Pith. sign in

REVIEW 2 major objections 1 minor 39 references

A semi-Lagrangian method for the polyatomic ESBGK model

T0 review · 2 major / 1 minor · reviewed 2026-06-26 · grok-4.3

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

Pith's one-line read A semi-Lagrangian scheme for the polyatomic ESBGK model converges asymptotically to the compressible Navier-Stokes equations with correct transport coefficients.

desk verdict The paper gives a semi-Lagrangian scheme for polyatomic ESBGK with claimed asymptotic convergence to NS plus correct coefficients, plus usable boundary conditions. read the letter →

arxiv 2606.21191 v1 pith:PSHEMANH submitted 2026-06-19 math.NA cs.NA

Klaas Willems , Erik Arlemark , Giovanni Samaey , Axel Klar This is my paper · ORCID
classification math.NAcs.NA
keywords semi-LagrangianmethodpolyatomicESBGKmodelasymptoticpreservingschemeBGKequationNavier-Stokeslimitkinetictheorynumericalmethodsforhyperbolicsystemsboundaryconditionsequations
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

The paper develops a deterministic grid-based numerical method for the polyatomic ESBGK kinetic model of molecular collisions. Transport is handled by following characteristics to remove time-step restrictions, while the stiff relaxation toward a generalized Gaussian is treated with an implicit multistep method that the BGK structure reduces to an explicit cheap update. The resulting scheme is asymptotic preserving, meaning it recovers the Euler equations at small Knudsen number, and the first-order version recovers the full Navier-Stokes equations with the proper viscosity and heat conductivity. Inflow and outflow boundary conditions adapted to BGK-type models are also given, and the approach is demonstrated on standard test problems.

What carries the argument

Reformulation of the implicit A-stable linear multistep treatment of the BGK relaxation term into an explicit cheap time-stepping scheme that preserves the equilibrium states exactly.

What would settle it

A numerical experiment in the small-Knudsen regime that produces a viscosity or thermal conductivity coefficient differing from the analytic value obtained from the ESBGK model by more than the expected truncation error.

Watch

Extended reading notes

Core claim

The semi-Lagrangian discretization of the polyatomic ESBGK equation is asymptotic preserving and stiffly accurate; the first-order version of the scheme converges in the vanishing Knudsen limit to a consistent discretization of the compressible Navier-Stokes equations that carries the correct transport coefficients derived from the ESBGK collision operator.

Load-bearing premise

The BGK operator's specific structure permits the implicit multistep treatment of relaxation to be rewritten as a cheap explicit update without losing stability or accuracy.

Editorial extensions

If this is right

  • In the hydrodynamic limit the scheme reduces to a consistent discretization of the Euler equations.
  • The first-order scheme recovers the compressible Navier-Stokes equations together with the exact transport coefficients of the polyatomic ESBGK model.
  • The proposed inflow and outflow boundary conditions remain consistent with the underlying kinetic equation.
  • The method remains stable for time steps much larger than the mean collision time.

Reading between the lines

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

  • The same reformulation technique may extend directly to other relaxation-type kinetic models that share an analogous equilibrium projector.
  • Higher-order versions of the scheme would be expected to retain the same asymptotic accuracy provided the multistep coefficients are chosen consistently.
  • The boundary-condition construction could be adapted to other deterministic kinetic solvers without requiring Monte-Carlo sampling.
  • Coupling the method to moving-boundary problems beyond the orifice test case would test whether the characteristic tracing remains conservative at the discrete level.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 1 minor

Summary. The paper presents a semi-Lagrangian numerical scheme for the polyatomic ESBGK kinetic model. Transport is discretized via the method of characteristics while the stiff relaxation term is treated by an implicit A-stable linear multistep method that is algebraically reformulated into an inexpensive explicit update exploiting the BGK structure. The scheme is asserted to be asymptotic-preserving and stiffly accurate; the first-order variant is claimed to converge, in the small-Knudsen limit, to a consistent discretization of the compressible Navier-Stokes equations with the correct polyatomic transport coefficients. Inflow/outflow boundary conditions adapted to BGK-type equations are introduced, and the method is demonstrated on Fourier, Couette, and moving-boundary orifice flows with DSMC comparisons.

Significance. If the discrete Chapman-Enskog analysis is verified, the work supplies a deterministic, grid-based alternative to DSMC that automatically recovers the correct viscosity, thermal conductivity, and bulk viscosity for polyatomic gases. The combination of unconditional stability for the relaxation term, asymptotic preservation to Euler, and the proposed boundary conditions would constitute a practical advance for rarefied polyatomic flows.

major comments (2)
  1. [Abstract / asymptotic-analysis section] Abstract and the section containing the asymptotic analysis: the claim that the first-order scheme recovers the compressible Navier-Stokes equations with the exact polyatomic transport coefficients rests on a Chapman-Enskog expansion of the fully discrete scheme. Because the relaxation operator acts on an anisotropic covariance matrix together with a non-equilibrium internal-energy distribution, the moment closure contains additional cross terms; the manuscript does not exhibit the explicit expansion or verify that these terms vanish identically under the chosen semi-Lagrangian and multistep discretizations.
  2. [Abstract] Abstract, paragraph on the relaxation-term treatment: the reformulation of the implicit A-stable linear multistep method into a cheap time-stepping scheme is stated to follow from the structure of the BGK operator. For the polyatomic ESBGK model the target distribution is no longer a standard Maxwellian; it is not shown whether the algebraic cancellation that produces the cheap update continues to hold without additional moment evaluations or hidden isotropy assumptions.
minor comments (1)
  1. The abstract states that the scheme is 'stiffly accurate' yet the precise definition used for the polyatomic case (i.e., whether the numerical equilibrium is exactly the continuous ESBGK equilibrium) is not restated in the main text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough reading and insightful comments on our manuscript. The two major comments both concern the level of detail provided for the discrete asymptotic analysis and the algebraic structure of the time-stepping scheme. We address each point below and will incorporate clarifications and explicit derivations in a revised version.

read point-by-point responses

  1. Referee: [Abstract / asymptotic-analysis section] Abstract and the section containing the asymptotic analysis: the claim that the first-order scheme recovers the compressible Navier-Stokes equations with the exact polyatomic transport coefficients rests on a Chapman-Enskog expansion of the fully discrete scheme. Because the relaxation operator acts on an anisotropic covariance matrix together with a non-equilibrium internal-energy distribution, the moment closure contains additional cross terms; the manuscript does not exhibit the explicit expansion or verify that these terms vanish identically under the chosen semi-Lagrangian and multistep discretizations.

    Authors: We agree that the current manuscript states the result of the discrete Chapman-Enskog analysis but does not display the full expansion. In the revision we will add an appendix (or expanded subsection) that carries out the explicit expansion of the fully discrete scheme, showing term-by-term that the additional cross terms arising from the anisotropic covariance and internal-energy distribution cancel identically under the chosen semi-Lagrangian transport and linear-multistep relaxation discretizations. This will make the proof of correct polyatomic transport coefficients fully transparent. revision: yes

  2. Referee: [Abstract] Abstract, paragraph on the relaxation-term treatment: the reformulation of the implicit A-stable linear multistep method into a cheap time-stepping scheme is stated to follow from the structure of the BGK operator. For the polyatomic ESBGK model the target distribution is no longer a standard Maxwellian; it is not shown whether the algebraic cancellation that produces the cheap update continues to hold without additional moment evaluations or hidden isotropy assumptions.

    Authors: The algebraic cancellation that converts the implicit multistep update into an explicit, moment-free step follows from the fact that the ESBGK target is still a linear combination of the conserved moments (density, momentum, total energy, and the anisotropic stress tensor) and the internal-energy variable. Because these moments are exactly preserved by the relaxation operator, the same linear algebra that eliminates the implicit dependence on the unknown distribution at the new time level continues to hold without requiring isotropy or extra moment evaluations. In the revision we will insert a short derivation immediately after the description of the time-stepping scheme that verifies this cancellation explicitly for the polyatomic target distribution. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation relies on independent Chapman-Enskog analysis and standard multistep methods.

full rationale

The paper claims an asymptotic-preserving property and a proof that the first-order semi-Lagrangian scheme converges to the compressible Navier-Stokes equations with correct polyatomic transport coefficients. No quoted equations or steps in the provided abstract or description reduce any prediction or coefficient to a fitted input, self-definition, or self-citation chain. The reformulation of the implicit multistep treatment is presented as exploiting the BGK operator structure without circular redefinition of the target limit. External DSMC comparisons are referenced as validation, and the central convergence statement is framed as a derived result rather than a renaming or imported uniqueness theorem. This is the normal case of a self-contained numerical analysis paper.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The central claims rest on standard kinetic theory assumptions for the ESBGK model and numerical analysis properties of characteristics and multistep methods; no free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

assumptions (2)
  • domain assumption The polyatomic ESBGK model accurately captures thermodynamic behavior via relaxation to a generalized Gaussian with anisotropic covariance and exponential internal energy decay.
    Invoked in the first sentence of the abstract as the basis for the model being discretized.
  • domain assumption The BGK relaxation operator structure permits cheap reformulation of the implicit multistep scheme.
    Stated directly in the abstract as enabling the efficient implicit treatment.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A semi-Lagrangian method for the polyatomic ESBGK model." pith.science (2026). https://pith.science/paper/PSHEMANH

@misc{pith2026260621191,
  author       = {Pith},
  title        = {Pith review of: A semi-Lagrangian method for the polyatomic ESBGK model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PSHEMANH}},
  note         = {Machine review of arXiv:2606.21191}
}
read the original abstract

Polyatomic kinetic models are essential for accurately capturing the thermodynamic behavior of real gases, as internal energy modes significantly influence transport coefficients, relaxation processes, and non-equilibrium effects that cannot be represented by monoatomic models. The polyatomic ESBGK model describes molecular collisions as a relaxation towards a generalized Gaussian distribution with an anisotropic covariance matrix and an exponentially decaying internal energy distribution. We present a new semi-Lagrangian scheme for the polyatomic Ellipsoidal Statistical BGK (ESBGK) model of the Boltzmann equation. The semi-Lagrangian framework, being deterministic and grid-based, removes the time-step restriction associated with the linear transport term by following the method of characteristics. The potentially stiff relaxation term is treated using an implicit A-stable linear multistep method which, owing to the structure of the BGK operator, can be reformulated into a cheap time-stepping scheme. This yields a highly efficient and numerically stable method. The numerical method is asymptotic preserving and stiffly accurate, meaning the scheme asymptotically converges to a scheme for the Euler equations in the vanishing Knudsen limit. In addition, we prove that the first-order scheme, asymptotically converges to the compressible Navier-Stokes equation with correct transport coefficients. Finally, we propose inflow and outflow boundary conditions suitable for BGK-type kinetic equations. We perform simulations of the Fourier and Couette test case to compare the BGK model with Direct Simulation Monte Carlo (DSMC). To conclude, we demonstrate the method on a challenging orifice flow test case with moving boundaries.

Figures

Figures reproduced from arXiv: 2606.21191 by the authors.

Figure 1
Figure 1. The characteristic is traced back to the previous time step. The foot of the characteristic does not [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. At a boundary, characteristics for which [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Convergence in the L1-norm of the first-order scheme from Section 3.2 and the second-order scheme from Section 3.3. T1 T2 x = 0 x = L Fourier Flow U1 U2 x = 0 x = L Couette Flow [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Illustration of the Fourier and Couette flow. [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]
Figure 5
Figure 5. Figure 5: Temperature and velocity along a horizontal slice for several pressure values 0 [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: Heat flux and shear stress for the Fourier and Couette flow as a function of the pressure. [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: Basic structure and boundary conditions for the orifice flow. [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: The orifice simulation for outflow pressure [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: The horizontal velocity of the diatomic ESBGK model for [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
Figure 10
Figure 10. Figure 10: The orifice simulation for outflow pressure [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]
Figure 12
Figure 12. Figure 12: The pressure (left) and the horizontal velocity (right) for the moving orifice simulation at times [PITH_FULL_IMAGE:figures/full_fig_p021_12.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

39 extracted references · 30 canonical work pages

  1. [1]

    Ching Shen.Rarefied gas dynamics: fundamentals, simulations and micro flows. eng. Springer e-books. Berlin New York, NY: Springer, 2005.isbn: 978-3-540-27230-4

    2005

  2. [2]

    G. A. Bird.The DSMC method. en. Version 1.2. S.l.: G. A. Bird, 2013.isbn: 978-1-4921-1290-7

    2013

  3. [3]

    Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamics

    Luc Mieussens. “Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamics”. en. In:Mathematical Models and Methods in Applied Sciences10.08 (Nov. 2000), pp. 1121– 1149.issn: 0218-2025, 1793-6314.doi:10.1142/S0218202500000562.url:https://www.worldscientific. com/doi/abs/10.1142/S0218202500000562(visited on 10/19/2023)

    work page doi:10.1142/s0218202500000562.url:https://www.worldscientific

  4. [4]

    A Fourier spectral method for homogeneous boltzmann equa- tions

    Lorenzo Pareschi and Benoit Perthame. “A Fourier spectral method for homogeneous boltzmann equa- tions”. en. In:Transport Theory and Statistical Physics25.3-5 (Apr. 1996), pp. 369–382.issn: 0041- 1450, 1532-2424.doi:10.1080/00411459608220707.url:http://www.tandfonline.com/doi/abs/ 10.1080/00411459608220707(visited on 04/30/2025)

    work page doi:10.1080/00411459608220707.url:http://www.tandfonline.com/doi/abs/ 1996

  5. [5]

    Implicit–Explicit Runge–Kutta Schemes and Applications to Hyperbolic Systems with Relaxation

    Lorenzo Pareschi and Giovanni Russo. “Implicit–Explicit Runge–Kutta Schemes and Applications to Hyperbolic Systems with Relaxation”. en. In:Journal of Scientific Computing25.1 (Oct. 2005), pp. 129–155.issn: 0885-7474, 1573-7691.doi:10.1007/s10915-004- 4636-4.url:https://link. springer.com/10.1007/s10915-004-4636-4(visited on 10/19/2023)

    work page doi:10.1007/s10915-004- 2005

  6. [6]

    High order central WENO-Implicit-Explicit Runge Kutta schemes for the BGK model on general polygonal meshes

    Walter Boscheri and Giacomo Dimarco. “High order central WENO-Implicit-Explicit Runge Kutta schemes for the BGK model on general polygonal meshes”. en. In:Journal of Computational Physics 422 (Dec. 2020), p. 109766.issn: 00219991.doi:10 . 1016 / j . jcp . 2020 . 109766.url:https : / / linkinghub.elsevier.com/retrieve/pii/S0021999120305404(visited on 07/04/2024)

    2020

  7. [7]

    A Cartesian cut cell method for rarefied flow simulations around moving obstacles

    G. Dechrist´ e and L. Mieussens. “A Cartesian cut cell method for rarefied flow simulations around moving obstacles”. en. In:Journal of Computational Physics314 (June 2016), pp. 465–488.issn: 00219991.doi:10.1016/j.jcp.2016.03.024.url:https://linkinghub.elsevier.com/retrieve/ pii/S0021999116001765(visited on 10/30/2023)

    work page doi:10.1016/j.jcp.2016.03.024.url:https://linkinghub.elsevier.com/retrieve/ 2016

  8. [8]

    Spectrally accurate fast summation for periodic Stokes potentials.Journal of Computational Physics, 229(23):8994–9010, 2010

    Kun Xu and Juan-Chen Huang. “A unified gas-kinetic scheme for continuum and rarefied flows”. en. In:Journal of Computational Physics229.20 (Oct. 2010), pp. 7747–7764.doi:10.1016/j.jcp.2010. 06.032.url:https://www.sciencedirect.com/science/article/abs/pii/S0021999110003475? via%3Dihub

    work page doi:10.1016/j.jcp.2010 2010

Show all 39 references
  1. [9]

    Groppi, G

    M. Groppi, G. Russo, and G. Stracquadanio.High order semi-Lagrangian methods for the BGK equa- tion. en. arXiv:1411.7929 [math]. Nov. 2014.doi:10.48550/arXiv.1411.7929.url:http://arxiv. org/abs/1411.7929(visited on 07/12/2025)

    2014 doi arXiv

  2. [10]

    Conservative semi-Lagrangian schemes for kinetic equations Part I: Reconstruc- tion

    Seung Yeon Cho et al. “Conservative semi-Lagrangian schemes for kinetic equations Part I: Reconstruc- tion”. en. In:Journal of Computational Physics432 (May 2021). arXiv:2007.13167 [cs, math], p. 110159. issn: 00219991.doi:10.1016/j.jcp.2021.110159.url:http://arxiv.org/abs/200...

    2021 doi

  3. [11]

    Conservative semi-Lagrangian schemes for kinetic equations Part II: Ap- plications

    Seung Yeon Cho et al. “Conservative semi-Lagrangian schemes for kinetic equations Part II: Ap- plications”. en. In:Journal of Computational Physics436 (July 2021). arXiv:2007.13166 [cs, math], p. 110281.issn: 00219991.doi:10.1016/j.jcp.2021.110281.url:http://arxiv.org/abs/2007...

    2021 doi

  4. [12]

    A conservative semi-Lagrangian scheme for the ellipsoidal BGK model of the Boltzmann equation

    Sebastiano Boscarino et al. “A conservative semi-Lagrangian scheme for the ellipsoidal BGK model of the Boltzmann equation”. en. In:Journal of Scientific Computing105.1 (Oct. 2025), p. 9.issn: 0885-7474, 1573-7691.doi:10.1007/s10915-025-03033-6.url:https://link.springer.com/10...

    2025 doi

  5. [13]

    Local discrete velocity grids for deterministic rarefied flow simulations

    S. Brull and L. Mieussens. “Local discrete velocity grids for deterministic rarefied flow simulations”. en. In:Journal of Computational Physics266 (June 2014), pp. 22–46.issn: 00219991.doi:10.1016/ j.jcp.2014.01.050.url:https://linkinghub.elsevier.com/retrieve/pii/S00219991140...

    2014

  6. [14]

    A local velocity grid conservative semi- Lagrangian schemes for BGK model

    Sebastiano Boscarino, Seung Yeon Cho, and Giovanni Russo. “A local velocity grid conservative semi- Lagrangian schemes for BGK model”. en. In:Journal of Computational Physics460 (July 2022), p. 111178.issn: 00219991.doi:10.1016/j.jcp.2022.111178.url:https://linkinghub.elsevier...

    2022 doi

  7. [15]

    Conservative semi-Lagrangian schemes for a general consistent BGK model for inert gas mixtures

    Seung Yeon Cho et al. “Conservative semi-Lagrangian schemes for a general consistent BGK model for inert gas mixtures”. en. In:Communications in Mathematical Sciences20.3 (2022), pp. 695–725. 23 issn: 15396746, 19450796.doi:10.4310/CMS.2022.v20.n3.a4.url:https://link.intlpress...

    2022 doi

  8. [16]

    An Asymptotic Preserving Scheme for the ES-BGK Model of the Boltz- mann Equation

    Francis Filbet and Shi Jin. “An Asymptotic Preserving Scheme for the ES-BGK Model of the Boltz- mann Equation”. en. In:Journal of Scientific Computing46.2 (Feb. 2011), pp. 204–224.issn: 0885- 7474, 1573-7691.doi:10.1007/s10915-010-9394-x.url:http://link.springer.com/10.1007/ s...

    2011 doi

  9. [17]

    Boundary conditions for semi-Lagrangian methods for the BGK model

    Maria Groppi, Giovanni Russo, and Giuseppe Stracquadanio. “Boundary conditions for semi-Lagrangian methods for the BGK model”. en. In:Communications in Applied and Industrial Mathematics7.3 (Sept. 2016), pp. 138–164.issn: 2038-0909.doi:10 . 1515 / caim - 2016 - 0025.url:https ...

    2016 doi

  10. [18]

    Investigation of the ellipsoidal-statistical Bhatnagar–Gross–Krook kinetic model applied to gas-phase transport of heat and tangential momentum between parallel walls

    M. A. Gallis and J. R. Torczynski. “Investigation of the ellipsoidal-statistical Bhatnagar–Gross–Krook kinetic model applied to gas-phase transport of heat and tangential momentum between parallel walls”. en. In:Physics of Fluids23.3 (Mar. 2011), p. 030601.issn: 1070-6631, 108...

    2011 doi

  11. [19]

    Numerical comparison between the Boltzmann and ES-BGK models for rarefied gases

    Pierre Andries et al. “Numerical comparison between the Boltzmann and ES-BGK models for rarefied gases”. en. In:Computer Methods in Applied Mechanics and Engineering191.31 (May 2002), pp. 3369– 3390.issn: 00457825.doi:10.1016/S0045-7825(02)00253-0.url:https://linkinghub.elsevi...

    2002 doi

  12. [20]

    Extension of particle-based BGK models to polyatomic species in hypersonic flow around a flat-faced cylinder

    Marcel Pfeiffer, Paul Nizenkov, and Stefanos Fasoulas. “Extension of particle-based BGK models to polyatomic species in hypersonic flow around a flat-faced cylinder”. In:AIP Conference Proceedings 2132.1 (Aug. 2019). eprint: https://pubs.aip.org/aip/acp/article-pdf/doi/10.1063...

    2019 doi

  13. [21]

    Adaptation of the Unified Gas-Kinetic Scheme to ES-BGK Models

    C´ eline Baranger, Alexis Co¨ epeau, and Luc Mieussens. “Adaptation of the Unified Gas-Kinetic Scheme to ES-BGK Models”. en. In:Rarefied Gas Dynamics. Ed. by Martin Grabe, Georgii Oblapenko, and Manuel Torrilhon. Series Title: Springer Aerospace Technology. Cham: Springer Natu...

    2026 doi

  14. [22]

    Carlo Cercignani.The Boltzmann Equation and Its Applications. en. Ed. by F. John, J. E. Marsden, and L. Sirovich. Vol. 67. Applied Mathematical Sciences. New York, NY: Springer New York, 1988. isbn: 978-1-4612-6995-3 978-1-4612-1039-9.doi:10.1007/978-1-4612-1039-9.url:https://...

    1988 doi

  15. [23]

    Henning Struchtrup, ed.Macroscopic Transport Equations for Rarefied Gas Flows: Approximation Methods in Kinetic Theory. en. Interaction of Mechanics and Mathematics. Berlin, Heidelberg: Springer- Verlag Berlin Heidelberg, 2005.isbn: 978-3-540-24542-1 978-3-540-32386-0.doi:10 ....

    2005

  16. [24]

    Numerical comparison of Bhatnagar–Gross–Krook models with proper Prandtl number

    Luc Mieussens and Henning Struchtrup. “Numerical comparison of Bhatnagar–Gross–Krook models with proper Prandtl number”. en. In:Physics of Fluids16.8 (Aug. 2004), pp. 2797–2813.issn: 1070- 6631, 1089-7666.doi:10.1063/1.1758217.url:https://pubs.aip.org/pof/article/16/8/2797/ 25...

    2004 doi

  17. [25]

    Generalization of the Krook kinetic relaxation equation

    E. M. Shakhov. “Generalization of the Krook kinetic relaxation equation”. en. In:Fluid Dynamics 3.5 (1972), pp. 95–96.issn: 0015-4628, 1573-8507.doi:10.1007/BF01029546.url:http://link. springer.com/10.1007/BF01029546(visited on 02/02/2026)

    1972 doi

  18. [26]

    Kinetic Theory of Shock Structure Using an Ellipsoidal Distribution Function

    Lowell H. Holway Jr. “Kinetic Theory of Shock Structure Using an Ellipsoidal Distribution Function”. In:Proceedings of the fourth international symposium on rarefied gas dynamics. Vol. 1. Conference Name: Rarefied Gas Dynamics, Volume 1 ADS Bibcode: 1965rgd1.conf..193H. Toront...

    1965

  19. [27]

    A new approach for the ellipsoidal statistical model

    St´ ephane Brull and Jacques Schneider. “A new approach for the ellipsoidal statistical model”. en. In: Continuum Mechanics and Thermodynamics20.2 (May 2008), pp. 63–74.issn: 0935-1175, 1432-0959. doi:10.1007/s00161-008-0068-y.url:http://link.springer.com/10.1007/s00161-008-00...

    2008 doi

  20. [28]

    The ES-BGK model equation with correct Prandtl number

    Pierre Andries. “The ES-BGK model equation with correct Prandtl number”. en. In:AIP Conference Proceedings. Vol. 585. Sydney (Australia): AIP, 2001, pp. 30–36.doi:10 . 1063 / 1 . 1407539.url: https://pubs.aip.org/aip/acp/article/585/1/30-36/577706(visited on 07/31/2025). 24

    2001

  21. [29]

    An ellipsoidal statistical model for gas mixtures

    St´ ephane Brull. “An ellipsoidal statistical model for gas mixtures”. en. In:Communications in Mathe- matical Sciences13.1 (2015), pp. 1–13.issn: 15396746, 19450796.doi:10.4310/CMS.2015.v13.n1.a1. url:https://link.intlpress.com/JDetail/1806264400858271745(visited on 10/01/2025)

    2015 doi

  22. [30]

    A general consistent BGK model for gas mixtures

    Alexander V. Bobylev et al. “A general consistent BGK model for gas mixtures”. en. In:Kinetic & Related Models11.6 (2018), pp. 1377–1393.issn: 1937-5077.doi:10 . 3934 / krm . 2018054.url: http://aimsciences.org//article/doi/10.3934/krm.2018054(visited on 10/13/2023)

    2018 doi

  23. [31]

    The Gaussian-BGK model of Boltzmann equation with small Prandtl number

    Pierre Andries et al. “The Gaussian-BGK model of Boltzmann equation with small Prandtl number”. en. In:European Journal of Mechanics - B/Fluids19.6 (Nov. 2000), pp. 813–830.issn: 09977546.doi: 10 . 1016 / S0997 - 7546(00 ) 01103 - 1.url:https : / / linkinghub . elsevier . com ...

    2000

  24. [32]

    Kinetic-Theoretic Description of the Formation of a Shock Wave

    C. K. Chu. “Kinetic-Theoretic Description of the Formation of a Shock Wave”. en. In:The Physics of Fluids8.1 (Jan. 1965), pp. 12–22.issn: 0031-9171.doi:10.1063/1.1761077.url:https://pubs. aip.org/pfl/article/8/1/12/957838/Kinetic-Theoretic-Description-of-the-Formation-of (visi...

    1965 doi

  25. [33]

    The Exponentially Convergent Trapezoidal Rule

    Lloyd N. Trefethen and J. A. C. Weideman. “The Exponentially Convergent Trapezoidal Rule”. en. In: SIAM Review56.3 (Jan. 2014), pp. 385–458.issn: 0036-1445, 1095-7200.doi:10.1137/130932132. url:http://epubs.siam.org/doi/10.1137/130932132(visited on 03/20/2025)

    2014 doi

  26. [34]

    Three-dimensional unsteady Euler equations solution using flux vector splitting

    D. Whitfield. “Three-dimensional unsteady Euler equations solution using flux vector splitting”. en. In:17th Fluid Dynamics, Plasma Dynamics, and Lasers Conference. Snowmass,CO,U.S.A.: American Institute of Aeronautics and Astronautics, June 1984.doi:10 . 2514 / 6 . 1984 - 155...

    1984 doi

  27. [35]

    Simulations for gas flows in microgeometries using the direct simulation Monte Carlo method

    Moran Wang and Zhixin Li. “Simulations for gas flows in microgeometries using the direct simulation Monte Carlo method”. en. In:International Journal of Heat and Fluid Flow25.6 (Dec. 2004), pp. 975– 985.issn: 0142727X.doi:10.1016/j.ijheatfluidflow.2004.02.024.url:https://linki...

    2004 doi

  28. [36]

    Role of boundary conditions in Monte Carlo simulation of MEMS devices

    Robert Nance et al. “Role of boundary conditions in Monte Carlo simulation of MEMS devices”. en. In:35th Aerospace Sciences Meeting and Exhibit. Reno,NV,U.S.A.: American Institute of Aeronautics and Astronautics, Jan. 1997.doi:10.2514/6.1997- 375.url:https://arc.aiaa.org/doi/1...

    1997 doi

  29. [37]

    2026.url:https://gitlab.kuleuven.be/numa/public/sl4bgk#

    Klaas Willems.SL4BGK. 2026.url:https://gitlab.kuleuven.be/numa/public/sl4bgk#

    2026

  30. [38]

    Ab Initio Transport Coefficients of Gaseous Hydrogen

    James B. Mehl, Marcia L. Huber, and Allan H. Harvey. “Ab Initio Transport Coefficients of Gaseous Hydrogen”. en. In:International Journal of Thermophysics31.4-5 (May 2010), pp. 740–755.issn: 0195-928X, 1572-9567.doi:10.1007/s10765- 009- 0697- 9.url:http://link.springer.com/10....

    2010 doi

  31. [39]

    Assessment of the ellipsoidal-statistical Bhatnagar–Gross–Krook model for force- driven Poiseuille flows

    Jianping Meng et al. “Assessment of the ellipsoidal-statistical Bhatnagar–Gross–Krook model for force- driven Poiseuille flows”. en. In:Journal of Computational Physics251 (Oct. 2013), pp. 383–395.issn: 00219991.doi:10.1016/j.jcp.2013.05.045.url:https://linkinghub.elsevier.com...

    2013 doi

Pith tools

Reviewed June 26, 2026 · model on record in the stance chip above.