Development of Efficient Techniques for Simulations of Kinetic Equations of Gas Dynamics

Numerical modeling of the behavior of gas flows over sharp leading edges of air vehicles, spacecraft plumes expanding into rarefied atmosphere, and micro channel flows are challenging because they require the solution of the Boltzmann equation. The Boltzmann equation is high dimensional, non-linear, and in regimes transitional from continuum to free molecular it leads to stiff problems. Methods have been proposed to enforce the conservation of mass, momentum, and energy, however no methods are known to provide a reliable high order convergence of the solution. In particular, high order convergence is difficult to achieve in the problem of gas-surface interaction such as the one that arises in heat transfer. We develop an efficient algorithm for high-order, high-accuracy solutions to gas-surface interaction problems using multiple time-stepping schemes and adaptive mesh refinement based on the discontinuous Galerkin (DG) approximation in both spatial and velocity variables. We implement both the Bhatnagar-Gross-Krook (BGK) and the Ellipsoidal Statistical Bhatnagar-Gross-Krook (ES-BGK) model Boltzmann equation in one spatial variable. The new method will be combined with existing DG solvers developed by Dr. Alekseenko.  We expect that the new method will have increased efficiency and achieve high-order convergence. These simulations will later be extended to multiple dimensions and will be used in simulations of microscale devices.