A multi-dimensional high-order discontinuous Galerkin method based on gas kinetic theory for viscous flow computations

This paper presents a high-order discontinuous Galerkin (DG) method based on a multi-dimensional gas kinetic evolution model for viscous flow computations. Generally, the DG methods for equations with higher order derivatives must transform the equations into a first order system in order to avoid the so-called “non-conforming problem”. In the traditional DG framework, the inviscid and viscous fluxes are numerically treated differently. Differently from the traditional DG approaches, the current method adopts a kinetic evolution model for both inviscid and viscous flux evaluations uniformly. By using a multi-dimensional gas kinetic formulation, we can obtain a spatial and temporal dependent gas distribution function for the flux integration inside the cell and at the cell interface, which is distinguishable from the Gaussian Quadrature point flux evaluation in the traditional DG method. Besides the initial higher order non-equilibrium states inside each control volume, a Linear Least Square (LLS) method is used for the reconstruction of smooth distributions of macroscopic flow variables around each cell interface in order to construct the corresponding equilibrium state. Instead of separating the space and time integrations and using the multistage Runge–Kutta time stepping method for time accuracy, the current method integrates the flux function in space and time analytically, which subsequently saves the computational time. Many test cases in two and three dimensions, which include high Mach number compressible viscous and heat conducting flows and the low speed high Reynolds number laminar flows, are presented to demonstrate the performance of the current scheme.