In this paper, we develop a new type of Runge--Kutta (RK) discontinuous Galerkin (DG) method for solving hyperbolic conservation laws. Compared with the original RKDG method, the new method features improved compactness and allows simple boundary treatment. The key idea is to hybridize two different spatial operators in an explicit RK scheme, utilizing local projected derivatives for inner RK stages and the usual DG spatial discretization for the final stage only. Limiters are applied only at the final stage for the control of spurious oscillations. We also explore the connections between our method and Lax--Wendroff DG schemes and ADER-DG schemes. Numerical examples are given to confirm that the new RKDG method is as accurate as the original RKDG method, while being more compact, for problems including two-dimensional Euler equations for compressible gas dynamics.

翻译：本文提出了一类新型的Runge-Kutta (RK) 间断Galerkin (DG) 方法，用于求解双曲守恒律。与原始RKDG方法相比，新方法具有更强的紧致性，并允许简化的边界处理。其核心思想是在显式RK格式中混合两种不同的空间算子：内部RK阶段使用局部投影导数，仅在最后阶段采用常规DG空间离散。抑制虚假振荡的限幅器仅在最终阶段施加。我们还探讨了本方法与Lax-Wendroff DG格式和ADER-DG格式之间的联系。数值算例证实，对于包含可压缩气体动力学二维Euler方程的问题，新RKDG方法在保持与原始RKDG方法同等精度的同时，具有更强的紧致性。