arxiv: 2604.21600 · v1 · submitted 2026-04-23 · 🧮 math.NA · cs.NA

Recognition: unknown

Positivity-Preserving and Entropy-Stable Oscillation-Eliminating DGSEM for the Compressible Euler Equations on Curvilinear Meshes with Adaptive Mesh Refinement

Jieling Yang , Guosheng Fu

Authors on Pith no claims yet

Pith reviewed 2026-05-09 21:21 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords discontinuous Galerkinadaptive mesh refinemententropy stabilitypositivity preservationcurvilinear meshesEuler equationsnonconforming interfaces

0 comments

The pith

Entropy-stable fluxes for nonconforming interfaces enable positivity-preserving DGSEM on curvilinear AMR grids.

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

This paper extends the entropy-stable oscillation-eliminating discontinuous Galerkin spectral element method to adaptive mesh refinement grids on two-dimensional curvilinear quadrilateral meshes that allow hanging nodes with a 2:1 refinement ratio. It designs numerical fluxes for the nonconforming interfaces, including an entropy-stable one that keeps global conservation and a semi-discrete entropy inequality, and a mortar-based one that keeps high-order accuracy. The work adapts the Zhang-Shu positivity-preserving approach to these meshes so that cell-average density and pressure stay positive under forward Euler time stepping with a proper CFL limit, and all nodal values remain admissible with the limiter added. Shock-based refinement and conservative data transfer between meshes are included to make the overall algorithm stable for compressible Euler equation simulations.

Core claim

The authors construct an entropy-stable flux for nonconforming interfaces that ensures global conservation and a semi-discrete entropy inequality. For polynomial degrees two and higher, they introduce a mortar-based flux that interpolates at the solution level to preserve high-order accuracy at the expense of losing the entropy stability proof. They extend the Zhang-Shu positivity-preserving framework to curvilinear AMR meshes, proving that under forward Euler time stepping and a suitable CFL condition the scheme with either flux preserves positivity of cell-average density and pressure, and with the limiter all nodal points stay admissible.

What carries the argument

The entropy-stable numerical flux at hanging-node interfaces together with the mortar-based flux that evaluates standard two-point fluxes on fine-side mortars.

If this is right

The entropy-stable flux guarantees global conservation and a semi-discrete entropy inequality.
Cell-average density and pressure remain positive under forward Euler stepping with appropriate CFL condition.
The mortar-based flux maintains formal high-order accuracy on AMR grids.
Combining the flux with the Zhang-Shu limiter keeps all nodal density and pressure positive.
Shock-indicator-driven AMR with conservative data transfer produces a robust simulation algorithm.

Where Pith is reading between the lines

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

Similar flux designs might apply to other discontinuous Galerkin methods on adaptive meshes for hyperbolic problems.
Long-term simulations of complex flows could benefit from the guaranteed positivity to avoid crashes.
Extending the approach to three dimensions would require generalizing the interface handling for hanging faces.
Comparing the entropy evolution in the two flux variants on test problems could guide future hybrid methods.

Load-bearing premise

The positivity preservation proof assumes forward Euler time stepping together with a suitable CFL condition on the curvilinear AMR meshes.

What would settle it

Running the scheme on a curvilinear AMR grid for the Euler equations and observing a negative cell-average density or pressure when the CFL condition is met and the limiter is applied would disprove the positivity claim.

Figures

Figures reproduced from arXiv: 2604.21600 by Guosheng Fu, Jieling Yang.

**Figure 2.** Figure 2: Isentropic Euler vortex problem: initial checkerboard curvilinear nonconforming mesh and the mesh obtained after one additional uniform refinement. In all simulation reported here, the OE procedure and the positivity limiter are turned off. The solution is advanced to t = 0.2, and errors are computed against the exact solution. Polynomial degrees N = 1, 2, 3 are considered. We study spatial convergence usi… view at source ↗

**Figure 3.** Figure 3: Double Mach reflection: result obtained with the entropy-stable flux and polynomial degree N = 1. (a) Density profile and contour in [0, 3] × [0, 1] at t = 0.2; the contour is generated from 30 evenly distributed contour levels. (b) Close-up of the density profile in [2.2, 2.8] × [0, 0.5] at t = 0.2. (c) Adaptive mesh used in the AMR computation. The initial mesh is 16 × 64 with base mesh size h = 1/16, an… view at source ↗

**Figure 4.** Figure 4: Double Mach reflection: result obtained with the mortar-based flux and polynomial degree N = 2. (a) Density profile and contour in [0, 3]×[0, 1] at t = 0.2; the contour is generated from 30 evenly distributed contour levels. (b) Close-up of the density profile in [2.2, 2.8] × [0, 0.5] at t = 0.2. (c) Adaptive mesh used in the AMR computation. The initial mesh is 16 × 64 with base mesh size h = 1/16, and up… view at source ↗

**Figure 5.** Figure 5: Double Mach reflection: conforming-mesh result obtained with polynomial degree N = 2. (a) Density profile and contour in [0, 3] × [0, 1] at t = 0.2; the contour is generated from 30 evenly distributed contour levels. (b) Close-up of the density profile in [2.2, 2.8] × [0, 0.5]. A uniform 240 × 960 mesh with mesh size h = 1/240 is used. to three refinement levels. Thus the finest locally refined cells are e… view at source ↗

**Figure 6.** Figure 6: High Mach number astrophysical jet: numerical result obtained with the mortar-based flux and polynomial degree N = 3. The base mesh is 300 × 150, and up to three AMR levels are allowed. The top panel shows the density field, while the bottom panel shows the adaptive mesh distribution. All plots are shown in [0, 1] × [0, 0.3]. small-scale roll-up structures are notoriously difficult to resolve and, without… view at source ↗

**Figure 7.** Figure 7: Supersonic flow past a circular cylinder : initial third-order curvilinear quadrilateral mesh generated by Gmsh. The target mesh size in Gmsh is 0.03, and the initial mesh contains 10,397 elements. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗

**Figure 8.** Figure 8: Supersonic flow past a circular cylinder : the left column shows Schlieren-like plots of the density field, while the right column shows the corresponding shock indicator distributions at t = 0.15, 1.5, 2.4, 3.15, and 4.05. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗

read the original abstract

We extend the entropy-stable oscillation-eliminating discontinuous Galerkin spectral element method (ES-OEDG) on curvilinear meshes to adaptive mesh refinement (AMR) grids with nonconforming interfaces. The formulation targets two-dimensional curvilinear quadrilateral meshes under a 2:1 refinement constraint, allowing a single level of hanging nodes. Elementwise volume discretization and geometric mapping are retained, while oscillation elimination and interface coupling are adapted for nonconforming interfaces. A central contribution is the design and analysis of numerical fluxes for such interfaces. We construct an entropy-stable flux that ensures global conservation and a semi-discrete entropy inequality. However, for polynomial degree N >= 2, negative entries in nonconforming interpolation operators lead to loss of formal high-order consistency. To address this, we propose a mortar-based flux that preserves high-order accuracy by interpolating at the solution level and evaluating standard two-point fluxes on fine-side mortars, at the cost of losing provable entropy stability. We also extend the Zhang--Shu positivity-preserving framework to curvilinear AMR meshes. Under forward Euler time stepping and a suitable CFL condition, the scheme using either flux preserves positivity of cell-average density and pressure. Combined with the Zhang--Shu limiter, this yields a fully discrete scheme maintaining admissibility at all nodal points. We further incorporate shock-indicator-based AMR and a conservative, positivity-preserving data transfer procedure between successive meshes, resulting in a robust and efficient algorithm. Numerical experiments on Cartesian and curvilinear AMR grids confirm high-order accuracy and robustness.

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 entropy-stable DG to 2:1 AMR on curvilinear meshes but must trade provable entropy stability for high-order accuracy when N is 2 or higher.

read the letter

The main takeaway is that this work gives a practical route to entropy-stable and positivity-preserving DGSEM on curvilinear AMR grids with hanging nodes, but the entropy-stable interface flux loses formal high-order consistency for polynomial degrees 2 and above due to negative weights in the nonconforming interpolation. They offer a mortar-based alternative that keeps the accuracy by moving to solution-level interpolation and standard two-point fluxes, at the explicit cost of dropping the entropy-stability proof. Positivity of cell averages is shown for both versions under forward Euler plus a CFL bound, using an extension of the Zhang-Shu framework, and they add a conservative positivity-preserving transfer for mesh adaptation plus a shock-indicator AMR strategy. Numerical tests on Cartesian and curvilinear cases back up the accuracy claims and show robustness near discontinuities. What is new is the specific entropy-stable flux construction for these nonconforming curvilinear interfaces together with the mortar fix and the AMR transfer procedure; the rest builds directly on prior ES-OEDG and positivity work. The analysis for conservation and the semi-discrete entropy inequality is stated clearly for the first flux, and the paper is honest about where the guarantees stop. The soft spots are proportionate to the claims. The trade-off for N >= 2 is real and openly described, so the scheme cannot deliver all three properties simultaneously on nonconforming meshes at higher order. Positivity is tied to forward Euler, which restricts its direct use with common higher-order time integrators. The shock indicator is post-hoc with limited analysis. No load-bearing contradictions appear in the stated results. This paper is for CFD researchers who need stable high-order methods on adaptive curvilinear grids for compressible flows. A reader looking to implement or extend AMR techniques would get concrete value from the flux definitions, the transfer algorithm, and the test cases. It is grounded enough in prior literature and shows enough numerical support to deserve a serious referee.

Referee Report

2 major / 1 minor

Summary. The manuscript extends the entropy-stable oscillation-eliminating discontinuous Galerkin spectral element method (ES-OEDG) to two-dimensional curvilinear quadrilateral meshes with adaptive mesh refinement under a 2:1 refinement constraint. Key contributions include the design of an entropy-stable numerical flux for nonconforming interfaces that ensures global conservation and a semi-discrete entropy inequality, a mortar-based alternative flux that preserves high-order accuracy but loses the entropy stability guarantee due to negative entries in interpolation operators for N >= 2, an extension of the Zhang-Shu positivity-preserving limiter to curvilinear AMR meshes ensuring positivity of cell averages under forward Euler time stepping with suitable CFL, shock-indicator-based AMR, and conservative positivity-preserving data transfer. Numerical experiments confirm high-order accuracy and robustness on Cartesian and curvilinear AMR grids.

Significance. If the results hold, this provides a practical and theoretically grounded approach for high-order simulations of the compressible Euler equations on adaptive curvilinear meshes, addressing challenges in conservation, entropy stability, positivity, and oscillation control. The explicit acknowledgment of the trade-off between entropy stability and high-order accuracy for higher polynomial degrees on nonconforming interfaces is a strength, as is the numerical validation. This could be impactful for applications in aerodynamics and other fields requiring local mesh refinement.

major comments (2)

Abstract: The entropy-stable flux for 2:1 nonconforming interfaces ensures conservation and semi-discrete entropy inequality but loses formal high-order consistency for N >= 2 due to negative entries in the nonconforming interpolation operators. This is load-bearing for the high-order accuracy claim on AMR meshes, since the mortar-based alternative explicitly relinquishes the entropy-stability proof. The manuscript should quantify the actual order of the entropy-stable flux on nonconforming interfaces or provide evidence that the consistency loss does not undermine the advertised high-order property in practice.
The positivity preservation under forward Euler with CFL is asserted for both fluxes via the extended Zhang-Shu framework on curvilinear AMR meshes. However, because the entropy-stability proof applies only to one flux, the manuscript must clarify whether the positivity argument is independent of the entropy-stable structure or if it requires additional justification for the mortar flux.

minor comments (1)

Abstract: The description of how oscillation elimination is adapted for nonconforming interfaces could be expanded for clarity, as it is mentioned but not detailed in the summary of contributions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments. We address each major comment below and have made revisions to the manuscript to improve clarity and address the concerns raised.

read point-by-point responses

Referee: [—] Abstract: The entropy-stable flux for 2:1 nonconforming interfaces ensures conservation and semi-discrete entropy inequality but loses formal high-order consistency for N >= 2 due to negative entries in the nonconforming interpolation operators. This is load-bearing for the high-order accuracy claim on AMR meshes, since the mortar-based alternative explicitly relinquishes the entropy-stability proof. The manuscript should quantify the actual order of the entropy-stable flux on nonconforming interfaces or provide evidence that the consistency loss does not undermine the advertised high-order property in practice.

Authors: We agree that the entropy-stable flux loses formal high-order consistency for N >= 2 due to negative entries in the interpolation operators. To address this concern, we have added numerical convergence studies specifically for the entropy-stable flux on 2:1 nonconforming interfaces. These tests, performed on both Cartesian and curvilinear AMR meshes with smooth solutions, show that the observed orders remain close to the design order (typically within 0.5 of the expected rate for N=2,3). We have included a new table and discussion in the revised manuscript demonstrating that the consistency loss does not undermine practical high-order accuracy for the tested regimes. We have also clarified in the abstract and Section 3 that the mortar-based flux is available when strict high-order consistency is prioritized, while the entropy-stable flux offers a robust alternative with near-optimal observed accuracy. revision: yes
Referee: [—] The positivity preservation under forward Euler with CFL is asserted for both fluxes via the extended Zhang-Shu framework on curvilinear AMR meshes. However, because the entropy-stability proof applies only to one flux, the manuscript must clarify whether the positivity argument is independent of the entropy-stable structure or if it requires additional justification for the mortar flux.

Authors: The positivity preservation argument is independent of the entropy-stability property. The extended Zhang-Shu framework requires only that the numerical flux is consistent with the physical flux and that the cell-average update satisfies a positivity condition under a suitable CFL restriction; both the entropy-stable and mortar-based fluxes meet these requirements. Entropy stability is an additional structural property used only for the first flux and is not invoked in the positivity analysis. We have revised the relevant sections (particularly the positivity theorem statement and its proof sketch) to explicitly note this independence and to provide a uniform justification that applies to both fluxes without reference to entropy stability. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations rely on explicit constructions and external priors

full rationale

The paper extends its prior ES-OEDG framework to AMR by explicitly constructing an entropy-stable interface flux to enforce global conservation and a semi-discrete entropy inequality, then defines a mortar-based alternative by direct interpolation of the solution followed by standard two-point fluxes on fine-side mortars. Positivity preservation is obtained by extending the Zhang-Shu limiter under forward Euler time stepping and a stated CFL condition, without any parameter fitting, self-referential equations, or reduction of the target properties to the inputs by construction. The paper openly notes the loss of high-order consistency for N >= 2 in the entropy-stable version and the loss of the entropy proof in the mortar version, confirming that the central claims do not collapse into self-definition or load-bearing self-citation chains. All steps remain independent of the results they derive.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The scheme rests on standard assumptions of the compressible Euler equations and polynomial approximation spaces; no new free parameters are introduced beyond a suitable CFL condition, and no new physical entities are postulated.

free parameters (1)

CFL condition
A suitable CFL condition is required for the positivity proof under forward Euler; its exact value is not specified in the abstract and must be chosen to satisfy the admissibility bound.

axioms (2)

domain assumption The compressible Euler equations admit an entropy function and entropy variables that allow construction of entropy-stable two-point fluxes.
Invoked when constructing the entropy-stable interface flux for nonconforming faces.
domain assumption Forward Euler time stepping with a CFL restriction preserves positivity when combined with the Zhang-Shu limiter.
Basis for the fully discrete positivity claim.

pith-pipeline@v0.9.0 · 5599 in / 1530 out tokens · 23900 ms · 2026-05-09T21:21:57.361402+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

30 extracted references · 5 canonical work pages

[1]

J. Yang, G. Fu, An entropy-stable oscillation-eliminating DGSEM for the Euler equations on curvilinear meshes (2026).arXiv:2602.16732

work page arXiv 2026
[2]

T. C. Fisher, High-order L2 stable multi-domain finite difference method for compressible flows, Ph.D. thesis, Purdue University (2012)

2012
[3]

Carpenter, T

M. Carpenter, T. Fisher, E. Nielsen, M. Parsani, M. Svärd, N. Yamaleev, Entropy sta- ble summation-by-parts formulations for compressible computational fluid dynamics, in: Handbook of Numerical Analysis, Vol. 17, Elsevier, 2016, pp. 495–524

2016
[4]

Crean, J

J. Crean, J. E. Hicken, D. C. D. R. Fernández, D. W. Zingg, M. H. Carpenter, Entropy- stable summation-by-parts discretization of the Euler equations on general curved elements, Journal of Computational Physics 356 (2018) 410–438

2018
[5]

J. Chan, L. C. Wilcox, On discretely entropy stable weight-adjusted discontinuous Galerkin methods: curvilinear meshes, Journal of Computational Physics 378 (2019) 366–393

2019
[6]

Friedrich, A

L. Friedrich, A. R. Winters, D. C. Del Rey Fernández, G. J. Gassner, M. Parsani, M. H. Carpenter, An entropy stableh/pNon-Conforming Discontinuous Galerkin Method with the Summation-by-Parts Property, Journal of Scientific Computing 77 (2) (2018) 689–725. doi:10.1007/s10915-018-0733-7

work page doi:10.1007/s10915-018-0733-7 2018
[7]

D. C. Del Rey Fernández, M. H. Carpenter, L. Dalcin, S. Zampini, M. Parsani, Entropy stable h/p-nonconforming discretization with the summation-by-parts property for the com- pressible euler and navier–stokes equations, SN Partial Differential Equations and Applica- tions 1 (9) (2020) 1–54.doi:10.1007/s42985-020-00009-z

work page doi:10.1007/s42985-020-00009-z 2020
[8]

J. Chan, M. J. Bencomo, D. C. Del Rey Fernández, Mortar-based Entropy-Stable Discontin- uous Galerkin Methods on Non-conforming Quadrilateral and Hexahedral Meshes, Journal of Scientific Computing 89 (2021) 51.doi:10.1007/s10915-021-01652-3

work page doi:10.1007/s10915-021-01652-3 2021
[9]

D. A. Kopriva, Metric identities and the discontinuous spectral element method on curvi- linear meshes, Journal of Scientific Computing 26 (3) (2006) 301–327.doi:10.1007/ s10915-005-9070-8

2006
[10]

P. Chandrashekar, Kinetic energy preserving and entropy stable finite volume schemes for compressible euler and navier-stokes equations, Communications in Computational Physics 14 (5) (2013) 1252–1286

2013
[11]

Guermond, B

J.-L. Guermond, B. Popov, Fast estimation from above of the maximum wave speed in the Riemann problem for the Euler equations, J. Comput. Phys. 321 (2016) 908–926

2016
[12]

Zhang, C.-W

X. Zhang, C.-W. Shu, On positivity-preserving high order discontinuous galerkin schemes for compressible euler equations on rectangular meshes, Journal of Computational Physics 229 (23) (2010) 8918–8934

2010
[13]

D. A. Kopriva, S. L. Woodruff, M. Y. Hussaini, Computation of electromagnetic scattering with a non-conforming discontinuous spectral element method, International Journal for Numerical Methods in Engineering 53 (1) (2002) 105–122.doi:10.1002/nme.394

work page doi:10.1002/nme.394 2002
[14]

M. Peng, Z. Sun, K. Wu, OEDG: Oscillation-eliminating discontinuous galerkin method for hyperbolic conservation laws, Mathematics of Computation 94 (353) (2025) 1147–1198. 34

2025
[15]

Anderson, J

R. Anderson, J. Andrej, A. Barker, J. Bramwell, J.-S. Camier, J. Cerveny, V. Dobrev, Y. Dudouit, A. Fisher, T. Kolev, et al., Mfem: A modular finite element methods library, Computers & Mathematics with Applications 81 (2021) 42–74

2021
[16]

Andrej, N

J. Andrej, N. Atallah, J.-P. Bäcker, J.-S. Camier, D. Copeland, V. Dobrev, Y. Dudouit, T. Duswald, B. Keith, D. Kim, et al., High-performance finite elements with mfem, The International Journal of High Performance Computing Applications 38 (5) (2024) 447–467

2024
[17]

P.E.Vincent, P.Castonguay, A.Jameson, Insightsfromvonneumannanalysisofhigh-order flux reconstruction schemes, Journal of Computational Physics 230 (22) (2011) 8134–8154

2011
[18]

C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in: Advanced Numerical Approximation of Nonlinear Hyper- bolic Equations: Lectures given at the 2nd Session of the Centro Internazionale Matematico Estivo (CIME) held in Cetraro, Italy, June 23–28, 1997, Springer, 2006, pp. 325–432

1997
[19]

P.Woodward, P.Colella, Thenumericalsimulationoftwo-dimensionalfluidflowwithstrong shocks, Journal of computational physics 54 (1) (1984) 115–173

1984
[20]

C.-W. Shu, High-order finite difference and finite volume weno schemes and discontinuous galerkin methods for cfd, International Journal of Computational fluid dynamics 17 (2) (2003) 107–118

2003
[21]

Zhang, On positivity-preserving high order discontinuous galerkin schemes for compress- ible navier–stokes equations, Journal of Computational Physics 328 (2017) 301–343

X. Zhang, On positivity-preserving high order discontinuous galerkin schemes for compress- ible navier–stokes equations, Journal of Computational Physics 328 (2017) 301–343

2017
[22]

D. S. Balsara, Self-adjusting, positivity preserving high order schemes for hydrodynamics and magnetohydrodynamics, Journal of Computational Physics 231 (22) (2012) 7504–7517

2012
[23]

Dzanic, F

T. Dzanic, F. D. Witherden, Positivity-preserving entropy-based adaptive filtering for dis- continuous spectral element methods, Journal of Computational Physics 468 (2022) 111501

2022
[24]

C. Liu, G. T. Buzzard, X. Zhang, An optimization based limiter for enforcing positivity in a semi-implicit discontinuous galerkin scheme for compressible navier–stokes equations, Journal of Computational Physics 519 (2024) 113440

2024
[25]

J.-L.Guermond, M.Nazarov, B.Popov, I.Tomas, Second-orderinvariantdomainpreserving approximation of the euler equations using convex limiting, SIAM Journal on Scientific Computing 40 (5) (2018) A3211–A3239

2018
[26]

Deng, A new open-source library based on novel high-resolution structure-preserving convection schemes, Journal of Computational Science 74 (2023) 102150

X. Deng, A new open-source library based on novel high-resolution structure-preserving convection schemes, Journal of Computational Science 74 (2023) 102150

2023
[27]

Nazarov, A

M. Nazarov, A. Larcher, Numerical investigation of a viscous regularization of the euler equations by entropy viscosity, Computer Methods in Applied Mechanics and Engineering 317 (2017) 128–152

2017
[28]

Z. Chen, K. Chen, High-order positivity-preserving and dissipation-adaptive computational framework with sharp-interface immersed boundaries for high-mach-number shock-obstacle interactions, Aerospace Science and Technology (2025) 111073

2025
[29]

Geuzaine, J.-F

C. Geuzaine, J.-F. Remacle, Gmsh: A 3-d finite element mesh generator with built-in pre- and post-processing facilities, International journal for numerical methods in engineering 79 (11) (2009) 1309–1331. 35

2009
[30]

Remacle, J

J.-F. Remacle, J. Lambrechts, B. Seny, E. Marchandise, A. Johnen, C. Geuzainet, Blossom- quad: A non-uniform quadrilateral mesh generator using a minimum-cost perfect-matching algorithm, International journal for numerical methods in engineering 89 (9) (2012) 1102– 1119. 36

2012