A High Order Discontinuous Galerkin Multi-Scale Approach for Kinetic-Hydrodynamic Simulations

该研究项目将开发新的数值方法来模拟稀薄气体的动力学。与广泛使用的蒙特卡罗方法相比，该算法能够在长时间模拟中更准确地捕获复杂的解结构。此外，物理守恒量，如质量、动量和能量，可以精确地保存在离散水平。新算法也有可能扩展到更广泛的应用领域，如等离子体物理学、天体物理学和半导体器件模拟。学生将通过参与研究项目而得到训练。该项目旨在开发一种基于高阶网格的多尺度数值方法来模拟动力和水动力之间的稀薄气体动力学。该方法基于所谓的微观-宏观动力学方程公式，将问题自然分解为平衡部分和非平衡部分。高阶空间精度采用节点不连续伽辽金（DG）有限元法实现，高阶时间精度采用全局刚性精度隐显龙格-库塔法实现。由于经过深思熟虑的设计和对水动力渐近性的考虑，所开发的方案成为在Knudsen数零极限下欧拉系统具有显式RK时间离散化的DG方法，以及在形式渐近分析中简化BGK碰撞算子的Navier-Stokes方程的局部DG离散化。这种局部DG方法在精神上类似于基于方程混合公式的经典方法。新方案将在动力学-水动力尺度上进行问题测试，并与简化BGK模型的结果、其椭球统计（ES-BGK）扩展以及宏观水动力模型的结果进行比较。该项目还将在考虑扩散水动力极限的情况下，对边界层进行数值模拟。

This research project will develop novel numerical methods for simulation of the dynamics of rarefied gas. Compared with the widely used Monte Carlo approach, the algorithm under development will be able to more accurately capture complicated solution structures in long-time simulations. Moreover, physically conserved quantities such as mass, momentum, and energy can be exactly preserved at the discrete level. The new algorithm also has the potential to be extended to a broader class of applications such as plasma physics, astrophysics, and semi-conductor device simulation. Students will be trained through involvement in the research project.This project aims to develop a very high order mesh-based multi-scale numerical approach to modeling rarified gas dynamics between the kinetic and hydrodynamic regimes. The approach is based on the so-called micro-macro formulation of the kinetic equation, which involves a natural decomposition of the problem into equilibrium and non-equilibrium parts. The high order spatial accuracy is achieved by a nodal discontinuous Galerkin (DG) finite element approach, and the high order temporal accuracy is achieved by globally stiffly accurate implicit-explicit Runge-Kutta methods. Due to deliberate design and considerations of the hydrodynamic asymptotics, the scheme under development becomes a DG method with explicit RK time discretizations for the Euler system in the zero limit of the Knudsen number, and a local DG discretization of the Navier-Stokes equations for a simplified BGK collision operator in a formal asymptotic analysis. Such a local DG method is similar in spirit to classical approaches based on a mixed formulation of the equations. The new scheme will be tested on problems at kinetic-hydrodynamic scales and compared with the results from the simplified BGK model, its ellipsoidal statistical (ES-BGK) extension, as well as with results from the macroscopic hydrodynamic models. The project will also study numerically the boundary layer for kinetic simulations when a diffusive hydrodynamic limit is considered.