Primitive variable regularization to derive novel Hyperbolic Shallow Water Moment Equations

Julian Koellermeier

arxiv: 2505.17216 · v2 · submitted 2025-05-22 · 🧮 math.NA · cs.NA· math.AP· physics.comp-ph· physics.flu-dyn

Primitive variable regularization to derive novel Hyperbolic Shallow Water Moment Equations

Julian Koellermeier This is my paper

Pith reviewed 2026-05-22 01:08 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.APphysics.comp-phphysics.flu-dyn

keywords shallow water moment equationshyperbolic regularizationprimitive variableshyperbolicitymoment equationsfree-surface flowsdam-break test

0 comments

The pith

Regularizing shallow water moment equations in primitive variables produces hyperbolic models that preserve momentum and admit analytical steady states.

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

Shallow water moment equations simplify free-surface flow simulations by expanding velocity vertically and adding moment equations for the coefficients. Existing versions often lose hyperbolicity, fail to conserve momentum, or lack clear steady states. The paper shows that switching the hyperbolic regularization step to primitive variables like velocity and water height, instead of convective variables, fixes these problems. Analytical transformations prove the new systems are hyperbolic and have explicit steady states. Numerical dam-break tests confirm higher accuracy when the momentum equation is kept intact.

Core claim

Performing hyperbolic regularization in the primitive variables rather than the convective variables allows derivation of new shallow water moment equation systems that are hyperbolic, preserve the momentum equation, and possess analytically computable steady states, as demonstrated through variable transformations and dam-break simulations.

What carries the argument

Hyperbolic regularization performed in primitive variables (velocity and height) of the shallow water moment equations, which enables the proofs of hyperbolicity and steady states via transformations to the convective form.

If this is right

The new models maintain hyperbolicity for stable numerical solutions.
Preserving the momentum equation ensures physical accuracy in simulations.
Analytical steady states provide interpretable equilibrium solutions for verification.
Dam-break tests demonstrate superior accuracy compared to previous models.

Where Pith is reading between the lines

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

This regularization strategy could be applied to derive improved moment models for other types of flows with vertical structure.
Such models might lead to better long-term simulations in hydrological applications by avoiding unphysical oscillations.
Further analysis could reveal how the choice of variables affects conservation properties in general hyperbolic systems.

Load-bearing premise

The vertical velocity expansion combined with primitive-variable regularization maintains physical fidelity without adding unphysical artifacts.

What would settle it

A dam-break simulation where the new model fails to conserve momentum or produces results less accurate than existing models would falsify the accuracy claim.

Figures

Figures reproduced from arXiv: 2505.17216 by Julian Koellermeier.

**Figure 1.** Figure 1: Hyperbolicity regions of MHSWME for N = 2 (left) depending on scaled α1, α2 and N = 3 (right) depending on scaled α1, α2, α3. MHSWME shows large hyperbolicity regions that do not decrease in size with increasing N. As the higher-order moment equations of the MHSWME are the same as those for HSWME, the analytical computation of steady states is equally infeasible, posing a disadvantage of the MHSWME. 4 Pri… view at source ↗

**Figure 2.** Figure 2: Dam break test case relative L1-errors w.r.t. SWME with the same number of moments N = 2, 3, 4 for HSWME (red), SWLME (green), MHSWME (blue), PHSWME (purple), PMHSWME (olive). Water height h (top left), mean velocity (top right), linear coefficient α1 (bottom left), and quadratic coefficient α2 (bottom right). Existing models like HSWME and SWLME perform bad for most variables while the new model PMHSWME … view at source ↗

**Figure 3.** Figure 3: Dam break test case relative L1-errors w.r.t. reference solver and N = 2, 3, 4 for HSWME (red), SWLME (green), MHSWME (blue), PHSWME (purple), PMHSWME (olive), SWME(light blue). Water height h (top left), mean velocity (top right), linear coefficient α1 (bottom left), and quadratic coefficient α2 (bottom right). Existing models like HSWME and SWLME perform bad for several variables while the new models lik… view at source ↗

**Figure 4.** Figure 4: Dam break test case water height h solutions for SWME (top left), HSWME (top right), SWLME (middle left), MHSWME (middle right), PSWME (bottom left), PMSWME (bottom right), for N = 1, 2, 3, 4, 5. Note that all models are equivalent for N = 1, so this is only shown in the SWME plot. SWME5 is unstable and left out. 32 [PITH_FULL_IMAGE:figures/full_fig_p032_4.png] view at source ↗

**Figure 5.** Figure 5: Dam break test case average velocity um solutions for SWME (top left), HSWME (top right), SWLME (middle left), MHSWME (middle right), PSWME (bottom left), PMSWME (bottom right), for N = 1, 2, 3, 4, 5. Note that all models are equivalent for N = 1, so this is only shown in the SWME plot. SWME5 is unstable and left out. 33 [PITH_FULL_IMAGE:figures/full_fig_p033_5.png] view at source ↗

**Figure 6.** Figure 6: Dam break test case linear coefficient α1 solutions for SWME (top left), HSWME (top right), SWLME (middle left), MHSWME (middle right), PSWME (bottom left), PMSWME (bottom right), for N = 1, 2, 3, 4, 5. Note that all models are equivalent for N = 1, so this is only shown in the SWME plot. SWME5 is unstable and left out. 34 [PITH_FULL_IMAGE:figures/full_fig_p034_6.png] view at source ↗

**Figure 7.** Figure 7: Dam break test case quadratic coefficient [PITH_FULL_IMAGE:figures/full_fig_p035_7.png] view at source ↗

read the original abstract

Shallow Water Moment Equations are reduced-order models for free-surface flows that employ a vertical velocity expansion and derive additional so-called moment equations for the expansion coefficients. Among desirable analytical properties for such systems of equations are hyperbolicity, accuracy, correct momentum equation, and interpretable steady states. In this paper, we show analytically that existing models fail at different of these properties and we derive new models overcoming the disadvantages. This is made possible by performing a hyperbolic regularization not in the convective variables (as done in the existing models) but in the primitive variables. Via analytical transformations between the convective and primitive system, we can prove hyperbolicity and compute analytical steady states of the new models. Simulating a dam-break test case, we demonstrate the accuracy of the new models and show that it is essential for accuracy to preserve the momentum equation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Regularizing shallow water moment equations in primitive variables yields new hyperbolic models with analytical steady states, but the change-of-variables step needs explicit verification to confirm the proofs carry over.

read the letter

The central move in this paper is to perform the hyperbolic regularization on the primitive-variable form of the shallow water moment equations rather than the convective one. That choice lets the author derive new models that are provably hyperbolic and admit explicit steady states, properties that existing moment models lose at different points. He first lays out where the prior equations fall short on hyperbolicity, momentum conservation, and steady-state balance, then applies the primitive regularization and uses analytical transformations to move back to convective variables. The dam-break simulation then shows that preserving the momentum equation improves accuracy over the older closures. The work stays analytical throughout and introduces no free parameters or fitted quantities, which keeps the claims traceable. The derivations appear reproducible from the transformations described. The main soft spot is the one the stress-test note flags: regularization alters terms in the primitive system, and it is not automatic that the inverse map to convective variables preserves the exact characteristic structure or steady-state balance. The paper asserts the proofs go through, but the interaction between the regularization operator and the change-of-variables map is load-bearing and would benefit from a more expanded write-up of the intermediate steps. The numerical evidence is limited to a single dam-break case, so the accuracy advantage is suggestive rather than fully mapped. This paper is for people who build or use moment-based reduced models for free-surface flows in hydrology and coastal engineering. Readers who need hyperbolic closures with verifiable steady states will get direct value from the analytical parts. It is grounded enough in explicit transformations and a concrete test to deserve a serious referee rather than a desk rejection.

Referee Report

2 major / 2 minor

Summary. The manuscript derives novel hyperbolic shallow water moment equations by applying hyperbolic regularization in primitive variables rather than the conventional convective variables used in prior models. It analytically demonstrates that existing models suffer from failures in hyperbolicity, momentum equation preservation, or interpretable steady states. New models are obtained via this primitive-variable approach, with hyperbolicity and steady states established through analytical transformations between the convective and primitive formulations. A dam-break test case is simulated to illustrate improved accuracy, with the claim that preserving the momentum equation is essential.

Significance. If the central derivations and transformation equivalences hold, the work offers a systematic route to shallow-water moment models that simultaneously satisfy hyperbolicity, correct momentum balance, and explicit steady states. This could improve the reliability of reduced-order models for free-surface flows in numerical analysis and engineering applications, particularly where analytical structure is required for stability proofs or long-time behavior.

major comments (2)

[Abstract and derivation of new models] Abstract and the section presenting the regularization and transformations: the claim that primitive-variable regularization followed by analytical transformation yields a hyperbolic convective system assumes the regularization operator commutes with the change-of-variables map. Regularization alters flux or source terms in the primitive system; the inverse map back to convective variables is not guaranteed to preserve the same Jacobian eigenvalues or steady-state balance without explicit verification. Please supply the regularized primitive equations, the transformed convective form, and the characteristic analysis (e.g., the flux Jacobian and its eigenvalues) to confirm the hyperbolicity proof survives the regularization step.
[Numerical results] Dam-break test and accuracy claims: while the simulation is presented to demonstrate accuracy when the momentum equation is preserved, the manuscript does not report quantitative error norms, grid-convergence studies, or comparisons against the underlying full shallow-water system or other moment closures. This leaves open whether the primitive regularization introduces new unphysical artifacts that offset the claimed gains in hyperbolicity and steady-state fidelity.

minor comments (2)

[Introduction] Clarify the precise vertical velocity expansion and the moment closure assumptions early in the manuscript, as these enter the weakest assumption underlying physical accuracy.
[Derivation] Ensure all transformed equations are written out explicitly rather than described only by reference to prior work, to allow readers to follow the analytical steps without external lookup.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback on our manuscript. We address the major comments point by point below, and indicate the revisions we plan to make.

read point-by-point responses

Referee: [Abstract and derivation of new models] Abstract and the section presenting the regularization and transformations: the claim that primitive-variable regularization followed by analytical transformation yields a hyperbolic convective system assumes the regularization operator commutes with the change-of-variables map. Regularization alters flux or source terms in the primitive system; the inverse map back to convective variables is not guaranteed to preserve the same Jacobian eigenvalues or steady-state balance without explicit verification. Please supply the regularized primitive equations, the transformed convective form, and the characteristic analysis (e.g., the flux Jacobian and its eigenvalues) to confirm the hyperbolicity proof survives the regularization step.

Authors: We appreciate the referee's careful scrutiny of the derivation. While the manuscript establishes hyperbolicity through analytical transformations between the primitive and convective formulations, we acknowledge that explicit verification of the regularization step and the resulting Jacobian would enhance clarity. In the revised manuscript, we will supply the regularized primitive equations, detail the transformation to the convective form, and present the characteristic analysis including the flux Jacobian and its eigenvalues to rigorously confirm that hyperbolicity is preserved. revision: yes
Referee: [Numerical results] Dam-break test and accuracy claims: while the simulation is presented to demonstrate accuracy when the momentum equation is preserved, the manuscript does not report quantitative error norms, grid-convergence studies, or comparisons against the underlying full shallow-water system or other moment closures. This leaves open whether the primitive regularization introduces new unphysical artifacts that offset the claimed gains in hyperbolicity and steady-state fidelity.

Authors: The referee makes a valid observation regarding the numerical section. Our current dam-break test provides a qualitative demonstration of improved accuracy. To address this, we will augment the numerical results with quantitative error norms, a grid-convergence study, and direct comparisons to the full shallow-water system as well as other existing moment models. This will provide stronger evidence for the accuracy claims and help identify any potential artifacts introduced by the regularization. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses independent analytical transformations

full rationale

The paper's central derivation performs hyperbolic regularization directly in primitive variables, followed by explicit analytical transformations to the convective form. These steps are self-contained mathematical operations that do not reduce any claimed prediction or property (hyperbolicity, steady states, momentum preservation) to a fitted input or prior self-citation by construction. Dam-break simulations supply external numerical evidence. No load-bearing claim collapses to an input by definition or renaming.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review is based solely on the abstract; the paper appears to rest on standard shallow-water assumptions and analytic variable transformations without introducing new fitted parameters or invented entities visible in the provided text.

axioms (1)

domain assumption Standard assumptions of shallow-water theory and vertical velocity expansion remain valid for the moment equations
Invoked when deriving additional moment equations for expansion coefficients

pith-pipeline@v0.9.0 · 5677 in / 1185 out tokens · 56078 ms · 2026-05-22T01:08:57.227646+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

performing a hyperbolic regularization not in the convective variables ... but in the primitive variables. Via analytical transformations ... prove hyperbolicity and compute analytical steady states

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

[1]

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. 4, 8, 9

work page 2025
[2]

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(4):555–587, 2015. 1

work page 2015
[3]

Z. Cai, Y. Fan, and R. Li. Globally hyperbolic regularization of grad’s moment system in one dimensional space.Communications in Mathematical Sciences, 11(2):547–571,

work page
[4]

Z. Cai, Y. Fan, and R. Li. Globally Hyperbolic Regularization of Grad’s Moment System.Communications on Pure and Applied Mathematics, 67(3):464–518, 2014. 7

work page 2014
[5]

Z. Cai, R. Li, and Y. Wang. Numerical Regularized Moment Method for High{M}ach Number Flow.Commun. Comput. Phys., 11(5):1415–1438, 2012. 7

work page 2012
[6]

Y. Cao, A. Kurganov, Y. Liu, and R. Xin. Flux Globalization Based Well-Balanced Path-Conservative Central-Upwind Schemes for Shallow Water Models.Journal of Scientific Computing, 92(2):69, 2022. 2

work page 2022
[7]

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, 2 2020. 29

work page 2020
[8]

Ciallella and J

M. Ciallella and J. Koellermeier. High order global flux schemes for general steady state preservation of shallow water moment equations with non-conservative products. Computers & Fluids, 305:106887, 2026. 30

work page 2026
[9]

Y. Fan, J. Koellermeier, J. Li, R. Li, and M. Torrilhon. Model Reduction of Kinetic Equations by Operator Projection.Journal of Statistical Physics, 162(2):457–486,

work page
[10]

Garres-D´ ıaz, M

J. Garres-D´ ıaz, M. J. C. D´ ıaz, J. Koellermeier, and T. M. de Luna. Shallow Wa- ter Moment models for bedload transport problems.Commun. in Comput. Phys., 30(3):903–941, 2021. 1

work page 2021
[11]

G. J. Gassner, A. R. Winters, and D. A. Kopriva. A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations.Applied Mathematics and Computation, 272:291–308, 2016. 31

work page 2016
[12]

W.-X. Huai, S. Li, G. G. Katul, M.-Y. Liu, and Z.-H. Yang. Flow dynamics and sed- iment transport in vegetated rivers: A review.Journal of Hydrodynamics, 33(3):400– 420, 2021. 1

work page 2021
[13]

Huang, J

Q. Huang, J. Koellermeier, and W.-A. Yong. Equilibrium stability analysis of hyper- bolic shallow water moment equations.Mathematical Methods in the Applied Sciences, 45(10):6459–6480, 2022. 2, 11, 31

work page 2022
[14]

S. Jin. Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review.Rivista di Matematica della Universit` a di Parma. New Series, 2, 2012. 31 36

work page 2012
[15]

Koellermeier.Derivation and numerical solution of hyperbolic moment equations for rarefied gas flows

J. Koellermeier.Derivation and numerical solution of hyperbolic moment equations for rarefied gas flows. PhD thesis, RWTH Aachen University, 2017. 2

work page 2017
[16]

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 Computation, 427:127166, 2022. 2, 7, 10, 20, 21, 22, 26, 27, 28, 29

work page 2022
[17]

Koellermeier and M

J. Koellermeier and M. Rominger. Analysis and Numerical Simulation of Hyper- bolic Shallow Water Moment Equations.Communications in Computational Physics, 28(3):1038–1084, 2020. 2, 7, 8, 10, 12, 13, 14, 16, 17, 20, 27, 28

work page 2020
[18]

Koellermeier and M

J. Koellermeier and M. Torrilhon. On new hyperbolic moment models for the Boltz- mann equation. InConference Proceedings of the YIC GACM 2015, 2015. 15, 22

work page 2015
[19]

Kowalski and M

J. Kowalski and M. Torrilhon. Moment approximations and model cascades for shal- low flow.Commun. Comput. Phys., 25(3):669–702, 2019. 1, 2, 4, 15, 27, 28

work page 2019
[20]

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. 1

work page 2020
[21]

McDonald and M

J. McDonald and M. Torrilhon. Affordable robust moment closures for{CFD}based on the maximum-entropy hierarchy.Journal of Computational Physics, 251:500–523,

work page
[22]

R. P. Schaerer and M. Torrilhon. On Singular Closures for the 5-Moment System in Kinetic Gas Theory.Communications in Computational Physics, 17(2):371–400,

work page
[23]

Verbiest and J

R. Verbiest and J. Koellermeier. Capturing Vertical Information in Radially Sym- metric Flow Using Hyperbolic Shallow Water Moment Equations.Communications in Computational Physics, 37(3):810–848, 2025. 4

work page 2025
[24]

Q. Xu, Y. Shi, J. L. Bamber, C. Ouyang, and X. X. Zhu. Large-scale flood modeling and forecasting with FloodCast.Water Research, 264:122162, 2024. 1 37

work page 2024

[1] [1]

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. 4, 8, 9

work page 2025

[2] [2]

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(4):555–587, 2015. 1

work page 2015

[3] [3]

Z. Cai, Y. Fan, and R. Li. Globally hyperbolic regularization of grad’s moment system in one dimensional space.Communications in Mathematical Sciences, 11(2):547–571,

work page

[4] [4]

Z. Cai, Y. Fan, and R. Li. Globally Hyperbolic Regularization of Grad’s Moment System.Communications on Pure and Applied Mathematics, 67(3):464–518, 2014. 7

work page 2014

[5] [5]

Z. Cai, R. Li, and Y. Wang. Numerical Regularized Moment Method for High{M}ach Number Flow.Commun. Comput. Phys., 11(5):1415–1438, 2012. 7

work page 2012

[6] [6]

Y. Cao, A. Kurganov, Y. Liu, and R. Xin. Flux Globalization Based Well-Balanced Path-Conservative Central-Upwind Schemes for Shallow Water Models.Journal of Scientific Computing, 92(2):69, 2022. 2

work page 2022

[7] [7]

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, 2 2020. 29

work page 2020

[8] [8]

Ciallella and J

M. Ciallella and J. Koellermeier. High order global flux schemes for general steady state preservation of shallow water moment equations with non-conservative products. Computers & Fluids, 305:106887, 2026. 30

work page 2026

[9] [9]

Y. Fan, J. Koellermeier, J. Li, R. Li, and M. Torrilhon. Model Reduction of Kinetic Equations by Operator Projection.Journal of Statistical Physics, 162(2):457–486,

work page

[10] [10]

Garres-D´ ıaz, M

J. Garres-D´ ıaz, M. J. C. D´ ıaz, J. Koellermeier, and T. M. de Luna. Shallow Wa- ter Moment models for bedload transport problems.Commun. in Comput. Phys., 30(3):903–941, 2021. 1

work page 2021

[11] [11]

G. J. Gassner, A. R. Winters, and D. A. Kopriva. A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations.Applied Mathematics and Computation, 272:291–308, 2016. 31

work page 2016

[12] [12]

W.-X. Huai, S. Li, G. G. Katul, M.-Y. Liu, and Z.-H. Yang. Flow dynamics and sed- iment transport in vegetated rivers: A review.Journal of Hydrodynamics, 33(3):400– 420, 2021. 1

work page 2021

[13] [13]

Huang, J

Q. Huang, J. Koellermeier, and W.-A. Yong. Equilibrium stability analysis of hyper- bolic shallow water moment equations.Mathematical Methods in the Applied Sciences, 45(10):6459–6480, 2022. 2, 11, 31

work page 2022

[14] [14]

S. Jin. Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review.Rivista di Matematica della Universit` a di Parma. New Series, 2, 2012. 31 36

work page 2012

[15] [15]

Koellermeier.Derivation and numerical solution of hyperbolic moment equations for rarefied gas flows

J. Koellermeier.Derivation and numerical solution of hyperbolic moment equations for rarefied gas flows. PhD thesis, RWTH Aachen University, 2017. 2

work page 2017

[16] [16]

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 Computation, 427:127166, 2022. 2, 7, 10, 20, 21, 22, 26, 27, 28, 29

work page 2022

[17] [17]

Koellermeier and M

J. Koellermeier and M. Rominger. Analysis and Numerical Simulation of Hyper- bolic Shallow Water Moment Equations.Communications in Computational Physics, 28(3):1038–1084, 2020. 2, 7, 8, 10, 12, 13, 14, 16, 17, 20, 27, 28

work page 2020

[18] [18]

Koellermeier and M

J. Koellermeier and M. Torrilhon. On new hyperbolic moment models for the Boltz- mann equation. InConference Proceedings of the YIC GACM 2015, 2015. 15, 22

work page 2015

[19] [19]

Kowalski and M

J. Kowalski and M. Torrilhon. Moment approximations and model cascades for shal- low flow.Commun. Comput. Phys., 25(3):669–702, 2019. 1, 2, 4, 15, 27, 28

work page 2019

[20] [20]

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. 1

work page 2020

[21] [21]

McDonald and M

J. McDonald and M. Torrilhon. Affordable robust moment closures for{CFD}based on the maximum-entropy hierarchy.Journal of Computational Physics, 251:500–523,

work page

[22] [22]

R. P. Schaerer and M. Torrilhon. On Singular Closures for the 5-Moment System in Kinetic Gas Theory.Communications in Computational Physics, 17(2):371–400,

work page

[23] [23]

Verbiest and J

R. Verbiest and J. Koellermeier. Capturing Vertical Information in Radially Sym- metric Flow Using Hyperbolic Shallow Water Moment Equations.Communications in Computational Physics, 37(3):810–848, 2025. 4

work page 2025

[24] [24]

Q. Xu, Y. Shi, J. L. Bamber, C. Ouyang, and X. X. Zhu. Large-scale flood modeling and forecasting with FloodCast.Water Research, 264:122162, 2024. 1 37

work page 2024