Pith. sign in

REVIEW 6 minor 131 references

Entropy-conservative SBP schemes converge to smooth solutions at the operators' accuracy order.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · grok-4.5

2026-07-30 12:41 UTC pith:ZOJCTWIU

load-bearing objection Solid relative-entropy convergence proof that actually covers Euler/shallow water and general diagonal-norm SBP; the rate is honest and the soft spots are already labeled.

arxiv 2607.27049 v1 pith:ZOJCTWIU submitted 2026-07-29 math.NA cs.NA

Convergence of entropy-conservative summation-by-parts discretizations to smooth solutions of hyperbolic conservation laws

Hendrik Ranocha This is my paper
classification math.NA cs.NA MSC 65M1265M0665M6065M7065M2035L65
keywords summation-by-parts operatorsentropy-conservative methodsrelative entropyhyperbolic conservation lawsconvergence analysisflux differencingsmooth solutions
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

High-order entropy-conservative discretizations are prized for robustness, yet almost no general proof said they actually converge. This paper supplies one: for smooth solutions of periodic hyperbolic conservation laws with a strictly convex entropy, the discrete relative entropy between the numerical solution and the exact nodal samples forces the error to shrink like h to the power p, where p is the accuracy order of the underlying diagonal-norm summation-by-parts operators. The argument covers general fluxes (including shallow water and compressible Euler), space-time sources, and a broad class of operators on curved meshes—finite differences, continuous and discontinuous Galerkin. The predicted rates are sharp for plain periodic finite differences; some special methods beat them in practice. A sympathetic reader cares because the same entropy structure that makes the schemes stable is shown to be enough, with consistency, to guarantee high-order convergence for smooth flows.

Core claim

Under stated assumptions, entropy-conservative flux-differencing semidiscretizations based on diagonal-norm SBP operators of order p admit a unique solution that stays admissible and satisfies an M-norm error bound of order h^p against any smooth solution on a finite time interval. The rate is exactly the pointwise accuracy order of the assembled operators; no homogeneity or global second-derivative bounds on the fluxes are required.

What carries the argument

The discrete relative entropy E_h—the quadrature of U(u^h)−U(u)−w(u)·(u^h−u)—is equivalent to the squared M-norm error. Entropy conservation cancels every term linear in the error; only quadratic flux remainders, truncation error, and a quadratic source remainder survive, and Gronwall then yields the rate.

Load-bearing premise

The proof needs the numerical solution to stay inside a fixed compact set of admissible states away from vacuum, which the paper obtains from an inverse estimate that requires the accuracy order to exceed half the space dimension.

What would settle it

Run the entropy-conservative scheme on a smooth manufactured solution (for example the 2-D Euler density wave or Burgers) with a family of diagonal-norm SBP operators whose assembled accuracy order is known; if the measured M-norm error fails to decay at least like h^p once the mesh is fine enough that states stay admissible, the central claim is false.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Compressible Euler and shallow-water entropy-conservative SBP schemes on periodic domains converge at order p for smooth solutions without positivity limiters on fine enough meshes.
  • The same a-priori rate applies to finite-difference, DGSEM, CGSEM, and multidimensional simplex SBP operators, including smoothly curved periodic grids once discrete metric identities hold.
  • State-independent manufactured sources are covered automatically, so method-of-manufactured-solutions tests inherit the theorem.
  • Observed superconvergence for even-degree DG or multi-block FD is outside the general bound and needs method-specific arguments.

Where Pith is reading between the lines

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

  • Removing the order-versus-dimension condition unconditionally would likely need a corrector expansion or an independent maximum principle, both left open.
  • The same relative-entropy balance should extend to entropy-dissipative interface terms once the dissipation on smooth data is controlled, giving a route to shock-capturing variants.
  • Bounded-domain entropy-stable boundary closures will add boundary residuals to E_h; whether those residuals still permit the full rate p is the natural next obstruction.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 6 minor

Summary. The paper proves that entropy-conservative flux-differencing semidiscretizations based on diagonal-norm SBP operators converge to smooth solutions of periodic hyperbolic balance laws at the accuracy order p of the assembled operators. The argument uses a discrete relative entropy E_h, establishes coercivity, exact discrete entropy balance, O(h^p) truncation, exact cancellation of all linear flux-error terms via Tadmor’s condition and its derivatives, quadratic control of flux remainders under a fixed-local-stencil hypothesis, a source remainder that is quadratic without entropy structure on the source, and a Gronwall-plus-L^∞ bootstrap. The framework covers general strictly convex entropies (including shallow water and compressible Euler), state-independent sources, and a broad class of operators (periodic FD, DGSEM, CGSEM, triangular SBP, and encapsulated curvilinear operators). Numerical experiments confirm the predicted rates and document known superconvergence for even-degree DG and multi-block FD.

Significance. This is a substantial and carefully scoped extension of Worku–Del Rey Fernández–Zingg: it removes the homogeneity/global-second-derivative restriction, admits general symmetrizable systems and manufactured sources, and works in an abstract diagonal-norm SBP setting that includes FD, CG, DG, and curved meshes. To the best of my knowledge it is the first a priori O(h^p) estimate for entropy-conservative high-order SBP flux differencing of nonlinear systems obtained by a discrete relative-entropy argument. Strengths include a complete, modular proof chain (Lemmas 4.1–4.8, Theorem 3.1), explicit isolation of the technical order condition p>d/2 (Props. 4.9–4.10), verification for Euler/shallow water with admissible compact sets away from vacuum, a detailed Appendix B on metric identities and encapsulated operators, and a public reproducibility repository. The limitations (smooth solutions, periodic BCs, entropy-conservative collocation only) are stated clearly and do not undermine the central claim as written.

minor comments (6)
  1. [Abstract] Abstract and several places in the extracted text contain run-together words (e.g. “sharpingeneral”, “An optimalanalysisisexpected…”). Please proofread the camera-ready abstract and introduction for spacing/hyphenation artifacts.
  2. [Section 6.1–6.2] In §6.1 the distinction between the guaranteed rate p (closure-limited for multi-block FD; p=k for element-based methods) and the higher observed EOCs is clear in the body but could be flagged once more when Tables 2 and 6 are introduced, so readers do not misread the tables as contradicting Theorem 3.1.
  3. [Section 3, Theorem 3.1] Assumption A8 / §4.8: the discussion is already excellent. A single forward pointer in the statement of Theorem 3.1 (“the condition p>d/2 is used only in the L^∞ bootstrap; see §4.8”) would help readers who jump to the theorem.
  4. [Section 5.4 / Appendix A] Corollary 5.7 and Appendix A: the explicit Hessian bounds are useful; a brief remark that the displayed c_U is not sharp near vacuum (already in Remark A.1) could also be echoed in the corollary statement so users do not treat (A.10) as sharp constants for mesh design.
  5. [Section 5.3] Notation: the dual use of H for water height and the occasional collision with mesh size h is handled by a footnote, but writing water height as η or h_w in displays (5.5)–(5.10) would reduce cognitive load.
  6. [Section 1 / Remark 5.3] References: the Worku et al. baseline is cited appropriately; if space permits, a one-sentence comparison of the p>1+d/2 requirement in that work versus the weaker p>d/2 (or p≥1 under global bounds) here would make the improvement fully explicit for readers of both papers.

Circularity Check

0 steps flagged

No significant circularity: convergence rate is derived from SBP structure, entropy conservation, and relative entropy, not fitted or assumed as the conclusion.

full rationale

Theorem 3.1 is proved in Section 4 from stated Assumptions A1–A8 via a self-contained chain: discrete relative-entropy coercivity (Lemma 4.1), exact entropy balance from Tadmor fluxes and skew-symmetric Q (Lemma 4.2), truncation of order p from operator accuracy (Lemma 4.3), exact cancellation of linear error terms by differentiating the entropy-conservation identity (Lemmas 4.5–4.6), quadratic remainder control under fixed local stencils (Lemma 4.8), source bound, then Gronwall plus L∞ bootstrap. None of these steps redefine the target rate as an input, fit constants to observed EOCs, or import a uniqueness theorem that secretly is the claim. Worku–Del Rey Fernández–Zingg is cited as the restricted baseline being extended (homogeneous fluxes, special CG-type operators), not used as an unexamined lemma that is the conclusion. Self-citations (fluxes, software, prior SBP work) supply concrete applications and operators, not the load-bearing convergence argument. Numerics in Section 6 corroborate sharpness and known superconvergence patterns; they do not calibrate the theorem. Score 0 is appropriate.

Axiom & Free-Parameter Ledger

0 free parameters · 9 axioms · 2 invented entities

Load-bearing content is a theorem under explicit assumptions A1–A8 plus standard SBP/entropy calculus. No parameters are fitted to obtain the rate. Invented objects are definitional (discrete relative entropy, structural remainder bound) rather than physical entities. Domain restrictions (smooth solutions, periodic BC, entropy-conservative collocation, diagonal norm, state-independent collocated sources) are declared up front and limit scope without circularly forcing the rate.

axioms (9)
  • domain assumption A1: strictly convex C^2 entropy pair compatible with the fluxes (symmetrizable hyperbolic system).
    Standard continuum structure (Godunov–Friedrichs–Lax–Mock); required for relative entropy and Tadmor fluxes.
  • domain assumption A2: classical C^{p+1} solution valued in a compact convex K⊂𝒜 with uniform Hessian bounds c_U I ≤ U'' ≤ C_U I (and U∈C^3 if sources present).
    Relative-entropy a priori theory is inherently smooth-solution theory; vacuum margins enter Euler/SWE constants.
  • domain assumption A3: family of diagonal-norm periodic SBP operators of order p with quasi-uniform masses ~ h^d.
    Defines the discretization class; excludes non-diagonal mass and N-dependent spectral p.
  • domain assumption A4: symmetric, consistent, C^{p+1} Tadmor entropy-conservative two-point fluxes.
    Enables discrete entropy equality and linear-error cancellation (Lemma 4.6).
  • ad hoc to paper A5/A5*: structural remainder bound, implied by fixed local stencil (locality, bounded nnz, |Q_ij|≲h^{d-1}).
    Only non-standard structural hypothesis beyond SBP+EC; essential so |w_i−w_j|=O(h) cancels mass/row-sum scaling (Lemma 4.8). Dense spectral operators fail it.
  • domain assumption A6–A7: continuous bounded state-independent collocated sources; consistent initial data with E_h(0)^{1/2}≤C_0 h^p.
    Exact collocation makes source contribution quadratic in the error (Lemma 4.7).
  • ad hoc to paper A8: p>d/2 for the L^∞ bootstrap into K.
    Technical inverse-estimate condition; paper supplies global/conditional escapes (Props. 4.9–4.10).
  • domain assumption Periodic boundary conditions only; no entropy-stable boundary closures in the theorem.
    Stated scope limit (§1.1); boundaries need case-by-case entropy estimates.
  • standard math Standard calculus facts: Taylor with integral remainder, discrete Gronwall, Picard–Lindelöf on R^N.
    Used throughout §4 without novelty claims.
invented entities (2)
  • Discrete relative entropy E_h (2.10) with mass-matrix quadrature independent evidence
    purpose: Lyapunov/error functional equivalent to ||e||_M^2 via entropy convexity
    Discrete analogue of Dafermos–DiPerna relative entropy; definitional tool, not a new physical law.
  • Structural remainder bound Assumption A5 no independent evidence
    purpose: Close the estimate on quadratic flux Taylor remainders contracted with (w_i−w_j) and Q
    Abstract hypothesis proved from fixed local stencil; could fail for non-local operators.

pith-pipeline@v1.2.0-daily-grok45 · 53111 in / 3707 out tokens · 67579 ms · 2026-07-30T12:41:38.080645+00:00 · methodology

0 comments
read the original abstract

Although entropy-based summation-by-parts (SBP) discretizations of hyperbolic conservation laws are widely used for their robustness and stability properties, there are very few results on their convergence. We extend a recent convergence analysis of Worku, Del Rey Fern\'andez, and Zingg (2026, DOI: 10.48550/arXiv.2603.18369) in two ways. First, instead of allowing only hyperbolic conservation laws whose fluxes are homogeneous and have globally bounded second derivatives (a restriction essentially to linear or quadratic fluxes), we consider general hyperbolic systems with strictly convex entropy and source terms depending on time and space. Second, instead of requiring a special class of SBP operators, we consider a general framework of diagonal-norm SBP operators on curved meshes, including finite differences, continuous and discontinuous Galerkin methods. Since the error analysis is based on a discrete relative entropy, it is restricted to smooth solutions. To enable a unified treatment of conservation laws, we restrict the analysis to periodic boundary conditions. Numerical results demonstrate that the predicted convergence rates are sharp in general, but can be improved for special cases such as discontinuous Galerkin methods with even polynomial degree and multi-block finite difference methods. An optimal analysis is expected to require more sophisticated arguments specialized to the class of methods instead of the general framework of SBP operators used in this work.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

131 extracted references · 41 canonical work pages · 3 internal anchors

  1. [1]

    A general framework to construct schemes satisfying additional conservation relations.Applicationtoentropyconservativeandentropydissipativeschemes

    R. Abgrall. “A general framework to construct schemes satisfying additional conservation relations.Applicationtoentropyconservativeandentropydissipativeschemes.”In:Journal of Computational Physics372 (2018), pp. 640–666.doi:10.1016/j.jcp.2018.06.031. arXiv: 1711.10358 [math.NA]

    Pith/arXiv arXiv 2018

  2. [2]

    AnalysisoftheSBP-SATStabilization for Finite Element Methods Part I: Linear problems

    R.Abgrall,J.Nordström,P.Öffner,andS.Tokareva.“AnalysisoftheSBP-SATStabilization for Finite Element Methods Part I: Linear problems.” In:Journal of Scientific Computing85.2 (2020), pp. 1–29.doi:10.1007/s10915-020-01349-z. arXiv:1912.08108 [math.NA]

    Pith/arXiv arXiv 2020

  3. [3]

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

    R. Abgrall, P. Öffner, and H. Ranocha. “Reinterpretation and Extension of Entropy Correc- tion Terms for Residual Distribution and Discontinuous Galerkin Schemes: Application to Structure Preserving Discretization.” In:Journal of Computational Physics453 (Mar. 2022), p. 110955.doi:10.1016/j.jcp.2022.110955. arXiv:1908.04556 [math.NA]

    arXiv 2022

  4. [4]

    A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems

    S. Adjerid, K. D. Devine, J. E. Flaherty, and L. Krivodonova. “A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems.” In:Computer Methods in Applied Mechanics and Engineering191.11–12 (2002), pp. 1097–1112.doi:10.1016/S0045- 7825(01)00318-8

    doi:10.1016/s0045- 2002

  5. [5]

    Encapsulated high order difference operators on curvilinear non-conforming grids

    O. Ålund and J. Nordström. “Encapsulated high order difference operators on curvilinear non-conforming grids.” In:Journal of Computational Physics385 (2019), pp. 209–224.doi: 10.1016/j.jcp.2019.02.007

    work page doi:10.1016/j.jcp.2019.02.007 2019

  6. [6]

    M.Anandan,K.R.Arun,A.Krishnamurthy,andM.Lukáčová-Medvidová.Erroranalysisof an asymptotic-preserving, energy-stable finite volume method for barotropic Euler equations. Mar. 2026.doi:10.48550/arXiv.2603.27421. arXiv:2603.27421 [math.NA]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2603.27421 2026

  7. [7]

    Stability and Convergence of a Class of Finite Element Schemes for Hyperbolic Systems of Conservation Laws

    C. Arvanitis, C. Makridakis, and A. E. Tzavaras. “Stability and Convergence of a Class of Finite Element Schemes for Hyperbolic Systems of Conservation Laws.” In:SIAM Journal on Numerical Analysis42.4 (2004), pp. 1357–1393.doi:10.1137/S0036142902420436

    work page doi:10.1137/s0036142902420436 2004

  8. [8]

    An entropy-stable and kinetic energy-preserving macro-element HDG method for compressible flows

    V. Badrkhani, M. F. P. ten Eikelder, and D. Schillinger.An entropy-stable and kinetic energy- preserving macro-element HDG method for compressible flows. 2025.doi:10 . 48550 / arXiv . 2507.22195. arXiv:2507.22195 [cs.CE]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2507.22195 2025

  9. [9]

    Sta- bility of the Active Flux Method in the Framework of Summation-by-Parts Operators

    W. Barsukow, C. Klingenberg, L. Lechner, J. Nordström, S. Ortleb, and H. Ranocha. “Sta- bility of the Active Flux Method in the Framework of Summation-by-Parts Operators.” In: BIT Numerical Mathematics66 (July 2026), p. 49.doi:10.1007/s10543-026-01144-6. arXiv: 2507.11068 [math.NA]

    Pith/arXiv arXiv 2026

  10. [10]

    Numerical methods for gasdynamic systems on unstructured meshes

    T. J. Barth. “Numerical methods for gasdynamic systems on unstructured meshes.” In: An Introduction to Recent Developments in Theory and Numerics for Conservation Laws. Vol. 5. Lecture Notes in Computational Science and Engineering. Berlin, Heidelberg: Springer, 1999, pp. 195–285.doi:10.1007/978-3-642-58535-7_5

    work page doi:10.1007/978-3-642-58535-7_5 1999

  11. [11]

    Julia:AFreshApproachtoNumer- ical Computing

    J.Bezanson,A.Edelman,S.Karpinski,andV.B.Shah.“Julia:AFreshApproachtoNumer- ical Computing.” In:SIAM Review59.1 (2017), pp. 65–98.doi:10.1137/141000671. arXiv: 1411.1607 [cs.MS]

    Pith/arXiv arXiv 2017

  12. [12]

    Boyd and L

    S. Boyd and L. Vandenberghe.Convex Optimization. Cambridge: Cambridge University Press, 2004.doi:10.1017/CBO9780511804441

    doi:10.1017/cbo9780511804441 2004

  13. [13]

    Error Estimate for Time-Explicit Finite Volume Approximation of Strong Solutions to Systems of Conservation Laws

    C. Cancès, H. Mathis, and N. Seguin. “Error Estimate for Time-Explicit Finite Volume Approximation of Strong Solutions to Systems of Conservation Laws.” In:SIAM Journal on NumericalAnalysis54.2(2016),pp.1263–1287.doi:10.1137/15M1029886.arXiv:1602.02128 [math.NA]

    Pith/arXiv arXiv 2016

  14. [14]

    Superconvergence of discontinuous Galerkin methods for linear hyperbolic equations

    W. Cao, Z. Zhang, and Q. Zou. “Superconvergence of discontinuous Galerkin methods for linear hyperbolic equations.” In:SIAM Journal on Numerical Analysis52.5 (2014), pp. 2555– 2573.doi:10.1137/130946873. 36

    work page doi:10.1137/130946873 2014

  15. [15]

    Efficient Compressible Turbulent Flow Simulations: Entropy Projection and Correction for an ILES in a Discontinuous Galerkin solver

    E Carnevali, A Crivellini, L Alberti, and A. Colombo. “Efficient Compressible Turbulent Flow Simulations: Entropy Projection and Correction for an ILES in a Discontinuous Galerkin solver.” In:Journal of Physics: Conference Series. Vol. 3173. 1. IOP Publishing. 2026, p. 012034.doi:10.1088/1742-6596/3173/1/012034

    work page doi:10.1088/1742-6596/3173/1/012034 2026

  16. [16]

    Entropy Stable Spectral Collocation Schemes for the Navier-Stokes Equations: Discontinuous Interfaces

    M. H. Carpenter, T. C. Fisher, E. J. Nielsen, and S. H. Frankel. “Entropy Stable Spectral Collocation Schemes for the Navier-Stokes Equations: Discontinuous Interfaces.” In:SIAM Journal on Scientific Computing36.5 (2014), B835–B867.doi:10.1137/130932193

    doi:10.1137/130932193 2014

  17. [17]

    Time-Stable Boundary Conditions for Finite-Difference Schemes Solving Hyperbolic Systems: Methodology and Application to High-Order Compact Schemes

    M. H. Carpenter, D. Gottlieb, and S. Abarbanel. “Time-Stable Boundary Conditions for Finite-Difference Schemes Solving Hyperbolic Systems: Methodology and Application to High-Order Compact Schemes.” In:Journal of Computational Physics111.2 (1994), pp. 220– 236.doi:10.1006/jcph.1994.1057

    arXiv 1994

  18. [18]

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

    J. Chan. “An artificial viscosity approach to high order entropy stable discontinuous Galerkin methods.” In:Journal of Computational Physics543 (2025), p. 114380.doi:10.1016/ j.jcp.2025.114380

    arXiv 2025

  19. [20]

    J. Chan, D. Knapp, M. McCallum, H. Ranocha, C. G. Taylor, P. Baasch, D. Doehring, J. Markert,J.Lampert,T.Montoya,andV.X.Wang.StartUpDG.jl:Initializesandsetsupreference elements and physical meshes for DG.https://github.com/jlchan/StartUpDG.jl. 2026. doi:10.5281/zenodo.19906078

    work page doi:10.5281/zenodo.19906078 2026

  20. [21]

    Ontheentropy projectionandtherobustnessofhighorderentropystablediscontinuousGalerkinschemes for under-resolved flows

    J.Chan,H.Ranocha,A.M.Rueda-Ramírez,G.Gassner,andT.Warburton.“Ontheentropy projectionandtherobustnessofhighorderentropystablediscontinuousGalerkinschemes for under-resolved flows.” In:Frontiers in Physics10 (2022), p. 898028.doi:10.3389/fphy. 2022.898028

    arXiv 2022

  21. [22]

    Kinetic Energy Preserving and Entropy Stable Finite Volume Schemes forCompressibleEulerandNavier-StokesEquations

    P. Chandrashekar. “Kinetic Energy Preserving and Entropy Stable Finite Volume Schemes forCompressibleEulerandNavier-StokesEquations.”In:CommunicationsinComputational Physics14.5 (2013), pp. 1252–1286.doi:10.4208/cicp.170712.010313a

    arXiv 2013

  22. [23]

    Means Generated by an Integral

    H. Chen. “Means Generated by an Integral.” In:Mathematics Magazine78.5 (2005), pp. 397– 399.doi:10.2307/30044201

    work page doi:10.2307/30044201 2005

  23. [24]

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

    T. Chen and C.-W. Shu. “Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws.” In:Journal of Computational Physics345 (2017), pp. 427–461.doi:10.1016/j.jcp.2017.05.025

    doi:10.1016/j.jcp.2017.05.025 2017

  24. [25]

    Review of Entropy Stable Discontinuous Galerkin Methods for Systems of Conservation Laws on Unstructured Simplex Meshes

    T. Chen and C.-W. Shu. “Review of Entropy Stable Discontinuous Galerkin Methods for Systems of Conservation Laws on Unstructured Simplex Meshes.” In:CSIAM Transactions on Applied Mathematics1.1 (2020), pp. 1–52.doi:10.4208/csiam-am.2020-0003

    doi:10.4208/csiam-am.2020-0003 2020

  25. [26]

    Superconvergence of discontinuous Galerkin and local discon- tinuous Galerkin schemes for linear hyperbolic and convection-diffusion equations in one space dimension

    Y. Cheng and C.-W. Shu. “Superconvergence of discontinuous Galerkin and local discon- tinuous Galerkin schemes for linear hyperbolic and convection-diffusion equations in one space dimension.” In:SIAM Journal on Numerical Analysis47.6 (2010), pp. 4044–4072.doi: 10.1137/090747701

    work page doi:10.1137/090747701 2010

  26. [28]

    Berlin Heidelberg: Springer, 2016.doi:10

    C.M.Dafermos.HyperbolicConservationLawsinContinuumPhysics.4thed.Vol.325.Grund- lehren der mathematischen Wissenschaften. Berlin Heidelberg: Springer, 2016.doi:10 . 1007/978-3-662-49451-6. 37

    2016

  27. [29]

    TheSecondLawofThermodynamicsandStability

    C.M.Dafermos.“TheSecondLawofThermodynamicsandStability.”In:ArchiveforRational Mechanics and Analysis70.2 (1979), pp. 167–179.doi:10.1007/BF00250353

    doi:10.1007/bf00250353 1979

  28. [30]

    A Posteriori Analysis of Fully Discrete Method of Lines Discontinuous Galerkin Schemes for Systems of Conservation Laws

    A. Dedner and J. Giesselmann. “A Posteriori Analysis of Fully Discrete Method of Lines Discontinuous Galerkin Schemes for Systems of Conservation Laws.” In:SIAM Journal on NumericalAnalysis54.6(2016),pp.3523–3549.doi:10.1137/15M1046265.arXiv:1510.05430 [math.NA]

    Pith/arXiv arXiv 2016

  29. [31]

    Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations

    D. C. Del Rey Fernández, J. E. Hicken, and D. W. Zingg. “Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations.” In:Computers & Fluids95 (2014), pp. 171–196.doi:10 . 1016 / j . compfluid.2014.02.016

    2014

  30. [32]

    Two methods of Galerkin type achieving optimum𝐿2 rates of convergence for first order hyperbolics

    J. E. Dendy. “Two methods of Galerkin type achieving optimum𝐿2 rates of convergence for first order hyperbolics.” In:SIAM Journal on Numerical Analysis11.3 (1974), pp. 637–653. doi:10.1137/0711052

    work page doi:10.1137/0711052 1974

  31. [33]

    Uniqueness of Solutions to Hyperbolic Conservation Laws

    R. J. DiPerna. “Uniqueness of Solutions to Hyperbolic Conservation Laws.” In:Indiana UniversityMathematicsJournal28.1(1979),pp.137–188.doi:10.1512/iumj.1979.28.28011

    doi:10.1512/iumj.1979.28.28011 1979

  32. [34]

    Doehring, J

    D. Doehring, J. Chan, H. Ranocha, M. Schlottke-Lakemper, M. Torrilhon, and G. Gassner. VolumeTermAdaptivityforDiscontinuousGalerkinMethods.Mar.2026.doi:10.48550/arXiv. 2603.24189. arXiv:2603.24189 [math.NA]

    doi:10.48550/arxiv.2603.24189 2026

  33. [35]

    Galerkin methods for first order hyperbolics: An example

    T. Dupont. “Galerkin methods for first order hyperbolics: An example.” In:SIAM Journal on Numerical Analysis10.5 (1973), pp. 890–899.doi:10.1137/0710074

    work page doi:10.1137/0710074 1973

  34. [36]

    Conservative correction procedures utilizing artificial dissipation operators

    A. K. Edoh. “Conservative correction procedures utilizing artificial dissipation operators.” In:JournalofComputationalPhysics504(2024),p.112880.doi:10.1016/j.jcp.2024.112880

    arXiv 2024

  35. [37]

    L. C. Evans.Partial Differential Equations. 2nd ed. Vol. 19. Graduate Studies in Mathematics. Providence, Rhode Island: American Mathematical Society, 2010.doi:10.1090/gsm/019

    doi:10.1090/gsm/019 2010

  36. [38]

    High-order entropy stable finite difference schemes for nonlinearconservationlaws:Finitedomains

    T. C. Fisher and M. H. Carpenter. “High-order entropy stable finite difference schemes for nonlinearconservationlaws:Finitedomains.”In:JournalofComputationalPhysics252(2013), pp. 518–557.doi:10.1016/j.jcp.2013.06.014

    doi:10.1016/j.jcp.2013.06.014 2013

  37. [39]

    Construction of Approximate Entropy Measure-Valued Solutions for Hyperbolic Systems of Conservation Laws

    U. S. Fjordholm, R. Käppeli, S. Mishra, and E. Tadmor. “Construction of Approximate Entropy Measure-Valued Solutions for Hyperbolic Systems of Conservation Laws.” In: Foundations of Computational Mathematics17.3 (2017), pp. 763–827.doi:10.1007/s10208- 015-9299-z

    doi:10.1007/s10208- 2017

  38. [40]

    EnergyPreservingandEnergyStableSchemes for the Shallow Water Equations

    U.S.Fjordholm,S.Mishra,andE.Tadmor.“EnergyPreservingandEnergyStableSchemes for the Shallow Water Equations.” In:Foundations of Computational Mathematics, Hong Kong

  39. [41]

    Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography

    U. S. Fjordholm, S. Mishra, and E. Tadmor. “Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography.” In:Journal of Computational Physics230.14 (2011), pp. 5587–5609.doi:10.1016/j.jcp.2011.03.042

    work page doi:10.1016/j.jcp.2011.03.042 2011

  40. [42]

    ArbitrarilyHigh-OrderAccurateEntropyStable Essentially Nonoscillatory Schemes for Systems of Conservation Laws

    U.S.Fjordholm,S.Mishra,andE.Tadmor.“ArbitrarilyHigh-OrderAccurateEntropyStable Essentially Nonoscillatory Schemes for Systems of Conservation Laws.” In:SIAM Journal on Numerical Analysis50.2 (2012), pp. 544–573.doi:10.1137/110836961

    work page doi:10.1137/110836961 2012

  41. [43]

    Systems of Conservation Equations with a Convex Ex- tension

    K. O. Friedrichs and P. D. Lax. “Systems of Conservation Equations with a Convex Ex- tension.” In:Proceedings of the National Academy of Sciences68.8 (1971), pp. 1686–1688.doi: 10.1073/pnas.68.8.1686

    doi:10.1073/pnas.68.8.1686 1971

  42. [44]

    Error estimates for a numerical ap- proximation to the compressible barotropic Navier-Stokes equations

    T. Gallouët, R. Herbin, D. Maltese, and A. Novotný. “Error estimates for a numerical ap- proximation to the compressible barotropic Navier-Stokes equations.” In:IMA Journal of Numerical Analysis36.2 (2016), pp. 543–592.doi:10.1093/imanum/drv028. 38

    work page doi:10.1093/imanum/drv028 2016

  43. [45]

    Stability Issues of Entropy-Stable and/or Split-form High-order Schemes

    G. J. Gassner, M. Svärd, and F. J. Hindenlang. “Stability Issues of Entropy-Stable and/or Split-form High-order Schemes.” In:Journal of Scientific Computing90.3 (2022), p. 79.doi: 10.1007/s10915-021-01720-8. arXiv:2007.09026 [math.NA]

    Pith/arXiv arXiv 2022

  44. [46]

    A Novel Robust Strategy for Discontinuous Galerkin Methods in Computational Fluid Mechanics: Why? When? What? Where?

    G. J. Gassner and A. R. Winters. “A Novel Robust Strategy for Discontinuous Galerkin Methods in Computational Fluid Mechanics: Why? When? What? Where?” In:Frontiers in Physics8 (2021), p. 612.doi:10.3389/fphy.2020.500690

    arXiv 2021

  45. [47]

    Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations

    G. J. Gassner, A. R. Winters, and D. A. Kopriva. “Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations.” In: Journal of Computational Physics327 (2016), pp. 39–66.doi:10.1016/j.jcp.2016.09.013

    doi:10.1016/j.jcp.2016.09.013 2016

  46. [48]

    ASkew-SymmetricDiscontinuousGalerkinSpectralElementDiscretization and Its Relation to SBP-SAT Finite Difference Methods

    G.J.Gassner.“ASkew-SymmetricDiscontinuousGalerkinSpectralElementDiscretization and Its Relation to SBP-SAT Finite Difference Methods.” In:SIAM Journal on Scientific Computing35.3 (2013), A1233–A1253.doi:10.1137/120890144

    doi:10.1137/120890144 2013

  47. [49]

    Awellbalancedandentropyconservative discontinuous Galerkin spectral element method for the shallow water equations

    G.J.Gassner,A.R.Winters,andD.A.Kopriva.“Awellbalancedandentropyconservative discontinuous Galerkin spectral element method for the shallow water equations.” In: Applied Mathematics and Computation272 (2016), pp. 291–308.doi:10.1016/j.amc.2015. 07.014

    doi:10.1016/j.amc.2015 2016

  48. [50]

    RelativeentropybasederrorestimatesfordiscontinuousGalerkinschemes

    J.Giesselmann.“RelativeentropybasederrorestimatesfordiscontinuousGalerkinschemes.” In:Bulletin of the Brazilian Mathematical Society, New Series47.1 (2016), pp. 359–372.doi: 10.1007/s00574-016-0144-z

    work page doi:10.1007/s00574-016-0144-z 2016

  49. [51]

    A Posteriori Analysis of Discontinuous Galerkin Schemes for Systems of Hyperbolic Conservation Laws

    J. Giesselmann, C. Makridakis, and T. Pryer. “A Posteriori Analysis of Discontinuous Galerkin Schemes for Systems of Hyperbolic Conservation Laws.” In:SIAM Journal on Numerical Analysis53.3 (2015), pp. 1280–1303.doi:10.1137/140970999. arXiv:1405.7616 [math.NA]

    Pith/arXiv arXiv 2015

  50. [52]

    Convergence of hyperbolic approximations to higher- order PDEs for smooth solutions

    J. Giesselmann and H. Ranocha. “Convergence of hyperbolic approximations to higher- order PDEs for smooth solutions.” In:The SMAI Journal of Computational Mathematics12 (Mar. 2026), pp. 75–102.doi:10.5802/jcm.144. arXiv:2508.04112 [math.NA]

    arXiv 2026

  51. [53]

    Glaubitz, A

    J. Glaubitz, A. Iske, J. Lampert, and P. Öffner.Why summation by parts is not enough. Feb. 2026.doi:10.48550/arXiv.2602.10786. arXiv:2602.10786 [math.NA]

    doi:10.48550/arxiv.2602.10786 2026

  52. [54]

    An interesting class of quasi-linear systems

    S. K. Godunov. “An interesting class of quasi-linear systems.” In:Doklady Akademii Nauk SSSR139.3 (1961), pp. 521–523

    1961

  53. [55]

    Theconvergenceratefordifferenceapproximationstomixedinitialbound- ary value problems

    B.Gustafsson.“Theconvergenceratefordifferenceapproximationstomixedinitialbound- ary value problems.” In:Mathematics of Computation29.130 (1975), pp. 396–406.doi:10. 1090/S0025-5718-1975-0386296-7

    1975

  54. [56]

    Gustafsson, H.-O

    B. Gustafsson, H.-O. Kreiss, and J. Oliger.Time-Dependent Problems and Difference Methods. 2nd ed. Hoboken, NJ: John Wiley & Sons, 2013.doi:10.1002/9781118548448

    doi:10.1002/9781118548448 2013

  55. [57]

    On the symmetric form of systems of conservation laws with entropy

    A. Harten. “On the symmetric form of systems of conservation laws with entropy.” In: Journal of Computational Physics49.1 (1983), pp. 151–164.doi:10 . 1016 / 0021 - 9991(83 ) 90118-3

    1983

  56. [58]

    A provably entropy stable subcell shock capturing approach for high order split form DG for the compressible Euler equations

    S. Hennemann, A. M. Rueda-Ramírez, F. J. Hindenlang, and G. J. Gassner. “A provably entropy stable subcell shock capturing approach for high order split form DG for the compressible Euler equations.” In:Journal of Computational Physics426 (2021), p. 109935. doi:10.1016/j.jcp.2020.109935. arXiv:2008.12044 [physics.comp-ph]

    arXiv 2021

  57. [59]

    Constructing stable, high-order finite-difference operators on point clouds over complex geometries

    J. Hicken, G. Yan, and S. Kaur. “Constructing stable, high-order finite-difference operators on point clouds over complex geometries.” In:Journal of Computational Physics532 (2025), p. 113940.doi:10.1016/j.jcp.2025.113940

    arXiv 2025

  58. [60]

    Entropy-stable, high-order summation-by-parts discretizations without in- terfacepenalties

    J. E. Hicken. “Entropy-stable, high-order summation-by-parts discretizations without in- terfacepenalties.”In:JournalofScientificComputing82.2(2020),p.50.doi:10.1007/s10915- 020-01154-8. 39

    doi:10.1007/s10915- 2020

  59. [61]

    MultidimensionalSummation-By- Parts Operators: General Theory and Application to Simplex Elements

    J.E.Hicken,D.C.DelReyFernández,andD.W.Zingg.“MultidimensionalSummation-By- Parts Operators: General Theory and Application to Simplex Elements.” In:SIAM Journal on Scientific Computing38.4 (2016), A1935–A1958.doi:10.1137/15M1038360

    doi:10.1137/15m1038360 2016

  60. [62]

    Entropy stable spacetime discontinuous Galerkin methods for the two-dimensional compressible Navier–Stokes equations

    A. Hiltebrand and S. May. “Entropy stable spacetime discontinuous Galerkin methods for the two-dimensional compressible Navier–Stokes equations.” In:Communications in Mathematical Sciences16.8 (2018), pp. 2095–2124

    2018

  61. [63]

    Entropy stable shock capturing space-time discontinuous Galerkinschemesforsystemsofconservationlaws

    A. Hiltebrand and S. Mishra. “Entropy stable shock capturing space-time discontinuous Galerkinschemesforsystemsofconservationlaws.”In:NumerischeMathematik126.1(2014), pp. 103–151.doi:10.1007/s00211-013-0558-0

    work page doi:10.1007/s00211-013-0558-0 2014

  62. [64]

    Error estimates to smooth solutions of semi-discrete discontin- uous Galerkin methods with quadrature rules for scalar conservation laws

    J. Huang and C.-W. Shu. “Error estimates to smooth solutions of semi-discrete discontin- uous Galerkin methods with quadrature rules for scalar conservation laws.” In:Numerical Methods for Partial Differential Equations33.2 (2017), pp. 467–488.doi:10.1002/num.22089

    work page doi:10.1002/num.22089 2017

  63. [65]

    A new finite element formulation for compu- tational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier-Stokes equations and the second law of thermodynamics

    T. J. R. Hughes, L. P. Franca, and M Mallet. “A new finite element formulation for compu- tational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier-Stokes equations and the second law of thermodynamics.” In:Computer Methods in Applied Me- chanics and Engineering54.2 (1986), pp. 223–234.doi:10.1016/0045-7825(86)90127-1

    work page doi:10.1016/0045-7825(86)90127-1 1986

  64. [66]

    A Flux Reconstruction Approach to High-Order Schemes Including Dis- continuous Galerkin Methods

    H. T. Huynh. “A Flux Reconstruction Approach to High-Order Schemes Including Dis- continuous Galerkin Methods.” In:18th AIAA Computational Fluid Dynamics Conference. American Institute of Aeronautics and Astronautics. 2007.doi:10.2514/6.2007-4079

    doi:10.2514/6.2007-4079 2007

  65. [67]

    Affordable, entropy-consistent Euler flux functions II: Entropy productionatshocks

    F. Ismail and P. L. Roe. “Affordable, entropy-consistent Euler flux functions II: Entropy productionatshocks.”In:JournalofComputationalPhysics228.15(2009),pp.5410–5436.doi: 10.1016/j.jcp.2009.04.021

    doi:10.1016/j.jcp.2009.04.021 2009

  66. [68]

    On a cell entropy inequality for discontinuous Galerkin meth- ods

    G. S. Jiang and C.-W. Shu. “On a cell entropy inequality for discontinuous Galerkin meth- ods.” In:Mathematics of Computation62.206 (1994), pp. 531–538.doi:10.1090/S0025-5718- 1994-1223232-7

    doi:10.1090/s0025-5718- 1994

  67. [69]

    An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation

    C. Johnson and J. Pitkäranta. “An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation.” In:Mathematics of Computation46.173 (1986), pp. 1–26.doi: 10.1090/S0025-5718-1986-0815828-4

    work page doi:10.1090/s0025-5718-1986-0815828-4 1986

  68. [70]

    ErrorEstimatesforFiniteVolumeApproximationsofClassical SolutionsforNonlinearSystemsofHyperbolicBalanceLaws

    V.JovanovićandC.Rohde.“ErrorEstimatesforFiniteVolumeApproximationsofClassical SolutionsforNonlinearSystemsofHyperbolicBalanceLaws.”In:SIAMJournalonNumerical Analysis43.6 (2006), pp. 2423–2449.doi:10.1137/S0036142903438136

    work page doi:10.1137/s0036142903438136 2006

  69. [71]

    Relaxation Runge-Kutta Methods: Conservation and Stability for Inner- Product Norms

    D. I. Ketcheson. “Relaxation Runge-Kutta Methods: Conservation and Stability for Inner- Product Norms.” In:SIAM Journal on Numerical Analysis57.6 (2019), pp. 2850–2870.doi: 10.1137/19M1263662. arXiv:1905.09847 [math.NA]

    Pith/arXiv arXiv 2019

  70. [72]

    D. A. Kopriva.Implementing Spectral Methods for Partial Differential Equations: Algorithms for ScientistsandEngineers.NewYork:SpringerScience&BusinessMedia,2009.doi:10.1007/ 978-90-481-2261-5

    2009

  71. [73]

    MetricIdentitiesandtheDiscontinuousSpectralElementMethodonCurvi- linear Meshes

    D.A.Kopriva.“MetricIdentitiesandtheDiscontinuousSpectralElementMethodonCurvi- linear Meshes.” In:Journal of Scientific Computing26.3 (2006), pp. 301–327.doi:10.1007/ s10915-005-9070-8

    2006

  72. [74]

    FiniteElementandFiniteDifferenceMethodsforHyperbolic PartialDifferentialEquations

    H.-O.KreissandG.Scherer.“FiniteElementandFiniteDifferenceMethodsforHyperbolic PartialDifferentialEquations.”In:MathematicalAspectsofFiniteElementsinPartialDifferential Equations. Ed. by C. de Boor. New York: Academic Press, 1974, pp. 195–212.doi:10.1016/ B978-0-12-208350-1.50012-1

    1974

  73. [75]

    H.-O.KreissandG.Scherer.OntheExistenceofEnergyEstimatesforDifferenceApproximations for Hyperbolic Systems. Tech. rep. Uppsala, Sweden: Department of Scientific Computing, Uppsala University, 1977. 40

    1977

  74. [76]

    A robust first order meshfree method for time-dependent nonlinear conservationlaws

    S. Kwan and J. Chan. “A robust first order meshfree method for time-dependent nonlinear conservationlaws.”In:AdvancesinComputationalScienceandEngineering6(2025),pp.1–24. doi:10.3934/acse.2025021

    doi:10.3934/acse.2025021 2025

  75. [77]

    FullyDiscrete,EntropyConservativeSchemes of Arbitrary Order

    P.G.LeFloch,J.-M.Mercier,andC.Rohde.“FullyDiscrete,EntropyConservativeSchemes of Arbitrary Order.” In:SIAM Journal on Numerical Analysis40.5 (2002), pp. 1968–1992.doi: 10.1137/S003614290240069X

    doi:10.1137/s003614290240069x 2002

  76. [78]

    Conditional a priori error estimates of finite volume and Runge-Kutta discontinuous Galerkin methods with abstract limiting for hyperbolic systems of conservation laws in 1D

    F. Leotta.Conditional a priori error estimates of finite volume and Runge-Kutta discontinuous Galerkin methods with abstract limiting for hyperbolic systems of conservation laws in 1D. June 2025.doi:10.48550/arXiv.2506.13221. arXiv:2506.13221 [math.NA]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2506.13221 2025

  77. [79]

    On the order of accuracy of finite difference operatorsondiagonalnormbasedsummation-by-partsform

    V. Linders, T. Lundquist, and J. Nordström. “On the order of accuracy of finite difference operatorsondiagonalnormbasedsummation-by-partsform.”In:SIAMJournalonNumerical Analysis56.2 (2018), pp. 1048–1063.doi:10.1137/17M1139333

    work page doi:10.1137/17m1139333 2018

  78. [80]

    Sub-optimal convergence of discontinuous Galerkin methods with central fluxes for linear hyperbolic equations with even degree polynomial approximations

    Y. Liu, C.-W. Shu, and M. Zhang. “Sub-optimal convergence of discontinuous Galerkin methods with central fluxes for linear hyperbolic equations with even degree polynomial approximations.” In:Journal of Computational Mathematics39.4 (2021), pp. 518–537.doi: 10.4208/jcm.2002-m2019-0305

    work page doi:10.4208/jcm.2002-m2019-0305 2021

  79. [81]

    ConvergenceofDiscontinuousGalerkinSchemesfor the Euler Equations via Dissipative Weak Solutions

    Lukácová-MedvidováandP.Öffner.“ConvergenceofDiscontinuousGalerkinSchemesfor the Euler Equations via Dissipative Weak Solutions.” In:Applied Mathematics and Computa- tion436(2023),p.127508.doi:10.1016/j.amc.2022.127508.arXiv:2202.10043 [math.NA]

    arXiv 2023

  80. [82]

    Error Estimates of the Godunov Method fortheMultidimensionalCompressibleEulerSystem

    M. Lukáčová-Medvidová, B. She, and Y. Yuan. “Error Estimates of the Godunov Method fortheMultidimensionalCompressibleEulerSystem.”In:JournalofScientificComputing91 (2022), p. 71.doi:10.1007/s10915-022-01843-6

    work page doi:10.1007/s10915-022-01843-6 2022

Showing first 80 references.