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Hierarchical Log-Gaussian Relaxation on a Fixed D3Q125 Velocity Set

T0 review · 3 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Decoupling moment-order relaxation cuts peak thermodynamic nonequilibrium by 6.565%

desk verdict Clear, honest numerical study of a new order-resolved collision construction, but the headline 6.565% reduction is a built-in monotonicity consequence of comparing En vs Etot sensors, not independent evidence of physical advantage. read the letter →

arxiv 2607.20846 v2 pith:IK5UAYRB submitted 2026-07-23 math.NA cs.NAphysics.comp-ph

Bj{o}rn Wu This is my paper · ORCID
classification math.NAcs.NAphysics.comp-ph MSC 76P0582C4065M08
keywords hierarchicalcollisionmodelorder-resolvedrelaxationthermodynamicnonequilibriumD3Q125velocitysetlog-GaussianspectrumeffectiveKnudsenindicatordiscrete-velocitykineticadaptiverarefactionsensor
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

Adaptive collision models in discrete-velocity kinetic methods usually compute one scalar rarefaction or nonequilibrium indicator and feed it to every retained moment order, coupling sectors that may be far from equilibrium at different times and places. This paper develops and tests a hierarchical alternative: a shared macroscopic-gradient background combined separately with second-, third-, and fourth-order thermodynamic nonequilibrium measures yields three effective indicators K2, K3, K4, each driving its own log-Gaussian relaxation spectrum on a fixed D3Q125 velocity set. The authors show that the resulting activation is selective (pure-order perturbations leave nonmatching sectors at roundoff), consistently lowers residual nonequilibrium relative to the common-sensor model in homogeneous and wave tests, and cuts peak total nonequilibrium by 6.565% in a smooth compression wave when only the nonequilibrium channels drive the sensor. The paper does not claim universal accuracy: the improvement is measured against its own comparator, the spectrum parameters are prescribed baselines, and independent kinetic-reference validation is declared necessary before a universal accuracy claim.

What carries the argument

The machinery is an order-resolved effective Knudsen indicator K_n = N_8(K_ρ, K_T, K_u, E_n), an unnormalized p-norm with p=8 that merges a shared macroscopic-gradient background (normalized density, temperature, and velocity gradients with scale λ) with an order-specific thermodynamic nonequilibrium measure E_n formed from the Frobenius norm of the nth-order raw-Hermite coefficient deviation from equilibrium, scaled by pressure and temperature. Each K_n enters a log-Gaussian scale-space spectrum w_{c,n}(K) = ½ erfc( ln( max(K,10⁻¹⁴)/K_{0,n} ) / (√2 σ_n) ), which blends a continuum-limit relaxation factor (s=1) with a kinetic-limit factor (s_kin = 0.20/0.10/0.05 for orders 2/3/4) and is mapp

What would settle it

Run the λ=0 smooth compression-wave benchmark with an independent high-resolution discrete-velocity or DSMC solver as reference: if the order-resolved peak total TNE moves closer to the reference than the common-sensor peak does, the central claim is supported; if it moves further away, the reduction is a numerical artifact. A second falsifier is a spectrum-parameter scan outside the tested range that reverses the sign of the total-TNE difference, which would break the claimed robustness of the correction's sign.

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Extended reading notes

Core claim

The paper's central claim is that order-resolved activation works as designed: replacing one shared nonequilibrium sensor with three order-specific effective Knudsen indicators removes the artificial coupling in which one strongly nonequilibrium sector suppresses relaxation in another. In the TNE-only sensor limit (λ=0) of a smooth periodic compression wave, peak total nonequilibrium intensity falls from 3.653046e-3 to 3.413224e-3 (6.565%), at every cell and in every retained moment sector, with the largest local relaxation correction in the fourth-order channel (Δs4 ~0.237). The correction's sign persists across timestep, transport-scheme, spectrum, boost, long-time, and shear-wave tests; i

Load-bearing premise

The comparison that supports the claimed advantage uses the paper's own common-sensor model as the only comparator and a hand-set log-Gaussian spectrum as the only relaxation schedule; if those prescribed parameters are arbitrary, the 6.565% reduction is likewise arbitrary, and without an independent kinetic reference the reduction cannot be tied to physical accuracy.

Editorial extensions

If this is right

  • In TNE-dominated flow regimes, collision models that use one scalar sensor will over-suppress relaxation of weakly nonequilibrium sectors; order-resolved activation removes that coupling, so similar smooth-flow tests should show systematically lower residual TNE.
  • The reported 6.565% and related percentages are configuration-dependent: timestep, transport scheme, and the prescribed spectrum all change the magnitude, so quantitative comparisons between studies must control these settings.
  • Because the method preserves conserved moments to floating-point accuracy, keeps populations positive without limiting in the main benchmarks, and adds only about 2% runtime in the reference implementation, it can be adopted into existing finite-volume discrete-velocity codes without changing the velocity set or transport stage.
  • As the macroscopic-gradient background strengthens, the common and order-resolved models converge (to order 10⁻⁷ in local factor differences at λ=0.04), meaning the hierarchical correction matters mainly in regions where nonequilibrium channels carry information the scalar sensor would discard.
  • The documented frame dependence of the raw-Hermite sensor under uniform boosts identifies central or locally centered Hermite moments as the natural next step; the paper shows the effect follows the analytical translation structure and is not an implementation artifact over the tested range.

Reading between the lines

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

  • If an independent kinetic reference confirms the direction of the correction, the same sensor-to-relaxation decoupling should transfer to other fixed velocity sets and to regularized or multiple-relaxation-time collision models, since the change is localized in the indicator construction, not in the velocity representation.
  • The fourth-order sector being the most sensitive suggests that retaining moments beyond fourth order would amplify the scalar-aggregation problem; the benefit of order-resolved activation may grow as more Hermite orders are kept.
  • A testable extension is coupling the hierarchical sensor with dynamic velocity-space adaptation (switching among D3Q125, D3Q343, and D3Q729); if the two improvements are independent their benefits should roughly add, and if they interact the coupling would reveal which mechanism actually governs accuracy.
  • Calibrating the log-Gaussian spectrum parameters (K0,n, σ_n, s_kin) against transport coefficients or reference solutions is the direct route to converting the qualitative robustness into a quantitative statement; absent that, the reported percentages should be read as properties of the chosen baseline schedule.
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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 an order-resolved hierarchical collision model on a fixed D3Q125 discrete-velocity set. A shared macroscopic-gradient rarefaction background is combined separately with second-, third-, and fourth-order thermodynamic nonequilibrium (TNE) measures to form three effective Knudsen indicators K2, K3, K4, each driving its own log-Gaussian relaxation spectrum. Numerical tests cover pure-order activation, homogeneous mixed-order relaxation, amplitude and composition scans, a temperature wave, a smooth compression wave, grid/timestep sensitivity, transport-discretization sensitivity, relaxation-spectrum sensitivity, uniform-boost frame checks, and a shear-wave modal-decay study. The headline result is that in the TNE-only sensor limit (λ=0) of the compression-wave benchmark, the order-resolved model reduces peak total TNE by 6.565% relative to the common-sensor model, with reductions in every retained moment sector. The paper explicitly disclaims universal accuracy improvement and identifies external kinetic-reference validation as the principal remaining step.

Significance. The paper is careful, well structured, and unusually honest about its limitations. The implementation checks are extensive and convincing as numerical verifications of the stated update rules: conservation to floating-point accuracy, positivity, long-time runs, pure-order selectivity, and quantitative boost diagnostics are all reported. However, the central numerical comparison is largely forced by the construction: with the chosen monotone relaxation schedule, splitting a common scalar sensor into component-wise sensors guarantees larger relaxation factors and hence lower residual TNE. Thus the 6.565% number and the sign of the reduction are not independent empirical evidence for a physical advantage of order resolution. The useful contribution is the explicit, well-tested hierarchical construction and the honest mapping of its regime dependence, not the quantitative accuracy claim. The paper would be acceptable as a mechanism/numerical-behavior study after the forced nature of the headline result is stated and the interpretation is reframed accordingly.

major comments (3)
  1. [§3.4 and §5.5.3] The λ=0 reduction is mathematically guaranteed by the definitions. With Kρ=KT=Ku=0, the sensors reduce to Kn=En and Kcommon=Etot. Since Etot>En whenever at least two TNE sectors are nonzero, and since s_n(K) is strictly decreasing in K (the erfc weight decreases with K and s_cont>s_kin), we have s_n^resolved>s_n^common for every n. Each sector is therefore relaxed more strongly, and the sector-wise and total TNE reductions follow automatically. The same monotonicity argument underlies the homogeneous results in §5.2–5.3 and the sign-robustness claims in §5.11–5.13. The paper should state this analytical guarantee explicitly and reframe the 6.565% result as a consequence of the comparator choice, not as an empirical discovery about the value of order resolution.
  2. [§3.5, Table 3, §5.13] The quantitative magnitude of the reported reductions is controlled by uncalibrated parameters (K0,n, σn, sn,kin) taken from the author's prior continuum-ballistic preprint [20], together with the selected norm order p=8 and sensor coefficients. The paper's own sensitivity scans show the reduction varying from 16.47% to 21.50% (spectrum scan, CFL=0.05) and from 2.418% to 11.523% (grid/timestep scan), so the 6.565% value is not a physical constant. Since no independent kinetic reference is provided, the numerical results cannot support any accuracy claim. The abstract and conclusions should consistently present these numbers as illustrative of the construction and should not imply that the model is validated by the internal common-sensor comparison.
  3. [§5.10 and §6.8] The laboratory-frame raw-Hermite sensor is not Galilean invariant. The uniform-boost tests show that a boost alone changes the third- and fourth-order TNE and the relaxation factors (for example, s4 drops from 1 to 0.489 at U0=0.4 in the homogeneous test, and transport-enabled peak TNE increases by 50.6% at U0=0.2). The paper acknowledges this and provides quantitative diagnostics, which is commendable. Nevertheless, for a kinetic model intended for compressible or high-speed flow this frame dependence is a genuine physical deficiency. At minimum, the claims should be restricted to the stated frame/mean-flow regime, and a central-moment or central-Hermite formulation should be identified as necessary for a frame-independent sensor.
minor comments (4)
  1. [§5.5.5, Table] The Common-model column in the grid-and-timestep table appears to have row-count digits accidentally prepended: the listed values like 11.226703×10−2, 26.394078×10−3, 43.653046×10−3, and 82.384581×10−3 contradict the text and the computed percentages. They should presumably read 1.226703×10−2, 6.394078×10−3, 3.653046×10−3, and 2.384581×10−3.
  2. [§5.9, Table 6] Text such as 'CFL= 0.4to0 .0015625' contains spacing and formatting errors; please fix for clarity.
  3. [§5.10] The notation Sym for normalized full index symmetrization is introduced without an explicit definition of the normalization factor. Please define it precisely.
  4. [General] The paper would benefit from an explicit statement about code and data availability. The text states that quadrature abscissae are generated by the implementation, but no repository or reproducibility artifact is mentioned.

Circularity Check

1 steps flagged · score 6.0 of 10

The λ=0 peak-TNE reduction is built into the sensor definition, not an independent discovery.

  1. self definitional [Sections 3.2, 3.4–3.5 and 5.5.3]
    "In this limit, the TNE channels dominate the effective indicators. The peak total TNE decreases from 3.653046×10−3 in the common-sensor model to 3.413224×10−3 in the order-resolved model, corresponding to a reduction of 6.565%. ... K_n = N_p(K_rho,K_T,K_u,E_n), n=2,3,4. The common-sensor formulation uses K_common = N_p(K_rho,K_T,K_u,E_tot) ... The total TNE indicator is the arithmetic sum, E_tot = E_2 + E_3 + E_4."

    At λ=0, K_rho=K_T=K_u=0, so K_n=E_n and K_common=E_tot=E_2+E_3+E_4. With any two E_n nonzero, K_common>K_n for every n. The log-Gaussian schedule is monotone: s_n,ref(K)=w_c,n(K)·1+(1−w_c,n(K))·s_n,kin, with w_c,n(K)=0.5 erfc(...) decreasing in K and s_n,kin<1, so s_n decreases as K increases. Hence s_n^resolved>s_n^common, and the post-collision factor (1−s_n) is smaller in the resolved model, forcing stronger relaxation and lower TNE in every sector after the same pre-collision state. The sign of the headline reduction is therefore a definitional consequence of comparing an additive total sensor with component-wise sensors; only the 6.565% magnitude depends on the dynamics and the hand-set spectrum.

full rationale

Most claims in the paper are self-contained numerical implementation checks rather than circular derivations: the finite-volume transport, projection/reconstruction, conservation diagnostics, positivity monitoring, and shear-wave effective-viscosity refinement compare two explicitly defined formulations on the same code. The paper also honestly disclaims physical accuracy ('A reduction in thermodynamic nonequilibrium is not by itself a proof of improved physical accuracy') and labels the relaxation-spectrum parameters as 'research-baseline values rather than universally calibrated physical constants.' The self-citation [20] is an openly adopted ansatz for the log-Gaussian schedule, not a load-bearing uniqueness theorem, and the spectrum-sensitivity study shows the sign is robust under perturbations. However, the central quantitative claim — that order-resolved activation reduces peak total TNE in the TNE-only sensor limit — is an analytical consequence of the definitions: E_tot is the arithmetic sum, so the common sensor always inputs a larger K into a monotone decreasing relaxation curve than each order-resolved sensor K_n=E_n. Thus the sign of the 6.565% reduction is guaranteed by construction; only the magnitude is a simulation result. Because this definitional forcing affects the paper's headline comparison while the implementation checks and regime-dependence studies retain independent content, the appropriate circularity score is 6 rather than 0 or 8.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The model introduces no new physical particles or forces. Its free parameters are the hand-prescribed log-Gaussian spectrum and sensor settings, which directly control the magnitude and even the sign of the reported corrections. The axioms are mostly standard quadrature facts plus the transferability of the author's earlier log-Gaussian blending idea to collision factors.

free parameters (4)
  • Transition scales K0,2=0.050, K0,3=0.030, K0,4=0.015 = 0.050, 0.030, 0.015
    Hand-set log-Gaussian transition locations for orders 2-4; called research-baseline values, not calibrated (Table 3).
  • Transition widths sigma_2=2.0, sigma_3=2.5, sigma_4=3.0 = 2.0, 2.5, 3.0
    Prescribed widths of the order-dependent log-Gaussian curves (Table 3).
  • Endpoint relaxation factors s_cont=1.0, s_kin,2=0.20, s_kin,3=0.10, s_kin,4=0.05 = 1.0, 0.20, 0.10, 0.05
    Continuum and kinetic limits of the collision strength for each moment order; chosen by hand (Table 3).
  • Rarefaction norm order p=8; sensor coefficients c2=c3=c4=1; default lambda=0.01; reference timestep 1.09381617e-3 = p=8, c=1, lambda=0.01, dt_ref=1.09381617e-3
    Sensor aggregation, gradient length scale, TNE multipliers, and collision scaling reference are prescribed, not derived (Sections 3.3-3.5, Table 4).
assumptions (4)
  • standard math D3Q125 tensor-product Gauss-Hermite quadrature is exact for Hermite products up to degree nine.
    Relied on in Section 2.1 and 2.5 so that fourth-order Hermite projection/reconstruction is an identity on the retained subspace.
  • domain assumption Fourth-order Hermite truncation is sufficient to represent the benchmark dynamics on D3Q125.
    The entire formulation is restricted to H0-H4; no evidence is given that truncation error is negligible for the reported tests (Section 2.4).
  • ad hoc to paper The log-Gaussian scale-space schedule from the author's prior work [20] transfers from continuum-ballistic flux blending to moment-dependent collision factors.
    Section 3.5 adopts [20] without independent validation or derivation; this is the core schedule that produces all reported relaxation behavior.
  • ad hoc to paper Laboratory-frame raw Hermite deviations are adequate TNE indicators despite their uniform-boost coupling.
    Section 3.2 and Section 5.10 acknowledge the raw-Hermite frame sensitivity and the lack of Galilean invariance, yet the central benchmarks use this sensor.

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Cite this review

Pith. "Pith review of Hierarchical Log-Gaussian Relaxation on a Fixed D3Q125 Velocity Set." pith.science (2026). https://pith.science/paper/IK5UAYRB

@misc{pith2026260720846,
  author       = {Pith},
  title        = {Pith review of: Hierarchical Log-Gaussian Relaxation on a Fixed D3Q125 Velocity Set},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IK5UAYRB}},
  note         = {Machine review of arXiv:2607.20846}
}
read the original abstract

We develop a hierarchical order-resolved relaxation model for a fixed D3Q125 discrete-velocity kinetic formulation. Conventional adaptive collision models often use one scalar rarefaction or nonequilibrium indicator for all retained moment orders, thereby coupling distinct kinetic sectors. Here, a shared macroscopic-gradient background is combined separately with second-, third-, and fourth-order thermodynamic nonequilibrium indicators to define effective measures K2, K3, and K4, each driving its own log-Gaussian relaxation spectrum. Pure-order perturbation tests verify selective activation, with nonmatching sectors remaining at roundoff level. Homogeneous mixed-order, amplitude, and composition tests show lower residual nonequilibrium than a common-sensor model while preserving positive populations. In a smooth periodic compression wave at the stated reference discretization and in the TNE-only sensor limit, the peak total nonequilibrium intensity is reduced by 6.565%, with reductions throughout the domain and in all retained moment sectors. Additional timestep, transport-discretization, relaxation-spectrum, uniform-boost, long-time, and shear-wave studies show that the sign of the hierarchical correction is robust over the tested configurations, while its magnitude depends on timestep, transport scheme, sensor frame, and relaxation spectrum. Quantitative boost checks identify the laboratory-frame raw-Hermite origin of the frame sensitivity without detecting an evident collision-path inconsistency over the tested range. The periodic benchmarks preserve the principal global invariants to floating-point accuracy and remain positive. These results establish the mechanism, selectivity, and numerical behavior of order-resolved activation on a fixed velocity set; independent kinetic-reference validation is still required before claiming universal accuracy improvement.

Figures

Figures reproduced from arXiv: 2607.20846 by the authors.

Figure 1
Figure 1. Pure-order activation of the hierarchical relaxation spectrum. Sep [PITH_FULL_IMAGE:figures/full_fig_p027_1.png] view at source ↗
Figure 2
Figure 2. Robustness of hierarchical activation to perturbation strength and [PITH_FULL_IMAGE:figures/full_fig_p030_2.png] view at source ↗
Figure 3
Figure 3. Order-resolved activation in a smooth periodic temperature wave. [PITH_FULL_IMAGE:figures/full_fig_p033_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Joint macroscopic-gradient activation and hierarchical relaxation [PITH_FULL_IMAGE:figures/full_fig_p036_4.png]
Figure 5
Figure 5. Figure 5: Spatial comparison of the common and order-resolved activation [PITH_FULL_IMAGE:figures/full_fig_p039_5.png]
Figure 6
Figure 6. Figure 6: Macroscopic-gradient sensor-scale dependence of the common and [PITH_FULL_IMAGE:figures/full_fig_p041_6.png]
Figure 7
Figure 7. Figure 7: Fixed-final-time grid-and-timestep sensitivity with timestep [PITH_FULL_IMAGE:figures/full_fig_p043_7.png]
Figure 8
Figure 8. Figure 8: Quantitative uniform-boost assessment of raw-Hermite TNE and re [PITH_FULL_IMAGE:figures/full_fig_p050_8.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Central-Hermite Sensing and Collision for Frame-Robust Order-Resolved Relaxation on D3Q125

    math.NA 2026-07 conditional novelty 4.5 of 10

    Fully central-Hermite sensing and collision on D3Q125 preserves modal purity and reduces post-transport cross-order frame discrepancy by a median 81% versus raw Hermite, without proving equal macroscopic Galilean gains.

Reference graph

Works this paper leans on

20 extracted references · 1 linked inside Pith · cited by 1 Pith paper

  1. [20]

    A log-gaussian scale-space limiter for hybrid continuum– ballistic gas dynamics.arXiv preprint arXiv:2607.12218, 2026

    Bjørn Wu. A log-gaussian scale-space limiter for hybrid continuum– ballistic gas dynamics.arXiv preprint arXiv:2607.12218, 2026. 63

    arXiv 2026

  2. [1]

    Santosh Ansumali and Iliya V. Karlin. Single relaxation time model for entropic lattice Boltzmann methods.Physical Review E, 65(5):056312, 2002

    2002

  3. [2]

    Recursive regularization step for high- order lattice Boltzmann methods.Physical Review E, 96(3):033306, 2017

    Christophe Coreixas, Gauthier Wissocq, Guillaume Puigt, Jean-François Boussuge, and Pierre Sagaut. Recursive regularization step for high- order lattice Boltzmann methods.Physical Review E, 96(3):033306, 2017. 61

    2017

  4. [3]

    Multiple-relaxation-time lattice Boltzmann models in three dimensions.Philosophical Transactions of the Royal Society of London

    Dominique d’Humières, Irina Ginzburg, Manfred Krafczyk, Pierre Lalle- mand, and Li-Shi Luo. Multiple-relaxation-time lattice Boltzmann models in three dimensions.Philosophical Transactions of the Royal Society of London. Series A, 360(1792):437–451, 2002

    2002

  5. [4]

    Discrete Boltzmann modeling of multiphase flows: Hydrodynamic and thermo- dynamic non-equilibrium effects.Soft Matter, 11(26):5336–5345, 2015

    Yanbiao Gan, Aiguo Xu, Guangcai Zhang, and Sauro Succi. Discrete Boltzmann modeling of multiphase flows: Hydrodynamic and thermo- dynamic non-equilibrium effects.Soft Matter, 11(26):5336–5345, 2015

    2015

  6. [5]

    On the kinetic theory of rarefied gases.Communications on Pure and Applied Mathematics, 2(4):331–407, 1949

    Harold Grad. On the kinetic theory of rarefied gases.Communications on Pure and Applied Mathematics, 2(4):331–407, 1949

    1949

  7. [6]

    Qinghong Guo, Yanbiao Gan, Bin Yang, Yanhong Wu, Huilin Lai, and Aiguo Xu. Thermodynamic nonequilibrium effects in three-dimensional high-speed compressible flows: Multiscale modeling and simulation via the discrete Boltzmann method.Physics of Fluids, 37(4):046117, 2025

    2025

  8. [7]

    Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability.Physical Review E, 61(6):6546–6562, 2000

    Pierre Lallemand and Li-Shi Luo. Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability.Physical Review E, 61(6):6546–6562, 2000

    2000

Show all 20 references
  1. [8]

    Theory of the lattice Boltzmann method: Acoustic and thermal properties in two and three dimensions

    Pierre Lallemand and Li-Shi Luo. Theory of the lattice Boltzmann method: Acoustic and thermal properties in two and three dimensions. Physical Review E, 68(3):036706, 2003

    2003

  2. [9]

    Linear hydrodynamics and stability of the discrete velocity Boltzmann equations.Journal of Fluid Mechanics, 897:A29, 2020

    Pierre-Alexandre Masset and Gauthier Wissocq. Linear hydrodynamics and stability of the discrete velocity Boltzmann equations.Journal of Fluid Mechanics, 897:A29, 2020

    2020

  3. [10]

    Patil and K

    Dhiraj V. Patil and K. N. Lakshmisha. Finite-volume TVD formulation of lattice Boltzmann simulation on unstructured mesh.Journal of Computational Physics, 228(14):5262–5279, 2009

    2009

  4. [11]

    Finite- volumeschemeforthelatticeBoltzmannmethodonunstructuredmeshes

    Gongwen Peng, Haowen Xi, Comer Duncan, and So-Hsiang Chou. Finite- volumeschemeforthelatticeBoltzmannmethodonunstructuredmeshes. Physical Review E, 59(4):4675–4682, 1999

    1999

  5. [12]

    Premnath and Sanjoy Banerjee

    Kannan N. Premnath and Sanjoy Banerjee. On the three-dimensional central moment lattice Boltzmann method.Journal of Statistical Physics, 143(4):747–794, 2011

    2011

  6. [13]

    General solution of lattices for cartesian lattice Bhatnagar–Gross–Krook models.Physical Review E, 81(3):036702, 2010

    Xiaowen Shan. General solution of lattices for cartesian lattice Bhatnagar–Gross–Krook models.Physical Review E, 81(3):036702, 2010. 62

    2010

  7. [14]

    Central-moment-based galilean-invariant multiple- relaxation-time collision model.Physical Review E, 100(4):043308, 2019

    Xiaowen Shan. Central-moment-based galilean-invariant multiple- relaxation-time collision model.Physical Review E, 100(4):043308, 2019

    2019

  8. [15]

    Discretization of the velocity space in the solution of the Boltzmann equation.Physical Review Letters, 80(1):65–68, 1998

    Xiaowen Shan and Xiaoyi He. Discretization of the velocity space in the solution of the Boltzmann equation.Physical Review Letters, 80(1):65–68, 1998

    1998

  9. [16]

    Kinetic theory representation of hydrodynamics: A way beyond the Navier–Stokes equation.Journal of Fluid Mechanics, 550:413–441, 2006

    Xiaowen Shan, Xue-Feng Yuan, and Hudong Chen. Kinetic theory representation of hydrodynamics: A way beyond the Navier–Stokes equation.Journal of Fluid Mechanics, 550:413–441, 2006

    2006

  10. [17]

    Accuracy of high-order lattice Boltzmann method for non-equilibrium gas flow.Journal of Fluid Mechanics, 907:A25, 2021

    Yangyang Shi, Lei Wu, and Xiaowen Shan. Accuracy of high-order lattice Boltzmann method for non-equilibrium gas flow.Journal of Fluid Mechanics, 907:A25, 2021

    2021

  11. [18]

    Karlin, and Hudong Chen

    Sauro Succi, Iliya V. Karlin, and Hudong Chen. Colloquium: Role of theH theorem in lattice Boltzmann hydrodynamic simulations.Reviews of Modern Physics, 74(4):1203–1220, 2002

    2002

  12. [19]

    Linear stability and isotropy properties of athermal regularized lattice Boltzmann methods.Physical Review E, 102(5):053305, 2020

    Gauthier Wissocq, Christophe Coreixas, and Jean-François Boussuge. Linear stability and isotropy properties of athermal regularized lattice Boltzmann methods.Physical Review E, 102(5):053305, 2020

    2020

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