumption BR-1 viscous discretization satisfies the localizable dissipation estimate in Lemma 1, adapted from Lemma 3.1 of [13].
    The stability of the viscosity term in the entropy balance (Lemma 2) rests on this estimate.
  • domain assumption Spurious gradient null-space modes of the BR-1 gradient are dissipated by upwind convective fluxes in practice, so the denominator in (24) does not become pathologically small relative to the numerator.
    Section 4.5 acknowledges the null modes and gives only an empirical argument that they are not a major issue; no proof is provided.
  • standard math For nodal collocation, the L2 projection operator equals the interpolation operator (ΠN = IN).
    Used in Section 4.3 to simplify the entropy projection at quadrature points.

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Cite this review

Pith. "Pith review of An artificial viscosity approach to high order entropy stable discontinuous Galerkin methods." pith.science (2026). https://pith.science/paper/CVFUZ3XW

@misc{pith2026250116529,
  author       = {Pith},
  title        = {Pith review of: An artificial viscosity approach to high order entropy stable discontinuous Galerkin methods},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CVFUZ3XW}},
  note         = {Machine review of arXiv:2501.16529}
}
read the original abstract

Entropy stable discontinuous Galerkin (DG) methods improve the robustness of high order DG simulations of nonlinear conservation laws. These methods yield a semi-discrete entropy inequality, and rely on an algebraic flux differencing formulation which involves both summation-by-parts (SBP) discretization matrices and entropy conservative two-point finite volume fluxes. However, explicit expressions for such two-point finite volume fluxes may not be available for all systems, or may be computationally expensive to compute. This paper proposes an alternative approach to constructing entropy stable DG methods using an entropy correction artificial viscosity, where the artificial viscosity coefficient is determined based on the local violation of a cell entropy inequality and the local entropy dissipation. The resulting method is a modification of the entropy correction introduced by Abgrall, Offner, and Ranocha (2022) in "Reinterpretation and Extension of Entropy Correction Terms for Residual Distribution and Discontinuous Galerkin Schemes: Application to Structure Preserving Discretization", and recovers the same global semi-discrete entropy inequality that is satisfied by entropy stable flux differencing DG methods. The entropy correction artificial viscosity coefficients are parameter-free and locally computable over each cell, and the resulting artificial viscosity preserves both high order accuracy and a hyperbolic maximum stable time-step size under explicit time-stepping.

Figures

Figures reproduced from arXiv: 2501.16529 by the authors.

Figure 1
Figure 1. Examples of spurious null space modes of the BR-1 gradient in 1D and 2D. [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Solutions (density) to the 2D Riemann problem using a matrix-weighted version of the local entropy [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Evolution of L2 error and entropy for the 1D density wave with amplitude A = 0.5. Here, “DG” refers to the standard DG method, “EC” refers to a flux differencing entropy stable DG method, and “AV” refers to DG with entropy correction artificial viscosity. We compute L 2 errors at final time T = 1.7 for amplitude A = 0.5 and show rates of convergence in [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Evolution of L2 error and entropy for the 1D density wave with amplitude A = 0.98. Here, “DG” refers to the standard DG method, “EC” refers to a flux differencing entropy stable DG method, and “AV” refers to DG with entropy correction artificial viscosity. (a) Flux dif…
Figure 5
Figure 5. Figure 5: Spectra of the linearized Jacobian for A = 0.98 with a nodal DG discretization with N = 7, K = 4 and a local Lax-Friedrichs interface flux. The plots are zoomed near the origin to highlight the presence of eigenvalues with positive real parts. Finally, we note that we …
Figure 6
Figure 6. Figure 6: Convergence of the volume entropy residual [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]
Figure 7
Figure 7. Figure 7: Comparison of density for entropy stable DG methods constructed using flux differencing and entropy [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]
Figure 8
Figure 8. Figure 8: Entropy stable DG solutions (density) of the Shu-Osher problem with [PITH_FULL_IMAGE:figures/full_fig_p024_8.png]
Figure 9
Figure 9. Figure 9: Entropy stable DG solutions (density) of the Shu-Osher problem with [PITH_FULL_IMAGE:figures/full_fig_p025_9.png]
Figure 10
Figure 10. Figure 10: Entropy stable DG solutions (density) of the Shu-Osher problem with [PITH_FULL_IMAGE:figures/full_fig_p025_10.png]
Figure 11
Figure 11. Figure 11: Solution of the 2D Riemann problem with degree [PITH_FULL_IMAGE:figures/full_fig_p026_11.png]
Figure 12
Figure 12. Figure 12: Solution (density) of the 2D long-time Kelvin-Helmholtz instability using DG with entropy correction [PITH_FULL_IMAGE:figures/full_fig_p027_12.png]

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