Discontinuous Galerkin BGK Method for Viscous Flow Equations: One-Dimensional Systems

Abstract

This paper is about the construction of a BGK Navier--Stokes (BGK-NS) solver in the discontinuous Galerkin (DG) framework. Since in the DG formulation the conservative variables and their slopes can be updated simultaneously, the flow evolution in each element involves only the flow variables in the nearest neighboring cells. Instead of using the semidiscrete approach in the Runge--Kutta discontinuous Galerkin (RKDG) method, the current DG-BGK method integrates the governing equations in time as well. Due to the coupling of advection and dissipative terms in the gas-kinetic formulation, the DG-BGK method solves the viscous governing equations directly. Numerical examples for the one-dimensional compressible Navier--Stokes solutions will be presented.