二维球坐标系辐射输运方程的数值方法研究

项目名称： 二维球坐标系辐射输运方程的数值方法研究

项目编号： No.11501041

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 刘元元

作者单位： 北京应用物理与计算数学研究所

项目金额： 17万元

中文摘要： 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。

中文关键词： 辐射输运；离散纵标法；保球对称；保正性；迭代方法

英文摘要： 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.

英文关键词： radiative transfer;discrete ordinates method;spherical symmetry-preserving;positivity-preserving;iteration method