可压缩多介质流体力学的广义黎曼问题数值方法

In recent years, the study of compressible multifluid problems receives much attention due to their engineering values and scientific significance. The analysis and numerical simulations of material interfaces have been one of research focuses in the field of scientific computations. This project aims to provide a set of reliable generalized Riemann problem (GRP) solvers for multifluid problems and corresponding high order finite volume schemes. The contents include: (1) the mathematical analysis of compressible multifluid models and their closure; (2) Fast Riemann solvers for multi-components with complex equations of states; (3) Spatial-temporal coupled GRP solvers adapted to the Arbitrary Lagrangian-Eulerian (ALE) framework; (4) arbitrarily high order GRP-based multi-dimensional finite volume methods. The final goal is to design a set of robust and practical algorithms for engineering problems such as Inertial Confinement Fusion (ICF).

由于可压缩多介质流体问题在工程领域的应用价值和科学上的重要意义，近年来受到诸多学者越来越多的关注；围绕着物质界面的分析和数值模拟已成为科学计算领域的研究热点问题之一。 基于该领域典型问题的物理特征， 本项目拟研究该领域广义黎曼解法器以及相应的时空耦合的高阶有限体积方法。 内容包括: (1) 可压缩多介质流体模型数学分析; (2) 关于多个组份复杂状态方程的快速黎曼解法器；(3) 适用于任意拉格朗日－欧拉框架的时空耦合广义黎曼解法器; (4)基于广义黎曼解法器的任意高阶精度、真正多维有限体积方法。研究目标在于建立一套鲁棒、适用于ICF等工程应用的实用算法。

本项目针对可压缩多介质流动问题的特点，发展时空高阶的广义黎曼问题解法器，设计时空耦合高精度的数值算法，并建立相关的数学理论。取得了如下主要结果：.1. 建立了含有间断解的数值方法的相容性和收敛性理论，澄清了高精度数值方法中时空耦合的重要性，完善前期发展的时空耦合两步四阶算法；.2. 针对多介质流动特点，以Baer-Nunziato多介质模型为基础，发展基于广义黎曼解法器的能量分裂和交错投影高阶多介质算法；.3. 对于多维可压缩跨音速流动，构造了一类超音区域的流动模块，探讨了音速线附近流动的正则性。.授本项目支持，课题组发表论文20多篇，其中Mathematics of Computation一篇，SIAM J. Scientific Computing一篇，Journal of Computational Physics五篇，Physics of Fluids 2篇， Archive Rational Mechanics and Analysis一篇。 其研究结果收到和很好评价，如俄罗斯研究人员 Nikolay M. Evstigneev在美国数学会的Mathscinet上（MR4166456 ）评价道：“ The paper can be recommended to all who are working in the field of numerical analysis and computational fluid dynamics. Results obtained in the paper are of vital importance with respect to the convergence analysis of practical high order schemes for balance laws.” ..部分结果在ICF等工程应用中已经得到体现。 .