辐射输运方程的渐近保持格式研究

The radiation transfer equation is very important in the researching field of astrophysics、weapon physics and inertial confinement fusion. Due to the complexity of the equation, it is very difficult to be solved in mathematics analytically. As the urgent to create some new methods to solve the problems in engineering, the research of the numerical solver for this equation is becoming a very hot topic. In computations, for high opacity region where the coefficient of material opacity is large, in order to keep the diffusive transport behavior of radiation photons, the computation meshes should be very small. Thus the computational times increasing greatly, it is the bottleneck problems in the physical problems’ simulations by using the radiation transfer equation. So how to construct a multi-scale numerical scheme (i.e., asymptotic preserving scheme) for radiation transfer equation that the mesh step is not depending on the material opacities but keeping diffusion transport behavior in high opacity material region is the key problem to be studied in this proposal. In this proposal, based on the previous researching works of the team, on one hand, by using the unified gas kinetic scheme (UGKS), construct the new numerical method which has the asymptotic preserving property for radiation transfer equation in cylindrical coordinate system, unstructured meshes, and the implicit method which can use the large time step at the same time; on the other hand, apply the numerical scheme we constructed to simulate the problems in inertial confinement fusion, et.al., to enhance the computing efficiency. So this researching work is very valuable in theory and application.

在天体物理、武器物理和惯性约束聚变研究中，辐射输运方程有着非常重要的应用。由于方程本身的复杂性，对其进行理论上的解析求解几乎是不可能的。迫切需要发展数值求解方法来解决工程中的应用问题，因此对其数值方法的研究已经逐渐成为一个热门的课题。在实际计算中，对不透明度系数大的光性厚介质区，要求空间网格步长非常小才能保持辐射的扩散传播性质。这极大增加了计算量，是制约辐射输运方程应用的瓶颈问题。设计空间网格步长不依赖不透明度系数,且在光性厚介质区保持扩散传播性质的多尺度辐射输运方程计算格式（也即渐近保持格式）是本项目研究的重点问题。本项目在课题组前期研究的基础上，一方面利用分子动理学方法(UGKS)发展柱坐标系、非结构网格以及隐式大时间步求解辐射输运方程的具有渐近保持性质的新计算格式；另一方面，把数值研究成果应用到惯性约束聚变等模拟中去，提高计算效率。因此，本项目的研究具有很好的理论和应用价值。

本课题研究辐射输运方程的渐近保持格式。在辐射输运方程的离散纵坐标研究方面，首先对多群辐射输运方程、带强散射项的辐射输运方程设计了渐近保持的统一气体分子动理学隐式格式(Implicit Unified gas kinetic scheme)；其次把矩形网格上辐射输运方程的渐近保持的统一气体分子动理学格式发展应用到一般四边形网格和非结构网格上;最后，结合流体力学的GKS(Gas Kinetic scheme)和辐射输运方程的UGKS算法，设计了整个辐射流体力学方程的渐近保持算法。在辐射输运方程的蒙特卡罗粒子模拟方面，通过引入宏观辅助方程的求解，改进了传统隐式蒙特卡罗方法中发射项的计算，设计了渐近保持的UGKP(Unified gas kinetic particle)方法，提高了传统隐式蒙特卡罗方法的计算效率。另外，该方面的研究结果也拓展应用到中子输运方程的计算中。该方面的研究极大提高了辐射输运方程的计算效率，促进了武器物理中辐射流体力学方程计算能力的发展。