This work explores the use of high-order finite-difference schemes along Hamiltonian lines to solve conservation laws on unstructured two-dimensional meshes. To demonstrate these procedures, the paper focuses on explicit and compact spatial discretizations of up to sixth-order coupled with implicit low-pass filters of up to tenth-order along the identified line-structures. The spatial discretization schemes are then extended for the simulation of high-speed flows to capture embedded shocks. A WENO-type switch is used to identify potential regions of shock, where two distinct shock-capturing approaches are investigated. The “adaptive filter" approach, which combines a sixth-order compact scheme with a locally reduced-order filter, is one method, while the second approach is a “hybrid compact-Roe" wherein a compact scheme is replaced with a third-order upwind-biased MUSCL scheme within the shock regions. These spatial discretization schemes are used with explicit time integration approaches to investigate canonical problems defined by the compressible Navier–Stokes equations. The applicabililty of the methodology is demonstrated by solving representative problems and for both inviscid and viscous flows. It is seen that even in the presence of significant grid discontinuities, the low-pass high-order filter preserves the advantages of the high-order technique. © 2022, American Institute of Aeronautics and Astronautics Inc, AIAA. All rights reserved.