  The Runge--Kutta discontinuous Galerkin (RKDG) method is a high-order
technique for addressing hyperbolic conservation laws, which has been refined
over recent decades and is effective in handling shock discontinuities. Despite
its advancements, the RKDG method faces challenges, such as stringent
constraints on the explicit time-step size and reduced robustness when dealing
with strong discontinuities. On the other hand, the Gas-Kinetic Scheme (GKS)
based on a high-order gas evolution model also delivers significant accuracy
and stability in solving hyperbolic conservation laws through refined spatial
and temporal discretizations. Unlike RKDG, GKS allows for more flexible CFL
number constraints and features an advanced flow evolution mechanism at cell
interfaces. Additionally, GKS' compact spatial reconstruction enhances the
accuracy of the method and its ability to capture stable strong discontinuities
effectively. In this study, we conduct a thorough examination of the RKDG
method using various numerical fluxes and the GKS method employing both compact
and non-compact spatial reconstructions. Both methods are applied under the
framework of explicit time discretization and are tested solely in inviscid
scenarios. We will present numerous numerical tests and provide a comparative
analysis of the outcomes derived from these two computational approaches.
