Front Tracking in Two and Three Dimensions

Front Tracking is the combination of two distinct computational traditions. Its purpose is to enable high quality, relatively coarse grid mesh solutions for problems complicated by the presence of fronts, interfaces, and other discontinuities, which may be of considerable geometric complexity. The rst of the two algorithmic ideas used in front tracking is upwind diierencing, for uids, materials, and other conservation law based problems arising in physics. The second is computational geometry, from which we draw methods to represent, track, and process the geometric structures deened by the discontinuities. This combination, which is called front tracking, is by now a mature capability in two dimensions 3], and has been used to achieve deenitive solutions to a number of problems which have otherwise resisted solution for decades. Here we report on recent extensions of front tracking to three dimensions 4], and we also present highlights from some recent two dimensional front tracking computations. Front Tracking applies to solutions which are conceptually piecewise smooth but not smooth. It thus applies to problems which contain important singularities, or jump discontinuities, concentrated on surfaces (curves in two dimensions). The location of the discontinuity surfaces are referred to as fronts. Front Tracking employs two grid systems. The traditional volume lling nite diierence grid supports the smooth solution, located in the regions between the fronts, and a lower dimensional grid (the front, or interface) deenes the location of the discontinuity and the jump in the solution variables across it. The dynamics of the front comes from the mathematical theory of Riemann solutions, which is an idealized solution of a single jump