Olof Runborg
Department of Numerical Analysis
School of Computer Science and Communication
KTH, Stockholm
We consider the propagation of an interface in a velocity field. The initial interface is described by a so-called normal mesh which is a multiresolution based way to describe curves and surfaces. This gives us a representation of the interface in terms of a set of wavelet vectors. Instead of tracking marker points on the interface we track the wavelet vectors, which like the markers satisfy ordinary differential equations. We show that the finer the spatial scale, the slower the wavelet vectors evolve. By designing a numerical method which takes longer time steps for finer spatial scales we are able to track the interface with the same overall accuracy as when directly tracking the markers, but at a computational cost of O(log N/Delta t) rather than O(N/Delta t) for N markers and timestep Delta t. We sketch the proof of this and show numerical examples supporting the theory. We also consider extensions to higher dimensions and co-dimensions.