有限容积方法数值求解声子玻尔兹曼方程中的偏置误差及其影响规律

冯相友; 陶文铨

doi:10.7652/xjtuxb202601009

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有限容积方法数值求解声子玻尔兹曼方程中的偏置误差及其影响规律

专题　“双碳”背景下的污染控制与资源化利用 | 更新时间：2026-01-15

- 有限容积方法数值求解声子玻尔兹曼方程中的偏置误差及其影响规律
- Offset Error and Its Influence in the Numerical Solution of Phonon Boltzmann Transport Equation Using the Finite Volume Method
- 西安交通大学学报 2026年60卷第1期页码：81-93
- 作者机构：
  
  西安交通大学热流科学与工程教育部重点实验室，710049，西安
- 作者简介：
  
  作者简介：冯相友（1997—），男，博士生；
  陶文铨（通信作者），男，教授，博士生导师，中国科学院院士。
- 基金信息：
  
  国家自然科学基金资助项目(51721004);西安市科技局基金资助项目。(2019218714SYS002CG024)
- DOI：10.7652/xjtuxb202601009
  中图分类号： TK124
- 收稿：2025-05-07，
  
  纸质出版：2026-01-10
- 稿件说明：
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冯相友, 陶文铨. 有限容积方法数值求解声子玻尔兹曼方程中的偏置误差及其影响规律[J]. 西安交通大学学报, 2026,60(1):81-93.

FENG Xiangyou, TAO Wenquan. Offset Error and Its Influence in the Numerical Solution of Phonon Boltzmann Transport Equation Using the Finite Volume Method[J]. Journal of Xi'an Jiaotong University, 2026, 60(1): 81-93.
冯相友, 陶文铨. 有限容积方法数值求解声子玻尔兹曼方程中的偏置误差及其影响规律[J]. 西安交通大学学报, 2026,60(1):81-93. DOI： 10.7652/xjtuxb202601009.

FENG Xiangyou, TAO Wenquan. Offset Error and Its Influence in the Numerical Solution of Phonon Boltzmann Transport Equation Using the Finite Volume Method[J]. Journal of Xi'an Jiaotong University, 2026, 60(1): 81-93. DOI： 10.7652/xjtuxb202601009.

摘要

为了进一步厘清有限容积方法（FVM）在数值求解声子玻尔兹曼方程（BTE）中的误差机制，在普遍认为的假散射和射线效应误差外，发现了一种新的误差来源——偏置误差，并分析了其影响规律。首先，基于两个热传导算例中的温度、热流计算偏差的分析，定义了偏置误差的概念。偏置误差是在采用迎风偏置格式离散对流项时，同一界面上不同方向声子能量所取上游节点不同，所产生的热流计算偏差。然后，对偏置误差的影响因素进行了分析。最后，探究了高阶格式特征线分布对偏置误差的影响规律。结果表明：影响偏置误差的因素主要有3个，即网格克努森数

（声子平均自由程与网格宽度的比值）、格式的偏置特性、结果的分布线型；在一般情况下，算例的1/

越大、所用离散格式特征线偏离零偏置线的程度越大、结果线型的非线性越强，其偏置误差越大；偏置误差的影响趋势可以由格式特征线偏离零偏置线的方向定性确定，若格式特征线在零偏置线之下，偏置误差会导致计算中高估热流，格式特征线在零偏置线之上则相反。该研究为声子BTE在求解微纳尺度传热问题中，合理选择对流项离散格式和结果的误差评估提供了理论支持。

Abstract

To further clarify the error mechanisms in the numerical solution of the phonon Boltzmann transport equation (BTE)using the finite volume method (FVM)，a new error source—offset error，is identified and its influence law is analyzed，in addition to the widely recognized false scattering and ray effect errors.First，based on the analysis of temperature and heat flux deviations in two heat conduction cases，the concept of offset error is defined：offset error refers to the heat flux calculation deviation arising from the use of upwind-biased schemes for discretizing the convection term，where phonon energies from different directions on the same interface adopt different upstream nodes.Subsequently，the factors influencing offset error are analyzed.Finally，the influence of higher-order scheme characteristic line distributions on offset error are investigated. The research results indicate that three main factors affect offset error：the grid Knudsen number

(ratio of phonon mean free path to grid width)，the offset characteristics of the scheme，and the distribution pattern of the results.Generally，larger 1/

，greater deviation of the discretization scheme's characteristic lines from the zero-offset line，and stronger nonlinearity of the result distribution pattern lead to larger offset errors.The influence trend of offset error can be qualitatively determined by the direction of the scheme's characteristic lines deviating from the zero-offset line.If the characteristic lines lie below the zero-offset line，offset error tends to overestimate the heat flux；conversely，if they lie above the zero-offset line，it tends to underestimate the heat flux. This study provides theoretical support for the selection of convection term discretization schemes and error evaluation of results in solving micro/nanoscale heat transfer problems using the phonon BTE.

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references

ZHANG Qingzhu,ZHANG Yongkui,LUO Yanna,et al.New structure transistors for advanced technology node CMOS ICs[J].National Science Review,2024, 11(3):nwae008.

CHEN G.Ballistic-diffusive heat-conduction equations[J]. Physical Review Letters,2001,86(11):2297-2300.

SVERDRUP P G,SINHA S,ASHEGHI M,et al. Measurement of ballistic phonon conduction near hotspots in silicon[J].Applied Physics Letters,2001 ,78 (21):3331-3333.

SUN Lin,MURTHY J Y.Domain size effects in molecular dynamics simulation of phonon transport in silicon[J].Applied Physics Letters,2006,89 (17):171919.

MINGO N,YANG Liu.Phonon transport in nanowires coated with an amorphous material:an atomistic Green's function approach[J].Physical Review:B,2003,68 (24):245406.

MAJUMDAR A.Microscale heat conduction in dielectric thin films[J].Journal of Heat Transfer,1993, 115(1):7-16.

黄昆,韩汝琦.固体物理学[M].北京:高等教育出版社,1988.

YANG Ronggui,CHEN Gang,LAROCHE M,et al. Simulation of nanoscale multidimensional transient heat conduction problems using ballistic-diffusive equations and phonon Boltzmann equation[J].Journal of Heat Transfer,2005,127(3):298-306.

MAZUMDER S,MAJUMDAR A.Monte Carlo study of phonon transport in solid thin films including dispersion and polarization[J].Journal of Heat Transfer, 2001 ,123(4):749-759.

MITTAL ARPIT M S.Prediction of non-equilibrium heat conduction in crystalline materials using the Boltzmann transport equation for phonons[D].Columbus, USA:The Ohio State University,2011.

MURTHY J Y,MATHUR S R.Computation of submicron thermal transport using an unstructured finite volume method[J].Journal of Heat Transfer,2002, 124(6):1176-1181.

MURTHY J Y,MATHUR S R.An improved computational procedure for sub-micron heat conduction[J]. Journal of Heat Transfer,2003,125(5):904-910.

NARUMANCHI S V J,MURTHY J Y,AMON C H.Submicron heat transport model in silicon accounting for phonon dispersion and polarization[J].Journal of Heat Transfer,2004,126(6):946-955.

GUO Yangyu, WANG Moran. Lattice Boltzmann modeling of phonon transport[J].Journal of Computational Physics,2016,315:1-15.

FIVELAND W A.Discrete-ordinates solutions of the radiative transport equation for rectangular enclosures[J].Journal of Heat Transfer,1984,106(4):699-706.

RAITHBY G D,CHUI E H.A finite-volume method for predicting a radiant heat transfer in enclosures with participating media[J].Journal of Heat Transfer, 1990,112(2):415-423.

CHUI E H,RAITHBY G D,HUGHES P M J.Prediction of radiative transfer in cylindrical enclosures with the finite volume method[J].Journal of Thermophysics and Heat Transfer,1992,6(4):605-611.

MODEST M F.Radiative heat transfer[M].3 rd ed. Boston,USA:Academic Press,2013.

CHAI J C,LEE H S,PATANKAR S V.Ray effect and false scattering in the discrete ordinates method[J]. Numerical Heat Transfer :Part B Fundamentals,1993 , 24(4):373-389.

陶文铨.数值传热学[M].2版.西安:西安交通大学出版社,2001.

WANG Tianjiao.Sub-micron thermal transport in ultra-scaled metal oxide semiconductor (MOS)devices[D]. West Lafayette,IN,USA:Purdue University,2007.

HSIEH T Y,LIN H,HSIEH T J,et al.Thermal conductivity modeling of periodic porous silicon with aligned cylindrical pores[J].Journal of Applied Physics,2012,111(12):124329.

HU Yue,SHEN Yongxing,BAO Hua.Ultra-efficient and parameter-free computation of submicron thermal transport with phonon Boltzmann transport equation[J]. Fundamental Research,2024,4(4):907-915.

SAURAV S, MAZUMDER S. Phonon Boltzmann transport equation based simulation of frequency domain thermoreflectance experiments[C]//ASME2022 International Mechanical Engineering Congress and Exposition.New York,NY,USA:ASME,2022:V008T11A062 .

NABOVATI A,SELLAN D P,AMON C H.On the lattice Boltzmann method for phonon transport[J]. Journal of Computational Physics,2011,230 (15):5864-5876.

ALI S A,KOLLU G,MAZUMDER S,et al.Largescale parallel computation of the phonon Boltzmann Transport Equation[J].International Journal of Thermal Sciences,2014,86:341-351.

罗小平.微纳导热多尺度模型及低维材料声子水动力学研究[D].哈尔滨:哈尔滨工业大学,2020.

ZHANG Chuang,GUO Zhaoli.Discrete unified gas kinetic scheme for multiscale heat transfer with arbitrary temperature difference[J].International Journal of Heat and Mass Transfer,2019,134:1127-1136.

BROIDO D A,MALORNY M,BIRNER G,et al.Intrinsic lattice thermal conductivity of semiconductors from first principles[J].Applied Physics Letters, 2007,91(23):231922.

GUO Yangyu,WANG Moran.Phonon hydrodynamics and its applications in nanoscale heat transport[J]. Physics Reports,2015,595:1-44.

NI Chunjian.Phonon transport models for heat conduction in sub-micron geometries with application to microelectronics[D]. West Lafayette, IN, USA:Purdue University,2009.

陶文铨.计算传热学的近代进展[M].北京:科学出版社,2000.

NARUMANCHI S V J,MURTHY J Y,AMON C H.Comparison of different phonon transport models for predicting heat conduction in silicon-on-insulator transistors[J].Journal of Heat Transfer,2005 ,127 (7):713-723.

GASKELL P H,LAU A K C.Curvature-compensated convective transport:SMART a new boundedness-preserving transport algorithm[J].International Journal for Numerical Methods in Fluids,1988,8(6):617-641.

宇波,焦开拓,陈宇杰,等.有限容积法对流项离散格式的稳定性与有界性研究[J].西安交通大学学报, 2023,57(4):107-121 .

YU Bo,JIAO Kaituo,CHEN Yujie,et al.Study on the stability and boundedness of discrete schemes for convection term based on finite volume method[J]. Journal of Xi'an Jiaotong University,2023,57 (4):107-121 .

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