Offset error and its influence in the numerical solution of phonon Boltzmann transport equation using finite volume method

FENG Xiang-You; TAO Wen-Quan

doi:xxxxxxxxxxxxxxxxxxxxx

您当前的位置：

首页 >

文章列表页 >

Offset error and its influence in the numerical solution of phonon Boltzmann transport equation using finite volume method

更新时间：2025-08-12

- Offset error and its influence in the numerical solution of phonon Boltzmann transport equation using finite volume method
- JOURNAL OF XI’AN JIAOTONG UNIVERSITY (2025)
- 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,陕西省西安市710049
- 作者简介：
- 基金信息：
- DOI：xxxxxxxxxxxxxxxxxxxxx
  CLC： TK124
- Received：07 May 2025，
  
  Revised：2025-08-04，
  
  Accepted：10 August 2025，
- 稿件说明：
移动端阅览
FENG Xiang-You, TAO Wen-Quan. Offset error and its influence in the numerical solution of phonon Boltzmann transport equation using finite volume method[J/OL]. JOURNAL OF XI’AN JIAOTONG UNIVERSITY, 2025.
DOI：

FENG Xiang-You, TAO Wen-Quan. Offset error and its influence in the numerical solution of phonon Boltzmann transport equation using finite volume method[J/OL]. JOURNAL OF XI’AN JIAOTONG UNIVERSITY, 2025. DOI： xxxxxxxxxxxxxxxxxxxxx.

摘要

为了进一步厘清有限容积方法（Finite Volume Method，FVM）在数值求解声子玻尔兹曼方程（Boltzmann Transport Equation，BTE）中的误差机制，在普遍认为的假散射和射线效应误差外，本文指出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。研究结果表明：影响偏置误差的因素主要有三个，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

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地确定，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

In order to further clarify the error mechanism of the Finite Volume Method (FVM) in numerically solving the Boltzmann Transport Equation (BTE)

this study points out a new error

namely offset error

in addition to the commonly recognized false scattering and radiation effect errors

and analyzes its influence. Firstly

based on the analysis of temperature and heat flux calculation deviations in two heat conduction cases

we defined the offset error as follows: the deviation in heat flux calculation caused by different upstream nodes of the phonon energy in different directions at the same interface when discretizing the ballistic term using an upwind scheme. Then the influencing factors of offset error were analyzed. Finally

the influence of the characteristic line of the high-order scheme on offset error was explored. The results showed that there are three main factors that affect the offset error: the Knudsen number based on the grid

(the ratio of the phonon mean free path to the grid width)

the offset characteristics of the scheme

and the distribution of the result. Generally

for a case

the larger the value of

the greater the deviation of the characteristic line of the discretization scheme from the zero-offset line

and the stronger the nonlinearity of the

distribution

the larger the offset error. The trend of the influence of the scheme off

set error can be qualitatively determined by the direction of the scheme characteristic line deviation from the zero-offset line. If the scheme’s characteristic line is below the zero-offset line

the offset error will lead to overestimation of heat flux in the calculation

and the opposite is true if the scheme’s characteristic line is above the zero-offset line. This study provides theoretical guidance for the choice of the discretization scheme of the ballistic term and the error assessment of the results in solving microscale/nanoscale heat transfer problems using phonon BTE.

关键词

Keywords

references

ZHANG Q , ZHANG Y , LUO Y , 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 L , 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 L . Phonon transport in nanowires coated with an amorphous material: An atomistic Green’s function approach [J ] . Physical Review B - Condensed Matter and Materials Physics , 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 .

HUANG K , HAN R . Solid State Physics [M ] . Beijing : Higher Education Press , 1988 .

YANG R , CHEN G , 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 ] . The Ohio State University , 2011 .

MURTHY J Y , MATHUR S R . Computation of sub-micron 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 Y , WANG M . 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-wolome 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 ] . Third ed . San Diego : Academic Press , 2013 .

CHAI J C , LEE H O 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 ] . 第二版 . 西安 : 西安交通大学出版社 , 2001 .

TAO W . Numerical Heat Transfer [M ] . Second ed. Xi’an : Xi’an Jiaotong University Press , 2001 .

WANG T . Sub-micron thermal transport in ultra-scaled metal oxide semiconductor(MOS) devices [D ] . 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 Y , SHEN Y , BAO H . Ultra-efficient and parameter-free computation of submicron thermal transport with phonon Boltzmann transport equation [J ] . Fundamental Research , 2024 , 4 ( 4 ): 907 - 915 .

NI C . Phonon transport models for the heat conduction in sub-micron geometries with application to microelectronics [D ] . Purdue University , 2009 .

SAURAV S , MAZUMDER S . Phonon Boltzmann transport equation based simulation of frequency domain thermoreflectance experiments [C ] . ASME International Mechanical Engineering Congress and Exposition, Proceedings (IMECE). Columbus , 2022 : IMECE2022 - 95630 .

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 . Large-scale 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 ] . 哈尔滨工业大学 , 2020 .

LUO X . Multiscale modeling of micro-nanoscale thermal conduction and research on phonon hydrodynamics in low-dimensional materials [D ] . Harbin Institute of Technology , 2020 .

ZHANG C , GUO Z . 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 Y , WANG M . Phonon hydrodynamics and its applications in nanoscale heat transport [J ] . Physics Reports , 2015 , 595 : 1 - 44 .

陶文铨 . 计算传热学的近代进展 [M ] . 北京 : 科学出版社 , 2000 .

TAO W . Modern Advances in Computational Heat Transfer [M ] . Beijing : Science Press , 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 B , JIAO K , CHEN Y , et al . Study on the stability and boundedness of discrete scheme for convection term based on finite volume method [J ] . Journal of Xi’an Jiaotong University , 2023 , 57 ( 4 ): 107 - 121 .

Views

下载量

CSCD

Alert me when the article has been cited

提交

Tools

Publicity Resources

Offset Error and Its Influence in the Numerical Solution of Phonon Boltzmann Transport Equation Using the Finite Volume Method

Numerical Simulation of the Energy Conversion Process in Thermoacoustic Engines with Different Stack Materials

Mechanism and Optimal Control Strategy for Hysteresis in Rapid Dehumidification Response of Indirect Thermal Systems in Electric Vehicles

Study on the Thermodynamic Properties in the Anomalous Region of Supercritical Water

Knowledge Graph Representation Learning Driven by Semantic Discrimination and Subgraph Integration

Related Author

FENG Xiangyou

TAO Wenquan

Related Institution

Key Laboratory of Thermo-Fluid Science & Engineering of MOE, Xi'an Jiaotong University

⁰