Entropy stable finite difference methods via entropy correction artificial viscosity and knapsack limiting

Brian Christner; Jesse Chan

REVIEW 4 major objections 4 minor 20 references

Entropy stable finite difference methods via entropy correction artificial viscosity and knapsack limiting

T0 review · 4 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read High-order finite difference schemes can be made entropy stable and positivity preserving by choosing diffusion coefficients from a tiny quadratic program.

desk verdict Solid entropy-stability extension to finite differences; positivity claim needs a real proof before it carries weight. read the letter →

arxiv 2508.21226 v2 pith:D3A32S65 submitted 2025-08-28 math.NA cs.NA

Brian Christner , Jesse Chan This is my paper

classification math.NAcs.NA MSC 68Q2568R1068U05

keywords entropystabilityfinitedifferencemethodsartificialviscosityknapsacklimitingpositivitypreservationcompressibleEulerequationsNavier-Stokessummation-by-partsoperators

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 transplants two stabilization techniques from discontinuous Galerkin methods to high-order finite difference discretizations of compressible flow: entropy-correction artificial viscosity and knapsack limiting. The resulting schemes satisfy a discrete entropy inequality, a discrete analog of the second law of thermodynamics, without any user-tuned hyperparameters. The knapsack-limited variant additionally provably keeps density and pressure positive for the compressible Euler and Navier-Stokes equations. If correct, this gives computational fluid dynamics a class of finite difference solvers that are simultaneously high-order accurate, entropy stable, positivity preserving, and computationally cheap.

What carries the argument

The load-bearing object is the parametrized flux fij(θ) = f^H_ij + θ(ui - uj), where f^H is the high-order central flux and θ is a diffusion coefficient proportional to added dissipation. Setting θ = (λij/2)θ̃ recasts this flux as a convex combination (1-θ̃)f^H + θ̃ f^L, the knapsack-blended flux, where f^L is a low-order entropy-stable and positivity-preserving flux. A per-node quadratic program selects θ; the unconstrained problem has a closed-form solution, while the knapsack-constrained problem is solved by a one-pass algorithm. Symmetrization θij = max{θ̂ij, θ̂ji} ensures conservation and suffices for the entropy inequality.

What would settle it

Run KL-FD with the stated limiting formula on a two-dimensional compressible Euler benchmark, such as the Kelvin-Helmholtz instability at order N=6 with 512 nodes per dimension and a timestep near the stability limit, and check whether any cell attains negative density or pressure; a single occurrence with the relative positivity parameter α > 0 would falsify the positivity-preservation claim as stated. Alternatively, verify the relative positivity inequality (4.3) cell-wise at every Runge-Kutta stage for a smooth periodic initial condition with N≥4; a violation would show the symmetrized limi

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Extended reading notes

Core claim

The central claim is that a high-order summation-by-parts finite difference discretization of a hyperbolic conservation law can be made entropy stable by adding diffusion coefficients θij to the central flux, with θij chosen as the minimal-norm solution of a per-node linear inequality a_i^T θ_i ≥ b_i. The paper proves in Lemma 3.2 that this optimization problem has a closed-form solution, so the scheme is hyperparameter free; symmetrization θij = max{θ̂ij, θ̂ji} preserves conservation while retaining entropy stability (Lemma 3.3). For the knapsack-limited scheme, the same flux is reinterpreted as a convex blend between the high-order central flux and a low-order positivity-preserving Lax-Fri

Load-bearing premise

The positivity-preservation claim rests on formula (5.1) and the cited positivity theory, neither derived nor proven inside this paper, holding for arbitrary high-order finite difference stencils; if that limiting-coefficient formula fails for a given stencil, the positivity guarantee collapses even though the entropy inequality still holds.

Editorial extensions

If this is right

Entropy-stable finite difference solvers for the compressible Euler and Navier-Stokes equations can be constructed without tunable parameters, at computational cost comparable to standard stabilized schemes.
Positivity of density and pressure is guaranteed for high-order finite difference approximations, removing a common failure mode for under-resolved or shock-dominated flows.
The knapsack-flux viewpoint unifies artificial viscosity and flux limiting: the same diffusion coefficient enforces the entropy inequality and blends the low- and high-order fluxes.
High spatial order, up to the stencil accuracy N, is preserved for smooth solutions, so the stabilization does not degrade accuracy.
The construction extends to multi-dimensional summation-by-parts operators and weakly enforced Dirichlet boundary conditions.

Reading between the lines

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

Because the machinery is written dimension-agnostically and the optimization is per node, the same per-node quadratic program could be applied directly to other summation-by-parts discretizations, such as discontinuous Galerkin spectral element methods, without modification; the paper gestures at this through its DG lineage but does not state it as a general recipe.
The entropy-stability proof only requires a_i ≥ 0 and symmetrization, so the argument likely extends to any convex entropy pair, not just the physical Euler entropy, suggesting applications to other hyperbolic systems such as magnetohydrodynamics provided a positivity-preserving low-order flux exists.
The relaxed entropy inequality in Section 8 introduces a dissipation-versus-oscillation trade-off that the paper documents numerically but does not rigorously bound; a natural next step would be to derive an a posteriori measure of entropy-dissipation error for the relaxed schemes.
The relative-positivity limiting formula (5.1) is imported from DG theory; a valuable test would be to verify it on genuinely multi-dimensional stencils and non-diagonal SBP norms, where the symmetrization step may interact differently with the limiting coefficients.

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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

The part that holds up is the entropy stability machinery. The derivation of the nodal entropy inequality (3.7), the explicit optimizer in Lemma 3.2, and the max-symmetrization that preserves both conservation and entropy stability (Lemma 3.3) are clean and self-contained. That is a genuine extension of ECAV to SBP finite differences, and it reads like a real result rather than an incremental tweak. The weak-boundary treatment is straightforward and useful.

The positivity story is the soft spot. The abstract promises provable positivity for Euler and Navier-Stokes, but the manuscript only sketches the knapsack machinery. Formula (5.1) for ℓ^c appears without derivation; the proof of the relative positivity condition is imported from the authors' DG papers [4, 11]. That transfer is not automatic: FD stencils couple nodes differently from DG subcell limiting, and the text itself notes the equivalence to the weighted diffusion form is only shown for LxF, while the numerics use HLLC. Also, the Navier-Stokes claim in the abstract has no discretization behind it in this paper. And the user-chosen α in the relative positivity constraint sits uneasily with the "hyperparameter free" label.

None of this kills the main contribution. The entropy-stable ECAV-FD part is provable and demonstrated. KL-FD is plausible and the experiments are encouraging, but the headline claim is ahead of the proof. The timing study mentioned in the introduction is also absent, though the paper does include convergence tables and several shock tests.

Who should read it: anyone building entropy stable finite difference codes or comparing stabilization strategies. It deserves a serious referee; the revision should be asked to derive or properly reference the positivity lemma for the FD case, either prove formula (5.1) or state it as a lemma with a proof, define the Navier-Stokes discretization or soften the claim, and report timings. I would send it to review rather than desk reject.

Referee Report

4 major / 4 minor

Summary. The manuscript proposes two entropy-stable finite difference schemes on SBP operators: ECAV-FD, which adds minimal artificial diffusion determined by solving a nodal quadratic optimization, and KL-FD, which blends a low-order entropy-stable positivity-preserving flux with a high-order flux via knapsack limiting. Section 3 gives a self-contained derivation of the discrete nodal entropy inequality and proves that symmetrized diffusion coefficients θij = max(θ̂ij, θ̂ji) preserve entropy stability and conservation. Section 4 introduces the knapsack-limited scheme and claims positivity preservation through a relative positivity condition, with a limiting-coefficient formula (5.1) used in the numerical experiments. The numerical section reports spatial convergence for N=2,...,5, Shu-Osher, Leblanc shock tube, Woodward-Colella blast wave, and two-dimensional Kelvin-Helmholtz instability, including a relaxed entropy-inequality variant in the appendix.

Significance. If the positivity-preservation claim is fully established, the paper makes a valuable contribution: it transfers two recently developed DG stabilization strategies (entropy-correction artificial viscosity and knapsack limiting) to finite difference methods, gives an explicit and cheap optimization solution, and demonstrates high-order convergence and robustness in challenging benchmarks. The Section 3 entropy-stability derivation is clean and largely self-contained, and the symmetrization argument is simple and convincing. However, the headline positivity claim for KL-FD is asserted rather than proved in this manuscript, and the Navier-Stokes claim and the HLLC-based numerical scheme are not covered by the presented theory. These gaps substantially affect the significance of the paper as a proof-of-concept, though the numerical evidence suggests the methods work in practice.

major comments (4)

[Section 4, Eqs. (4.3)–(4.4) and Eq. (5.1)] The positivity-preservation guarantee for KL-FD is asserted, not derived. The text states that limiting coefficients ℓc can be derived so that θ ∈ [ℓc,1] implies the relative positivity condition (4.3), but no existence proof is given, and the computational formula (5.1) is introduced in Section 5 without derivation. The formula uses nodal totals r_i^H, r_i^L, while the actual scheme uses edgewise coefficients that are symmetrized and then re-optimized in (4.4); it is not shown that the nodal construction survives these steps for the SBP finite-difference stencils of Section 2. This is the core of the abstract's 'provably preserves positivity' claim and must be proved or explicitly reduced to a precise theorem.
[Abstract and Section 4] The abstract claims positivity for the compressible Euler and Navier-Stokes equations, but the manuscript only defines the hyperbolic conservation law (2.1). No Navier-Stokes or viscous discretization, viscous flux, or corresponding boundary treatment is presented anywhere. Unless the claim is strictly inherited from the low-order scheme of [11], the transfer to the high-order KL-FD scheme is not established. The claim should be restricted to the compressible Euler equations or supported by a concrete viscous discretization and proof.
[Section 4, HLLC paragraph] The key identity f_KL(θ) = (1−θ)f_H + θf_L is proved only for the Lax-Friedrichs flux, using Eq. (2.7). The paper then states that HLLC can be used and that 'the same strategies apply,' and all numerical positivity tests use KL-FD-HLLC. For HLLC, f_L − f_H is not proportional to u_i − u_j, so neither the convex blending used in (4.3) nor the nonnegativity of a_ij = ||n_ij||(v_j−v_i)^T(f_H−f_L) is automatic. The theory therefore does not currently cover the scheme that is actually tested.
[Section 4 and Appendix (8.1)–(8.2)] The positivity proof and the one-iteration convergence of the quadratic knapsack algorithm are delegated to [4] and [11], both of which are discontinuous Galerkin/subcell-limiting papers. The transfer from DG subcell limiting to flux-differenced finite-difference SBP discretizations is not automatic and is not argued in the manuscript. Since the abstract's central claim is a provable positivity guarantee, the relevant theorem should be stated and proved here, or the claim should be marked as conditional on the cited DG theory.

minor comments (4)

[Eq. (5.1)] The limiting-coefficient formula (5.1) is not clamped to [0,1]. If the formula can produce values outside this range, the subsequent interpretation of θ + ℓc as a convex blending coefficient is unclear; please state the range or add clamping.
[Table 3] The entry '2.53¨ 105' should presumably be '2.53×10^{-5}'. Please correct the exponent formatting.
[Section 8.1] The text refers to the 'Sod shocktube problem' and to Figure (1), but Figure (1) displays the Shu-Osher shock tube from Section 5.2. Please clarify whether the experiments in Section 8.1 are Shu-Osher or Sod, and adjust the figure references accordingly.
[Abstract and Section 5.3] The abstract describes both schemes as 'hyperparameter free,' but KL-FD with positivity uses the relative positivity parameter α, and the experiments use α = 0.5 and α = 0.1. Please either clarify that α is a user-specified safety factor rather than a fitted hyperparameter, or modify the hyperparameter-free claim.

Circularity Check

2 steps flagged · score 4.0 of 10

Entropy stability is constructed in-paper, but the positivity and relaxed-optimality claims are imported from same-authors' preprints.

self citation load bearing [Section 4, Positivity Preservation (around Eqs. (4.3)-(4.4)); cf. end of Section 4 for the HLLC caveat]
"Limiting coefficients ℓc can be derived such that if θij P rℓcij, 1s, the resulting scheme satisfies a relative positivity condition ... An efficient, linear-time algorithm for this quadratic knapsack problem was proved in [4]. At this point, symmetrizing the diffusion coefficients to θij “ max ! ˆθij, ˆθji ) , the scheme (3.13) is provably positivity preserving when applied with the diffusion coefficients θij` ℓcij."

The claimed positivity preservation is not proved in this paper. The existence of ℓc satisfying (4.3), the one-iteration solution of the quadratic knapsack problem (4.4), and the positivity of the low-order scheme are all taken from [4] (the present authors' own arXiv preprint) and [11] (co-authored by J. Chan). No theorem in this paper transfers those DG subcell-limiting results to SBP flux-differenced finite-difference stencils. The text even concedes that for the HLLC flux actually used in all KL-FD simulations, f^KL(θ) is not equivalent to the weighted flux (3.5), so the convex-blend reduction on which (4.3)-(4.4) rely is not established for the simulated scheme. Thus the headline 'provably preserves positivity' is carried by a same-author citation chain rather than an in-paper derivat
self citation load bearing [Appendix, Section 8 (Relaxed Entropy Inequality), around Eq. (8.1)]
"It is shown in [4] that Algorithm 3.1 converges in exactly one iteration to the solution for such a problem. Therefore, an explicit formula can be derived for the optimal solution for the optimization problem (8.1)"

The relaxed schemes RECAV-FD/RKL-FD are validated numerically using an explicit optimal solution whose derivation is outsourced to [4], another preprint by the same authors; the one-iteration convergence theorem and the resulting formula are not proved here. This is a secondary load-bearing self-citation: the claimed optimality of the relaxed diffusion coefficients is exactly the content of [4]'s algorithm, and no independent verification is supplied. It does not undermine the main entropy-stability theorem, but it contributes to the circularity burden.

full rationale

The core entropy-stability derivation (Section 3) is self-contained: the discrete entropy inequality (3.7) is manipulated into a linear feasibility condition a_i^T θ_i ≥ b_i, and Lemmas 3.1-3.3 prove existence and verify that the symmetrized max choice satisfies it. This is a constructive verification, not a prediction that reduces to its inputs. The conservation lemmas and boundary treatment are also in-paper. No fitted parameter is renamed as a prediction; the diffusion coefficients are optimization variables, not data fits. The circularity burden is concentrated in Section 4 and the Appendix. Section 4's positivity-preservation guarantee is stated as 'provably' but its proof is not present: the limiting-coefficient existence, the knapsack algorithm's one-iteration convergence, and the low-order positivity are cited to [4] and [11], both involving the present authors ([4] being the same authors' preprint). Moreover, the only low-order flux for which the f_KL(θ)= (1-θ)f_H+θ f_L reduction is demonstrated is LxF (2.7), while all KL-FD numerics use HLLC; the paper explicitly disclaims equivalence in that case. This makes the positivity theorem's application to the headline scheme unsupported. The appendix similarly relies on [4] for the explicit solution of the relaxed problem. These are self-citations that are load-bearing for secondary claims, but the primary entropy-stability result retains independent mathematical content. Hence score 4, not higher: no equation-level reduction by construction was found.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The main stabilization coefficient θ is state-dependent and computed by optimization, so it is not a fitted free parameter. The user-chosen α in the positivity-preserving variant is a real knob, contradicting the 'hyperparameter free' label. Several axioms are standard SBP and admissibility assumptions, but the positivity theorem and the relaxed solver rest on the authors' prior preprint [4], which is not independently verified here.

free parameters (1)

Relative positivity parameter α = 0.5 (Leblanc), 0.1 (Woodward-Colella), no universal value
User-specified bound in (4.3) and (5.1). The abstract calls the schemes hyperparameter free, but the positivity-preserving KL-FD variant requires choosing α and a low-order flux.

assumptions (6)

standard math Summation-by-parts finite difference operators satisfy D_k + D_k^T = B_k and are skew-symmetric under periodic boundary conditions.
Used throughout Section 2 and in Lemmas 3.3 and 3.4 to prove conservation and derive the flux-differencing form.
domain assumption There exists a convex entropy-entropy flux pair (η,F) with entropy variables v = ∇uη, and the solution is admissible (positive density and pressure).
Needed for the entropy inequality (3.1), the chain rule, and Lemma 3.1's nonnegativity of a_ij.
domain assumption At admissible states, the entropy projection du/dv is symmetric positive definite, so (v_j - v_i)^T (u_j - u_i) >= 0.
Used in Lemma 3.1 to guarantee a_i >= 0, which is necessary for the explicit solution of the optimization problem.
domain assumption The time integrator is strong stability preserving (SSP), so it can be written as a convex combination of forward Euler updates.
Section 4 relies on SSP time stepping to combine the low-order positivity-preserving update with high-order accuracy.
domain assumption The low-order scheme (4.1) with LxF or HLLC flux is positivity preserving for the compressible Euler and Navier-Stokes equations under a suitable timestep.
The positivity guarantee of KL-FD is built directly on this property; the paper cites [11] rather than proving it.
ad hoc to paper Algorithm 3.1 of [4] converges in exactly one iteration and yields the explicit solution of the relaxed two-variable knapsack problem (8.1).
Load-bearing for the relaxed ECAV-FD and KL-FD variants; the proof is not included here and [4] is the authors' own preprint.

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Cite this review

Pith. "Pith review of Entropy stable finite difference methods via entropy correction artificial viscosity and knapsack limiting." pith.science (2026). https://pith.science/paper/D3A32S65

@misc{pith2026250821226,
  author       = {Pith},
  title        = {Pith review of: Entropy stable finite difference methods via entropy correction artificial viscosity and knapsack limiting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D3A32S65}},
  note         = {Machine review of arXiv:2508.21226}
}

read the original abstract

Entropy stable methods have become increasingly popular in the field of computational fluid dynamics. They often work by satisfying some form of a discrete entropy inequality: a discrete form of the 2nd law of thermodynamics. Schemes which satisfy a (semi-)discrete entropy inequality typically behave much more robustly, and do so in a way that is hyperparameter free. Recently, a new strategy was introduced to construct entropy stable discontinuous Galerkin methods: knapsack limiting, which blends together a low order, positivity preserving, and entropy stable scheme with a high order accurate scheme, in order to produce a high order accurate, entropy stable, and positivity preserving scheme. Another recent strategy introduces an entropy correction artificial viscosity into a high order scheme, aiming to satisfy a cell entropy inequality. In this work, we introduce the techniques of knapsack limiting and artificial viscosity for finite difference discretizations. The proposed schemes preserve high order accuracy in sufficiently smooth conditions, are entropy stable, and are hyperparameter free. Moreover, the proposed knapsack limiting scheme provably preserves positivity for the compressible Euler and Navier-Stokes equations. Both schemes achieve this goal without significant performance tradeoffs compared to state of the art stabilized schemes.

Figures

Figures reproduced from arXiv: 2508.21226 by the authors.

**Figure 2.** KL-FD-HLLC solution to the Leblanc shock tube with 4000 nodes and relative [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** KL-FD-HLLC solution to the Woodward-Colella blast wave with 1200 nodes [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: ECAV-FD solution to Kelvin-Helmholtz instability problem. [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

**Figure 5.** Figure 5: KL-FD-HLLC solution to Kelvin-Helmholtz instability problem. [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]

**Figure 6.** Figure 6: Density of the Shu-Osher shock tube solution at final time [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]

**Figure 7.** Figure 7: Density of the Shu-Osher shock tube solution at final time [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]

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Works this paper leans on

20 extracted references · 17 canonical work pages

[11]

A positivity preserving strategy for entropy stable discontinuous Galerkin discretizations of the compressible Euler and Navier-Stokes equa- tions

Yimin Lin, Jesse Chan, and Ignacio Tomas. A positivity preserving strategy for entropy stable discontinuous Galerkin discretizations of the compressible Euler and Navier-Stokes equa- tions. Journal of Computational Physics, 475:111850, 2023

work page 2023
[4]

Entropy stable nodal discontinuous Galerkin methods via quadratic knapsack limiting

Brian Christner and Jesse Chan. Entropy stable nodal discontinuous Galerkin methods via quadratic knapsack limiting. arXiv preprint arXiv:2507.14488, 2025

arXiv 2025
[1]

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

Jesse Chan. An artificial viscosity approach to high order entropy stable discontinuous Galerkin methods. arXiv preprint arXiv:2501.16529, 2025

arXiv 2025
[2]

Rueda-Ram ´ ırez, Gregor Gassner, and Tim War- burton

Jesse Chan, Hendrik Ranocha, Andr´ es M. Rueda-Ram ´ ırez, Gregor Gassner, and Tim War- burton. On the Entropy Projection and the Robustness of High Order Entropy Stable Discontinuous Galerkin Schemes for Under-Resolved Flows. Frontiers in Physics, Volume 10 - 2022, 2022

work page 2022
[3]

Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws

Tianheng Chen and Chi-Wang Shu. Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws. Journal of Computational Physics, 345:427–461, 2017

2017
[5]

High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains

Travis C Fisher and Mark H Carpenter. High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains. Journal of Computational Physics, 252:518– 557, 2013

work page 2013
[6]

Gassner, Andrew R

Gregor J. Gassner, Andrew R. Winters, and David A. Kopriva. Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations. Journal of Computational Physics, 327:39–66, 2016

work page 2016
[7]

Numerical approximation of hyperbolic systems of conservation laws, volume 118

Edwige Godlewski and Pierre-Arnaud Raviart. Numerical approximation of hyperbolic systems of conservation laws, volume 118. Springer Science & Business Media, 2013. 20 B. CHRISTNER AND J. CHAN (a) n “ 500 nodes (b) n “ 1500 nodes, ECA V-FD (c) n “ 1500 nodes, KL-FD (d) n “ 500 nodes (e) n “ 1500 nodes, RECA V-FD (f) n “ 1500 nodes, RKL-FD Fig. 7: Density ...

work page 2013

Show all 20 references

[8]

Invariant domain preserving discretization-independent schemes and convex limiting for hyperbolic systems

Jean-Luc Guermond, Bojan Popov, and Ignacio Tomas. Invariant domain preserving discretization-independent schemes and convex limiting for hyperbolic systems. Computer Methods in Applied Mechanics and Engineering, 347:143–175, 2019

2019
[9]

Lax, and Bram van Leer

Amiram Harten, Peter D. Lax, and Bram van Leer. On Upstream Differencing and Godunov- Type Schemes for Hyperbolic Conservation Laws. SIAM Review, 25(1):35–61, 1983

1983
[10]

High order entropy stable discontinuous Galerkin spectral element methods through subcell limiting

Yimin Lin and Jesse Chan. High order entropy stable discontinuous Galerkin spectral element methods through subcell limiting. Journal of Computational Physics, 498:112677, 2024

2024
[12]

Sparse invariant domain preserving discontinuous Galerkin methods with subcell convex limiting

Will Pazner. Sparse invariant domain preserving discontinuous Galerkin methods with subcell convex limiting. Computer Methods in Applied Mechanics and Engineering, 382:113876, 2021

2021
[13]

DISPATCH methods: An approximate, entropy-based Riemann solver for ideal magnetohydrodynamics

Andrius Popovas. DISPATCH methods: An approximate, entropy-based Riemann solver for ideal magnetohydrodynamics. Astronomy Astrophysics, 698:A69, June 2025

2025
[14]

DifferentialEquations.jl – A Performant and Feature- Rich Ecosystem for Solving Differential Equations in Julia

Christopher Rackauckas and Qing Nie. DifferentialEquations.jl – A Performant and Feature- Rich Ecosystem for Solving Differential Equations in Julia. The Journal of Open Research Software, 5(1), 2017. Exported from https://app.dimensions.ai on 2019/05/05

2017
[15]

Adaptive numerical simulations with Trixi.jl: A case study of Julia for scientific computing

Hendrik Ranocha, Michael Schlottke-Lakemper, Andrew Ross Winters, Erik Faulhaber, Jesse Chan, and Gregor Gassner. Adaptive numerical simulations with Trixi.jl: A case study of Julia for scientific computing. Proceedings of the JuliaCon Conferences, 1(1):77, 2022

2022
[16]

Mono- lithic convex limiting for Legendre-Gauss-Lobatto discontinuous Galerkin spectral-element methods

Andr´ es M Rueda-Ram ´ ırez, Benjamin Bolm, Dmitri Kuzmin, and Gregor J Gassner. Mono- lithic convex limiting for Legendre-Gauss-Lobatto discontinuous Galerkin spectral-element methods. Communications on Applied Mathematics and Computation, 6(3):1860–1898, 2024

2024
[17]

Trixi.jl: Adaptive high-order numerical simulations of hyperbolic PDEs in Julia

Michael Schlottke-Lakemper, Gregor J Gassner, Hendrik Ranocha, Andrew R Winters, Jesse Chan, and Andr´ es Rueda-Ram ´ ırez. Trixi.jl: Adaptive high-order numerical simulations of hyperbolic PDEs in Julia. https://github.com/trixi-framework/Trixi.jl, 2025

2025
[18]

Entropy stable boundary conditions for the Euler equations

Magnus Sv¨ ard. Entropy stable boundary conditions for the Euler equations. Journal of Com- putational Physics, 426:109947, 2021

2021
[19]

Entropy-stable in-and outflow boundary conditions for the compressible Navier-Stokes equations

Magnus Sv¨ ard and Anita Gjesteland. Entropy-stable in-and outflow boundary conditions for the compressible Navier-Stokes equations. arXiv preprint arXiv:2506.21065, 2025

2025 arXiv
[20]

Local subcell monolithic DG/FV convex property preserving scheme on un- structured grids and entropy consideration

Fran¸ cois Vilar. Local subcell monolithic DG/FV convex property preserving scheme on un- structured grids and entropy consideration. Journal of Computational Physics, 521:113535, 2025

2025

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [11]

A positivity preserving strategy for entropy stable discontinuous Galerkin discretizations of the compressible Euler and Navier-Stokes equa- tions

Yimin Lin, Jesse Chan, and Ignacio Tomas. A positivity preserving strategy for entropy stable discontinuous Galerkin discretizations of the compressible Euler and Navier-Stokes equa- tions. Journal of Computational Physics, 475:111850, 2023

work page 2023

[2] [4]

Entropy stable nodal discontinuous Galerkin methods via quadratic knapsack limiting

Brian Christner and Jesse Chan. Entropy stable nodal discontinuous Galerkin methods via quadratic knapsack limiting. arXiv preprint arXiv:2507.14488, 2025

arXiv 2025

[3] [1]

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

Jesse Chan. An artificial viscosity approach to high order entropy stable discontinuous Galerkin methods. arXiv preprint arXiv:2501.16529, 2025

arXiv 2025

[4] [2]

Rueda-Ram ´ ırez, Gregor Gassner, and Tim War- burton

Jesse Chan, Hendrik Ranocha, Andr´ es M. Rueda-Ram ´ ırez, Gregor Gassner, and Tim War- burton. On the Entropy Projection and the Robustness of High Order Entropy Stable Discontinuous Galerkin Schemes for Under-Resolved Flows. Frontiers in Physics, Volume 10 - 2022, 2022

work page 2022

[5] [3]

Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws

Tianheng Chen and Chi-Wang Shu. Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws. Journal of Computational Physics, 345:427–461, 2017

2017

[6] [5]

High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains

Travis C Fisher and Mark H Carpenter. High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains. Journal of Computational Physics, 252:518– 557, 2013

work page 2013

[7] [6]

Gassner, Andrew R

Gregor J. Gassner, Andrew R. Winters, and David A. Kopriva. Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations. Journal of Computational Physics, 327:39–66, 2016

work page 2016

[8] [7]

Numerical approximation of hyperbolic systems of conservation laws, volume 118

Edwige Godlewski and Pierre-Arnaud Raviart. Numerical approximation of hyperbolic systems of conservation laws, volume 118. Springer Science & Business Media, 2013. 20 B. CHRISTNER AND J. CHAN (a) n “ 500 nodes (b) n “ 1500 nodes, ECA V-FD (c) n “ 1500 nodes, KL-FD (d) n “ 500 nodes (e) n “ 1500 nodes, RECA V-FD (f) n “ 1500 nodes, RKL-FD Fig. 7: Density ...

work page 2013