辐射输运方程的统一气体动理学格式  

Unified gas kinetic schemes for the radiation transfer equations

作　　者：江松[1] 徐昆[2] 孙文俊[1,3] 许小静 Song Jiang;Kun Xu;Wenjun Sun;Xiaojing Xu

机构地区：[1]北京应用物理与计算数学研究所,北京100094 [2]香港科技大学数学系,中国香港 [3]北京大学应用物理与技术研究中心,北京100871 [4]西南科技大学理学院,绵阳621010

出　　处：《中国科学：数学》2021年第6期799-832,共34页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11631008,11772281,91852114和11671048);中德中心(批准号:GZ 1465);中以合作项目(批准号:2519/17);香港研究基金会(批准号:16206617);中国工程物理研究院基金(批准号:CX20200026);国家重大计划(批准号:GJXM92579);科学挑战计划(批准号:TZ2016002)资助项目。

摘　　要：辐射输运方程的数值模拟在天体物理、武器物理和惯性约束与磁约束聚变等研究中都起着非常重要的作用.在实际问题中,背景介质的不透明度系数决定了辐射光子在其中的传输行为.光性薄(不透明度系数小)的介质对辐射光子是透明的,光子与背景介质的相互作用弱,光子传输具有输运传播性质;而光性厚(不透明度系数大)的介质对辐射光子是不透明的,光子与背景介质的相互作用强,光子传输具有扩散性质.因此在辐射输运方程的计算中,如何设计既能得到光子输运传播性质又能捕捉光子扩散传播性质的渐近保持离散格式是目前一个非常活跃和前沿的研究方向.本文简要介绍近几年在辐射输运方程的渐近保持统一气体动理学格式(unified gas kinetic scheme,UGKS)研究方面的进展.本文主要以灰体辐射输运方程为例,详细介绍UGKS的构造方法并给出其渐近分析.同时,结合角度有限元方法和球谐函数展开的方法,介绍如何减弱/去除基于离散纵标法的UGKS具有射线效应的问题,以及相应的改进渐近保持格式.此外,也介绍了将渐近保持的UGKS应用拓展到考虑流体运动的完全辐射流体力学方程组.最后,用一些数值例子验证了格式的渐近保持性和保正性等性质.The equations of radiation transfer have played an important role in the fields such as astrophysics and inertial confinement fusion.For practical radiative transfer problems,the properties of the background material greatly influence the behavior of radiation transfer.For a low opacity material(small opacity coefficient),the interaction between the radiation and material is weak,and the radiation propagates in a transparent way.However,for a high opacity material(large opacity coefficient),there is severe interaction between radiation and material with a diminishing photon mean free path.As a result,the diffusive radiative behavior will emerge.It is an active topic now in the numerical solution of radiation transfer equations to construct a numerical scheme which can not only obtain the transport behavior but also capture the diffusion behavior.In this paper,we mainly review the recent progress in the extension of the unified gas kinetic scheme(UGKS)to radiative transfer problems.Based on the example of the grey radiation transfer equations,we give the details of the construction of the UGKS and the asymptotic analysis.At the same time,combining the UGKS with the finite element and sphere harmonic expansion methods for the angular variable,we describe some techniques on how to reduce/eleminate the ray effect which is an inherent shortcoming in the discrete ordinate method.Furthermore,we extend this asymptotic preserving UGKS to the equations of radiation hydrodynamics.Finally,numerical examples are presented to show the asymptotic and positive preserving properties of the proposed schemes.

关 键 词：辐射输运方程 渐近保持 保正 统一气体动理学格式 辐射流体力学方程组 

分 类 号：O175[理学—数学]