磁流体动力学（MHD）方程的高阶约束输运有限元方法

The MHD equations describe the dynamics of a charged system under an interaction with a magnetic field and the conservation of the mass, momentum, and energy for the plasma system. Numerical modeling of plasmas has shown that the observance of the zero divergence of the magnetic field plays an important role in reproducing the correct physics in the plasma fluid. In this proposal, we will develop a high order constrained-transport method for the MHD equations using our recently constructed hierachical H(Div) basis functions in tedrahehral elements. A H(Div) conforming finite element method will be used for the magnetic field B while a high order discontinuous Galerkin method will be used for the hydrodynamic equations of the magnetic fluid. The enforcement of the divergence free condition of the magneticfield will be done algebraically using an interior bubble functions in the new H(div) basis elementwise. Also, the DG method will be based on an orthogonal basis functions developed in our group on tetrahedral elements.Various properties of the proposed numerical methods such as accuracy and stability, conditioning, and boundary condition treatment will be investigated in this project. Numerical simulation of several MHD problems will also be conducted.

磁流体动力学（MHD）方程描述了等离子体系统的质量守恒、动量守恒和能量守恒三大定律及简化的Maxwell方程，可用于刻画磁场中带电系统的动力学。已有数值试验表明满足磁场无散度条件对于准确地数值重现等离子体相关的物理现象至关重要。基于本组最新发展的四面体元上的分级H(div) 基函数，本项目将设计数值求解MHD方程的高阶约束输运方法，主要包括求解磁场方程的H(div) 协调有限元方法和求解流体动力学方程的高阶间断Galerkin方法。利用新H(div) 基函数内的泡型内部函数，我们将用代数方法使得磁场全局无散度条件得到满足，同时，间断Galerkin方法也将采用新设计的四面体L2正交分级基函数。我们还将深入研究所得高阶数值方法的精度、稳定性、条件数、边界处理和限制器设计等方面，并应用于模拟几个典型的磁流体动力学问题。

本项目设计求解方程MHD方程的高阶约束输运数值方法，主要由以下两部分数值方法组成：..•.求解MHD方程中磁场方程的无散度的H(div) 协调有限元方法.将采用一类新的H(div) 基函数离散磁场 的方程，每个时间步内，通过在每个单元上加入一个高阶泡型内部函数来校正磁场使得校正后的磁场是全局无散度的H(div) 向量场。..•.求解MHD方程中Navier-Stokes型方程的高阶间断Galerkin方法.这里的间断Galerkin方法将采用最新发展的四面体元上的正交多项式基函数。..研究进展:已完成以下三年内预期的任务：..(1).测试新设计的四面体元基函数的精度和所得质量矩阵的条件数； . (2) 测试高阶基函数的数值积分；. (3) 新引入的泡形基函数对精度和刚性的影响。. (4) 设计并程序实现数值求解磁场方程的算法，包括精确满足无散度条件和基于H(div) 分级有限元基函数离散后所得矩阵方程的快速迭代求解器；.(5) 基于四面体元上的正交分级基函数设计并程序实现MHD方程中流体动力学方程的高阶间断Galerkin 方法。.(6) We have tested new H(div) basis with free divergence for 3-D magnetic equation..(7) We have implemented 2-D MHD equations with H(Div) basis for magnetic field and DG for Navier-Stokes equations