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.

翻译：Runge-Kutta间断Galerkin（RKDG）方法是一种求解双曲守恒律的高阶数值技术，近几十年来不断发展完善，能有效处理激波间断。尽管取得了显著进展，RKDG方法仍面临显式时间步长严格受限以及处理强间断时鲁棒性下降等挑战。另一方面，基于高阶气体演化模型的气体动理学格式（GKS）通过精细的空间和时间离散，在求解双曲守恒律时同样展现出显著的精度和稳定性。与RKDG不同，GKS允许更灵活的CFL数约束，并具有先进的单元界面流动演化机制。此外，GKS的紧致空间重构提升了方法的精度及其有效捕捉稳定强间断的能力。本研究系统考察了采用多种数值通量的RKDG方法，以及采用紧致与非紧致空间重构的GKS方法。两种方法均在显式时间离散框架下应用，并仅在无粘场景中进行测试。我们将展示大量数值试验，并对这两种计算方法所得结果进行对比分析。