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

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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

更新时间：2025-08-12

- 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 finite volume method
- 西安交通大学学报 2025年
- 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：
  
  国家自然科学基金创新群体基金(No. 51721004);西安市科技局基金(20192187 14SYS002CG024)
- DOI：xxxxxxxxxxxxxxxxxxxxx
  中图分类号： TK124
- 收稿：2025-05-07，
  
  修回：2025-08-04，
  
  录用：2025-08-10，
- 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/OL]. 西安交通大学学报, 2025.

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.
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/OL]. 西安交通大学学报, 2025. DOI： xxxxxxxxxxxxxxxxxxxxx.

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.

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