A Fixed Grid Method for Capturing the Motion of Self-Intersecting Wavefronts and Related PDEs

Moving surfaces that self-intersect arise naturally in the geometric optics model of wavefront motion. Standard ray tracing techniques can be used to compute these motions, but they lose resolution as rays diverge. In this paper we develop numerical methods that maintain uniform spatial resolution of the front at all times. Our approach is a fixed grid, wavefront capturing formulation based on the Dynamic Surface Extension method of Steinhoff and Fan (Technical report, University of Tennessee Space Institute). The new methods can treat arbitrarily complicated self intersecting fronts, as well as refraction, reflection, and focusing. We also develop methods for curvature-dependent front motions and the motion of filaments. We validate our methods with numerical experiments.