We describe a three-dimensional front tracking algorithm, discuss its numerical implementation, and present studies to validate the correctness of this approach. Based on the results of the two-dimensional computations, we expect three-dimensional front tracking to significantly improve computational efficiencies for problems dominated by discontinuities. In some cases, for which the interface computations display considerable numerical sensitivity, we expect a greatly enhanced capability. AMS subject classifications. 35L65, 35L67, 65M99PII. S1064827595293600 1. Introduction. Front tracking is a numerical method in which surfaces of discontinuity are given explicit computational degrees of freedom; these degrees of freedom are supplemented by degrees of freedom representing continuous solution values at regular grid points. This method is ideal for solutions in which discontinuities are an important feature, and especially where their accurate computation is difficult by other methods. Computational continuum mechanics abounds in such problems, which include phase transition boundaries, flame fronts, material boundaries, slip surfaces, shear bands, and shock waves. The method was initiated by Richtmyer and Morton [55] and was used for high quality aerodynamic computations by Moretti, Grossman, and Marconi [50,51,52].A systematic development of front tracking in two dimensions has been carried out by the authors and their coworkers [20,12,21,22,19,10,29]. See [40,61,42,11,2,45,46] and additional references in the survey [37] for other approaches to front tracking in two dimensions. Special purpose front tracking codes have also been developed, for example, for simulation of the deposition and etching process for the manufacture of semiconductors [31]. Computer-aided design packages for solid geometry use similar concepts, under the terminology of nonmanifold geometry. There are also a number of one-dimensional front tracking codes [41,33,60,9,45,56,34].The first conclusion to emerge from this body of work is that it is possible to apply front tracking in a systematic fashion to complex shock or wave front interaction problems, including problems with bifurcations, with changes of wave front topology, as occurs after interaction, or crossing of one wave (tracked discontinuity) by another. In other words, the first conclusion is that front tracking is a feasible

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We study the large time behavior in L' of the compressible, isentropic, viscous I-D flow. Under the assumption that the initial data are smooth and small, we show that the solutions are approximated by the solutions of a parabolic system, and in turn by diffusion waves, which are solutions of Burgers equations. Decay rates in L' are obtained. Our method is based on the study of pointwise properties in the physical space of the fundamental solution to the linearized system.

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