This study enables the use of explicit and compact high-order finite-difference schemes with a line-based solver to solve conservation laws on unstructured grids. A quadrilateral subdivision process is used to identify unique line structures (also known as Hamiltonian loops) before formulating stencil-based discretizations. To demonstrate the methodology for canonical flows represented by the Navier–Stokes equations, up to sixth-order spatial discretization and a maximum of tenth-order low-pass filters are implemented along with the Hamiltonian loops. The filter restores the benefits of the high-order approach even in the presence of abrupt grid discontinuities that cause abrupt changes in loop curvature. Isentropic vortex convection, lid-driven cavity, double periodic shear layer, and inviscid and viscous flow past a cylinder and NACA 0012 airfoil are all used to demonstrate the formulation. The ability of some schemes, particularly the fourth-order explicit, to nearly achieve the formal order of accuracy and successfully predict the flow following available results in the literature is one of the key findings. Compact schemes have been found to be more sensitive to loop curvature than explicit schemes. © 2022 Elsevier Ltd