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

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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.