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

TL;DR

This paper introduces a new method for discretizing advection equations on non-cartesian grids and graphs, effectively tracking oscillations caused by rough velocity fields and revealing structural conditions for solution regularity.

Contribution

The paper presents a novel approach to handle oscillations in advection discretizations on complex grids, addressing longstanding challenges in the field.

Findings

Method effectively tracks oscillations in rough velocity fields.

Identifies structural conditions on meshes for solution regularity.

Improves stability of advection discretizations on non-cartesian grids.

Abstract

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