  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.
