三维多群辐射扩散方程组的保极值数值格式研究

Multigroup radiation diffusion equations are the important part of radiation hydrodynamics, and are widely used in many fields, such as inertial confinement fusion (ICF) and astronomic physics. This equations involve many coupled physical processes, which have characters of strong discontinuity, highly nonlinearity, and have to be solved on distorted meshes in the framework of multi-material arbitrary Lagrangian-Eulerian (ALE). The discrete energy density often has negative parts or numerical oscillation. How to obtain the discrete solution with the second-order accuracy with preserving the non-negative constraints and extremum principle is one of the important and challenging problems in designing diffusion schemes. Aimed at satisfying the need of 3D simulations for applied problems, this project studies the second-order nonlinear positive-preserving scheme and extremum-principle preserving scheme for 3D multigroup radiation diffusion equations on the distorted meshes: (1) a second-order nonlinear lower bound preserving (positive-preserving) finite volume (FV) scheme suitable for different kinds of largely distorted meshes; 2) a second-order nonlinear extremum-principle-preserving(lower and upper bound) FV scheme suitable for more general discontinuous coefficients; 3) a numerical program will be developed and integrated to simulate the typical radiation hydrodynamics problems. This project will improve the accuracy, robustness and conservation of multi-material ALE simulations for 3D radiation hydrodynamics. The improved multi-material ALE simulation indeed will promote the relevant physical research, and has important theoretical creativity and practical application values.

多群辐射扩散方程组是辐射流体力学的重要组成部分，在惯性约束聚变(ICF)、天体物理等领域中有广泛应用。多群辐射扩散方程组具有强间断、强耦合、强非线性等特点，在流体拉氏变形网格上求解时往往出现能量密度出负或者数值震荡等非物理现象，如何使得数值解保证精度(二阶)的同时保持解的正性或者上下界是扩散计算方法研究中重要且富有挑战的内容之一。本项目针对三维应用问题需求，开展三维大变形网格上多群辐射扩散方程组的二阶精度的保极值有限体积格式研究：(1)研究适应多种大变形网格的二阶精度的非线性保下界(保正)有限体积格式；(2)研究适应任意间断系数的二阶精度的非线性保极值(保上下界)有限体积格式；(3)研制程序模块，对典型辐射流体力学问题开展数值模拟研究。本项目的研究将提升三维辐射流体力学多介质ALE模拟的精度、鲁棒性和守恒性，也会促进相关物理问题研究的深入,具有重要的理论研究意义和实际应用价值。

多群辐射扩散方程组是辐射流体力学的重要组成部分，在惯性约束聚变(ICF)、天体物理等领域中有广泛应用。多群辐射扩散方程组具有强间断、强耦合、强非线性等特点，在流体拉氏变形网格上求解时往往出现数值解部分出负或者数值震荡等现象。为了抑制这些非物理现象，需要研究辐射扩散方程的保极值数值格式，提升辐射流体力学数值模拟的置信度及保物理性质。. 本项目首先研究了三维辐射扩散方程的保正有限体积格式，改进了传统三维扩散格式中的节点插值算法的计算复杂度、精度及鲁棒性，格式精度达到二阶且保持数值解的正性。然后，通过对机器学习科学计算领域的大量调研，研究了采用机器学习求解二维、三维各向异性辐射扩散方程的保极值数值解法，基于加权一阶微分形式设计了新的损失函数形式，大大提升了机器学习求解间断各向异性辐射扩散问题的精度，同时保证了数值解的正性及上下界约束。最后，研究了将新设计的保物理约束格式应用到ICF典型实验的三维模拟中，开展了格式的适应性研究，新的格式在时间步长及计算精度上均具有良好的表现，数值模拟结果有力支撑了ICF应用研究。. 总之，我们圆满完成了项目研究目标，改进了已有的保正格式，设计了基于机器学习的保正、保极值数值算法，成功将格式集成应用于ICF研究领域，在高水平杂志上发表和投稿了多篇学术论文。