Pith. sign in

REVIEW 4 major objections 4 minor 48 references

UGKWP and IUGKP methods for Multi-Scale Phonon Transport with Dispersion and Polarization

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

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

Pith's one-line read The paper claims that two wave-particle schemes, UGKWP and IUGKP, solve the frequency-dependent phonon Boltzmann equation across all Knudsen regimes, with IUGKP converging in 4 to 320 iterations and running 3D multiscale heat conduction…

desk verdict A useful extension of UGKWP to multi-frequency phonon BTE with a real efficiency claim, but the IUGKP validation relies on a variable-relaxation-time algorithm that the paper only references, not derives. read the letter →

arxiv 2506.16203 v1 pith:EBCBLRYS submitted 2025-06-19 physics.comp-ph

Hongyu Liu , Xiaojian Yang , Chuang Zhang , Xing Ji , Kun Xu This is my paper · ORCID
classification physics.comp-ph MSC 65M0880A1982C70 PACS 44.10.+i63.20.-e
keywords unifiedgas-kineticwaveparticlemethodphononBoltzmanntransportequationmultiscaleheatconductiondispersionpolarizationimplicitgaskineticfrequency-spaceadaptivesamplingKnudsennumber
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that the frequency-dependent phonon Boltzmann equation, with full dispersion and polarization effects, can be solved efficiently across every transport regime by two wave-particle schemes. The explicit UGKWP method adapts automatically: in the diffusive limit its flux collapses to Fourier's law, while in the ballistic limit free-streaming particles carry the non-equilibrium flux. The steady-state IUGKP method adds a macroscopic prediction equation so that convergence is fast at both small and large Knudsen numbers, and adaptive sampling in frequency space keeps the particle count per cell at the level of a single-frequency gray method. If these claims hold, large three-dimensional multiscale heat-conduction simulations become laptop-scale rather than supercomputer-scale problems.

What carries the argument

The carrier of the argument is the wave-particle decomposition of each frequency group's distribution into an equilibrium 'wave' flux, computed analytically from the integral solution of the unsteady BGK equation, and a free-transport 'particle' flux. For IUGKP, the load-bearing new object is the macroscopic prediction equation (Eq. 76), $-\nabla\cdot(\gamma\kappa_{\mathrm{eff}}\nabla\delta T)=\nabla\cdot q$, an inexact-Newton acceleration that dominates convergence at small Knudsen numbers while automatically shutting off in the ballistic limit. Adaptive sampling maps particles to frequency intervals by cumulative energy fraction, reducing the total particle count from order $N_B\times N_{\mathrm{ref}}$ to order $N_{\mathrm{ref}}$.

What would settle it

Run the one-dimensional cross-plane silicon film tests at L = 10 nm, 100 nm, 1 $\mu$m, and 100 $\mu$m with Terris relaxation times using only the constant-tau IUGKP derivation given in Section 4; if the results fail to match the DUGKS reference solutions or lose the reported convergence ratios of 400, 180, 20, and 38, the IUGKP validation collapses.

Watch

Extended reading notes

Core claim

The central discovery is that treating each discrete phonon frequency as its own gray BGK group, joined only by a frequency-integrated collision compatibility condition, lets a unified gas-kinetic wave-particle solver handle dispersion and polarization without extra memory growth. The IUGKP variant replaces explicit time-marching with a steady-state iteration in which particles free-stream a distance drawn from the physical mean free path, and a newly constructed macroscopic prediction equation of the form $-\nabla\cdot(\gamma\kappa_{\mathrm{eff}}\nabla\delta T)=\nabla\cdot q$ supplies the missing small-Knudsen acceleration. The paper reports that IUGKP converges in 4 to 320 iterations on one-dimensional films from 10 nm to 100 $\mu$m, up to two orders of magnitude faster than UGKWP, and completes three-dimensional cubic and heat-dissipation simulations in 40 and 100 minutes respectively while using about 2 GB of memory.

Load-bearing premise

The IUGKP algorithm is derived only for a constant relaxation time per frequency group, but every numerical test uses a relaxation time that depends on temperature and frequency, so the validation leans on a variable-tau formulation the paper does not present.

Editorial extensions

If this is right

  • If correct, steady-state multiscale heat conduction is no longer memory-bound: full three-dimensional runs fit in about 2 GB and finish on one core.
  • UGKWP retains accuracy for unsteady problems, so the method pair covers transient and steady simulations with the same machinery.
  • Frequency-adaptive sampling should make the cost of adding dispersion and polarization nearly independent of the number of frequency groups used.
  • The reported IUGKP iteration counts, from 4 in the diffusive limit to 320 in the ballistic regime, suggest the method needs no regime-specific algorithmic switching.
  • The automatic collapse to Fourier's law in the diffusive limit means no separate diffusion solver is needed for low-Knudsen regimes.

Reading between the lines

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

  • The constant-tau derivation gap implies that the paper's validation rests on a variable-relaxation-time formulation drawn from the companion gray IUGKP work; a reader applying the method to real silicon must obtain that formulation.
  • Because adaptive sampling allocates particles by equilibrium energy fraction, the same idea could reduce memory in deterministic discrete-ordinate or unified gas-kinetic solvers by weighting angular quadrature per frequency.
  • The macroscopic prediction equation is effectively a Poisson solve for the temperature correction, so its amplification factor could be viewed as a tunable preconditioner that may extend to nonlinear or strongly temperature-dependent materials.
  • If the 40-to-100-minute three-dimensional runs hold, routine device-scale thermal simulation with resolved dispersion becomes a desktop task, enabling fast optimization of micro- and nano-scale electronics cooling.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The manuscript develops two particle-based multiscale methods for the frequency-dependent phonon Boltzmann transport equation with dispersion and polarization: the UGKWP method for unsteady problems and the IUGKP method for steady-state problems. Both methods solve multiple groups of BGK equations at discrete frequency points, construct multiscale fluxes from integral solutions of the BGK equation, and use an adaptive frequency-space sampling technique to control particle counts. The IUGKP method adds a macroscopic diffusion-synthetic prediction equation to accelerate convergence at small Knudsen numbers. Numerical validation covers 1D, 2D, and 3D silicon heat conduction cases against deterministic reference solutions, with reported speedups of one to two orders of magnitude and 3D simulations running in 40--100 minutes on a single-core laptop.

Significance. If the methods perform as claimed, they offer a practical route to large-scale phonon transport simulations that are currently intractable for deterministic DOM-type solvers. The constant-relaxation-time derivations are self-contained, the algorithms are clearly structured, and the efficiency results in Sections 5.4 and 5.5 are impressive if reproducible. However, the validation and the headline efficiency claims rest on an unstated variable-relaxation-time formulation and an unspecified tuning parameter, so the significance cannot be fully assessed from the manuscript in its present form. The paper does not ship code, but the algorithmic descriptions are sufficiently detailed for the constant-τ case to be reimplemented.

major comments (4)
  1. [Section 4.2, Eq. (57)] The IUGKP algorithm is derived only for constant relaxation time per group, with the text explicitly stating 'we only present the algorithm formulation for the case where τg is constant' and referring to [44] for the spatially varying case. However, all numerical tests use Terris' relaxation rule (Table 2), in which τ depends on temperature and frequency; because temperature varies in every test, τg varies spatially in every run. The steady-state integral solution in Eq. (52) and the CDF in Eq. (53) are written with spatially varying τ, but the particle evolution step and the free-streaming length sampling in Eq. (57) use a constant τg. The manuscript does not provide the variable-τ particle resampling, compatibility conditions, or the corresponding form of the macroscopic prediction equation. As a result, the numerical validations in Sections 5.1--5.5 exercise an algorithm whose specification is absent from this paper, and the central convergence claim for IUGKP at small Knudsen numbers is not directly supported by the derivation presented here. Please include the variable-τ formulation or restrict the tests and claims to the constant-τ case.
  2. [Section 3, Eqs. (20)--(25)] The same constant-τ limitation affects the UGKWP derivation: the integral solution (20), the expansion (21)--(23), and the flux coefficients (25) assume a constant τg along the characteristic and across the time step. With the temperature-dependent Terris rule used in Sections 5.1 and 5.2, τg varies with x and t, and the manuscript does not state how τg is evaluated in the integral solution, in the coefficients, or in the compatibility condition (11). This is load-bearing for the UGKWP validation as well, since the transient tests use the same relaxation model. Please clarify the treatment of spatially varying τ in the UGKWP flux construction, or restrict the claims accordingly.
  3. [Section 4.4, Eq. (76) and Algorithm 2] The amplification factor γ in the macroscopic prediction equation is a free parameter, but its value or selection criterion is never reported in the numerical tests or in Algorithm 2. Because the convergence speed of IUGKP in the diffusive regime depends directly on this factor (as explained in Remark 3), the iteration counts in Table 3 and the CPU times in Tables 4 and 6 cannot be reproduced without this information. Please state the γ values used for each test and the criterion for choosing them.
  4. [Section 4.1, Eqs. (53)--(56)] The transition from the integral solution to the particle-based representation uses the identity P_{g,c} = dP_{g,f}/dx, which the text states holds for constant τg. For variable τg, this identity does not hold in the form needed for the cumulative weighting in Eq. (56). Since the tests use variable τg, the derivation of Eq. (56) is not applicable to the tested settings. This is a specific instance of the gap identified in the first major comment and should be addressed in the revision.
minor comments (4)
  1. [Eq. (45)] In the second term of F^{fr,wave}_{g,ij}, 'c2' appears to be a typo and should read 'V_g^2'.
  2. [Table 3] The iteration count for UGKWP at L = 100 μm is reported as '12,0000' (120,000) and the ratio as 38; please check the typography and arithmetic consistency.
  3. [Section 5.3, Tables 4 and 6] The captions and text do not consistently distinguish the UGKWP variant without frequency adaptive sampling (Table 4) from the variant with it (Table 6); please label the variants identically in the text and in the table captions.
  4. [Section 5.4] The text states that the IUGKP results for the 3D cubic domain are 'shown in Fig .11', but the contour plot for that test is Fig. 8; the figure cross-references should be corrected.

Circularity Check

1 steps flagged · score 3.0 of 10

Core derivations are self-contained and no quantity is obtained by fitting; the main concern is a load-bearing same-author citation: the variable-tau IUGKP formulation used in all validation runs is deferred to ref [44], not derived here.

  1. self citation load bearing [Section 4.2, Eq. (52)-(57); Section 5, Table 2]
    "we only present the algorithm formulation for the case where τg is constant. For the case where τg varies with space, please refer to our previous work [44]."

    This is an omitted-proof and load-bearing self-citation concern rather than a definitional reduction. The IUGKP derivation in this paper gives the integral solution (52), CDF (53), PDF (54), and free-streaming length (57) only for constant τg, and the macro prediction (76) uses κeff = Σ Cg τg |Vg|^2/3. All Section 5 tests use Terris' relaxation rule (Table 2), which depends on T and ω, so τg is spatially varying in every run. The variable-τ particle resampling and compatibility weights are neither derived nor reproduced; they are simply deferred to [44], a parallel preprint by the same authors. The numerical validations therefore test an algorithm not fully specified in this manuscript. This is a missing-support gap that touches the central IUGKP efficiency claim, not a claim that Eq.

full rationale

None of the derivation steps reduces to its own inputs. The temperature relation (17) is obtained from the compatibility condition (11); the multiscale interface flux (24) and coefficients (25) are algebraic consequences of the BGK integral solution; the wave-particle split (41)-(45) is derived, not fitted; the macro prediction equation (76) is a standard diffusion-synthetic acceleration built from the same relaxation-time model, with κeff being the conventional conductivity and no fitted parameter being renamed as a prediction. The adaptive frequency-space sampling (65)-(67) lowers particle count by design rather than by an independent prediction, so it is a design choice, not a circular claim. The reference solutions in [48] share first author Chuang Zhang, but [48] is a separately published DUGKS/synthetic-iteration benchmark, so the overlap is a caution for independence rather than circularity. The one substantive concern is the Section 4.2 limitation quoted above: the IUGKP algorithm is derived only for constant τg, while validation runs use a temperature- and frequency-dependent relaxation time, with the variable-τ formulation deferred to same-author ref [44]. That makes the validation gap load-bearing for the IUGKP speed and accuracy claims, but it is not a case of the paper's predictions being equivalent to its inputs. Hence the score is 3 rather than 6+.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The methods rest on five modeling assumptions: linearization of the equilibrium distribution (Eq. 6), the BGK relaxation approximation (Eq. 1), isotropic dispersion equal to the [100] direction (Section 5), an IUGKP derivation limited to constant relaxation time (Section 4.2), and adaptive sampling by energy fraction (Eq. 65). One hand-chosen parameter, the amplification factor γ (Eq. 76), is left unspecified. No new physical entities are postulated; the macro prediction operator is a numerical preconditioner.

free parameters (1)
  • γ (amplification factor in macro prediction equation)
    Introduced in Eq. (76) and Remark 3 to reduce statistical noise and stiffness in the IUGKP macro prediction; no value or selection rule is given, and it directly affects the converged solution and iteration path.
assumptions (5)
  • domain assumption Bose-Einstein equilibrium is linearized around a fixed reference temperature Tref (Eq. (6)).
    Used to write the energy-form BGK equation (Section 2, Eqs. (5)-(7)); conclusion says extension to arbitrary temperature differences is future work, so current validity is limited to small ΔT.
  • domain assumption Phonon transport is modeled by a single-mode BGK relaxation approximation per frequency group (Eqs. (1), (9)).
    Standard approximation in the phonon BTE literature; the paper does not justify it beyond prior practice.
  • domain assumption Dispersion in all crystal directions equals the [100] direction for silicon (Section 5).
    Stated in Section 5 before numerical tests; this isotropic approximation defines group velocities and phonon density of states used in every test.
  • ad hoc to paper IUGKP particle evolution is derived for constant τg per group (Section 4.2), while tests use temperature-dependent τ.
    Section 4.2 explicitly limits the derivation to constant τg and defers the variable-τ case to ref [44]; the validation nevertheless uses Terris' temperature-dependent relaxation rule (Table 2).
  • domain assumption Adaptive frequency sampling allocates particles by equilibrium energy fraction Fg (Eq. (65)).
    Section 4.3 argues low-energy groups are negligible regardless of Knudsen number; this assumes energy fraction is a sufficient proxy for a group's contribution to temperature and flux.

how reviews work

0 comments
Cite this review

Pith. "Pith review of UGKWP and IUGKP methods for Multi-Scale Phonon Transport with Dispersion and Polarization." pith.science (2026). https://pith.science/paper/EBCBLRYS

@misc{pith2026250616203,
  author       = {Pith},
  title        = {Pith review of: UGKWP and IUGKP methods for Multi-Scale Phonon Transport with Dispersion and Polarization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EBCBLRYS}},
  note         = {Machine review of arXiv:2506.16203}
}
read the original abstract

This paper presents two novel methods for solving multi-scale phonon transport problems with dispersion and polarization effects: the unified gas-kinetic wave-particle (UGKWP) method and the implicit unified gas-kinetic particle (IUGKP) method. Both approaches are based on solving multiple groups of BGK equations at discrete frequency points. The UGKWP method constructs multiscale macroscopic fluxes at cell interfaces through the integral solution of the unsteady BGK equation and efficiently captures non-equilibrium transport using statistical particles. Its wave-particle adaptive framework ensures computational efficiency across different regimes: in the diffusive limit, it matches the cost of explicit diffusion equation solutions, while in the ballistic limit, it performs comparably to pure particle methods. The IUGKP method, specifically designed for steady-state problems, determines the particle evolution scale based on the physical mean free path. This approach enables rapid convergence at both large and small Knudsen numbers, with the latter facilitated by a newly constructed macroscopic prediction equation. Both methods incorporate an adaptive frequency-space sampling technique that maintains particle counts per cell comparable to single-frequency methods, significantly improving computational efficiency and memory usage. The accuracy and efficiency of both methods are validated through various numerical tests, including large-scale three-dimensional conduction heat transfer simulations. Results demonstrate their effectiveness in handling complex phonon transport phenomena across multiple scales.

Figures

Figures reproduced from arXiv: 2506.16203 by the authors.

Figure 1
Figure 1. Dispersion relationship. boundary is imposed with a temperature of TL. The reference solutions can be obtained in [48]. To minimize statistical noise, the reference number for sampling particles in each cell is set to 100,000. The one-dimensional computational domain is discretized into 40 uniform cells, and the CFL number is set to 0.8. Fig.3 illustrates a comparison among the computed results of the IUGKP method, … view at source ↗
Figure 2
Figure 2. A schematic diagram of one-dimensional heat conduction in a dielectric film. [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Comparison of 1D heat conduction results across a film at [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Comparison of 1D transient heat conduction results across a film. [PITH_FULL_IMAGE:figures/full_fig_p023_4.png]
Figure 5
Figure 5. Figure 5: Computational domain and boundary conditions of heat transfer in the 2D square domain. [PITH_FULL_IMAGE:figures/full_fig_p024_5.png]
Figure 6
Figure 6. Figure 6: Comparison of 2D square heat transfer results, respectively corresponding: [PITH_FULL_IMAGE:figures/full_fig_p025_6.png]
Figure 7
Figure 7. Figure 7: Computational domain and boundary conditions of Heat transfer in the 3D cubic domain. [PITH_FULL_IMAGE:figures/full_fig_p026_7.png]
Figure 8
Figure 8. Figure 8: Contour of the heat conduction in the 3D cubic. Left: 3D contour; Right: slice of [PITH_FULL_IMAGE:figures/full_fig_p027_8.png]
Figure 9
Figure 9. Figure 9: Quantitative results comparison. Left: Temperature distribution from [0, 0.5, 0.5] to [1, 0.5, 0.5]; [PITH_FULL_IMAGE:figures/full_fig_p027_9.png]
Figure 10
Figure 10. Figure 10: Computational domain and boundary conditions for Heat dissipation simulation in three [PITH_FULL_IMAGE:figures/full_fig_p028_10.png]
Figure 11
Figure 11. Figure 11: Contour of the heat dissipation simulation in three-dimensional structures. Left: 3D contour; [PITH_FULL_IMAGE:figures/full_fig_p028_11.png]
Figure 12
Figure 12. Figure 12: Quantitative results comparison, temperature distribution from [0.5, 0.5, 1.0] to [0.5, 0.5, 0]. [PITH_FULL_IMAGE:figures/full_fig_p029_12.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

48 extracted references · 46 canonical work pages

  1. [44]

    Implicit unified gas kinetic particle method for steady-state solution of multiscale phonon transport

    Hongyu Liu, Xiaojian Yang, Chuang Zhang, Xing Ji, and Kun Xu. Implicit unified gas kinetic particle method for steady-state solution of multiscale phonon transport. arXiv preprint arXiv:2506.09465, 2025

    work page Pith review arXiv 2025

  2. [1]

    Stettler, Stephen M

    Mark A. Stettler, Stephen M. Cea, Sayed Hasan, Lei Jiang, Patrick H. Keys, Colin D. Landon, Prabakar Marepalli, Daniel Pantuso, and Cory E. Weber. Industrial TCAD: Modeling atoms to chips. IEEE Transactions on Electron Devices, 68(11):5350–5357, 2021

    work page 2021

  3. [2]

    Review of multiscale simulation in submicron heat transfer

    Jayathi Y Murthy, Sreekant VJ Narumanchi, Pascual-Gutierrez Jose’A, Tianjiao Wang, Chunjian Ni, and Sanjay R Mathur. Review of multiscale simulation in submicron heat transfer. International Journal for Multiscale Computational Engineering, 3(1), 2005

    work page 2005

  4. [3]

    C. Prasad. A review of self-heating effects in advanced CMOS technologies. IEEE Transactions on Electron Devices, 66(11):4546–4555, 2019. 30

    work page 2019

  5. [4]

    Chapter eight - near-junction thermal managements of electronics

    Yu-Chao Hua, Yang Shen, Zheng-Lai Tang, Dao-Sheng Tang, Xin Ran, and Bing-Yang Cao. Chapter eight - near-junction thermal managements of electronics. Advances in Heat Transfer, 56:355–434, 2023

    work page 2023

  6. [5]

    Nanoscale energy transport and conversion a parallel treatment of electrons molecules phonons and photons mit pappalardo series in mechanical engineering by gang chen

    Parallel Treatment Of Electrons Molecules Phonons. Nanoscale energy transport and conversion a parallel treatment of electrons molecules phonons and photons mit pappalardo series in mechanical engineering by gang chen

    work page

  7. [6]

    Colloquium: phononic thermal properties of two-dimensional materials

    Xiaokun Gu, Yujie Wei, Xiaobo Yin, Baowen Li, and Ronggui Yang. Colloquium: phononic thermal properties of two-dimensional materials. Rev. Mod. Phys., 90:041002, Nov 2018

    work page 2018

  8. [7]

    Interfacial thermal resistance: Past, present, and future

    Jie Chen, Xiangfan Xu, Jun Zhou, and Baowen Li. Interfacial thermal resistance: Past, present, and future. Rev. Mod. Phys., 94:025002, Apr 2022

    work page 2022

Show all 48 references
  1. [8]

    Electrons and phonons: the theory of transport phenomena in solids

    John M Ziman. Electrons and phonons: the theory of transport phenomena in solids. Oxford university press, 2001

    2001

  2. [9]

    A comparative study of submicron phonon transport using the boltzmann transport equation and the lattice boltzmann method

    Ankur Chattopadhyay and Arvind Pattamatta. A comparative study of submicron phonon transport using the boltzmann transport equation and the lattice boltzmann method. Numerical Heat Transfer, Part B: Fundamentals, 66(4):360–379, 2014

    2014

  3. [10]

    Boltzmann transport equation for thermal transport in electronic materials and devices

    Matthew C Barry, Nitish Kumar, and Satish Kumar. Boltzmann transport equation for thermal transport in electronic materials and devices. Annual Review of Heat Transfer, 24:131–172, 2022

    2022

  4. [11]

    Boltzmann transport equation based modeling of phonon heat conduction: Progress and challenges

    Sandip Mazumder. Boltzmann transport equation based modeling of phonon heat conduction: Progress and challenges. Annual Review of Heat Transfer, 24:71–130, 2022

    2022

  5. [12]

    Progress of discrete unified gas-kinetic scheme for multiscale flows

    Zhaoli Guo and Kun Xu. Progress of discrete unified gas-kinetic scheme for multiscale flows. Adva. Aerodyn., 3(1):6, Feb 2021

    2021

  6. [13]

    Peraud, Colin D

    Jean-Philippe M. Peraud, Colin D. Landon, and Nicolas G. Hadjiconstantinou. Monte carlo methods for solving the Boltzmann equation. Annu. Rev. Heat Transfer, 17:205–265, 2014

    2014

  7. [14]

    Monte Carlo transient phonon transport in sil- icon and germanium at nanoscales

    David Lacroix, Karl Joulain, and Denis Lemonnier. Monte Carlo transient phonon transport in sil- icon and germanium at nanoscales. Physical Review B—Condensed Matter and Materials Physics, 72(6):064305, 2005

    2005

  8. [15]

    Lattice Boltzmann method and its applications in engineering, volume 3

    Zhaoli Guo and Chang Shu. Lattice Boltzmann method and its applications in engineering, volume 3. World Scientific, 2013

    2013

  9. [16]

    Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media

    Knut Stamnes, S-Chee Tsay, Warren Wiscombe, and Kolf Jayaweera. Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media. Applied optics, 27(12):2502–2509, 1988

    1988

  10. [17]

    Finite volume method for radiative heat transfer using unstructured meshes

    Jayathi Y Murthy and SR Mathur. Finite volume method for radiative heat transfer using unstructured meshes. Journal of thermophysics and heat transfer, 12(3):313–321, 1998

    1998

  11. [18]

    Sadayappan, and Arpit Mittal

    Syed Ashraf Ali, Gautham Kollu, Sandip Mazumder, P. Sadayappan, and Arpit Mittal. Large-scale parallel computation of the phonon Boltzmann transport equation. Int. J. Therm. Sci, 86:341 – 351, 2014

    2014

  12. [19]

    V. A. Fiveland and J. P. Jessee. Acceleration schemes for the discrete ordinates method.J. Thermophys. Heat Transfer., 10(3):445–451, 1996

    1996

  13. [20]

    Discrete unified gas kinetic scheme for multiscale heat transfer based on the phonon boltzmann transport equation

    Zhaoli Guo and Kun Xu. Discrete unified gas kinetic scheme for multiscale heat transfer based on the phonon boltzmann transport equation. International Journal of Heat and Mass Transfer, 102:944–958, 2016

    2016

  14. [21]

    A discrete unified gas kinetic scheme for phonon boltzmann trans- port equation accounting for phonon dispersion and polarization

    Xiao-Ping Luo and Hong-Liang Yi. A discrete unified gas kinetic scheme for phonon boltzmann trans- port equation accounting for phonon dispersion and polarization. International Journal of Heat and Mass Transfer, 114:970–980, 2017

    2017

  15. [22]

    Discrete unified gas kinetic scheme for multiscale heat transfer with arbitrary temperature difference

    Chuang Zhang and Zhaoli Guo. Discrete unified gas kinetic scheme for multiscale heat transfer with arbitrary temperature difference. International Journal of Heat and Mass Transfer, 134:1127–1136, 2019

    2019

  16. [23]

    Unified implicit kinetic scheme for steady multiscale heat transfer based on the phonon Boltzmann transport equation

    Chuang Zhang, Zhaoli Guo, and Songze Chen. Unified implicit kinetic scheme for steady multiscale heat transfer based on the phonon Boltzmann transport equation. Phys. Rev. E, 96:063311, Dec 2017

    2017

  17. [24]

    Acceler- ation strategy of source iteration method for the stationary phonon boltzmann transport equation

    Chuang Zhang, Samuel Huberman, Xinliang Song, Jin Zhao, Songze Chen, and Lei Wu. Acceler- ation strategy of source iteration method for the stationary phonon boltzmann transport equation. International Journal of Heat and Mass Transfer, 217:124715, 2023. 31

    2023

  18. [25]

    Synthetic iterative scheme for thermal applications in hotspot systems with large temperature variance

    Chuang Zhang, Qin Lou, and Hong Liang. Synthetic iterative scheme for thermal applications in hotspot systems with large temperature variance. International Journal of Heat and Mass Transfer, 236:126374, 2025

    2025

  19. [26]

    Lattice bgk models for navier-stokes equation

    Yue-Hong Qian, Dominique d’Humi` eres, and Pierre Lallemand. Lattice bgk models for navier-stokes equation. EPL (Europhysics Letters), 17(6):479, 1992

    1992

  20. [27]

    Molecular Gas Dynamics And The Direct Simulation Of Gas Flows

    G A Bird. Molecular Gas Dynamics And The Direct Simulation Of Gas Flows. Oxford University Press, 05 1994

    1994

  21. [28]

    R. B. Peterson. Direct Simulation of Phonon-Mediated Heat Transfer in a Debye Crystal. Journal of Heat Transfer, 116(4):815–822, 11 1994

    1994

  22. [29]

    Monte Carlo study of phonon transport in solid thin films including dispersion and polarization

    Sandip Mazumder and Arunava Majumdar. Monte Carlo study of phonon transport in solid thin films including dispersion and polarization. J. Heat Transfer, 123(4):749–759, 2001

    2001

  23. [30]

    Monte carlo study of phonon heat conduction in silicon thin films including contributions of optical phonons

    Arpit Mittal and Sandip Mazumder. Monte carlo study of phonon heat conduction in silicon thin films including contributions of optical phonons. J. Heat Transfer, 132(5):052402, 2010

    2010

  24. [31]

    P´ eraud and Nicolas G

    Jean-Philippe M. P´ eraud and Nicolas G. Hadjiconstantinou. Efficient simulation of multidimen- sional phonon transport using energy-based variance-reduced monte carlo formulations. Phys. Rev. B, 84:205331, Nov 2011

    2011

  25. [32]

    Adjoint-based deviational Monte Carlo methods for phonon transport calculations

    Jean-Philippe M P´ eraud and Nicolas G Hadjiconstantinou. Adjoint-based deviational Monte Carlo methods for phonon transport calculations. Physical Review B, 91(23):235321, 2015

    2015

  26. [33]

    Silva, D

    B.H. Silva, D. Lacroix, M. Isaiev, and L. Chaput. Monte carlo simulation of phonon transport from ab-initio data with nano-k. Computer Physics Communications, 294:108954, 2024

    2024

  27. [34]

    Aref, Gary F

    Abhishek Pathak, Avinash Pawnday, Aditya Prasad Roy, Amjad J. Aref, Gary F. Dargush, and Dipan- shu Bansal. MCBTE: A variance-reduced monte carlo solution of the linearized boltzmann transport equation for phonons. Comput. Phys. Commun., 265:108003, 2021

    2021

  28. [35]

    Unified gas-kinetic wave-particle methods

    Yajun Zhu, Chang Liu, Chengwen Zhong, and Kun Xu. Unified gas-kinetic wave-particle methods. ii. multiscale simulation on unstructured mesh. Physics of Fluids, 31(6), 2019

    2019

  29. [36]

    Unified gas-kinetic wave-particle methods iv: multi-species gas mixture and plasma transport

    Chang Liu and Kun Xu. Unified gas-kinetic wave-particle methods iv: multi-species gas mixture and plasma transport. Advances in Aerodynamics, 3:1–31, 2021

    2021

  30. [37]

    Unified gas-kinetic wave-particle method for multiscale flow simulation of partially ionized plasma

    Zhigang Pu and Kun Xu. Unified gas-kinetic wave-particle method for multiscale flow simulation of partially ionized plasma. Journal of Computational Physics, page 113918, 2025

    2025

  31. [38]

    Unified gas kinetic wave particle method for polydisperse gas-solid particle multiphase flow

    Xiaojian Yang, Wei Shyy, and Kun Xu. Unified gas kinetic wave particle method for polydisperse gas-solid particle multiphase flow. Journal of Fluid Mechanics, 983:A37, 2024

    2024

  32. [39]

    Unified gas-kinetic wave-particle method for frequency-dependent radiation transport equation

    Xiaojian Yang, Yajun Zhu, Chang Liu, and Kun Xu. Unified gas-kinetic wave-particle method for frequency-dependent radiation transport equation. Journal of Computational Physics, 522:113587, 2025

    2025

  33. [40]

    Wave-Particle Based Multiscale Modeling and Simulation of Non- equilibrium Turbulent Flows

    Xiaojian Yang and Kun Xu. Wave-Particle Based Multiscale Modeling and Simulation of Non- equilibrium Turbulent Flows. arXiv preprint arXiv:2503.07207, 2025

    2025 arXiv

  34. [41]

    Unified gas-kinetic wave-particle method for multi-scale phonon transport

    Hongyu Liu, Xiaojian Yang, Chuang Zhang, Xing Ji, and Kun Xu. Unified gas-kinetic wave-particle method for multi-scale phonon transport. arXiv preprint arXiv:2505.09297, 2025

    2025 arXiv

  35. [42]

    Heat transfer physics

    Massoud Kaviany. Heat transfer physics. Cambridge University Press, 2008

    2008

  36. [43]

    A model for collision processes in gases

    Prabhu Lal Bhatnagar, Eugene P Gross, and Max Krook. A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Physical Review, 94(3):511, 1954

    1954

  37. [45]

    Adams and Edward W

    Marvin L. Adams and Edward W. Larsen. Fast iterative methods for discrete-ordinates particle trans- port calculations. Prog. Nucl. Energ., 40(1):3 – 159, 2002

    2002

  38. [46]

    An implicit kinetic scheme for multiscale heat transfer problem accounting for phonon dispersion and polarization

    Chuang Zhang, Zhaoli Guo, and Songze Chen. An implicit kinetic scheme for multiscale heat transfer problem accounting for phonon dispersion and polarization. International Journal of Heat and Mass Transfer, 130:1366–1376, 2019

    2019

  39. [47]

    Modeling semiconductor nanos- 32 tructures thermal properties: The dispersion role

    Damian Terris, Karl Joulain, Denis Lemonnier, and David Lacroix. Modeling semiconductor nanos- 32 tructures thermal properties: The dispersion role. Journal of Applied Physics, 105(7), 2009

    2009

  40. [48]

    A fast synthetic iterative scheme for the stationary phonon boltzmann transport equation

    Chuang Zhang, Songze Chen, Zhaoli Guo, and Lei Wu. A fast synthetic iterative scheme for the stationary phonon boltzmann transport equation. International Journal of Heat and Mass Transfer, 174:121308, 2021. 33

    2021

Pith tools

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