非匹配网格上强间断辐射扩散方程的高效数值方法研究

With the physical modeling in the fields of inertial confinement fusion becoming more and more precise, slip calculation is introduced into the Lagrange method of multi-medium radiation hydrodynamics, resulting in the formation of non-conforming meshes. Meanwhile, the radiation diffusion equations are strongly anisotropic and nonlinear and the coefficients are strongly discontinuous. Therefore, the robustness of numerical simulation of radiation diffusion equations on distorted and non-conforming meshes has always been a key problem to be solved. The aim of this project is to propose tailored finite point method to solve the radiation diffusion equations on non-conforming meshes. The flux will be expressed as a linear combination of exact solutions for the constant coefficient equation in each cell. It will be convenient to construct new schemes according to interface conditions of the continuous solution and flux on each edge. The algorithm will be more robust on distorted and non-conforming meshes due to the local stencil and independence of the topological structure of the cell. The programs will be simple and can be easily realized. For strong discontinuity, the rapid transitions of solution can be grasped by imposing the continuity condition of the flux on the edge and taking the advantages provided by local exact solutions even on coarse meshes. Furthermore, we will rewrite the strong nonlinear radiation diffusion equation by using asymptotic analysis. We will develop an asymptotic preserving scheme that can accurately approximate the diffusion even when the nonlinearity is extremely strong. With a proper special discretization, the derived numerical scheme will be stable, preserve positivity. The study on algorithm of this project will provide a new technical approach with high-precision and high-resolution numerical simulation for radiation diffusion equations.

惯性约束聚变等领域的物理建模越来越精细，在多介质辐射流体力学Lagrange方法中引入滑移计算，导致形成非匹配网格，且相应的辐射扩散方程常常是高度非线性的、强各向异性的，系数是强间断的。因此，非匹配网格的辐射扩散方程的数值方法的健壮性问题一直是需要解决的核心问题之一。本课题拟将通量由单元网格内方程精确解组成的基函数的线性组合显示表示，根据网格单元边界界面连接条件“解相等，流连续”设计非匹配网格上辐射扩散方程的有限点格式，其具有局部模板且不依赖于网格的拓扑结构，处理扭曲和非匹配网格过程中算法更加健壮，程序简洁容易实现；对含有强间断的扩散通过流连续和局部精确解使得即使在粗网格条件下也能抓住解的快速变化。对强非线性扩散方程，拟通过渐近分析改写原系统，并设计稳定、保正的渐近保持格式，以提高强非线性扩散的计算效率。本课题的算法研究将为辐射扩散方程的高精度、高分辨的数值模拟提供新的技术途径。

辐射扩散方程的数值计算作为辐射流体力学数值模拟的主要部分，在天体物理、核物理、生物医学以及其他一些高科技领域也有重要应用，在这些应用领域中，高精度、高分辨的数值模拟是其主要的研究手段之一。因此，在扭曲严重多边形、非匹配网格上，对含有强间断、强各向异性张量的扩散方程设计高精度计算格式显得尤为重要。本项目执行三年期间主要的研究内容.1).完成设计扭曲网格、非匹配网格上有限点数值计算格式。通过基于局部网格常系统扩散方程的真解的线性组合形式构建近似解和通量的表达式，根据流连续条件建立离散格式。.2).有效求解强间断辐射扩散问题。局部网格常系统扩散方程真解线性组合与界面连接条件处理强间断问题，得到原系统满足局部守恒律的计算格式，解决在大变形网格上多介质扩散计算精度低，数值结果通常会出现非物理振荡、难以捕捉到界面层的快速变化等问题。.3).算法进行推广并设计快速算法。使得算法能够处理基于非结构网格、扭曲网格强含有界面层边界层的多尺度粒子输运方程，解决辐射输运方程计算量大，迭代格式计算效率低等问题。.本项目通量的构造对于扭曲网格和非匹配网格已有的有限体积的计算方法来说，具有很好的参考价值；处理间断的扩散问题可构造不需要增设虚拟点，保持精度和不破坏矩阵结构，通过局部基函数可以得到局部网格的任意点的值和各偏导数值显示表示，这种技术手段可以有效解决辐射扩散强间断数值计算难以捕捉或者含有非物理震荡问题；算法推广到辐射输运方程，构建渐近保持格式，这为辐射流体力学中的辐射输运方程的高效能数值模拟提供了新的思路。