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

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关键词：声子玻尔兹曼方程；有限容积方法；偏置误差；离散格式；微纳尺度传热中图分类号：TK124文献标志码：ADOI：10.7652/xjtuxb202601009 文章编号：0253-987X（2026）01-0081-13

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

FENG Xiangyou，TAO Wenquan （KeyLaboratoryof Thermo-Fluid Science&Engineering of MOE，Xi'an Jiaotong University，Xi'an 710049，China）

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 diferent directions on the same interface adopt different upstream nodes. Subsequently， the factors influencing ofset error are analyzed. Finall，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 KnΔ （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/KnΔ ，greater deviation of the discretization scheme's characteristic lines from the zero-ofset 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.

Keywords： phonon Boltzmann transport equation； finite volume method；ofset error；discretization scheme；microscale/nanoscale heat transfer

热失效是半导体器件的主要失效形式之一，器件的热分析对其安全稳定运行至关重要。（剩余18734字）