In this paper, we develop bound-preserving techniques for the Runge--Kutta (RK) discontinuous Galerkin (DG) method with compact stencils (cRKDG method) for hyperbolic conservation laws. The cRKDG method was recently introduced in [Q. Chen, Z. Sun, and Y. Xing, SIAM J. Sci. Comput., 46: A1327--A1351, 2024]. It enhances the compactness of the standard RKDG method, resulting in reduced data communication, simplified boundary treatments, and improved suitability for local time marching. This work improves the robustness of the cRKDG method by enforcing desirable physical bounds while preserving its compactness, local conservation, and high-order accuracy. Our method is extended from the seminal work of [X. Zhang and C.-W. Shu, J. Comput. Phys., 229: 3091--3120, 2010]. We prove that the cell average of the cRKDG method at each RK stage preserves the physical bounds by expressing it as a convex combination of three types of forward-Euler solutions. A scaling limiter is then applied after each RK stage to enforce pointwise bounds. Additionally, we explore RK methods with less restrictive time step sizes. Because the cRKDG method does not rely on strong-stability-preserving RK time discretization, it avoids its order barriers, allowing us to construct a four-stage, fourth-order bound-preserving cRKDG method. Numerical tests on challenging benchmarks are provided to demonstrate the performance of the proposed method.

翻译：本文针对双曲守恒律的紧致模板龙格-库塔（RK）间断伽辽金（DG）方法（cRKDG方法）发展了保界技术。cRKDG方法由[Q. Chen, Z. Sun, and Y. Xing, SIAM J. Sci. Comput., 46: A1327--A1351, 2024]近期提出。该方法增强了标准RKDG方法的紧致性，从而减少了数据通信、简化了边界处理，并提高了局部时间推进的适用性。本研究通过强制满足期望的物理界，同时保持其紧致性、局部守恒性和高阶精度，提升了cRKDG方法的鲁棒性。我们的方法扩展自[X. Zhang and C.-W. Shu, J. Comput. Phys., 229: 3091--3120, 2010]的开创性工作。我们通过将cRKDG方法在每个RK阶段的单元平均值表达为三类前向欧拉解的凸组合，证明了其保持物理界。随后在每个RK阶段后应用缩放限制器以强制逐点有界。此外，我们探索了时间步长限制更宽松的RK方法。由于cRKDG方法不依赖于强稳定保持（SSP）的RK时间离散，从而避免了其阶数障碍，使我们能够构建一个四阶段、四阶的保界cRKDG方法。本文提供了在具有挑战性的基准测试上的数值实验，以展示所提方法的性能。