My current work aims at providing accurate numerical solutions for aerodynamic analysis at practical engineering scales, with minimal modeling at the sub-grid level, which requires the development of high-order numerical methods suitable for the current generation of High Performance Computers, and poses modeling, algorithmic, and implementation challenges.

I will summarize our current approach and report the results of some recent efforts on the implementation of implicit time integration of a discontinuous Galerkin discretization of the Navier-Stokes equations. I will also present a newly developed method for the numerical solution of the Boltzmann equation using a Bhatnagar–Gross–Krook (BGK) collision operator, where internal degrees of freedom are included to extend the model to polyatomic molecules and general constitutive laws. The promises of this approach will be illustrated by a series of multi-scale numerical experiments of high-speed flows.