Pith. sign in

REVIEW 3 major objections 4 minor 62 references

High order global flux schemes for general steady state preservation of shallow water moment equations with non-conservative products

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read High-order WENO finite volume schemes preserve all steady states of shallow water moment equations by reconstructing a global flux that absorbs source terms and non-conservative products, with no closed-form equilibria needed.

desk verdict Useful extension of flux globalization to non-conservative products, but the 'fully well-balanced' claim overreaches: only lake-at-rest is exactly preserved, while moving equilibria are kept to design-order accuracy. read the letter →

arxiv 2507.00573 v1 pith:T2DZ4Q4R submitted 2025-07-01 math.NA cs.NA

Mirco Ciallella , Julian Koellermeier This is my paper
classification math.NAcs.NA MSC 65M0835L6565M20
keywords globalfluxmethodWENOwell-balancedmovingequilibriashallowwatermomentequationsnon-conservativeproductslakeatrestfinitevolume
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that a numerical scheme can preserve every steady state of the shallow water moment equations — reduced free-surface flow models whose velocity varies with depth — without knowing any steady state in advance. The extra moment equations contain non-conservative products, which make closed-form steady states impossible to derive for the second-order and hyperbolic variants, and practically important flows are usually small perturbations of such steady states, so a scheme that destroys them produces spurious waves. The proposed mechanism is flux globalization: source terms and non-conservative products are absorbed into one global flux whose constancy characterizes every steady state, and the scheme reconstructs that global flux with high-order WENO instead of reconstructing the conserved variables. If the claim is right, a single code, unchanged, keeps lake-at-rest at machine precision and moving equilibria at the reconstruction's nominal order across the whole model family, including SWME2 and HSWME2, for which no analytical steady state exists.

What carries the argument

The central object is the global flux $G(U,x) = F(U) + R(U,x)$, in which the piece $R$ collects the integrals of all source terms and non-conservative products. In the quasi-conservative system $\partial_t U + \partial_x G = 0$, a steady state is exactly a state with $G$ constant, so a numerical flux built from $G$ alone has zero dissipation at equilibrium and the scheme is well-balanced automatically, whatever the steady state is. Three pieces carry the argument: a high-order Gauss quadrature that evaluates $R$ from WENO-reconstructed variables, with the derivative in $B(U)\partial_x U$ approximated by differentiating the Lagrange interpolant of those variables; interface jumps of $R$ fixed along a linear path (Equations (59) and (63)), the unique choice the paper shows to yield exact lake-at-rest preservation; and WENO reconstruction applied to cell averages of $G$ rather than of $U$, so that at equilibrium all reconstructed values coincide and the dissipation vanishes.

What would settle it

Run the supercritical SWME1 configuration of Section 6.2, but take as the reference not the analytic equilibrium from Equation (23) but the steady state computed by the same scheme on a much finer mesh; the coarse-mesh errors should converge at the nominal WENO order. Apply the same fine-mesh-reference test to SWME2 or HSWME2, for which no analytic equilibrium exists: if WENO5 errors on nested meshes drop at roughly fifth order, moving equilibria of models without closed-form steady states are indeed preserved to the reconstruction order, and if the error stagnates at a low-order plateau, they are not. A second probe targets the path choice directly: evolve a genuinely discontinuous steady state and check whether the discrete state the scheme preserves is the physically expected one, since the linear path is then the only convention available.

Watch

Extended reading notes

Core claim

The paper claims that flux globalization — rewriting the balance law $\partial_t U + \partial_x F(U) = B(U)\partial_x U + S(U,x)$ as the quasi-conservative system $\partial_t U + \partial_x G = 0$ with $G(U,x) = F(U) + R(U,x)$ — extends to systems with non-conservative products and yields fully well-balanced high-order WENO finite volume schemes for the whole family of shallow water moment models, with no analytical steady states required. The integral parts $R$ absorb the non-conservative products and the source terms; they are computed by a high-order quadrature that uses the WENO-reconstructed variables inside the products, and their jumps across cell interfaces are defined by integrating along the linear path (Equations (59) and (63)), the same convention used by path-conservative schemes. Since a steady state is exactly a state with $G = \mathrm{const}$, any numerical flux that depends only on the global flux has vanishing dissipation at equilibrium, and the paper supplies an upwind flux and a cheaper central flux with this property. The scheme is proved to preserve lake-at-rest exactly (Proposition 4.4), converges at nominal order on supercritical and subcritical moving equilibria of SWME1 against the analytic solution obtained from Equation (23), and applies without modification to SWLME2, HSWME2 and SWME2, capturing small perturbations even though their steady states have no closed form.

Load-bearing premise

The load-bearing assumption is that every steady state is exactly a state of constant global flux, and that joining the left and right states across each cell interface by a straight line is the physically correct path for all steady states — a property proved in Appendix A only for lake-at-rest, while for the moving equilibria of the models without closed-form steady states it rests on the perturbation tests of Section 6.3.

Editorial extensions

If this is right

  • Lake-at-rest solutions are preserved to machine precision for every reconstruction order tested (WENO1, WENO3, WENO5) and for both numerical global fluxes, as predicted by Proposition 4.4.
  • On smooth moving equilibria of SWME1, where the exact equilibrium follows from Equation (23), the scheme reaches its nominal order of accuracy, with GF-WENO5 close to fifth order and even the piecewise-constant GF-WENO1 superconverging to second order on stationary problems.
  • The same scheme, with only the model flux and matrices exchanged, applies to SWLME2, HSWME2 and SWME2, where no closed-form steady states exist, and resolves small perturbations of these equilibria — including the double-bump waves of HSWME2 — with errors comparable to those of the first-order model.
  • Because the numerical flux depends only on the global flux, the well-balanced property is independent of the reconstruction order, the numerical flux, and the averaged state used for dissipation, as stated in Remark 4.1.

Reading between the lines

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

  • The 'absorb everything into G, reconstruct G, jump along a linear path' recipe is model-agnostic, so the same construction should transfer to other balance laws with non-conservative products — two-layer shallow water, bedload transport, and multi-phase or kinetic moment models — without re-deriving their equilibria, whenever steady states are characterized by constant G.
  • The paper proves exact preservation only for lake-at-rest; the moving-equilibrium evidence for SWME2 and HSWME2 is perturbation experiments against numerically computed equilibria, so a manufactured steady state on smooth bathymetry, checked on nested meshes, would confirm the preservation order for non-lake equilibria of models without closed-form equilibria.
  • The linear interface path is the point where physics enters as a numerical convention: Equation (59) is the unique jump that yields lake-at-rest well-balancing, which means the scheme preserves the steady states selected by the linear path, and a genuinely discontinuous equilibrium could in principle be preserved with the wrong discrete state if that path is not the physical one.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes a high-order WENO finite volume method for shallow water moment equations (SWME1, SWME2, HSWME2, SWLME2) based on the flux-globalization approach. The system is rewritten in quasi-conservative form with a global flux G that includes integrals of both source terms and non-conservative products, and the numerical fluxes are defined only in terms of reconstructed values of G. The authors claim that this family of schemes is fully well-balanced and preserves all steady states without a priori knowledge of analytical steady states. They prove exact preservation for the lake-at-rest solution in Appendix A, verify it to machine precision in Tables 1 and 2, and demonstrate high-order convergence for SWME1 supercritical and subcritical moving equilibria in Tables 3 through 6. For SWME2 and HSWME2, where no closed-form steady states exist, only perturbation tests are provided in Section 6.3.

Significance. The intended contribution is significant: if the full 'preserve all steady states' claim were established, the method would constitute a general recipe for constructing well-balanced high-order schemes for balance laws with non-conservative products, extending the earlier global-flux framework of [29] to a much broader class of models. The provided lake-at-rest preservation proof, the machine-precision results of Tables 1 and 2, and the clean design-order convergence of Tables 3 to 6 are valuable and constitute a solid numerical contribution. The perturbation experiments across four moment models (Section 6.3) further demonstrate the flexibility of the approach. However, as detailed in the major comments, the exactness claim is currently supported only for lake-at-rest; for moving equilibria the numerical evidence is consistent with high-order asymptotic well-balancedness, not exact preservation. With the claims appropriately revised and some additional analysis or evidence, the paper would be a useful contribution to the well-balanced scheme literature.

major comments (3)
  1. [Abstract and §4.2 (Remark 4.3), Tables 3–6] The phrase 'fully well-balanced' and the assertion that the method 'preserves all steady states' are stronger than what is proven. The only exact-preservation proof is for lake-at-rest (Appendix A), which relies on the special reconstruction of η and b and the interface jump (59). For smooth moving equilibria, the global flux quadrature in Eqs. (44)–(45) is only asymptotically exact: on an exact steady state, the computed cell averages of G oscillate at truncation level, the WENO reconstruction of G is not exactly constant, and the numerical flux difference in Eq. (39) does not vanish. The paper's own convergence tables confirm this: in Tables 3 and 4, the L2 error in h for GF-WENO5 at Ne=100 is about 8.5e-9 and decreases at the design order, while only hum, which is directly set by the first component of G, is at round-off. The scheme is therefore exactly well-balanced for lake-at-rest and only asymptotically well-balanced of order p for smooth moving equilibria. I recommend rewording the claims accordingly and, if possible, adding a quantitative statement about the size of the discrete residual at a steady state.
  2. [§4.4, Eq. (63)] The treatment of non-conservative products at cell interfaces uses the linear path, leading to the jump formula in Eq. (63). This path choice is a genuine modeling assumption for discontinuous steady states and is not addressed by the lake-at-rest proof in Appendix A. Since the paper claims preservation of 'all steady states' in general hyperbolic balance laws, the path-conservative ambiguity is load-bearing. The authors should either restrict the claim to smooth steady states, or discuss the path dependence (e.g., with reference to the path-conservative framework [20]) and demonstrate, for at least one discontinuous equilibrium, that the linear path gives the physically relevant discrete solution.
  3. [§6.3] For SWME2 and HSWME2, no analytical steady states are available, and the numerical experiments only show evolution of a perturbation around a numerically computed equilibrium. This does not directly test whether the scheme preserves the discrete steady state over time. To substantiate the claim that the method is applicable without prior knowledge of steady states, I suggest adding a self-convergence test: compute the steady state on a sequence of refined meshes and compare the resulting discrete equilibria, or monitor the time residual of an unperturbed simulation to show it remains at the expected truncation level rather than growing.
minor comments (4)
  1. [Eq. (60) and the surrounding text] There is a typo in Eq. (60): the integral expression contains a stray 'dx' after the second term, and the path definition reads 'Ψ(0; ηL, ηR) = ηL, Ψ(0; ηL, ηR) = ηR' where the second occurrence should be 'Ψ(1; ηL, ηR) = ηR'.
  2. [§4.1] The sentence 'we use a piecewise polynomial reconstruction of the integral term R' is misleading, because in Eq. (45) the reconstruction is of the conservative variables at quadrature points and R is computed by quadrature, not independently reconstructed. Please rephrase to describe the actual procedure.
  3. [Remark 4.2] The phrase 'we starts the integration from the ghost cells' contains a subject-verb agreement error; it should be 'we start the integration'.
  4. [§6.2, final paragraph] The sentence 'the reader is refer to [29]' should be 'the reader is referred to [29]'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the well-balanced property is a designed consequence of flux globalization, and the moving-equilibrium tests are checked against an independently derived analytical steady state.

full rationale

The derivation chain is not circular. The scheme is built on the flux-globalization identity G=F+R, which is defined so that any smooth steady state of the balance law satisfies ∂xG=0, and the finite volume update (39) then has vanishing residual whenever the reconstructed global flux is constant; Remark 4.3 makes this explicit. This is a designed sufficient condition, not an assumption that already contains the target conclusion 'the method preserves all steady states'. The nontrivial content, exact preservation of lake-at-rest, is proved in Appendix A by a telescoping identity using the reconstructed η and b and the jump definition (59), and it is verified at round-off level in Tables 1-2. For moving equilibria the paper does not prove exact preservation; Tables 3-6 show convergence of h and hα1 to the analytical steady state at design order rather than at machine precision, so the Abstract's wording 'preserve all steady states' is broader than what is established. That is an overclaim or rigor issue, not circularity. The reference moving equilibria are obtained independently from the model equations via Eqs. (20)-(23), not from the numerical scheme, and no parameters are fitted to the validation data; WENO weights, Gauss-Legendre quadrature, and DeC time integration are standard ingredients. Self-citations [29] and [43] provide the earlier global-flux framework and the SWME steady-state characterization, but the lake-at-rest preservation proof is reproduced in the paper and the SWME1 equilibrium formula is a closed-form consequence of the model rather than an unverified premise. The 'only path' uniqueness claim in Remark 4.5 is asserted and would benefit from a fuller proof, but that is a path-consistency ambiguity, not a reduction of the paper's central claims to their inputs. Overall, no step in the claimed derivation is equivalent to its own input by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The scheme introduces no fitted constants and no new physical entities. It relies on the model equations taken from prior literature, standard WENO and DeC background, and a path-conservative interpretation of non-conservative products. The test parameters (gravity g, friction parameters nu and lambda, bathymetry b(x)) are model inputs and not tuned to achieve the reported results.

assumptions (5)
  • domain assumption The shallow water moment models SWME1, SWME2, HSWME2, SWLME2 as recalled in Section 3 are valid reduced-order models and are hyperbolic for the states used in the tests.
    The paper builds on model derivations from [47,44,43] and notes that SWME2 may lose hyperbolicity for large moments (Section 3.2, Figure 3); the upwind numerical flux (40) requires a real eigenstructure, so the tests implicitly assume the states lie in the hyperbolic region.
  • domain assumption At steady state the global flux G in Equations (24), (28), (32), and (36) is constant, and this constancy characterizes equilibria of the original system including non-conservative products.
    This equivalence is central to Remark 4.3 and to the well-balanced construction. It is exact for smooth solutions of the quasi-conservative form (7), but for discontinuous states it depends on the chosen path, which is discussed in Section 4.4.
  • domain assumption The linear path used for interface jumps in Equations (59) and (63) is the correct path, and it is the only path that preserves the lake-at-rest equilibrium.
    Section 4.4 and Remark 4.5 adopt path-conservative theory from [20] with a linear path for eta and for the moment variables; Appendix A proves the well-balanced property for lake-at-rest only for this path choice.
  • standard math Standard WENO reconstruction properties hold: a constant field is reconstructed exactly and the quadrature for cell-averaged global fluxes is consistent and high-order.
    Section 4.2 uses classical WENO from [40,6] with linear weights and smoothness indicators; Remark 4.3 states that constants are reproduced exactly, which is the mechanism behind the well-balanced property.
  • standard math The DeC time integration attains the same order as the spatial discretization and does not pollute the steady-state preservation results.
    Section 5 uses the DeC method from [30,29]; the convergence tables confirm the expected orders, so the time discretization is not limiting in the reported tests.

how reviews work

0 comments
Cite this review

Pith. "Pith review of High order global flux schemes for general steady state preservation of shallow water moment equations with non-conservative products." pith.science (2026). https://pith.science/paper/T2DZ4Q4R

@misc{pith2026250700573,
  author       = {Pith},
  title        = {Pith review of: High order global flux schemes for general steady state preservation of shallow water moment equations with non-conservative products},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T2DZ4Q4R}},
  note         = {Machine review of arXiv:2507.00573}
}
read the original abstract

Shallow water moment equations are reduced-order models for free-surface flows that allow to represent vertical variations of the velocity profile at the expense of additional evolution equations for a number of additional variables, so called moments. This introduces non-linear non-conservative products in the system, which make the analytical characterization of steady states much harder if not impossible. The lack of analytical steady states poses a challenge for the design of well-balanced schemes, which aim at preserving such steady states as crucial in many applications. In this work, we present a family of fully well-balanced, high-order WENO finite volume methods for general hyperbolic balance laws with non-conservative products like the shallow water moment equations, for which no analytical steady states are available. The schemes are based on the flux globalization approach, in which both source terms and non-conservative products are integrated with a tailored high order quadrature in the divergence term. The resulting global flux is then reconstructed instead of the conservative variables to preserve all steady states. Numerical tests show the optimal convergence of the method and a significant error reduction for steady state solutions. Furthermore, we provide a numerical comparison of perturbed steady states for different families of shallow water moment equations, which illustrates the flexibility of our method that is valid for general equations without prior knowledge of steady states.

Figures

Figures reproduced from arXiv: 2507.00573 by the authors.

Figure 1
Figure 1. Shallow water equations: model variables. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Vertically varying velocity profiles with a change of sign (left), and without a change of [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Second order shallow water moment model: loss of hyperbolicity (blue) and hyperbolic [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Small perturbation of lake at rest computed with GF-WENO1 (green), GF-WENO3 [PITH_FULL_IMAGE:figures/full_fig_p024_4.png]
Figure 5
Figure 5. Figure 5: Small perturbation of lake at rest computed with GF-WENO1 (green), GF-WENO3 [PITH_FULL_IMAGE:figures/full_fig_p025_5.png]
Figure 6
Figure 6. Figure 6: Supercritical case without friction computed with GF-WENO5 for the SWME1 model: [PITH_FULL_IMAGE:figures/full_fig_p027_6.png]
Figure 7
Figure 7. Figure 7: Subcritical case without friction computed with GF-WENO5 for the SWME1 model: [PITH_FULL_IMAGE:figures/full_fig_p027_7.png]
Figure 8
Figure 8. Figure 8: Convergence analysis for supercritical (left) and subcritical (right) moving equilibria [PITH_FULL_IMAGE:figures/full_fig_p031_8.png]
Figure 9
Figure 9. Figure 9: Small perturbation of supercritical flow without friction computed with GF-WENO1 [PITH_FULL_IMAGE:figures/full_fig_p032_9.png]
Figure 10
Figure 10. Figure 10: Small perturbation of supercritical flow without friction computed with GF-WENO1 [PITH_FULL_IMAGE:figures/full_fig_p033_10.png]
Figure 11
Figure 11. Figure 11: Supercritical case with friction computed with GF-WENO5 for the SWME2 model: [PITH_FULL_IMAGE:figures/full_fig_p034_11.png]
Figure 12
Figure 12. Figure 12: Small perturbation of supercritical steady state with friction computed with the GF [PITH_FULL_IMAGE:figures/full_fig_p035_12.png]
Figure 13
Figure 13. Figure 13: Small perturbation of supercritical steady state with friction computed with the GF [PITH_FULL_IMAGE:figures/full_fig_p036_13.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

62 extracted references · 61 canonical work pages

  1. [29]

    Ciallella, D

    M. Ciallella, D. Torlo, and M. Ricchiuto. Arbitrary high order weno finite volume scheme with flux globalization for moving equilibria preservation. Journal of Scientific Computing, 96(2):53, 2023

    work page 2023

  2. [20]

    Castro, T

    M. Castro, T. Morales de Luna, and C. Par´ es. Chapter 6 - Well-Balanced Schemes and Path-Conservative Numerical Methods. In R. Abgrall and C.-W. Shu, editors, Handbook of Numerical Methods for Hyperbolic Problems , volume 18 of Handbook of Numerical Analysis, pages 131 – 175. Elsevier, 2017

    work page 2017

  3. [1]

    R. Abgrall. High order schemes for hyperbolic problems using globally continuous ap- proximation and avoiding mass matrices. Journal of Scientific Computing , 73(2-3):461– 494, 2017

    work page 2017

  4. [2]

    Abgrall, E

    R. Abgrall, E. Le M´ el´ edo, P.¨Offner, and D. Torlo. Relaxation Deferred Correction Methods and their Applications to Residual Distribution Schemes. The SMAI Journal of computational mathematics , 8:125–160, 2022

    work page 2022

  5. [3]

    Alcrudo and F

    F. Alcrudo and F. Benkhaldoun. Exact solutions to the riemann problem of the shallow water equations with a bottom step. Computers & Fluids , 30(6):643–671, 2001

    work page 2001

  6. [4]

    Arpaia and M

    L. Arpaia and M. Ricchiuto. r- adaptation for shallow water flows: conservation, well balancedness, efficiency. Computers & Fluids , 160:175–203, 2018

    work page 2018

  7. [5]

    Audusse, F

    E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein, and B. t. Perthame. A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM Journal on Scientific Computing , 25(6):2050–2065, 2004

    work page 2004

  8. [6]

    D. S. Balsara and C.-W. Shu. Monotonicity preserving weighted essentially non- oscillatory schemes with increasingly high order of accuracy. Journal of Computational Physics, 160(2):405–452, 2000

    work page 2000

Show all 62 references
  1. [7]

    Barsukow, M

    W. Barsukow, M. Ciallella, M. Ricchiuto, and D. Torlo. Genuinely multi-dimensional stationarity preserving global flux finite volume formulation for nonlinear hyperbolic pdes. arXiv preprint arXiv:2506.21700 , 2025

    2025

  2. [8]

    Barsukow, M

    W. Barsukow, M. Ricchiuto, and D. Torlo. Structure preserving nodal continuous fi- nite elements via global flux quadrature. Numerical Methods for Partial Differential Equations, 41(1):e23167, 2025

    2025

  3. [9]

    Bauerle, A

    M. Bauerle, A. J. Christlieb, M. Ding, and J. Huang. On the rotational invariance and hyperbolicity of shallow water moment equations in two dimensions. SIAM Journal on Mathematical Analysis, 57(1):1039–1085, 2025. 39

    2025

  4. [10]

    Bermudez and M

    A. Bermudez and M. E. Vazquez. Upwind methods for hyperbolic conservation laws with source terms. Computers & Fluids , 23(8):1049–1071, 1994

    1994

  5. [11]

    Bernetti, V

    R. Bernetti, V. A. Titarev, and E. F. Toro. Exact solution of the riemann problem for the shallow water equations with discontinuous bottom geometry.Journal of Computational Physics, 227(6):3212–3243, 2008

    2008

  6. [12]

    Berthon and C

    C. Berthon and C. Chalons. A fully well-balanced, positive and entropy-satisfying Godunov-type method for the shallow-water equations. Mathematics of Computation , 85(299):1281–1307, 2016

    2016

  7. [13]

    Bi and E

    Q. Bi and E. A. Toorman. Mixed-sediment transport modelling in scheldt estuary with a physics-based bottom friction law. Ocean Dynamics, 65:555–587, 2015

    2015

  8. [14]

    Bollermann, G

    A. Bollermann, G. Chen, A. Kurganov, and S. Noelle. A well-balanced reconstruction of wet/dry fronts for the shallow water equations. Journal of Scientific Computing , 56(2):267–290, 2013

    2013

  9. [15]

    Caballero-C´ ardenas, I

    C. Caballero-C´ ardenas, I. G´ omez-Bueno, A. Del Grosso, J. Koellermeier, and T. M. de Luna. A semi-implicit exactly fully well-balanced relaxation scheme for the shal- low water linearized moment equations. Computer Methods in Applied Mechanics and Engineering, 437:117788, 2025

    2025

  10. [16]

    Canestrelli, A

    A. Canestrelli, A. Siviglia, M. Dumbser, and E. F. Toro. Well-balanced high-order centred schemes for non-conservative hyperbolic systems. applications to shallow water equations with fixed and mobile bed. Advances in Water Resources , 32(6):834–844, 2009

    2009

  11. [17]

    Y. Cao, Q. Huang, J. Koellermeier, A. Kurganov, and Y. Liu. Flux globalization based well-balanced path-conservative central-upwind schemes for shallow water linearized moment equations. arXiv preprint arXiv:2505.23144 , 2025

    2025 arXiv

  12. [18]

    M. G. Carlino and E. Gaburro. Well balanced finite volume schemes for shallow water equations on manifolds. Applied Mathematics and Computation , 441:127676, 2023

    2023

  13. [19]

    Caselles, R

    V. Caselles, R. Donat, and G. Haro. Flux-gradient and source-term balancing for certain high resolution shock-capturing schemes. Computers & Fluids , 38(1):16–36, 2009

    2009

  14. [21]

    M. J. Castro, J. A. Garcıa-Rodrıguez, J. M. Gonz´ alez-Vida, J. Macıas, C. Par´ es, and M. E. V´ azquez-Cend´ on. Numerical simulation of two-layer shallow water flows through channels with irregular geometry. Journal of Computational Physics , 195(1):202–235, 2004. 40

    2004

  15. [22]

    M. J. Castro and C. Par´ es. Well-balanced high-order finite volume methods for systems of balance laws. Journal of Scientific Computing , 82(2):48, 2020

    2020

  16. [23]

    Cheng, A

    Y. Cheng, A. Chertock, M. Herty, A. Kurganov, and T. Wu. A new approach for designing moving-water equilibria preserving schemes for the shallow water equations. Journal of Scientific Computing , 80(1):538–554, 2019

    2019

  17. [24]

    Chertock, S

    A. Chertock, S. Cui, A. Kurganov, S ¸. N.¨Ozcan, and E. Tadmor. Well-balanced schemes for the Euler equations with gravitation: Conservative formulation using global fluxes. Journal of Computational Physics , 358:36–52, 2018

    2018

  18. [25]

    Chertock, A

    A. Chertock, A. Kurganov, X. Liu, Y. Liu, and T. Wu. Well-balancing via flux globaliza- tion: Applications to shallow water equations with wet/dry fronts. Journal of Scientific Computing, 90(1):1–21, 2022

    2022

  19. [26]

    Christlieb, B

    A. Christlieb, B. Ong, and J.-M. Qiu. Integral deferred correction methods constructed with high order Runge-Kutta integrators. Mathematics of Computation , 79(270):761– 783, 2010

    2010

  20. [27]

    Ciallella, L

    M. Ciallella, L. Micalizzi, V. Michel-Dansac, P. ¨Offner, and D. Torlo. A high-order, fully well-balanced, unconditionally positivity-preserving finite volume framework for flood simulations. GEM-International Journal on Geomathematics , 16(1):1–33, 2025

    2025

  21. [28]

    Ciallella, L

    M. Ciallella, L. Micalizzi, P. ¨Offner, and D. Torlo. An arbitrary high order and positivity preserving method for the shallow water equations. Computers & Fluids , 247:105630, 2022

    2022

  22. [30]

    J. W. Daniel, V. Pereyra, and L. L. Schumaker. Iterated deferred corrections for initial value problems. Acta Cient. Venezolana, 19:128–135, 1968

    1968

  23. [31]

    Delestre, C

    O. Delestre, C. Lucas, P.-A. Ksinant, F. Darboux, C. Laguerre, T.-N.-T. Vo, F. James, and S. Cordier. Swashes: a compilation of shallow water analytic solutions for hydraulic and environmental studies. International Journal for Numerical Methods in Fluids , 72(3):269–300, 2013

    2013

  24. [32]

    M. C. D ´ ıaz, J. A. L´ opez-Garc ´ ıa, and C. Par´ es. High order exactly well-balanced numeri- cal methods for shallow water systems. Journal of Computational Physics , 246:242–264, 2013

    2013

  25. [33]

    Donat and A

    R. Donat and A. Mart ´ ınez-Gavara. Hybrid second order schemes for scalar balance laws. Journal of Scientific Computing , 48:52–69, 2011. 41

    2011

  26. [34]

    A. Dutt, L. Greengard, and V. Rokhlin. Spectral Deferred Correction Methods for Ordinary Differential Equations. BIT Numerical Mathematics , 40(2):241–266, 2000

    2000

  27. [35]

    J. M. Gallardo, C. Par´ es, and M. Castro. On a well-balanced high-order finite vol- ume scheme for shallow water equations with topography and dry areas. Journal of Computational Physics , 227(1):574–601, 2007

    2007

  28. [36]

    Garres-D ´ ıaz, M

    J. Garres-D ´ ıaz, M. Castro, J. Koellermeier, and T. Morales de Luna. Shallow water moment models for bedload transport problems. Communications in Computational Physics, 30(3):903–941, 2021

    2021

  29. [37]

    Gasc´ on and J

    L. Gasc´ on and J. Corber´ an. Construction of second-order TVD schemes for nonhomoge- neous hyperbolic conservation laws. Journal of Computational Physics , 172(1):261–297, 2001

    2001

  30. [38]

    Han Veiga, P

    M. Han Veiga, P. ¨Offner, and D. Torlo. DeC and ADER: Similarities, Differences and a Unified Framework. Journal of Scientific Computing , 87(1):1–35, 2021

    2021

  31. [39]

    Izgin and P

    T. Izgin and P. ¨Offner. A study of the local dynamics of modified patankar dec and higher order modified patankar–rk methods. ESAIM: Mathematical Modelling and Nu- merical Analysis, 57(4):2319–2348, 2023

    2023

  32. [40]

    Jiang and C.-W

    G.-S. Jiang and C.-W. Shu. Efficient implementation of weighted ENO schemes. Journal of Computational Physics , 126(1):202–228, 1996

    1996

  33. [41]

    Kazolea, C

    M. Kazolea, C. Par´ es, and M. Ricchiuto. Approximate well-balanced weno finite dif- ference schemes using a global-flux quadrature method with multi-step ode integrator weights. Computers & Fluids , 296:106646, 2025

    2025

  34. [42]

    Kim and R

    J. Kim and R. J. LeVeque. Two-layer shallow water system and its applications. Hy- perbolic problems: theory, numerics and applications , 67:737–743, 2009

    2009

  35. [43]

    Koellermeier and E

    J. Koellermeier and E. Pimentel-Garc ´ ıa. Steady states and well-balanced schemes for shallow water moment equations with topography. Applied Mathematics and Computa- tion, 427:127166, 2022

    2022

  36. [44]

    Koellermeier and M

    J. Koellermeier and M. Rominger. Analysis and numerical simulation of hyperbolic shal- low water moment equations. Communications in Computational Physics , 28(3):1038– 1084, 2020

    2020

  37. [45]

    Koellermeier, R

    J. Koellermeier, R. P. Schaerer, and M. Torrilhon. A framework for hyperbolic approx- imation of kinetic equations using quadrature-based projection methods. Kinetic and Related Models, 7(3):531–549, 2014

    2014

  38. [46]

    Koellermeier and M

    J. Koellermeier and M. Torrilhon. Numerical Study of Partially Conservative Moment Equations in Kinetic Theory. Communications in Computational Physics , 21(4):981– 1011, 2017. 42

    2017

  39. [47]

    Kowalski and M

    J. Kowalski and M. Torrilhon. Moment approximations and model cascades for shallow flow. Communications in Computational Physics , 25(3):669–702, 2019

    2019

  40. [48]

    G. A. Lawrence. The hydraulics of steady two-layer flow over a fixed obstacle. Journal of Fluid Mechanics , 254:605–633, 1993

    1993

  41. [49]

    Liu, C.-W

    Y. Liu, C.-W. Shu, and M. Zhang. Strong stability preserving property of the deferred correction time discretization. Journal of Computational Mathematics , pages 633–656, 2008

    2008

  42. [50]

    Mac ´ ıas, M

    J. Mac ´ ıas, M. J. Castro, S. Ortega, and J. M. Gonz´ alez-Vida. Performance assessment of tsunami-hysea model for nthmp tsunami currents benchmarking. field cases. Ocean Modelling, 152:101645, 2020

    2020

  43. [51]

    Mantri, P

    Y. Mantri, P. ¨Offner, and M. Ricchiuto. Fully well-balanced entropy controlled discon- tinuous galerkin spectral element method for shallow water flows: global flux quadrature and cell entropy correction. Journal of Computational Physics , 498:112673, 2024

    2024

  44. [52]

    Micalizzi, M

    L. Micalizzi, M. Ricchiuto, and R. Abgrall. Novel well-balanced continuous interior penalty stabilizations. Journal of Scientific Computing , 100(1):14, 2024

    2024

  45. [53]

    Micalizzi and D

    L. Micalizzi and D. Torlo. A new efficient explicit deferred correction framework: anal- ysis and applications to hyperbolic pdes and adaptivity. Communications on Applied Mathematics and Computation , 6(3):1629–1664, 2024

    2024

  46. [54]

    Micalizzi, D

    L. Micalizzi, D. Torlo, and W. Boscheri. Efficient iterative arbitrary high-order meth- ods: an adaptive bridge between low and high order. Communications on Applied Mathematics and Computation , 7(1):40–77, 2025

    2025

  47. [55]

    Michel-Dansac, C

    V. Michel-Dansac, C. Berthon, S. Clain, and F. Foucher. A well-balanced scheme for the shallow-water equations with topography. Computers & Mathematics with Applications, 72(3):568–593, 2016

    2016

  48. [56]

    M. L. Minion. Semi-implicit spectral deferred correction methods for ordinary differen- tial equations. Commun. Math. Sci. , 1(3):471–500, 09 2003

    2003

  49. [57]

    Noelle, N

    S. Noelle, N. Pankratz, G. Puppo, and J. R. Natvig. Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows.Journal of Computational Physics, 213(2):474–499, 2006

    2006

  50. [58]

    ¨Offner and D

    P. ¨Offner and D. Torlo. Arbitrary high-order, conservative and positivity preserving Patankar-type deferred correction schemes. Applied Numerical Mathematics , 153:15– 34, 2020

    2020

  51. [59]

    Pimentel-Garc ´ ıa

    E. Pimentel-Garc ´ ıa. Fully well-balanced methods for shallow water linearized moment model with friction. In XVI International Conference on Hyperbolic Problems: Theory, Numerics, Applications, pages 195–208. Springer, 2022. 43

    2022

  52. [60]

    P. L. Roe. Approximate Riemann solvers, parameter vectors, and difference schemes. Journal of Computational Physics , 43(2):357–372, 1981

    1981

  53. [61]

    M. H. Veiga, L. Micalizzi, and D. Torlo. On improving the efficiency of ader methods. Applied Mathematics and Computation , 466:128426, 2024

    2024

  54. [62]

    Xing and C.-W

    Y. Xing and C.-W. Shu. High order finite difference WENO schemes with the exact con- servation property for the shallow water equations. Journal of Computational Physics , 208(1):206–227, 2005. 44

    2005

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.