多介质流体力学方程保物理特性的计算方法研究

Multi-material fluid dynamics is widely seen in practical problems,such as atomic bomb simulation, Inertial Confinement Fusion (ICF), planetary physics etc. The challenges of this kind of problems are involved of multi-material, moving interface, large deformation and sharp discontinuities coupling together. The scheme should be positivity-preserving, which update new states with positive density, internal energy and other physically positive variables. In this project, we have three parts: (1) Based on traditional spatial grid and material grid, the MMALE technique introduces the mixed grid to improve the simulation for large deformation. (2) In order to capture the multi-material and sharp discontinuities, the weak Galerkin method is applied for the spatial discretization of multi-material fluid dynamics. Meanwhile, we shall design positive-preserving scheme base on approximate Riemann solver. (3) Multi-material fluid dynamics with stochastic interface problem is considered for describing the moving interface efficiently. The shape-derivative and perturbation analysis are applied to get high order approximation for expectation and variance of numerical solution.The proposed theoretical and computational research will significantly strengthen our ability in dealing with challenging issues within Multi-material fluid dynamics, which is valuable and has long lasting impact in advancing science and technology.

多介质流体力学方程在武器物理、惯性约束聚变(ICF)、天体物理等很多科学与工程领域有着广泛的应用背景。这类问题的难点在于多介质、运动交界面、大变形和强间断等复杂流动特征的耦合，以及在计算求解时物理性质的保持。在本项目中，我们拟从以下三方面对这类问题进行研究：（1）采用多介质任意Lagrangian Eulerian 方法，在传统的空间网格和物质网格基础上引入混合网格，以更好处理大变形问题。（2）将弱Galerkin方法运用到多介质流体的空间方向，用于更好刻画多介质和强间断等现象。同时，基于近似Riemann解法器，设计与弱有限元相关的保正性算法。（3）为了能够更好地刻画运动界面，我们拟尝试处理随机交界面的多介质流体力学方程，采取形状导数和摄动分析得到数值解均值和方差的高阶逼近。这些理论与数值计算方法的研究将有助于人们对多介质流体力学问题中相关现象的认识，有着重要的实用价值和现实意义。

本项目主要围绕多介质流体力学方程保物理特性的计算方法进行的研究。经过三年的集中攻关，主要成果包括：1）首次给出了一般具一阶对流项拟线性反应扩散方程的有限元的离散极值原理，并相应给出了我们的离散极值原理成立所需要的有限元剖分满足的几何约束条件; 2）系统地研究了弱有限元方法的理论框架和应用，针对流体力学中的典型方程设计了一系列稳定、高效的数值格式; 3）采用了弱有限元方法对计算流体动力学中的线性双曲方程进行了数值模拟；4）在超收敛方面，给出了新的弱有限元解决方案超精确到拉格朗日型插值解，这为开发一个高效的后处理技术提供了更好的逼近解梯度。

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