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arxiv: 2605.01774 · v1 · submitted 2026-05-03 · ⚛️ physics.flu-dyn

Recognition: unknown

Entropic lattice Boltzmann method for general anisotropic advection--diffusion

Jingsen Feng , Jing Leng , Jingchao Jiang , Xu Chu
Authors on Pith no claims yet

Pith reviewed 2026-05-09 16:54 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords entropic lattice Boltzmannanisotropic advection-diffusiontensorial relaxationChapman-Enskog analysisporous mediaRayleigh-Benard convectionTaylor dispersion
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The pith

A local entropic lattice Boltzmann scheme recovers the general anisotropic advection-diffusion equation via tensorial flux relaxation.

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

The paper presents a lattice Boltzmann discretization for the full-tensor anisotropic advection-diffusion equation that works even when principal diffusion directions are rotated relative to the computational grid. Non-equilibrium populations are split into a first-order flux sector and a residual ghost sector; the diffusion tensor is imposed through local tensorial relaxation of the flux while an ADE-corrected entropic stabilizer with positivity fallback controls higher-order content. Chapman-Enskog analysis establishes that the scheme recovers the target macroscopic equation together with a discrete-time relation between the physical tensor and the relaxation matrix. The resulting update is local and matrix-free, which matters for transport problems in porous media, convection, and particle dispersion where standard discretizations suffer from grid-orientation errors under large anisotropy.

Core claim

The central claim is that the proposed entropic lattice Boltzmann method recovers the target full-tensor anisotropic advection-diffusion equation. Non-equilibrium populations are partitioned into a flux sector whose tensorial relaxation sets the physical diffusion tensor and a ghost sector whose content is controlled by an ADE-corrected entropic stabilizer. Chapman-Enskog analysis supplies the discrete-time diffusivity relation that links the relaxation matrix to the imposed tensor, and the scheme remains stable for rotated, spatially varying, and heterogeneous tensors.

What carries the argument

Split of non-equilibrium populations into first-order flux sector and residual ghost sector, with tensorial relaxation of the flux and ADE-corrected entropic stabilization.

If this is right

  • The update applies without modification to rotated, spatially varying, heterogeneous, and dynamically coupled tensors.
  • Benchmarks confirm stability and accuracy for anisotropy ratios of order 10^4, local contrasts of 3x10^4:1, and high-Peclet advection.
  • The method quantifies orientation-induced Taylor dispersion enhancement for Brownian rods under shear.
  • Anisotropic Rayleigh-Benard convection runs remain stable across seven decades of anisotropy ratio and reveal systematic changes in plume shape and heat transfer.

Where Pith is reading between the lines

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

  • The strictly local update structure would support straightforward GPU parallelization for large-scale 3D anisotropic transport simulations.
  • The same splitting and stabilization strategy could be tested on time-dependent or nonlinear diffusion tensors without reformulating the core algorithm.
  • If the positivity fallback remains robust, the approach may extend to other kinetic schemes that currently struggle with strong off-diagonal diffusion terms.

Load-bearing premise

The flux-ghost population split together with the entropic stabilizer sufficiently suppresses higher-order kinetic modes so that the scheme recovers the full anisotropic equation without uncontrolled errors or instabilities.

What would settle it

A 3D simulation of rotated high-anisotropy advection-diffusion in which the measured effective diffusion tensor deviates from the imposed tensor by more than discretization error, or in which the Rayleigh-Benard convection run becomes unstable at anisotropy ratios of order 10^4.

Figures

Figures reproduced from arXiv: 2605.01774 by Jingchao Jiang, Jing Leng, Jingsen Feng, Xu Chu.

Figure 1
Figure 1. Figure 1: Centroid-centered three-dimensional concentration distributions at the largest normalized time shown in view at source ↗
Figure 2
Figure 2. Figure 2: Evolution of centroid-centered concentration profiles along the three principal axes. Rows (a)–(d) correspond to the four diffusivity ratios in view at source ↗
Figure 3
Figure 3. Figure 3: Sinusoidal-decay validation for the four rotated constant-tensor cases. Panel (a) shows the normalized modal amplitudes 𝐴(𝑡)∕𝐴(0) against the normalized time 𝜏 = 𝛼theory𝑡 for the four anisotropy ratios 104 ∶ 104 ∶ 1, 104 ∶ 103 ∶ 1, 104 ∶ 102 ∶ 1, and 104 ∶ 101 ∶ 1. The dashed line denotes the exact decay law 𝑒 −𝜏 . Panel (b) reports the relative decay-rate error |𝛼ELBM − 𝛼theory|∕𝛼theory for the same four cases view at source ↗
Figure 4
Figure 4. Figure 4: Sensitivity of the sinusoidal-decay benchmark to the Euler angle 𝛽𝐸 for the fixed anisotropy ratio 𝑑1 ∶ 𝑑2 ∶ 𝑑3 = 104 ∶ 102 ∶ 1. Panel (a) presents the normalized decay curves for 𝛽𝐸 = 0◦ , 30◦ , 60◦ , and 90◦ together with the exact law 𝑒 −𝜏 . Panel (b) shows the corresponding relative decay-rate errors view at source ↗
Figure 5
Figure 5. Figure 5: Numerical and analytical solutions for the three-dimensional anisotropic advection–diffusion equation with constant velocity and variable diffusion tensor at 𝑃 𝑒 = 106 . Rows (a)–(d) correspond to the four baseline diffusivity ratios 𝑑1 ∶ 𝑑2 ∶ 𝑑3 = 104 ∶ 104 ∶ 1, 104 ∶ 103 ∶ 1, 104 ∶ 102 ∶ 1, and 104 ∶ 101 ∶ 1. Columns show the 𝑥𝑦 section at 𝑧 = 0.25 and 𝑡 = 1, the 𝑦𝑧 section at 𝑥 = 0.25 and 𝑡 = 2, and the… view at source ↗
Figure 6
Figure 6. Figure 6: Schematic of Taylor dispersion of elongated rods in a plane Poiseuille flow. A Gaussian plug is initially localized in the streamwise direction and uniformly distributed across the channel width. Shear-induced rotation and anisotropic diffusion redistribute the rods across the shear layers. At long times, this coupling yields an effective axial transport with modified mean speed and enhanced longitudinal d… view at source ↗
Figure 7
Figure 7. Figure 7: Geometric and transport quantities for an elongated Brownian rod in a plane Poiseuille flow [16]. The local shear rate ̇𝛾(𝑦) drives Jeffery rotation, 𝜃 is the in-plane orientation angle, 𝐷∥ and 𝐷⟂ are the translational diffusivities parallel and perpendicular to the rod axis, and 𝐷𝜃 is the rotational diffusivity. The semimajor and semiminor axes are denoted by 𝑎𝑝 and 𝑏𝑝 , so the aspect ratio is 𝑝 = 𝑎𝑝∕𝑏𝑝 .… view at source ↗
Figure 8
Figure 8. Figure 8: Comparison between the semi-analytic closure and ELBM for Taylor dispersion of elongated rods at 𝑃 𝑒 = 1000. Panel (a) shows the dimensionless mean particle speed 𝑢𝑚 as a function of 𝑃 𝑒𝑟 ; panel (b) shows the normalized effective dispersion 𝜅∕𝜅𝑠 . Solid lines denote the semi-analytic predictions obtained from Eqs. (76)–(84), and open circles are ELBM measurements extracted from the long-time growth of the… view at source ↗
Figure 9
Figure 9. Figure 9: Anisotropic porous cube-array geometries used for the effective-conductivity measurement. Panel (a) shows the ordered array, where the dispersed cubes remain aligned through successive streamwise layers. Panel (b) shows the cross array, where alternating layers are shifted in the transverse plane. The dispersed cube phase is embedded in a continuous conducting matrix between the hot and cold plates, and th… view at source ↗
Figure 10
Figure 10. Figure 10: Temperature distributions in the ordered cube array for the four Euler-angle cases in view at source ↗
Figure 11
Figure 11. Figure 11: Temperature distributions in the cross cube array for the same material tensors as view at source ↗
Figure 12
Figure 12. Figure 12: Temperature departures from the conductive profile in anisotropic Rayleigh–Bénard convection at 𝑅𝑎 = 109 and 𝑃 𝑟𝑣 = 1. Panels (a)–(h) show increasing values of 𝑟 = 𝑑2∕𝑑1 from 10−4 to 103 . The vertical diffusivity 𝑑2 is fixed in all cases, while the horizontal diffusivity decreases as 𝑟 increases view at source ↗
Figure 13
Figure 13. Figure 13: Heat transport and plume morphology as functions of 𝑟 = 𝑑2∕𝑑1 . Panel (a) compares the bulk Nusselt number with the flux-plateau estimate. Panel (b) reports two bulk plume statistics computed from 𝜃 − 𝜃cond over 0.2𝐻 ≤ 𝑧 ≤ 0.8𝐻: the horizontal spectral centroid 𝑛𝑐 and the fraction of spectral energy in modes 𝑛 ≥ 8 view at source ↗
Figure 14
Figure 14. Figure 14: Late-time vertical profiles for the anisotropic Rayleigh–Bénard sweep. Panel (a) shows the cross-section￾averaged heat-flux profile 𝑁𝑢(𝑧) = ⟨𝑞𝑧 ⟩𝑥∕𝑞0 . Panel (b) shows the horizontally averaged temperature profile, with the dashed line denoting the conductive solution view at source ↗
read the original abstract

Many transport processes exhibit direction-dependent diffusion, described macroscopically by the full-tensor anisotropic advection--diffusion equation (ADE). Numerical discretization is demanding when the principal axes are rotated relative to the mesh, since mixed derivatives and oblique fluxes amplify grid-orientation errors under large tensor contrasts. This paper develops a local entropic lattice Boltzmann discretization for the general anisotropic ADE. The non-equilibrium population is split into a first-order flux sector and a residual ghost sector. The diffusion tensor is imposed through local tensorial relaxation of the flux, while higher-order kinetic content is controlled by an ADE-corrected entropic stabilizer with positivity fallback. Chapman--Enskog analysis shows the scheme recovers the target full-tensor equation with a discrete-time diffusivity relation between the physical tensor and the flux-relaxation matrix. The update is local, matrix-free, and applies to rotated, spatially varying, heterogeneous, and dynamically coupled tensor transport. We validate it on 3D benchmarks--advected Gaussian plumes, decay of rotated Fourier modes, and source-driven transport with varying tensors--covering off-diagonal diffusion, high-P\'eclet advection, anisotropy ratios of O(104)O(10^4) O(104), and local contrasts up to $3\times10^4:1$. It is then applied to orientation-induced Taylor dispersion of Brownian rods, quantifying enhancement from shear-driven rotation. Heat-conduction tests include rotated thermal-conductivity measurements and effective conduction in heterogeneous porous media with anisotropy up to $10^4:1. Finally, anisotropic Rayleigh--B\'enard convection is simulated to examine how plume morphology and heat transfer change over seven decades of anisotropy ratios, demonstrating an accurate, stable local solver for strongly anisotropic advection--diffusion.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 2 minor

Summary. The paper develops a local entropic lattice Boltzmann discretization for the general anisotropic advection-diffusion equation (ADE) with arbitrary diffusion tensor, including rotated, heterogeneous, and dynamically varying cases. Non-equilibrium populations are split into a first-order flux sector (tensorially relaxed to impose the target diffusion tensor) and a residual ghost sector; higher-order content is controlled via an ADE-corrected entropic stabilizer with positivity fallback. Chapman-Enskog analysis is claimed to recover the full-tensor ADE with a discrete-time diffusivity relation between the physical tensor and the flux-relaxation matrix. The scheme is validated on 3D benchmarks (advected Gaussians, rotated Fourier modes, source-driven transport, Taylor dispersion of rods, heat conduction in porous media, and anisotropic Rayleigh-Bénard convection) at anisotropy ratios up to 10^4 and local contrasts of 3×10^4:1.

Significance. If the recovery and stability claims hold, the method supplies a matrix-free, fully local solver for strongly anisotropic transport that avoids grid-orientation errors common in finite-difference or standard LBM approaches under large tensor contrasts and rotations. The multi-benchmark validation suite, covering off-diagonal diffusion, high-Péclet advection, and coupled problems such as orientation-induced dispersion and anisotropic convection, provides concrete evidence of practical utility across seven decades of anisotropy ratios. The explicit discrete-time diffusivity relation and local update are strengths for implementation in complex geometries.

major comments (1)
  1. [Chapman-Enskog analysis section] The Chapman-Enskog analysis (abstract and the multiscale section) recovers the target ADE from the linear tensorial relaxation of the flux sector but does not explicitly derive the O(Kn) contribution of the nonlinear ADE-corrected entropic stabilizer (or the positivity fallback) to the macroscopic equation. Under the reported anisotropy ratios of O(10^4) and local contrasts of 3×10^4:1, the widely differing relaxation rates across tensor eigenvalues can source higher moments from the ghost sector that back-react at first order, potentially violating the stated discrete-time diffusivity relation; a concrete bound or additional term in the expansion is required to confirm absence of spurious corrections.
minor comments (2)
  1. [validation sections] The abstract and validation sections report anisotropy ratios up to 10^4 and contrasts of 3×10^4:1 but do not tabulate quantitative L2 or L∞ errors, convergence rates with grid refinement, or direct comparisons against reference solutions (e.g., finite-element or spectral methods) for the rotated Fourier-mode and heterogeneous-tensor cases; adding these would strengthen the quantitative support.
  2. [method section] Notation for the flux-relaxation matrix and the ADE-corrected entropic stabilizer should be introduced with explicit definitions (including how the correction term is constructed) before the Chapman-Enskog expansion to improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The single major comment identifies a legitimate gap in the explicit multiscale derivation that we will address directly in the revision.

read point-by-point responses

  1. Referee: [Chapman-Enskog analysis section] The Chapman-Enskog analysis (abstract and the multiscale section) recovers the target ADE from the linear tensorial relaxation of the flux sector but does not explicitly derive the O(Kn) contribution of the nonlinear ADE-corrected entropic stabilizer (or the positivity fallback) to the macroscopic equation. Under the reported anisotropy ratios of O(10^4) and local contrasts of 3×10^4:1, the widely differing relaxation rates across tensor eigenvalues can source higher moments from the ghost sector that back-react at first order, potentially violating the stated discrete-time diffusivity relation; a concrete bound or additional term in the expansion is required to confirm absence of spurious corrections.

    Authors: We agree that the original multiscale section focuses on the linear tensorial relaxation of the flux sector and does not explicitly expand the nonlinear contributions of the ADE-corrected entropic stabilizer or the positivity fallback to O(Kn). In the revised manuscript we will augment the Chapman-Enskog analysis to include these terms. We will demonstrate that the stabilizer is constructed to leave the first-order flux moments unchanged, so that its nonlinear action enters the macroscopic equation only at O(Kn²) or higher. We will also derive a bound on the ghost-sector back-reaction that holds when the tensorial relaxation rates satisfy the reported discrete-time diffusivity relation, showing that the back-reaction remains negligible for anisotropy ratios up to 10^4 and local contrasts up to 3×10^4:1. These additions will be placed in the multiscale section and will not alter the core recovery statement or the numerical validation results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via standard multiscale analysis

full rationale

The paper defines a novel splitting of non-equilibrium populations into flux and ghost sectors, imposes the diffusion tensor via local tensorial relaxation of the flux sector, and controls higher moments with an ADE-corrected entropic stabilizer plus positivity fallback. It then applies the standard Chapman-Enskog multiscale expansion directly to this defined scheme to derive the target full-tensor ADE and the discrete-time diffusivity relation. The relaxation matrix is constructed from the physical tensor as an input, not fitted to any output quantity. No step reduces by construction to its own result, no parameter is renamed as a prediction, and the recovery proof does not rely on a self-citation chain for its load-bearing content. Prior entropic LBM citations supply the stabilizer technique but are not invoked as a uniqueness theorem or to smuggle the anisotropic extension; the central claim remains independently verifiable by repeating the CE expansion on the given update rule.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard kinetic-theory assumptions plus the specific numerical splitting and stabilization strategy introduced here; no new physical entities are postulated.

free parameters (1)
  • elements of the flux-relaxation matrix
    Set locally from the physical diffusion tensor to satisfy the discrete-time diffusivity relation recovered by Chapman-Enskog analysis.
axioms (2)
  • standard math Chapman-Enskog multiscale expansion recovers the target macroscopic ADE from the kinetic scheme
    Invoked to demonstrate that the discretization produces the correct full-tensor equation.
  • domain assumption The ADE-corrected entropic stabilizer with positivity fallback adequately controls higher-order moments and maintains stability
    Used to manage residual ghost-sector content without introducing uncontrolled errors.

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Reference graph

Works this paper leans on

65 extracted references · 57 canonical work pages

  1. [1]

    Torquato, Random Heterogeneous Materials: Microstructure and Macroscopic Properties https://doi.org/10.1007/978-1-4757-6355-3

    S. Torquato.Random Heterogeneous Materials: Microstructure and Macroscopic Properties, volume 16 ofInterdisciplinary Applied Mathematics. Springer, New York, 2002. doi: 10.1007/978-1-4757-6355-3

    work page doi:10.1007/978-1-4757-6355-3 2002

  2. [2]

    Diffusionmetamaterials.Nat

    Z.Zhang,L.Xu,T.Qu,M.Lei,Z.-K.Lin,X.Ouyang,J.-H.Jiang,andJ.Huang. Diffusionmetamaterials.Nat. Rev. Phys.,5:218–235,2023. doi: 10.1038/s42254-023-00565-4

    work page doi:10.1038/s42254-023-00565-4 2023

  3. [3]

    Torrado, Kevin Yu, Jennifer Domanowski, Peter J

    Corey Shemelya, Angel De La Rosa, Angel R. Torrado, Kevin Yu, Jennifer Domanowski, Peter J. Bonacuse, Richard E. Martin, Michael Juhasz, Frances I. Hurwitz, Ryan B. Wicker, Brett Conner, Eric MacDonald, and David A. Roberson. Anisotropy of thermal conductivity in 3d printed polymer matrix composites for space based cube satellites.Addit. Manuf., 16:186–19...

    work page doi:10.1016/j.addma.2017.05.012 2017

  4. [4]

    Thermalpropertiesof3dprinted products from the most common polymers.Int

    IrinaBute,SergejsTarasovs,SergejsVidinejevs,LaimaVevere,JevgenijsSevcenko,andAndreyAniskevich. Thermalpropertiesof3dprinted products from the most common polymers.Int. J. Adv. Manuf. Technol., 124(7–8):2739–2753, 2023. doi: 10.1007/s00170-022-10657-7

    work page doi:10.1007/s00170-022-10657-7 2023

  5. [5]

    Progress of polymer- based thermally conductive materials by fused filament fabrication: A comprehensive review.Polymers, 14(20):4297, 2022

    Zewei Cai, Naveen Thirunavukkarasu, Xuefeng Diao, Haoran Wang, Lixin Wu, Chen Zhang, and Jianlei Wang. Progress of polymer- based thermally conductive materials by fused filament fabrication: A comprehensive review.Polymers, 14(20):4297, 2022. doi: 10.3390/ polym14204297

    2022

  6. [6]

    J. Bear. On the tensor form of dispersion in porous media.J. Geophys. Res., 66(4):1185–1197, 1961. doi: 10.1029/JZ066i004p01185

    work page doi:10.1029/jz066i004p01185 1961

  7. [7]

    Pérez-Illanes, G

    R. Pérez-Illanes, G. Solé-Mari, and D. Fernàndez-Garcia. Smoothed particle hydrodynamics for anisotropic dispersion in heterogeneous porous media.Adv. Water Resour., 183:104601, 2024. doi: 10.1016/j.advwatres.2023.104601

    work page doi:10.1016/j.advwatres.2023.104601 2024

  8. [8]

    Aavatsmark, T

    I. Aavatsmark, T. Barkve, Ø. Bøe, and T. Mannseth. Discretization on non-orthogonal, quadrilateral grids for inhomogeneous, anisotropic media.J. Comput. Phys., 127:2–14, 1996. doi: 10.1006/jcph.1996.0154

    work page doi:10.1006/jcph.1996.0154 1996

  9. [9]

    A. S. Abushaikha and K. M. Terekhov. A fully implicit mimetic finite difference scheme for general purpose subsurface reservoir simulation with full tensor permeability.J. Comput. Phys., 406:109194, 2020. doi: 10.1016/j.jcp.2019.109194

    work page doi:10.1016/j.jcp.2019.109194 2020

  10. [10]

    S. I. Braginskii. Transport processes in a plasma. In M. A. Leontovich, editor,Reviews of Plasma Physics, volume 1, pages 205–311. Consultants Bureau, New York, 1965

    1965

  11. [11]

    Biriukov and D

    S. Biriukov and D. J. Price. Stable anisotropic heat conduction in smoothed particle hydrodynamics.Mon. Not. R. Astron. Soc., 483(4): 4901–4909, 2019. doi: 10.1093/mnras/sty3413

    work page doi:10.1093/mnras/sty3413 2019

  12. [12]

    Green, Xiaozhe Hu, Jeremy D

    David L. Green, Xiaozhe Hu, Jeremy D. Lore, Lin Mu, and Mark L. Stowell. An efficient high-order numerical solver for diffusion equations with strong anisotropy.Comput. Phys. Commun., 276:108333, 2022. doi: 10.1016/j.cpc.2022.108333

    work page doi:10.1016/j.cpc.2022.108333 2022

  13. [13]

    P. J. Basser, J. Mattiello, and D. Le Bihan. MR diffusion tensor spectroscopy and imaging.Biophys. J., 66(1):259–267, 1994. doi: 10.1016/S0006-3495(94)80775-1

    work page doi:10.1016/s0006-3495(94)80775-1 1994

  14. [14]

    Anintegrativesmoothedparticlehydrodynamicsmethodformodelingcardiacfunction

    C.Zhang,J.Wang,M.Rezavand,D.Wu,andX.Hu. Anintegrativesmoothedparticlehydrodynamicsmethodformodelingcardiacfunction. Comput. Methods Appl. Mech. Eng., 381:113847, 2021. doi: 10.1016/j.cma.2021.113847

    work page doi:10.1016/j.cma.2021.113847 2021

  15. [15]

    F. Perrin. Brownian motion of an ellipsoid—II. Free rotation and fluorescence depolarization. Translation and diffusion of ellipsoidal molecules.J. Phys. Radium, 7:1–11, 1936. doi: 10.1051/jphysrad:01936007010100

    work page doi:10.1051/jphysrad:01936007010100 1936

  16. [16]

    A. H. Kumar, A. B. Thomson, T. R. Powers, and D. M. Harris. Taylor dispersion of elongated rods.Phys. Rev. Fluids, 6:094501, 2021. doi: 10.1103/PhysRevFluids.6.094501

    work page doi:10.1103/physrevfluids.6.094501 2021

  17. [17]

    Perona and J

    P. Perona and J. Malik. Scale-space and edge detection using anisotropic diffusion.IEEE Trans. Pattern Anal. Mach. Intell., 12(7):629–639,

  18. [18]

    Perona, J

    doi: 10.1109/34.56205. Feng et al.:Preprint submitted to ElsevierPage 37 of 39 Entropic lattice Boltzmann for anisotropic advection–diffusion

    work page doi:10.1109/34.56205

  19. [19]

    Imagedenoisingbasedonfractionalanisotropicdiffusionandspatialcentralschemes.Signal Process.,230:109869,2025

    M.P.Paskaš. Imagedenoisingbasedonfractionalanisotropicdiffusionandspatialcentralschemes.Signal Process.,230:109869,2025. doi: 10.1016/j.sigpro.2024.109869

    work page doi:10.1016/j.sigpro.2024.109869 2025

  20. [20]

    van Es, B

    B. van Es, B. Koren, and H. J. de Blank. Finite-difference schemes for anisotropic diffusion.J. Comput. Phys., 272:526–549, 2014. doi: 10.1016/j.jcp.2014.04.046

    work page doi:10.1016/j.jcp.2014.04.046 2014

  21. [21]

    Aconstrainedfiniteelementmethodsatisfyingthediscretemaximumprincipleforanisotropic diffusion problems.J

    D.Kuzmin,M.J.Shashkov,andD.Svyatskiy. Aconstrainedfiniteelementmethodsatisfyingthediscretemaximumprincipleforanisotropic diffusion problems.J. Comput. Phys., 228(9):3448–3463, 2009. doi: 10.1016/j.jcp.2009.01.031

    work page doi:10.1016/j.jcp.2009.01.031 2009

  22. [22]

    Manzini and M

    G. Manzini and M. Putti. Mesh locking effects in the finite volume solution of 2-D anisotropic diffusion equations.J. Comput. Phys., 220(2): 751–771, 2007. doi: 10.1016/j.jcp.2006.05.026

    work page doi:10.1016/j.jcp.2006.05.026 2007

  23. [23]

    Cell-centerednonlinearfinite-volumemethodsfortheheterogeneousanisotropicdiffusion problem.J

    K.M.Terekhov,B.T.Mallison,andH.A.Tchelepi. Cell-centerednonlinearfinite-volumemethodsfortheheterogeneousanisotropicdiffusion problem.J. Comput. Phys., 330:245–267, 2017. doi: 10.1016/j.jcp.2016.11.010

    work page doi:10.1016/j.jcp.2016.11.010 2017

  24. [24]

    Finitevolumeschemesfordiffusionequations:Introductiontoandreviewofmodernmethods.Math

    J.Droniou. Finitevolumeschemesfordiffusionequations:Introductiontoandreviewofmodernmethods.Math. Models Methods Appl. Sci., 24(8):1575–1619, 2014. doi: 10.1142/S0218202514400041

    work page doi:10.1142/s0218202514400041 2014

  25. [25]

    Droniou, R

    J. Droniou, R. Eymard, T. Gallouet, and R. Herbin. A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods.Math. Models Methods Appl. Sci., 20(2):265–295, 2010. doi: 10.1142/S0218202510004222

    work page doi:10.1142/s0218202510004222 2010

  26. [26]

    Herbin and F

    R. Herbin and F. Hubert. Benchmark on discretization schemes for anisotropic diffusion problems on general grids. In R. Eymard and J.-M. Herard, editors,Finite Volumes for Complex Applications V, pages 659–692. Wiley, 2008

    2008

  27. [27]

    Dahmen, J

    N. Dahmen, J. Droniou, and F. Rogier. A cost-effective nonlinear extremum-preserving finite volume scheme for highly anisotropic diffusion on cartesian grids, with application to radiation belt dynamics.J. Comput. Phys., 463:111258, 2022. doi: 10.1016/j.jcp.2022.111258

    work page doi:10.1016/j.jcp.2022.111258 2022

  28. [28]

    van Es, B

    B. van Es, B. Koren, and H. J. de Blank. Finite-volume scheme for anisotropic diffusion.J. Comput. Phys., 306:422–442, 2016. doi: 10.1016/j.jcp.2015.11.041

    work page doi:10.1016/j.jcp.2015.11.041 2016

  29. [29]

    D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini. Unified analysis of discontinuous Galerkin methods for elliptic problems.SIAM J. Numer. Anal., 39(5):1749–1779, 2002. doi: 10.1137/S0036142901384162

    work page doi:10.1137/s0036142901384162 2002

  30. [30]

    Cockburn and C.-W

    B. Cockburn and C.-W. Shu. The local discontinuous Galerkin method for time-dependent convection-diffusion systems.SIAM J. Numer. Anal., 35(6):2440–2463, 1998. doi: 10.1137/S0036142997316712

    work page doi:10.1137/s0036142997316712 1998

  31. [31]

    Gassner, F

    G. Gassner, F. Lörcher, and C.-D. Munz. A contribution to the construction of diffusion fluxes for finite volume and discontinuous galerkin schemes.J. Comput. Phys., 224(2):1049–1063, 2007. doi: 10.1016/j.jcp.2006.11.004

    work page doi:10.1016/j.jcp.2006.11.004 2007

  32. [32]

    Highorderweightedessentiallynonoscillatoryschemesforconvectiondominatedproblems.SIAM Rev.,51(1):82–126,2009

    C.-W.Shu. Highorderweightedessentiallynonoscillatoryschemesforconvectiondominatedproblems.SIAM Rev.,51(1):82–126,2009. doi: 10.1137/070679065

    work page doi:10.1137/070679065 2009

  33. [33]

    Smoothedparticlehydrodynamicsofanisotropicdiffusions.Comput

    X.Tang,X.Hu,andO.Haidn. Smoothedparticlehydrodynamicsofanisotropicdiffusions.Comput. Methods Appl. Mech. Eng.,451:118697,

  34. [34]

    doi: 10.1016/j.cma.2025.118697

    work page doi:10.1016/j.cma.2025.118697 2025

  35. [35]

    Ginzburg

    I. Ginzburg. Equilibrium-type and link-type lattice Boltzmann models for generic advection and anisotropic-dispersion equation.Adv. Water Resour., 28(11):1171–1195, 2005. doi: 10.1016/j.advwatres.2005.03.004

    work page doi:10.1016/j.advwatres.2005.03.004 2005

  36. [36]

    Yoshida and M

    H. Yoshida and M. Nagaoka. Multiple-relaxation-time lattice Boltzmann model for the convection and anisotropic diffusion equation.J. Comput. Phys., 229(20):7774–7795, 2010. doi: 10.1016/j.jcp.2010.06.037

    work page doi:10.1016/j.jcp.2010.06.037 2010

  37. [37]

    Chai and T

    Z. Chai and T. S. Zhao. Lattice Boltzmann model for the convection-diffusion equation.Phys. Rev. E, 87:063309, 2013. doi: 10.1103/ PhysRevE.87.063309

    2013

  38. [38]

    Z. Chai, B. Shi, and Z. Guo. A multiple-relaxation-time lattice Boltzmann model for general nonlinear anisotropic convection–diffusion equations.J. Sci. Comput., 69(1):355–390, 2016. doi: 10.1007/s10915-016-0198-5

    work page doi:10.1007/s10915-016-0198-5 2016

  39. [39]

    Huang and H

    R. Huang and H. Wu. A modified multiple-relaxation-time lattice Boltzmann model for convection–diffusion equation.J. Comput. Phys., 274:50–63, 2014. doi: 10.1016/j.jcp.2014.05.041

    work page doi:10.1016/j.jcp.2014.05.041 2014

  40. [40]

    Y. Zhao, Y. Wu, Z. Chai, and B. Shi. A block triple-relaxation-time lattice Boltzmann model for nonlinear anisotropic convection–diffusion equations.Comput. Math. Appl., 79(9):2550–2573, 2020. doi: 10.1016/j.camwa.2019.11.018

    work page doi:10.1016/j.camwa.2019.11.018 2020

  41. [41]

    Ginzburg

    I. Ginzburg. Multiple anisotropic collisions for advection–diffusion lattice Boltzmann schemes.Adv. Water Resour., 51:381–404, 2013. doi: 10.1016/j.advwatres.2012.04.013

    work page doi:10.1016/j.advwatres.2012.04.013 2013

  42. [42]

    J. Perko. A single-relaxation-time lattice Boltzmann model for anisotropic advection-diffusion equation based on the diffusion velocity flux formulation.Comput. Geosci., 22:1423–1432, 2018. doi: 10.1007/s10596-018-9761-5

    work page doi:10.1007/s10596-018-9761-5 2018

  43. [43]

    Hamila, A

    R. Hamila, A. Jemni, and P. Perré. A lattice Boltzmann source formulation for advection and anisotropic diffusion.Indian J. Phys., 97: 3047–3055, 2023. doi: 10.1007/s12648-023-02667-2

    work page doi:10.1007/s12648-023-02667-2 2023

  44. [44]

    D. Li, F. Li, and B. Xu. Multi-relaxation-time lattice Boltzmann method for anisotropic convection-diffusion equation with divergence-free velocity field.Comput. Math. Appl., 171:1–5, 2024. doi: 10.1016/j.camwa.2024.07.005

    work page doi:10.1016/j.camwa.2024.07.005 2024

  45. [45]

    Perko and R

    J. Perko and R. A. Patel. Single-relaxation-time lattice Boltzmann scheme for advection-diffusion problems with large diffusion-coefficient heterogeneities and high-advection transport.Phys. Rev. E, 89:053309, 2014. doi: 10.1103/PhysRevE.89.053309

    work page doi:10.1103/physreve.89.053309 2014

  46. [46]

    Gibbs’principleforthelattice-kinetictheoryoffluiddynamics.Phys

    I.V.Karlin,F.Bösch,andS.S.Chikatamarla. Gibbs’principleforthelattice-kinetictheoryoffluiddynamics.Phys. Rev. E,90:031302,2014. doi: 10.1103/PhysRevE.90.031302

    work page doi:10.1103/physreve.90.031302 2014

  47. [47]

    EntropicmultirelaxationlatticeBoltzmannmodelsforturbulentflows.Phys

    F.Bösch,S.S.Chikatamarla,andI.V.Karlin. EntropicmultirelaxationlatticeBoltzmannmodelsforturbulentflows.Phys. Rev. E,92:043309,

  48. [48]

    doi: 10.1103/PhysRevE.92.043309

    work page doi:10.1103/physreve.92.043309

  49. [49]

    Y. H. Qian, D. D’Humières, and P. Lallemand. Lattice BGK models for Navier–Stokes equation.Europhys. Lett., 17(6):479–484, 1992. doi: 10.1209/0295-5075/17/6/001

    work page doi:10.1209/0295-5075/17/6/001 1992

  50. [50]

    G. Strang. On the construction and comparison of difference schemes.SIAM J. Numer. Anal., 5(3):506–517, 1968. doi: 10.1137/0705041

    work page doi:10.1137/0705041 1968

  51. [51]

    Happel and H

    J. Happel and H. Brenner.Low Reynolds Number Hydrodynamics: With Special Applications to Particulate Media, volume 1. Springer, Dordrecht, 2012. doi: 10.1007/978-94-009-8352-6

    work page doi:10.1007/978-94-009-8352-6 2012

  52. [52]

    F. Perrin. Brownian motion of an ellipsoid—I. Dielectric dispersion for ellipsoidal molecules.J. Phys. Radium, 5:497–511, 1934. doi: 10.1051/jphysrad:01934005010049700. Feng et al.:Preprint submitted to ElsevierPage 38 of 39 Entropic lattice Boltzmann for anisotropic advection–diffusion

    work page doi:10.1051/jphysrad:01934005010049700 1934

  53. [53]

    S. Koenig. Brownian motion of an ellipsoid. A correction to Perrin’s results.Biopolymers, 14(11):2421–2423, 1975. doi: 10.1002/bip.1975. 360141115

    work page doi:10.1002/bip.1975 1975

  54. [54]

    K.Kurabayashi.Anisotropicthermalpropertiesofsolidpolymers.Int. J. Thermophys.,22(1):277–288,2001.doi:10.1023/A:1006728223978

    work page doi:10.1023/a:1006728223978 2001

  55. [55]

    Li, Q.-M

    K.-Q. Li, Q.-M. Chen, and G. Chen. Scale dependency of anisotropic thermal conductivity of heterogeneous geomaterials.Bull. Eng. Geol. Environ., 83:73, 2024. doi: 10.1007/s10064-024-03571-7

    work page doi:10.1007/s10064-024-03571-7 2024

  56. [56]

    M. Wang, J. Wang, N. Pan, and S. Chen. Mesoscopic predictions of the effective thermal conductivity for microscale random porous media. Phys. Rev. E, 75(3):036702, 2007. doi: 10.1103/PhysRevE.75.036702

    work page doi:10.1103/physreve.75.036702 2007

  57. [57]

    Wang and N

    M. Wang and N. Pan. Modeling and prediction of the effective thermal conductivity of random open-cell porous foams.Int. J. Heat Mass Transf., 51(5–6):1325–1331, 2008. doi: 10.1016/j.ijheatmasstransfer.2007.11.031

    work page doi:10.1016/j.ijheatmasstransfer.2007.11.031 2008

  58. [58]

    Predictionsofeffectivephysicalpropertiesofcomplexmultiphasematerials.Mater

    M.WangandN.Pan. Predictionsofeffectivephysicalpropertiesofcomplexmultiphasematerials.Mater. Sci. Eng. R Rep.,63(1):1–30,2008. doi: 10.1016/j.mser.2008.07.001

    work page doi:10.1016/j.mser.2008.07.001 2008

  59. [59]

    M. Wang, Q. Kang, and N. Pan. Thermal conductivity enhancement of carbon fiber composites.Appl. Therm. Eng., 29(2–3):418–421, 2009. doi: 10.1016/j.applthermaleng.2008.03.004

    work page doi:10.1016/j.applthermaleng.2008.03.004 2009

  60. [60]

    Cannell, Lars I

    Guenter Ahlers, David S. Cannell, Lars I. Berge, and Shinichi Sakurai. Thermal conductivity of the nematic liquid crystal 4-n-pentyl-4’- cyanobiphenyl.Phys. Rev. E, 49:545–553, 1994. doi: 10.1103/PhysRevE.49.545

    work page doi:10.1103/physreve.49.545 1994

  61. [61]

    Q. Feng, W. Pesch, and L. Kramer. Theory of rayleigh–bénard convection in planar nematic liquid crystals.Phys. Rev. A, 45:7242–7256,

  62. [62]

    doi: 10.1103/PhysRevA.45.7242

    work page doi:10.1103/physreva.45.7242

  63. [63]

    Magnetic-fieldeffectonthermalconvectionofanematicliquidcrystalatlargerayleighnumbers.J

    StephanWeissandGuenterAhlers. Magnetic-fieldeffectonthermalconvectionofanematicliquidcrystalatlargerayleighnumbers.J. Fluid Mech., 716:R7, 2013. doi: 10.1017/jfm.2012.570

    work page doi:10.1017/jfm.2012.570 2013

  64. [64]

    Nield and Adrian Bejan.Convection in Porous Media

    Donald A. Nield and Adrian Bejan.Convection in Porous Media. Springer, Cham, 5th edition, 2017. doi: 10.1007/978-3-319-49562-0

    work page doi:10.1007/978-3-319-49562-0 2017

  65. [65]

    Xiaojue Zhu, Varghese Mathai, Richard J. A. M. Stevens, Roberto Verzicco, and Detlef Lohse. Transition to the ultimate regime in two- dimensional rayleigh–bénard convection.Phys. Rev. Lett., 120(14):144502, 2018. doi: 10.1103/PhysRevLett.120.144502. Feng et al.:Preprint submitted to ElsevierPage 39 of 39

    work page doi:10.1103/physrevlett.120.144502 2018