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We investigate the properties of discretizations of advection equations onnon-cartesian grids and graphs in general. Advection equations discretized onnon-cartesian grids have remained a long-standing challenge as the structure ofthe grid can lead to strong oscillations in the solution, even for otherwiseconstant velocity fields. We introduce a new method to track oscillations ofthe solution for rough velocity fields on any graph. The method in particularhighlights some inherent structural conditions on the mesh for propagatingregularity on solutions.

Discretizing advection equations with rough velocity fields on non-cartesian grids