Uniformly High Order Discontinuous Galerkin Gas Kinetic Scheme for Compressible flows

In this paper, a uniformly high-order discontinuous Galerkin gas kinetic scheme (DG-HGKS) is proposed to solve the Euler equations of compressible flows. The new scheme is an extension of the one-stage compact and efficient high-order GKS (CEHGKS, Li et al. , 2021. J. Comput. Phys. 447, 110661) in the finite volume framework. The main ideas of the new scheme consist of two parts. Firstly, starting from a fully discrete DG formulation, the numerical fluxes and volume integrals are expanded in time. Secondly, the time derivatives are replaced by spatial derivatives using the techniques in CEHGKS. To suppress the non-physical oscillations in the discontinuous regions while minimizing the number of "troubled cells", an effective limiter strategy compatible with the new scheme has been developed by combining the KXRCF indicator and the SHWENO reconstruction technique. The new scheme can achieve arbitrary high-order accuracy in both space and time, thereby breaking the previous limitation of no more than third-order accuracy in existing one-stage DG-HGKS schemes. Numerical tests in 1D and 2D have demonstrated the robustness and effectiveness of the scheme.