arxiv: 2604.03158 · v1 · submitted 2026-04-03 · 🧮 math.NA · cs.NA

Recognition: 2 theorem links

· Lean Theorem

Entropy correction artificial viscosity for high order DG methods using multiple artificial viscosities

Raymond Park , Jesse Chan

Authors on Pith no claims yet

Pith reviewed 2026-05-13 18:41 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords discontinuous Galerkinentropy stabilityartificial viscosityhigh-order methodsshock capturingentropy correctionnumerical dissipation

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The pith

Multiple artificial viscosities with analytical parameters enable precise targeting of physical phenomena in entropy-stable DG methods.

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

The paper develops an entropy correction artificial viscosity for discontinuous Galerkin methods and extends it to multiple viscosity types, including separate terms for momentum and thermal diffusion. It derives simple analytical expressions for the optimal values of these parameters. Tests on one- and two-dimensional problems with shocks and under-resolved features show that the multi-parameter version lets users focus dissipation on chosen physical effects more accurately than a single monolithic viscosity parameter. This keeps the entropy inequality that improves robustness without needing expensive entropy-conservative fluxes. The result is greater flexibility while preserving stability for general settings.

Core claim

Extending the entropy correction artificial viscosity to multiple types such as viscosity and thermal diffusivity, together with analytically derived optimal parameter values, allows more precise targeting of specific physical phenomena than a single viscosity parameter while retaining robustness across general problem settings, as shown by comparisons on one- and two-dimensional test problems.

What carries the argument

The entropy correction artificial viscosity formulation extended to multiple viscosity types, with closed-form analytical expressions for the optimal parameters of each type.

If this is right

Users can apply dissipation selectively to thermal effects separately from momentum diffusion.
Robustness for shocks, turbulence, and under-resolved features is preserved without retuning.
Entropy stability is achieved without evaluating computationally expensive entropy-conservative fluxes.
Simple analytical formulas replace ad-hoc parameter choices for the viscosity terms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same multi-parameter construction could be applied to three-dimensional problems or to additional diffusion mechanisms such as species transport.
Targeted lower dissipation in some regions may improve resolution of fine-scale turbulence features compared with monolithic viscosity.
The analytical parameter expressions could be adapted to other high-order discretizations that use artificial viscosity for stabilization.

Load-bearing premise

That the derived analytical expressions for optimal viscosity parameters remain effective and generalizable beyond the specific 1D and 2D test problems shown, without requiring problem-specific retuning.

What would settle it

A three-dimensional shock-turbulence interaction test in which the multi-viscosity parameters fail to enforce the entropy inequality or produce non-robust solutions without retuning.

Figures

Figures reproduced from arXiv: 2604.03158 by Jesse Chan, Raymond Park.

**Figure 2.** Figure 2: Receding Flow 𝑁 = 2, 𝑀 = 130 5.4 Kelvin Helmholtz instability We demonstrate the impact of incorporating the SVV model through long-time simulations of the Kelvin–Helmholtz instability (KHI). Problem setup can be found in [7]. As KHI exhibits shear stresses and features vorticular movements, we expect inclusion of SVV will better model the physics that dictate the phenomena, and the coefficient of SVV tha… view at source ↗

**Figure 3.** Figure 3: Solution to Kelvin Helmholtz instability using SVV and Laplacian viscosity and [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

**Figure 4.** Figure 4: 𝑁 = 3, 64 × 64, 𝑇 = 0.14, log scale. To highlight how small the SVV coefficient is relative to the Laplacian coefficient, we show the heat map of the log scale of the coefficients [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗

read the original abstract

Entropy stable discontinuous Galerkin (DG) methods display improved robustness for problems with shocks, turbulence, and under-resolved features by enforcing an entropy inequality. Such methods have traditionally relied on entropy conservative (EC) fluxes that are computationally expensive to evaluate. An alternative approach for enforcing an entropy inequality is through a minimally dissipative ``entropy correction" artificial viscosity. We review how to construct such an artificial viscosity formulation and extend this approach to multiple types of viscosity (e.g., viscosity and thermal diffusivity). We determine simple analytical expressions for optimal viscosity parameters. We compare this to the case of a single monolithic viscosity parameter for different 1D and 2D problems, and show that the proposed method allows users to more precisely target specific physical phenomena while retaining robustness for general problem settings.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends single entropy-correction viscosity to multiple independent terms with closed-form optimal parameters, giving more targeted control on 1D/2D tests but resting on untested assumptions about local smoothness and flux form.

read the letter

The main takeaway is that this work generalizes entropy correction artificial viscosity to multiple independent viscosities with closed-form optimal parameters, offering more targeted dissipation control in high-order DG simulations. They start by reviewing how to build a single entropy correction viscosity that enforces the entropy inequality without expensive entropy-conservative fluxes. Then they extend it to separate terms, like viscosity and thermal diffusivity, and derive analytical expressions for the parameters by minimizing a local dissipation measure. In the 1D and 2D examples, this multi-version performs better than the single-parameter baseline for hitting specific physical behaviors while staying robust. The approach is practical and builds directly on existing entropy-stable DG ideas. The comparisons are straightforward and illustrate the flexibility gain. A potential soft spot is the generality of those analytical parameters. The derivation likely relies on local assumptions about the solution or flux that may not hold uniformly across all regimes, such as strong shocks or varying mesh resolutions. Without a broader sensitivity analysis, it's unclear if the expressions remain near-optimal without adjustment for new problems or higher dimensions. The citation pattern looks appropriate, referencing the single-viscosity literature. The numerical evidence supports the claims on the presented cases. This is for people in the DG community focused on entropy stability for fluid dynamics problems. Readers looking for ways to improve robustness in under-resolved simulations would get immediate value from the method. It deserves a serious referee because the idea is well-motivated, the math is accessible, and the results show a clear practical benefit. I recommend putting it through peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes an extension of entropy-correction artificial viscosity for high-order discontinuous Galerkin methods by introducing multiple distinct viscosity terms (e.g., separate viscosity and thermal diffusivity). It derives simple analytical expressions for the optimal parameters of these terms, compares the multi-viscosity formulation against a single monolithic parameter on selected 1D and 2D test problems, and claims that the new approach permits more precise targeting of physical phenomena while preserving robustness for general settings.

Significance. If the analytical expressions remain near-optimal without retuning across a broader range of regimes, the method supplies a low-cost route to entropy stability that avoids the expense of entropy-conservative fluxes, thereby increasing flexibility for DG simulations of shocks, turbulence, and under-resolved flows.

major comments (2)

[Derivation of optimal multi-viscosity parameters] The derivation of the closed-form optimal parameters (obtained by minimizing a local dissipation functional or solving an algebraic system tied to the entropy correction) is not accompanied by an explicit statement of the underlying assumptions, such as local smoothness for Taylor expansion of entropy variables or a particular numerical-flux structure. If these assumptions are violated by strong shocks or under-resolved turbulence, the claimed optimality may not hold, undermining the central claim of parameter-free generality.
[Numerical experiments section] The numerical comparisons on 1D and 2D problems demonstrate improvement over a single viscosity parameter, yet no sensitivity study is reported that perturbs polynomial degree, mesh aspect ratio, or Mach number while retaining the same analytical expressions. This omission leaves the robustness claim load-bearing on the specific test suite rather than on a general proof of near-optimality.

minor comments (2)

[Abstract] The abstract states that comparisons were performed on 1D and 2D problems but does not list the specific test cases or quantitative metrics (e.g., error norms or entropy-production rates) used to support the improvement claim.
[Notation and formulation] Notation for the separate viscosity coefficients should be introduced once and used consistently; occasional reuse of symbols for the single-viscosity case creates minor ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment point by point below, indicating the changes we will make in the revised version.

read point-by-point responses

Referee: The derivation of the closed-form optimal parameters (obtained by minimizing a local dissipation functional or solving an algebraic system tied to the entropy correction) is not accompanied by an explicit statement of the underlying assumptions, such as local smoothness for Taylor expansion of entropy variables or a particular numerical-flux structure. If these assumptions are violated by strong shocks or under-resolved turbulence, the claimed optimality may not hold, undermining the central claim of parameter-free generality.

Authors: We appreciate this observation. In the revised manuscript, we will add an explicit subsection detailing the assumptions, including the local smoothness requirement for the Taylor expansion of entropy variables and the specific numerical-flux structure employed. We acknowledge that strong shocks or highly under-resolved turbulence may locally violate the smoothness assumption. Nevertheless, the entropy correction is formulated to enforce the inequality in a global sense, and our numerical results on problems containing shocks demonstrate that the method retains robustness even when the local optimality is approximate. The analytical expressions are intended to provide effective, untuned parameters rather than exact optimality in every regime. revision: yes
Referee: The numerical comparisons on 1D and 2D problems demonstrate improvement over a single viscosity parameter, yet no sensitivity study is reported that perturbs polynomial degree, mesh aspect ratio, or Mach number while retaining the same analytical expressions. This omission leaves the robustness claim load-bearing on the specific test suite rather than on a general proof of near-optimality.

Authors: We agree that additional sensitivity information would strengthen the robustness claim. In the revised manuscript we will augment the numerical experiments section with a short sensitivity study that perturbs polynomial degree (P=1 to P=3) and mesh aspect ratio on the existing 1D and 2D test cases while keeping the analytical parameters fixed. Regarding Mach number, the derivation is independent of Mach number because it follows directly from the entropy correction; we will add a brief discussion of this independence together with one supplementary high-Mach test to illustrate retention of performance. A general proof of near-optimality across all regimes lies outside the present scope, but the combination of the analytical derivation and the expanded numerical evidence supports the practical utility of the expressions. revision: partial

Circularity Check

0 steps flagged

No significant circularity; analytical expressions derived independently from entropy inequality

full rationale

The paper reviews the construction of entropy-correction artificial viscosity from prior entropy-stable DG literature, then extends the formulation to separate viscosity and thermal diffusivity parameters. It states that simple analytical expressions for the optimal parameters are determined by minimizing a local dissipation functional tied directly to the entropy correction term. No step reduces by construction to a fitted input renamed as prediction, nor does any load-bearing claim rest solely on a self-citation chain whose own justification is unverified. The 1D/2D comparisons serve as validation rather than the source of the closed-form expressions. The derivation chain remains self-contained against the stated entropy inequality and dissipation minimization, with no evidence of self-definitional loops or ansatz smuggling.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that entropy inequalities can be enforced via minimally dissipative artificial viscosity and that analytical expressions for optimal multi-viscosity parameters can be derived from the underlying DG discretization without additional fitting.

axioms (1)

domain assumption Entropy inequality can be enforced through artificial viscosity in DG methods
Invoked in the review of the entropy-correction formulation.

pith-pipeline@v0.9.0 · 5421 in / 1155 out tokens · 26991 ms · 2026-05-13T18:41:25.467973+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation washburn_uniqueness_aczel echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

min ½ εᵀ W ε s.t. rᵀ ε = σ_k(u_h) … ε_i = σ r_i / Σ r_j²
IndisputableMonolith/Foundation/AbsoluteFloorClosure reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

entropy correction artificial viscosity … provably dissipative BR1 discretization

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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