A Cartesian adaptive level set method

Simulations of °ows involving free surfaces have become ubiquitous in the literature. The evolution of both simulation methods and computer speed have allowed the inves-tigation of many problems of increasing complexity and engineering relevance. For 2D simulations, it is reasonably common to use grid resolutions of 512 £ 512 or larger. See for example the 2D planar breaking wave simulations of Chen et al. (1999), or the ax-isymmetric drop breakup simulations of Han and Tryggvason (1999a, 1999b). Many such free surface problems are fundamentally three dimensional and the absence of three-dimensional e®ects such as the pinch-o ® of stretched ¯laments by surface tension makes it di±cult to make valid physical conclusions or use 2D results to develop models. Achiev-ing similar grid resolution in 3D, however, is outside the computational reach of most research centers. For many problems of interest, a ¯ne grid is required in only a relatively small portion of the domain (e.g. around regions of high surface curvature) and much coarser grids are su±cient to capture the smoothly evolving velocity ¯elds far from the surface. It is this type of problem that motivates the present Cartesian adaptive method, where the ¯nes