A compact fourth-order gas-kinetic scheme for the Euler and Navier–Stokes equations

Abstract

In this paper, a fourth-order compact gas-kinetic scheme (GKS) is developed for the compressible Euler and Navier–Stokes equations under the framework of two-stage fourth-order temporal discretization and Hermite WENO (HWENO) reconstruction. Due to the high-order gas evolution model, the GKS provides a time dependent gas distribution function at a cell interface. This time evolution solution can be used not only for the flux evaluation across a cell interface and its time derivative, but also time accurate flow variables at a cell interface. As a result, besides updating the conservative flow variables inside each control volume, the cell averaged slopes inside each control volume through the differences of flow variables at the cell interfaces can be updated as well in GKS. So, with the updated flow variables and their slopes inside each cell, the HWENO techniques can be naturally implemented for the compact high-order reconstruction at the beginning of each time step. Therefore, a compact higher-order GKS, such as the two-stage fourth-order compact scheme can be constructed. This fourth-order compact GKS has the same stencil and as robust as the second-order scheme. In comparison with the fourth-order DG method, they have the same stencil. In order to get a fourth-order temporal accuracy, the GKS uses two stages and DG needs four stages in the time stepping methods. The CFL number used in GKS is on the order of 0.5 instead of 0.11 in the DG. This research concludes that beyond the first-order Riemann solver the use of high-order time evolution model at a cell interface is extremely helpful in the design of robust, accurate, and efficient higher-order compact schemes for the compressible flow simulations.