pith. sign in

arxiv: 2605.27327 · v1 · pith:KKWYCKURnew · submitted 2026-05-26 · 🧮 math.NA · cs.NA

A collocation scheme that is equivalent to discontinuous Galerkin discretizations

Jason Hicken This is my paper

Pith reviewed 2026-06-29 15:19 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords collocationdiscontinuous Galerkinsummation-by-partssemi-discretizationequivalencehigh-order methodsnumerical PDEs
0
0 comments X

The pith

A summation-by-parts collocation operator produces solutions identical to discontinuous Galerkin discretizations on the same quadrature.

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

The paper shows that semi-discretizations built from Chan's summation-by-parts collocation operator are algebraically equivalent to DG semi-discretizations that employ the identical quadrature rule. Equivalence is independent of how many degrees of freedom the collocation scheme uses and remains valid even when some quadrature weights are non-positive. The work verifies the result on the advection equation and Burgers' equation, demonstrates how nullspace modes stay zero, and recovers equivalence for entropy-stable splittings via residual projection. A reader would care because the result unifies two families of high-order methods, letting implementers choose whichever formulation is more convenient while preserving the stability and accuracy properties of the other.

Core claim

Semi-discretizations based on the summation-by-parts collocation operator produce solutions that are equivalent to solutions of a DG semi-discretization using the same underlying quadrature. The equivalence holds regardless of the number of degrees of freedom in the collocation scheme and when the quadrature is not strictly positive. Extraneous degrees of freedom lie in the nullspace of the operator and remain zero throughout an unsteady simulation. Equivalence can be restored for entropy-stable schemes by projecting the collocation residual onto the relevant polynomial space. The collocation operator also yields semi-discretizations with smaller spectral radii than a standard SBP constructi

What carries the argument

The summation-by-parts collocation operator (as defined in Chan 2018) that satisfies a discrete integration-by-parts identity exactly and is used to build the semi-discretization.

Load-bearing premise

The collocation operator must satisfy the summation-by-parts property exactly as defined in the cited reference; without it the algebraic steps that establish equivalence would not hold.

What would settle it

A side-by-side computation on the constant-coefficient advection equation on triangular elements in which the collocation solution and the DG solution on identical quadrature nodes differ at any time step would falsify the claimed equivalence.

read the original abstract

A spectral collocation operator with the summation-by-parts property was introduced by Chan to develop entropy-stable discontinuous Galerkin (DG) semi-discretizations (https://doi.org/10.1016/j.jcp.2018.02.033). The present work shows that semi-discretizations based on this collocation operator produce solutions that are equivalent to solutions of a DG semi-discretization using the same underlying quadrature. The equivalence holds regardless of the number of degrees of freedom in the collocation scheme and when the quadrature is not strictly positive. Extraneous degrees of freedom in the collocation scheme are associated with the nullspace of the operator and remain zero throughout an unsteady simulation. If necessary, nullspace consistency can be recovered by introducing projection-based numerical dissipation that targets only the extraneous modes. The equivalence between collocation and DG solutions is verified for the constant-coefficient advection equation and Burgers' equation on triangular meshes. The numerical results show that equivalence breaks down for entropy-stable semi-discretizations of Burgers' equation based on a skew-symmetric splitting, but that equivalence can be recovered by projecting the collocation scheme's residual onto the relevant polynomial space. In addition to investigating equivalence, the results demonstrate that the collocation operator produces semi-discretizations with favorable spectral radii compared with a commonly used summation-by-parts operator construction.

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 / 2 minor

Summary. The paper claims that semi-discretizations based on Chan's summation-by-parts (SBP) collocation operator are algebraically equivalent to DG semi-discretizations that employ the same underlying quadrature rule. Equivalence holds for arbitrary numbers of degrees of freedom, including when quadrature weights are not strictly positive; extraneous collocation modes lie in the operator nullspace and remain zero in unsteady simulations. The claim is supported by an algebraic derivation from the SBP property, with numerical verification on constant-coefficient advection and Burgers' equation on triangular meshes. Equivalence fails for skew-symmetric entropy-stable Burgers discretizations but is recovered by projecting the collocation residual onto the relevant polynomial space. The collocation operator is also shown to yield smaller spectral radii than a standard SBP construction.

Significance. If the algebraic equivalence holds, the work provides a rigorous bridge between collocation and DG methods that preserves DG properties (including for non-positive quadratures) while offering implementation simplicity and improved spectral radii. The algebraic (rather than fitted) nature of the proof, explicit handling of the nullspace, and recovery mechanism for entropy-stable cases are strengths. Numerical confirmation on both linear and nonlinear problems adds credibility. This could aid analysis of high-order entropy-stable schemes for hyperbolic conservation laws.

major comments (2)
  1. [§3 (equivalence derivation)] The central algebraic equivalence is stated to follow directly from the SBP property as defined in Chan (2018). The manuscript should explicitly cite the precise SBP identity (e.g., the discrete integration-by-parts relation) used in the derivation and confirm that no additional assumptions on the quadrature or polynomial degree are introduced beyond those in the reference.
  2. [§5.2 (entropy-stable Burgers)] For the skew-symmetric Burgers case, the paper correctly identifies breakdown of equivalence and recovers it via projection. However, the load-bearing step—why the projection restores equivalence only for the collocation residual and not the DG residual—requires a short explicit argument or counter-example showing that the projected operator remains consistent with the DG flux differencing.
minor comments (2)
  1. [§2] Notation for the collocation degrees of freedom versus the DG polynomial space should be unified (e.g., consistent use of subscripts for nodal vs. modal representations) to avoid reader confusion when comparing the two schemes.
  2. [§6] Figure captions for the spectral-radius comparisons should state the exact polynomial degree and mesh type used, and include a brief note on how the radii were computed (e.g., maximum eigenvalue of the semi-discrete operator).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the supportive review and constructive suggestions. We address each major comment below.

read point-by-point responses

  1. Referee: [§3 (equivalence derivation)] The central algebraic equivalence is stated to follow directly from the SBP property as defined in Chan (2018). The manuscript should explicitly cite the precise SBP identity (e.g., the discrete integration-by-parts relation) used in the derivation and confirm that no additional assumptions on the quadrature or polynomial degree are introduced beyond those in the reference.

    Authors: We agree that an explicit citation improves clarity. In the revised manuscript we will directly reference the discrete integration-by-parts relation (Q + Q^T = B) from Chan (2018) that underpins the algebraic steps. The derivation uses only this identity together with the definition of the collocation operator; no further assumptions on quadrature positivity or polynomial degree are introduced beyond those already required by the SBP operator in the cited reference. revision: yes

  2. Referee: [§5.2 (entropy-stable Burgers)] For the skew-symmetric Burgers case, the paper correctly identifies breakdown of equivalence and recovers it via projection. However, the load-bearing step—why the projection restores equivalence only for the collocation residual and not the DG residual—requires a short explicit argument or counter-example showing that the projected operator remains consistent with the DG flux differencing.

    Authors: We appreciate the request for additional detail. The projection is applied only to the collocation residual to enforce membership in the underlying polynomial space of degree at most N; the DG residual is already formulated inside this space, so no projection is required. In the revision we will add a short paragraph showing that the projected collocation operator is algebraically identical to the DG flux-differencing operator on the polynomial component, because the projection annihilates only the extraneous nullspace modes while leaving the action on the DG subspace unchanged. This algebraic identity guarantees consistency without needing a separate counter-example. revision: yes

Circularity Check

0 steps flagged

Algebraic equivalence from external SBP property; no circularity

full rationale

The derivation establishes an algebraic equivalence between the collocation semi-discretization and a DG scheme by direct manipulation of the summation-by-parts (SBP) property of the collocation operator, as defined externally in Chan (2018). This property is an input assumption, not derived within the paper, and the equivalence steps (including handling of nullspace modes and projection for consistency) follow from operator algebra without fitting, self-definition, or renaming. The paper explicitly identifies where equivalence fails (skew-symmetric Burgers) and recovers it via projection, with numerical checks on advection and Burgers. No load-bearing self-citation or ansatz smuggling occurs; the central claim is self-contained against the stated external SBP definition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the summation-by-parts property of the collocation operator from prior literature and standard properties of quadrature rules; no new free parameters or invented entities are introduced.

axioms (1)
  • domain assumption The collocation operator satisfies the summation-by-parts property as defined by Chan (2018).
    This property is invoked to establish the algebraic equivalence between collocation and DG operators.

pith-pipeline@v0.9.1-grok · 5761 in / 1135 out tokens · 32485 ms · 2026-06-29T15:19:52.075072+00:00 · methodology

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

29 extracted references · 19 canonical work pages

  1. [1]

    In: Boor, C

    Kreiss, H.O., Scherer, G.: Finite element and finite difference methods for hyper- bolic partial differential equations. In: Boor, C. (ed.) Mathematical Aspects of Finite Elements in Partial Differential Equations. Academic Press, New York (1974). Mathematics Research Center, the University of Wisconsin

    1974

  2. [2]

    Journal of Computational Physics252, 518–557 (2013) https://doi.org/10.1016/j.jcp.2013.06.014 24

    Fisher, T.C., Carpenter, M.H.: High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains. Journal of Computational Physics252, 518–557 (2013) https://doi.org/10.1016/j.jcp.2013.06.014 24

    work page doi:10.1016/j.jcp.2013.06.014 2013

  3. [3]

    SIAM Jour- nal on Scientific Computing35(3), 1233–1253 (2013) https://doi.org/10.1137/ 120890144

    Gassner, G.J.: A skew-symmetric discontinuous Galerkin spectral element dis- cretization and its relation to SBP-SAT finite difference methods. SIAM Jour- nal on Scientific Computing35(3), 1233–1253 (2013) https://doi.org/10.1137/ 120890144

    2013

  4. [4]

    SIAM Journal on Scientific Computing36(5), 835–867 (2014) https://doi

    Carpenter, M.H., Fisher, T.C., Nielsen, E.J., Frankel, S.H.: Entropy stable spectral collocation schemes for the Navier–Stokes equations: Discontinuous inter- faces. SIAM Journal on Scientific Computing36(5), 835–867 (2014) https://doi. org/10.1137/130932193

    work page doi:10.1137/130932193 2014

  5. [5]

    SIAM Journal on Scientific Computing38(4), 1935–1958 (2016) https: //doi.org/10.1137/15m1038360

    Hicken, J.E., Del Rey Fern´ andez, D.C., Zingg, D.W.: Multidimensional summation-by-parts operators: General theory and application to simplex ele- ments. SIAM Journal on Scientific Computing38(4), 1935–1958 (2016) https: //doi.org/10.1137/15m1038360

    work page doi:10.1137/15m1038360 1935

  6. [6]

    Journal of Computational Physics362, 346–374 (2018) https: //doi.org/10.1016/j.jcp.2018.02.033

    Chan, J.: On discretely entropy conservative and entropy stable discontinuous Galerkin methods. Journal of Computational Physics362, 346–374 (2018) https: //doi.org/10.1016/j.jcp.2018.02.033

    work page doi:10.1016/j.jcp.2018.02.033 2018

  7. [7]

    Journal of Scientific Computing81(1), 459–485 (2019)

    Chan, J.: Skew-symmetric entropy stable modal discontinuous galerkin formula- tions. Journal of Scientific Computing81(1), 459–485 (2019)

    2019

  8. [8]

    SIAM Journal on Scientific Computing46(4), 2270–2297 (2024) https://doi.org/10

    Montoya, T., Zingg, D.W.: Efficient tensor-product spectral-element operators with the summation-by-parts property on curved triangles and tetrahedra. SIAM Journal on Scientific Computing46(4), 2270–2297 (2024) https://doi.org/10. 1137/23M1573963

    2024

  9. [9]

    Journal of Computational Physics397, 108819 (2019) https: //doi.org/10.1016/j.jcp.2019.07.018

    Sv¨ ard, M., Nordstr¨ om, J.: On the convergence rates of energy-stable finite- difference schemes. Journal of Computational Physics397, 108819 (2019) https: //doi.org/10.1016/j.jcp.2019.07.018

    work page doi:10.1016/j.jcp.2019.07.018 2019

  10. [10]

    https://arxiv.org/abs/2602.10786

    Glaubitz, J., Iske, A., Lampert, J., ¨Offner, P.: Why summation by parts is not enough (2026). https://arxiv.org/abs/2602.10786

    work page arXiv 2026

  11. [11]

    Journal of Computational and Applied Mathematics237(1), 111–125 (2013) https://doi.org/10.1016/j.cam.2012.07.015

    Hicken, J.E., Zingg, D.W.: Summation-by-parts operators and high-order quadra- ture. Journal of Computational and Applied Mathematics237(1), 111–125 (2013) https://doi.org/10.1016/j.cam.2012.07.015

    work page doi:10.1016/j.cam.2012.07.015 2013

  12. [12]

    Journal of Computational Physics266, 214–239 (2014)

    Del Rey Fern´ andez, D.C., Boom, P.D., Zingg, D.W.: A generalized framework for nodal first derivative summation-by-parts operators. Journal of Computational Physics266, 214–239 (2014)

    2014

  13. [13]

    Journal of Computational Physics356, 410–438 (2018) https://doi.org/10.1016/j.jcp.2017.12.015 25

    Crean, J., Hicken, J.E., Del Rey Fern´ andez, D.C., Zingg, D.W., Carpenter, M.H.: Entropy-stable summation-by-parts discretization of the Euler equations on gen- eral curved elements. Journal of Computational Physics356, 410–438 (2018) https://doi.org/10.1016/j.jcp.2017.12.015 25

    work page doi:10.1016/j.jcp.2017.12.015 2018

  14. [14]

    Calcolo38(4), 173–199 (2001) https: //doi.org/10.1007/s10092-001-8180-4

    Becker, R., Braack, M.: A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo38(4), 173–199 (2001) https: //doi.org/10.1007/s10092-001-8180-4

    work page doi:10.1007/s10092-001-8180-4 2001

  15. [15]

    Journal of Scientific Computing82(50), 35 (2020) https: //doi.org/10.1007/s10915-020-01154-8

    Hicken, J.E.: Entropy-stable, high-order summation-by-parts discretizations with- out interface penalties. Journal of Scientific Computing82(50), 35 (2020) https: //doi.org/10.1007/s10915-020-01154-8

    work page doi:10.1007/s10915-020-01154-8 2020

  16. [16]

    Journal of Computational Physics335, 283–310 (2017) https://doi.org/10.1016/j.jcp.2017.01.042

    Mattsson, K.: Diagonal-norm upwind sbp operators. Journal of Computational Physics335, 283–310 (2017) https://doi.org/10.1016/j.jcp.2017.01.042

    work page doi:10.1016/j.jcp.2017.01.042 2017

  17. [17]

    Journal of Computational Physics422, 109784 (2020) https: //doi.org/10.1016/j.jcp.2020.109784

    Lundgren, L., Mattsson, K.: An efficient finite difference method for the shallow water equations. Journal of Computational Physics422, 109784 (2020) https: //doi.org/10.1016/j.jcp.2020.109784

    work page doi:10.1016/j.jcp.2020.109784 2020

  18. [18]

    Journal of Computational Physics523, 113624 (2025) https://doi.org/10

    Hew, J.K.J., Duru, K., Roberts, S., Zoppou, C., Ricardo, K.: Strongly stable dual-pairing summation by parts finite difference schemes for the vector invariant nonlinear shallow water equations – i: Numerical scheme and validation on the plane. Journal of Computational Physics523, 113624 (2025) https://doi.org/10. 1016/j.jcp.2024.113624

    work page arXiv 2025

  19. [19]

    Journal of Computational Physics527, 113796 (2025) https://doi.org/10.1016/j.jcp.2025.113796

    Worku, Z.A., Hicken, J.E., Zingg, D.W.: Tensor-product split-simplex summation-by-parts operators. Journal of Computational Physics527, 113796 (2025) https://doi.org/10.1016/j.jcp.2025.113796

    work page doi:10.1016/j.jcp.2025.113796 2025

  20. [20]

    Statistical inverse problems: discretization, model reduction and inverse crimes

    Worku, Z.A., Hicken, J.E., Zingg, D.W.: Very high-order symmetric positive- interior quadrature rules on triangles and tetrahedra. Journal of Computational and Applied Mathematics472, 116782 (2025) https://doi.org/10.1016/j.cam. 2025.116782

    work page doi:10.1016/j.cam 2025

  21. [21]

    Journal of Computational Physics 411, 109380 (2020) https://doi.org/10.1016/j.jcp.2020.109380

    Marchildon, A.L., Zingg, D.W.: Optimization of multidimensional diagonal-norm summation-by-parts operators on simplices. Journal of Computational Physics 411, 109380 (2020) https://doi.org/10.1016/j.jcp.2020.109380

    work page doi:10.1016/j.jcp.2020.109380 2020

  22. [22]

    Journal of Scientific Computing6(4), 345–390 (1991) https://doi.org/10.1007/bf01060030

    Dubiner, M.: Spectral methods on triangles and other domains. Journal of Scientific Computing6(4), 345–390 (1991) https://doi.org/10.1007/bf01060030

    work page doi:10.1007/bf01060030 1991

  23. [23]

    Comptes Rendus Hebdomadaires des Seances de l‘Academie des Sciences 245(26), 2459–2461 (1957)

    Proriol, J.: Sur une famille de polynomes ` a deux variables orthogonaux dans un triangle. Comptes Rendus Hebdomadaires des Seances de l‘Academie des Sciences 245(26), 2459–2461 (1957)

    1957

  24. [24]

    In: Theory and Application of Special Functions, pp

    Koornwinder, T.: Two-variable analogues of the classical orthogonal polynomials. In: Theory and Application of Special Functions, pp. 435–495. Academic Press, New York (1975)

    1975

  25. [25]

    Journal of Computational 26 Physics532, 113940 (2025) https://doi.org/10.1016/j.jcp.2025.113940

    Hicken, J.E., Kaur, S., Yan, G.: Constructing stable, high-order finite-difference operators on point clouds over complex geometries. Journal of Computational 26 Physics532, 113940 (2025) https://doi.org/10.1016/j.jcp.2025.113940

    work page doi:10.1016/j.jcp.2025.113940 2025

  26. [26]

    NASA Technical Memorandum, 109112 (1994)

    Carpenter, M., Kennedy, C.: Fourth-order 2N-storage Runge-Kutta schemes. NASA Technical Memorandum, 109112 (1994)

    1994

  27. [27]

    Journal of Open Research Software5(1) (2017)

    Rackauckas, C., Nie, Q.: DifferentialEquations.jl–a performant and feature-rich ecosystem for solving differential equations in Julia. Journal of Open Research Software5(1) (2017)

    2017

  28. [28]

    Journal of Computational Physics 140(1), 122–147 (1998) https://doi.org/10.1006/jcph.1998.5884

    Liu, Y., Vinokur, M.: Exact integrations of polynomials and symmetric quadra- ture formulas over arbitrary polyhedral grids. Journal of Computational Physics 140(1), 122–147 (1998) https://doi.org/10.1006/jcph.1998.5884

    work page doi:10.1006/jcph.1998.5884 1998

  29. [29]

    Springer, New York (2008) 27

    Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algo- rithms, Analysis, and Applications. Springer, New York (2008) 27

    2008