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

FENG Xiangyou; TAO Wenquan

doi:10.7652/xjtuxb202601009

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

Special Topic Pollution Control and Resource Utilization in the “Dual Carbon”Context | 更新时间：2026-01-15

- Offset Error and Its Influence in the Numerical Solution of Phonon Boltzmann Transport Equation Using the Finite Volume Method
- Journal of Xi'an Jiaotong University Vol. 60, Issue 1, Pages: 81-93(2026)
- 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：
- 基金信息：
- DOI：10.7652/xjtuxb202601009
  CLC： TK124
- Received：07 May 2025，
  
  Published：10 January 2026
- 稿件说明：
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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：

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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