Abstract/Summary

A moving-mesh finite difference scheme based on local conservation is presented for a class of scale-invariant second-order nonlinear diffusion problems with moving boundaries that (a) preserves the scaling properties and (b) is exact at the nodes for initial conditions sampled from similarity solutions. Details are presented for one-dimensional problems, the extension to multidimensions is described, and the exactness property is confirmed for two radially symmetric moving boundary problems, the porous medium equation and a simplistic glacier equation. In addition, the accuracy of the scheme is also tested for non self-similar initial conditions by computing relative errors in the approximate solution (in the norm) and the approximate boundary position, indicating superlinear convergence.