When simulating hyperbolic conservation laws with discontinuous solutions, high-order linear numerical schemes often produce undesirable spurious oscillations. In this paper, we propose a jump filter within the discontinuous Galerkin (DG) method to mitigate these oscillations. This filter operates locally based on jump information at cell interfaces, targeting high-order polynomial modes within each cell. Besides its localized nature, our proposed filter preserves key attributes of the DG method, including conservation, $L^2$ stability, and high-order accuracy. We also explore its compatibility with other damping techniques, and demonstrate its seamless integration into a hybrid limiter. In scenarios featuring strong shock waves, this hybrid approach, incorporating this jump filter as the low-order limiter, effectively suppresses numerical oscillations while exhibiting low numerical dissipation. Additionally, the proposed jump filter maintains the compactness of the DG scheme, which greatly aids in efficient parallel computing. Moreover, it boasts an impressively low computational cost, given that no characteristic decomposition is required and all computations are confined to physical space. Numerical experiments validate the effectiveness and performance of our proposed scheme, confirming its accuracy and shock-capturing capabilities.

翻译：在模拟具有间断解的双曲守恒律时，高阶线性数值格式常会产生不良的伪振荡。本文提出一种基于间断Galerkin（DG）方法的跳跃滤波器以抑制此类振荡。该滤波器基于单元界面处的跳跃信息进行局部操作，针对每个单元内的高阶多项式模态。除了其局部特性外，所提出的滤波器保留了DG方法的关键属性，包括守恒性、$L^2$稳定性以及高阶精度。我们还探讨了其与其他阻尼技术的兼容性，并展示了其与混合限制器的无缝集成。在包含强冲击波的场景中，这种将跳跃滤波器作为低阶限制器的混合方法能有效抑制数值振荡，同时表现出较低的数值耗散。此外，所提出的跳跃滤波器保持了DG格式的紧致性，这极大地有助于高效并行计算。而且，由于无需进行特征分解且所有计算均局限于物理空间，该滤波器具有极低的计算成本。数值实验验证了所提方案的有效性与性能，证实了其精度和激波捕捉能力。