二维球坐标系辐射输运方程的数值方法研究

The radiative transfer equation is widely used in many applications such as inertial confinement fusion, astrophysics and weapon physics. Analytical solutions are not available for most radiative transfer equations; therefore the study of numerical methods is very important for solving radiative transfer equations. For the capsule implosion problems with axis-symmetric properties, the critical issues for the schemes are to keep conservative and one-dimensional spherical symmetry. The capsule implosion is a large compression deformation problem; so that we can apply Eulerian methods which have fixed meshes and the distinguished advantage in dealing with great deformation problems, to simulate and analyze the radiation hydrodynamics efficiently. In general, Eulerian methods can easily preserve symmetry property by choosing the coordinates for the computational spaces, for instance, Eulerian methods in spherical coordinates can be applied in spherical symmetry problems. In this project we will study numerical methods for the conservative radiative transfer equation simulated by Eulerian methods in two-dimensional spherical coordinates, in which we use implicit time difference, multigroup discretization of the energy variable, discrete ordinates method of the angular variable and finite difference spatial discretization. We then analyze the properties of the scheme theoretically such as the compatible, conservative, spherical symmetry-preserving and positivity-preserving properties. Finally we design the radiative transfer code in two-dimensional spherical coordinates, in order to achieve high order accurate and reliable simulations for the capsule implosion problems.

辐射输运方程在惯性约束聚变、天体物理、武器物理等领域中有着重要的应用，方程只在极简单的少数情况下才有解析解，对一般实际问题，必须进行数值求解。惯性约束聚变靶丸内爆数值模拟中，几何区域和解是轴对称的，格式的守恒性和一维球对称性是物理问题的基本要求。靶丸内爆是一个大收缩比压缩变形问题，采用固定网格的欧拉方法描述，可以有效地分析整个辐射流体力学现象。欧拉方法一般通过计算物理空间的坐标系选取来保持计算格式的对称性，如计算球对称问题，一般采用球坐标系。本项目研究欧拉方法下的二维球坐标系辐射输运方程守恒形式的数值方法，其中对时间采用隐式差分离散，对能量采用多群近似，对角变量采用离散纵标方法离散，对空间变量采用有限差分方法离散；并理论分析格式的相容性、守恒性、球对称性和保正性等性质，编制二维球坐标系辐射输运程序；最后应用于靶丸内爆问题的研究，实现靶丸内爆过程的高精度、高置信度的数值模拟。

在间接驱动惯性约束聚变ICF中，靶丸内爆是核心过程，其中的关键是创造接近一维球对称的内爆条件，并有效控制流体力学不稳定性的发展，以确保点火热斑的形成。辐射是压缩靶丸形成热斑的驱动源，辐射对靶丸的烧蚀过程在ICF整个过程中占有非常重要的作用。靶丸外层的等离子体辐射自由程很大，辐射对靶丸的烧蚀应当采用辐射输运模型。LARED-S程序是主要基于二维球坐标系的多介质辐射流体欧拉内爆模拟程序，是当前点火靶内爆二维计算的主力工具，在点火靶设计与评估中正发挥着重要作用，目前程序采用多群辐射扩散模型，由于辐射扩散模型不能准确描述辐射烧蚀过程中的不对称性匀滑作用，因此迫切需要发展LARED-S程序的二维球坐标系辐射输运计算功能。. 本项目发展了二维球坐标系辐射输运方程的离散纵标方法，并理论分析证明了格式性质，包括相容性、守恒性、一维球对称性和步格式保证性，并撰写了系统完整的二维球坐标系辐射输运离散纵标方法的数学方案；接着编制了辐射输运SN程序并进行了考核，验证了格式的精度、程序保持一维球对称性和计算驱动不对称性问题的正确性等；最后将辐射输运SN程序与LARED-S程序进行了集成，实现了LARED-S程序的辐射输运并行计算功能。完成了LARED-S程序辐射输运建模多个典型算例的考核应用，数值结果表明，辐射输运建模提高了LARED-S程序靶内爆计算置信度。LARED-S程序辐射输运建模目前已应用于点火靶设计与评估中，为ICF点火靶内爆精密模拟提供了有力工具。