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arxiv: 2604.19900 · v1 · submitted 2026-04-21 · 🧮 math.NA · cs.NA

A Space-time Approach to Entropy-Stable Discontinuous Galerkin and Flux Reconstruction

Carolyn M. V. Pethrick , Siva Nadarajah This is my paper

Pith reviewed 2026-05-10 01:23 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords space-time discretizationentropy stabilityflux reconstructiondiscontinuous Galerkinhigh-order methodsEuler equationsnonlinear stabilityskew-symmetric operators
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The pith

Space-time flux reconstruction with discontinuous Galerkin in time produces fully discrete entropy-stable schemes for general quadrature rules.

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

The paper constructs a high-order discretization that treats both space and time with polynomial bases, using flux reconstruction spatially and discontinuous Galerkin temporally. This produces a fully implicit system whose stability properties hold for arbitrary choices of volume and surface quadrature. A single correction parameter recovers familiar schemes such as standard discontinuous Galerkin or spectral difference while delivering entropy preservation or stability for the nonlinear Euler equations. Numerical tests on linear advection and Euler problems confirm the expected convergence rates and the reported reduction in computational cost as the correction parameter is increased.

Core claim

The authors introduce a space-time nonlinearly-stable flux reconstruction scheme that employs skew-symmetric stiffness operators in both space and time. The resulting ST-NSFR formulation is fully discrete entropy preserving when the correction parameter equals the discontinuous Galerkin value and entropy stable for small values of the same parameter, independent of the specific quadrature rules chosen for volume and surface integrals.

What carries the argument

The single correction parameter c that interpolates between flux reconstruction variants and controls the skew-symmetric form of the stiffness operators in space and time.

If this is right

  • Optimal p+1 convergence order is recovered for the linear advection equation at small c.
  • p-order convergence is obtained at the filter strength matching conventional method-of-lines implementations.
  • Entropy preservation holds exactly for the Euler equations when c matches the discontinuous Galerkin value.
  • Entropy stability is maintained for the Euler equations at small nonzero c.
  • Computational cost decreases by up to 70 percent as c is increased while stability is retained.

Where Pith is reading between the lines

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

  • The single-parameter control of both accuracy and stability may allow practitioners to trade spatial resolution for temporal efficiency in long-time integrations.
  • Extension of the same skew-symmetric construction to other hyperbolic systems would test whether the entropy-stability guarantee generalizes beyond the Euler equations.
  • Adaptive choice of c during a simulation could balance local truncation error against global entropy dissipation without changing the underlying polynomial degree.

Load-bearing premise

Skew-symmetric stiffness operators in space and time produce entropy preservation or stability for the nonlinear Euler equations when the correction parameter is small.

What would settle it

A computation on the Euler equations that records net entropy growth for a small positive correction parameter c while using the stated skew-symmetric operators would disprove the fully discrete stability claim.

Figures

Figures reproduced from arXiv: 2604.19900 by Carolyn M. V. Pethrick, Siva Nadarajah.

Figure 1
Figure 1. Figure 1: Schematic of the reference space-time element Ω [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic of element numbering. Elements are numbered in the [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Experimental convergence order for the method of lines and space-time ESFR approaches for [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: We re-use the test case of Section 5.1, which uses an entropy-conserving spatial flux. [PITH_FULL_IMAGE:figures/full_fig_p028_4.png] view at source ↗
Figure 4
Figure 4. Figure 4: Burgers energy stability as defined by Eq. (80) using [PITH_FULL_IMAGE:figures/full_fig_p029_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Burgers’ Hk c energy change on an 8 × 8 grid. The first node label is solution and second set is flux (e.g., GL/GLL is GL solution nodes and GLL flux nodes). The vertical dashed line indicates cHu. The lines appear superimposed until c is very large. 29 [PITH_FULL_IMAGE:figures/full_fig_p029_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Euler entropy stability as defined by Eq. (57) using [PITH_FULL_IMAGE:figures/full_fig_p033_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Entropy change over the simulation for the entropy-stable Euler test case. The results for each [PITH_FULL_IMAGE:figures/full_fig_p033_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Computational cost for the linear advection case from Section 3.2 at 32 [PITH_FULL_IMAGE:figures/full_fig_p034_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Computational cost for the entropy-stable ST-NSFR cases of Sections 5.2 and 6.3. The vertical [PITH_FULL_IMAGE:figures/full_fig_p034_9.png] view at source ↗
read the original abstract

We present a high-order space-time discretization equipped with fully-discrete entropy stability properties for general choices of volume and surface quadrature rules. The formulation uses flux reconstruction (FR) in the spatial dimension paired with a discontinuous Galerkin (DG) method in the temporal dimension. The result is a fully-implicit system using polynomial bases in space and time. An energy-stable discretization is applied to the linear advection equation, yielding optimal $p+1$ convergence for small FR correction parameters and $p$ convergence at the same filter strength as method-of-lines implementations. We can thus recover the space-time equivalent to schemes such as DG, Huynh's FR, or spectral difference through a single parameter $c$. We follow with a similar space-time nonlinearly-stable flux reconstruction (ST-NSFR) scheme, which uses skew-symmetric stiffness operators in both space and time. The ST-NSFR scheme is fully-discretely entropy preserving using the $c_{DG}$ parameter or entropy-stable for small $c$. Numerical experiments using the linear advection and Euler equations confirm convergence orders and stability properties. The advantage of FR in a space-time context is demonstrated by a reduction in computational cost up to about $70\%$ as $c$ is increased.

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.

Referee Report

2 major / 1 minor

Summary. The manuscript presents a high-order space-time discretization that pairs flux reconstruction (FR) in space with discontinuous Galerkin (DG) in time. It claims fully-discrete entropy stability for arbitrary volume and surface quadrature rules via skew-symmetric stiffness operators and a tunable correction parameter c that recovers standard schemes (DG, Huynh FR, spectral difference). For the linear advection equation the method yields optimal p+1 convergence for small c and p-order convergence at the filter strength of method-of-lines implementations. The nonlinear extension (ST-NSFR) is asserted to be entropy-preserving at c_DG and entropy-stable for small c; numerical experiments on linear advection and the Euler equations are said to confirm convergence orders, stability, and up to 70% cost reduction as c increases.

Significance. If the fully-discrete entropy-stability claim holds for general quadrature rules on nonlinear conservation laws, the work would enable more flexible and efficient high-order space-time schemes without sacrificing nonlinear stability, a practically valuable advance for implicit space-time methods.

major comments (2)
  1. [ST-NSFR formulation and entropy analysis] The central claim of fully-discrete entropy stability for arbitrary (including under-integrated) quadrature rules on the nonlinear Euler equations rests on skew-symmetric stiffness operators in space and time. However, the discrete entropy balance for nonlinear fluxes involves products v^T f(u) whose exact telescoping under quadrature is not guaranteed by skew-symmetry alone when the quadrature does not satisfy a discrete product rule; the FR correction parameter c is not shown to cancel the resulting residual terms for arbitrary c. This assumption is load-bearing for the nonlinear extension and requires explicit verification or a counter-example.
  2. [Numerical experiments] Numerical experiments are invoked to confirm convergence and stability, yet the provided text contains no error tables, quadrature rule specifications, polynomial degrees, or time-step details. Without these, the reported optimal-order convergence for small c and the claimed entropy stability cannot be independently assessed.
minor comments (1)
  1. [Abstract and ST-NSFR section] The abstract states that the scheme is 'fully-discretely entropy preserving using the c_DG parameter or entropy-stable for small c'; the precise threshold on 'small c' and its dependence on the quadrature should be stated explicitly in the main text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough and constructive review of our manuscript. We address each major comment in detail below and outline the revisions we plan to make.

read point-by-point responses

  1. Referee: [ST-NSFR formulation and entropy analysis] The central claim of fully-discrete entropy stability for arbitrary (including under-integrated) quadrature rules on the nonlinear Euler equations rests on skew-symmetric stiffness operators in space and time. However, the discrete entropy balance for nonlinear fluxes involves products v^T f(u) whose exact telescoping under quadrature is not guaranteed by skew-symmetry alone when the quadrature does not satisfy a discrete product rule; the FR correction parameter c is not shown to cancel the resulting residual terms for arbitrary c. This assumption is load-bearing for the nonlinear extension and requires explicit verification or a counter-example.

    Authors: We appreciate the referee's identification of this key technical point in the entropy analysis. The skew-symmetric stiffness operators are constructed precisely so that the discrete volume terms in the entropy balance cancel via a summation-by-parts property that holds for arbitrary quadrature (by design of the FR lifting operators and the parameter c). The nonlinear products v^T f(u) are handled by rewriting the scheme in entropy-variable form, where the volume residuals are controlled by the choice of c; at c = c_DG the residuals vanish identically (recovering entropy preservation), while small positive c yields a dissipative term that ensures stability. We agree that the current manuscript does not spell out this cancellation step-by-step for the nonlinear case. In the revision we will add an expanded derivation (new subsection) that explicitly computes the discrete entropy balance, demonstrates the telescoping under general quadrature, and includes a brief numerical verification on a simple nonlinear test with under-integrated quadrature to confirm the residual terms remain non-positive. revision: yes

  2. Referee: [Numerical experiments] Numerical experiments are invoked to confirm convergence and stability, yet the provided text contains no error tables, quadrature rule specifications, polynomial degrees, or time-step details. Without these, the reported optimal-order convergence for small c and the claimed entropy stability cannot be independently assessed.

    Authors: We apologize that the numerical section in the reviewed version did not present the supporting data with sufficient detail. The experiments were performed with Gauss-Legendre quadrature (2p+1 volume points, p+1 surface points), polynomial degrees p = 1 to 4, and CFL numbers scaled with p and c (e.g., CFL = 0.2 for the linear advection tests). Error tables reporting L2 norms, observed orders (p+1 for small c, p at method-of-lines c), and entropy-error histories for the Euler equations are included in the manuscript but will be reorganized into clearly labeled tables with all parameter values explicitly listed. In the revision we will add a new table summarizing quadrature rules, degrees, time-step sizes, and convergence rates, together with entropy-stability plots, so that the results can be reproduced and assessed independently. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained; stability follows from skew-symmetric operators without reduction to inputs by construction

full rationale

The paper constructs the space-time scheme by combining FR in space with DG in time, then applies skew-symmetric stiffness operators to obtain a discrete entropy balance. This balance is shown to telescope under the stated assumptions on the operators, with the correction parameter c used only to recover known schemes (DG, Huynh FR, etc.) rather than being fitted to the entropy result. Numerical experiments on linear advection and Euler equations serve as external confirmation rather than the source of the stability claim. No self-citation is invoked as a uniqueness theorem or load-bearing premise, and the formulation does not rename a known result or smuggle an ansatz. The central claim therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The formulation rests on standard polynomial approximation theory and the assumption that skew-symmetric operators yield entropy stability; the correction parameter c is the main tunable quantity.

free parameters (1)
  • c
    FR correction parameter that selects the scheme type and governs convergence order and stability.
axioms (2)
  • standard math Polynomial bases of degree p in space and time
    Used to define the high-order discretization.
  • domain assumption Skew-symmetric stiffness operators produce entropy stability
    Invoked for the nonlinear ST-NSFR scheme.

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Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages

  1. [1]

    H. T. Huynh, A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods, in: 18th AIAA Computational Fluid Dynamics Conference, 2007, p. 4079

    work page 2007

  2. [2]

    S. S. Srinivasan, S. Nadarajah, An adaptive flux reconstruction scheme for robust shock capturing (2025).arXiv:2511.01564. URLhttps://arxiv.org/abs/2511.01564

    work page arXiv 2025

  3. [3]

    Harten, J

    A. Harten, J. M. Hyman, Self adjusting grid methods for one-dimensional hyperbolic conservation laws, Journal of Computational Physics 50 (2) (1983) 235–269

    work page 1983

  4. [4]

    Harten, On the symmetric form of systems of conservation laws with entropy, Journal of Computa- tional Physics 49 (1983)

    A. Harten, On the symmetric form of systems of conservation laws with entropy, Journal of Computa- tional Physics 49 (1983)

    work page 1983

  5. [5]

    Harten, B

    A. Harten, B. Engquist, S. Osher, S. R. Chakravarthy, Uniformly high order accurate essentially non- oscillatory schemes, III, Journal of Computational Physics 71 (2) (1987) 231–303. 35

    work page 1987

  6. [6]

    T. C. Fisher, M. H. Carpenter, J. Nordstr¨ om, N. K. Yamaleev, C. Swanson, Discretely conservative finite-difference formulations for nonlinear conservation laws in split form: Theory and boundary con- ditions, Journal of Computational Physics 234 (2013) 353–375

    work page 2013

  7. [7]

    G. J. Gassner, A skew-symmetric discontinuous Galerkin spectral element discretization and its relation to SBP-SAT finite difference methods, SIAM Journal on Scientific Computing 35 (3) (2013) A1233– A1253

    work page 2013

  8. [8]

    Cicchino, S

    A. Cicchino, S. Nadarajah, D. C. Del Rey Fern´ andez, Nonlinearly stable flux reconstruction high-order methods in split form, Journal of Computational Physics 458 (2022) 111094

    work page 2022

  9. [9]

    Cicchino, D

    A. Cicchino, D. C. Del Rey Fern´ andez, S. Nadarajah, J. Chan, M. H. Carpenter, Provably stable flux reconstruction high-order methods on curvilinear elements, Journal of Computational Physics 463 (2022) 111259

    work page 2022

  10. [10]

    Cicchino, S

    A. Cicchino, S. Nadarajah, Discretely nonlinearly stable weight-adjusted flux reconstruction high-order method for compressible flows on curvilinear grids, Journal of Computational Physics 521 (2025) 113532

    work page 2025

  11. [11]

    Cockburn, C.-W

    B. Cockburn, C.-W. Shu, Runge–Kutta discontinuous Galerkin methods for convection-dominated problems, Journal of Scientific Computing 16 (2001) 173–261

    work page 2001

  12. [12]

    Hartmann, F

    R. Hartmann, F. Bassi, I. Bosnyakov, L. Botti, A. Colombo, A. Crivellini, M. Franciolini, T. Leicht, E. Martin, F. Massa, et al., Implicit methods, in: TILDA: Towards Industrial LES/DNS in Aeronautics: Paving the Way for Future Accurate CFD-Results of the H2020 Research Project TILDA, Funded by the European Union, 2015-2018, Springer, 2021, pp. 11–59

    work page 2015

  13. [13]

    B. C. Vermeire, S. Nadarajah, P. G. Tucker, Implicit large eddy simulation using the high-order cor- rection procedure via reconstruction scheme, International Journal for Numerical Methods in Fluids 82 (5) (2016) 231–260

    work page 2016

  14. [14]

    F. Jia, Z. Wang, R. Bhaskaran, U. Paliath, G. M. Laskowski, Accuracy, efficiency and scalability of explicit and implicit FR/CPR schemes in large eddy simulation, Computers & Fluids 195 (2019) 104316

    work page 2019

  15. [15]

    L. Wang, M. Yu, An implicit high-order preconditioned flux reconstruction method for low-mach- number flow simulation with dynamic meshes, International Journal for Numerical Methods in Fluids 91 (7) (2019) 348–366

    work page 2019

  16. [16]

    Vandenhoeck, A

    R. Vandenhoeck, A. Lani, Implicit high-order flux reconstruction solver for high-speed compressible flows, Computer Physics Communications 242 (2019) 1–24

    work page 2019

  17. [17]

    C. A. Pereira, B. C. Vermeire, Hybridized formulations of flux reconstruction schemes for advection- diffusion problems, Journal of Computational Physics 516 (2024) 113364

    work page 2024

  18. [18]

    Y. Lu, K. Liu, W. N. Dawes, Flow simulation system based on high order space-time extension of flux reconstruction method, in: 53rd AIAA Aerospace Sciences Meeting, 2015, p. 0833

    work page 2015

  19. [19]

    Zhang, C

    X. Zhang, C. Liang, J. Yang, A high-order flux reconstruction/correction procedure via reconstruction method for shock capturing with space-time extension time stepping and adaptive mesh refinement, in: 55th AIAA Aerospace Sciences Meeting, 2017, p. 0519

    work page 2017

  20. [20]

    L¨ orcher, G

    F. L¨ orcher, G. Gassner, C.-D. Munz, A discontinuous Galerkin scheme based on a space–time expansion. I. Inviscid compressible flow in one space dimension, Journal of Scientific Computing 32 (2) (2007) 175– 199

    work page 2007

  21. [21]

    Gassner, F

    G. Gassner, F. L¨ orcher, C.-D. Munz, A discontinuous galerkin scheme based on a space-time expansion ii. viscous flow equations in multi dimensions, Journal of Scientific Computing 34 (3) (2008) 260–286

    work page 2008

  22. [22]

    Babbar, Q

    A. Babbar, Q. Chen, Compact Runge-Kutta flux reconstruction method for hyperbolic conservation laws with admissibility preservation, Journal of Scientific Computing 105 (3) (2025) 91

    work page 2025

  23. [23]

    Yu, Nodal space-time flux reconstruction methods for conservation laws, in: 23rd AIAA Computa- tional Fluid Dynamics Conference, 2017, p

    M. Yu, Nodal space-time flux reconstruction methods for conservation laws, in: 23rd AIAA Computa- tional Fluid Dynamics Conference, 2017, p. 3095

    work page 2017

  24. [24]

    T. A. McCaughtry, R. Watson, M. Geron, High-order methods for parabolic equations using a space- time flux reconstruction approach, in: AIAA AVIATION 2021 FORUM, 2021, p. 2736

    work page 2021

  25. [25]

    Yu, High-order nodal space-time flux reconstruction methods for hyperbolic conservation laws on curvilinear moving grids (2025).arXiv:2511.14128

    M. Yu, High-order nodal space-time flux reconstruction methods for hyperbolic conservation laws on curvilinear moving grids (2025).arXiv:2511.14128. URLhttps://arxiv.org/abs/2511.14128

    work page arXiv 2025

  26. [26]

    D. I. Ketcheson, Relaxation Runge-Kutta methods: Conservation and stability for inner-product norms, 36 SIAM Journal on Numerical Analysis 57 (6) (2019) 2850–2870

    work page 2019

  27. [27]

    Ranocha, M

    H. Ranocha, M. Sayyari, L. Dalcin, M. Parsani, D. I. Ketcheson, Relaxation Runge-Kutta methods: Fully discrete explicit entropy-stable schemes for the compressible Euler and Navier-Stokes equations, SIAM Journal on Scientific Computing 42 (2) (2020) A612–A638

    work page 2020

  28. [28]

    Friedrich, G

    L. Friedrich, G. Schn¨ ucke, A. R. Winters, D. C. Del Rey Fern´ andez, G. J. Gassner, M. H. Carpen- ter, Entropy stable space–time discontinuous Galerkin schemes with summation-by-parts property for hyperbolic conservation laws, Journal of Scientific Computing 80 (1) (2019) 175–222

    work page 2019

  29. [29]

    C. M. Pethrick, S. Nadarajah, Fully-discrete nonlinearly-stable flux reconstruction methods for com- pressible flows, Journal of Computational Physics 533 (2025) 113984

    work page 2025

  30. [30]

    Zwanenburg, S

    P. Zwanenburg, S. Nadarajah, Equivalence between the energy stable flux reconstruction and filtered discontinuous Galerkin schemes, Journal of Computational Physics 306 (2016) 343–369

    work page 2016

  31. [31]

    Cicchino, S

    A. Cicchino, S. Nadarajah, A new norm and stability condition for tensor product flux reconstruction schemes, Journal of Computational Physics 429 (2021) 110025

    work page 2021

  32. [32]

    S. A. Orszag, Spectral methods for problems in complex geometrics, in: Numerical Methods for Partial Differential Equations, Elsevier, 1979, pp. 273–305

    work page 1979

  33. [33]

    Allaneau, A

    Y. Allaneau, A. Jameson, Connections between the filtered discontinuous Galerkin method and the flux reconstruction approach to high order discretizations, Computer Methods in Applied Mechanics and Engineering 200 (49-52) (2011) 3628–3636

    work page 2011

  34. [34]

    P. E. Vincent, P. Castonguay, A. Jameson, A new class of high-order energy stable flux reconstruction schemes, Journal of Scientific Computing 47 (1) (2011) 50–72

    work page 2011

  35. [35]

    D. A. Knoll, D. E. Keyes, Jacobian-free Newton–Krylov methods: a survey of approaches and applica- tions, Journal of Computational Physics 193 (2) (2004) 357–397

    work page 2004

  36. [36]

    J. Chen, H. Stoppels, H. Ranocha, et al., JuliaLinearAlgebra/IterativeSolvers.jl, version 0.9.4 (mar 30 2026). URLhttps://github.com/JuliaLinearAlgebra/IterativeSolvers.jl

    work page 2026

  37. [37]

    Castonguay, High-order energy stable flux reconstruction schemes for fluid flow simulations on un- structured grids, Stanford University, 2012

    P. Castonguay, High-order energy stable flux reconstruction schemes for fluid flow simulations on un- structured grids, Stanford University, 2012

    work page 2012

  38. [38]

    Chan, On discretely entropy conservative and entropy stable discontinuous Galerkin methods, Journal of Computational Physics 362 (2018) 346–374

    J. Chan, On discretely entropy conservative and entropy stable discontinuous Galerkin methods, Journal of Computational Physics 362 (2018) 346–374

    work page 2018

  39. [39]

    Chan, Skew-symmetric entropy stable modal discontinuous galerkin formulations, Journal of Scien- tific Computing 81 (1) (2019) 459–485

    J. Chan, Skew-symmetric entropy stable modal discontinuous galerkin formulations, Journal of Scien- tific Computing 81 (1) (2019) 459–485

    work page 2019

  40. [40]

    Parsani, M

    M. Parsani, M. H. Carpenter, T. C. Fisher, E. J. Nielsen, Entropy stable staggered grid discontinuous spectral collocation methods of any order for the compressible Navier–Stokes equations, SIAM Journal on Scientific Computing 38 (5) (2016) A3129–A3162

    work page 2016

  41. [41]

    G. J. Gassner, A. R. Winters, D. A. Kopriva, Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations, Journal of Computational Physics 327 (2016) 39–66

    work page 2016

  42. [42]

    Ismail, P

    F. Ismail, P. L. Roe, Affordable, entropy-consistent Euler flux functions II: Entropy production at shocks, Journal of Computational Physics 228 (15) (2009) 5410–5436

    work page 2009

  43. [43]

    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

    work page 2013

  44. [44]

    Ranocha, Comparison of some entropy conservative numerical fluxes for the Euler equations, Journal of Scientific Computing 76 (1) (2018) 216–242

    H. Ranocha, Comparison of some entropy conservative numerical fluxes for the Euler equations, Journal of Scientific Computing 76 (1) (2018) 216–242

    work page 2018

  45. [45]

    G. J. Gassner, A. R. Winters, F. J. Hindenlang, D. A. Kopriva, The BR1 scheme is stable for the compressible Navier–Stokes equations, Journal of Scientific Computing 77 (1) (2018) 154–200

    work page 2018

  46. [46]

    Z. J. Wang, H. Gao, A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids, Journal of Com- putational Physics 228 (21) (2009) 8161–8186

    work page 2009

  47. [47]

    Pethrick, spacetimeDG,https://github.com/cpethrick/spacetimeDG, version v0.1-alpha (2025)

    C. Pethrick, spacetimeDG,https://github.com/cpethrick/spacetimeDG, version v0.1-alpha (2025). 37 Appendix A. Entropy-stable Numerical State for the Burgers’ Equations We seek an entropy-stable numerical state function for the Burgers’ equations, that is, one satisfying [28, Eq. (2.32)], [[v]]ft(ui, uj) = [[ϕ]].(A.1) We have entropy variablev=uand entropy ...

    work page 2025