Multiscale cell-based coarsening for discontinuous problems

Whether tracking the eye of a storm, the leading edge of a wildfire, or the front of a chemical reaction, one finds that significant change occurs at the thin edge of an advancing line. The tracking of such change-fronts comes in myriad forms with a wide variety of applications expressible as PDEs. Expanding on Ami Harten's ideas, we construct an alternative to wavelet-based grid refinement, a multiresolution coarsening method that is capable of capturing sharp gradients across different scales, thus improving PDE-based simulations by concentrating computational resources where the solution varies sharply. We present this alternative grid coarsening method and compare its performance to other multiresolution methods by means of several examples.