A Compact Fourth-order Gas-kinetic Scheme for the Euler and Navier-Stokes Solutions

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 evolution solution at a cell interface. As a result, besides updating the conservative flow variables inside each control volume, the GKS can get the cell averaged slopes inside each control volume as well through the differences of flow variables at the cell interfaces. So, with the updated flow variables and their slopes inside each cell, the HWENO reconstruction can be naturally implemented for the compact high-order reconstruction at the beginning of next step. Therefore, a compact higher-order GKS, such as the two-stages fourth-order compact scheme can be constructed. This scheme is as robust as second-order one, but more accurate solution can be obtained. In comparison with compact fourth-order DG method, the current scheme has only two stages instead of four within each time step for the fourth-order temporal accuracy, and the CFL number used here can be on the order of $0.5$ instead of $0.11$ for the DG method. Through this research, it concludes that the use of high-order time evolution model rather than the first order Riemann solution is extremely important for the design of robust, accurate, and efficient higher-order schemes for the compressible flows.