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

本研究课题将开发用于稀薄气体动力学模拟的新型数值方法。 与广泛使用的蒙特卡罗方法相比，正在开发的算法将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持在离散水平。新的算法也有可能被扩展到更广泛的应用，如等离子体物理，天体物理和半导体器件模拟。 学生将通过参与研究项目进行培训。该项目旨在开发一种基于网格的高阶多尺度数值方法，用于模拟动力学和流体动力学之间的稀薄气体动力学。该方法是基于所谓的微观宏观制定的动力学方程，它涉及到一个自然的分解成平衡和非平衡部分的问题。采用节点间断Galerkin（DG）有限元方法实现空间高阶精度，采用全局刚性精度隐-显Runge-Kutta方法实现时间高阶精度。 由于精心设计和考虑的流体动力学渐近性，该计划的发展成为一个DG方法与显式RK时间离散的欧拉系统的克努森数的零极限，和一个本地DG离散的Navier-Stokes方程的简化BGK碰撞算子在正式的渐近分析。这种局部DG方法在精神上类似于基于混合公式化方程的经典方法。新方案将在动力-水动力尺度的问题上进行测试，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时，对动力学模拟的边界层进行数值研究。

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.