3 Shock Capturing Methods

8 were performed by Mark Sussman, see ref 29], 30]. The algorithm is based on the level set method and is quite simple to implement. It should be stressed that no special work is needed at pinchoo, merging, development of kinks, etc. and the algorithm is very easy to implement on a xed, uniform grid. All these gures should be read left to right, top to bottom. Figure 1 shows the evolution of a 2D rising bubble with high Reynolds number and low surface tension. Figure 2 shows the evolution of a large water drop with no surface tension which deforms as it hits the base. Figure 3 shows the evolution with surface tension. The drop remains circular as it hits the base. Figure 4 shows the evolution of two water drops colliding with each other. The combined drop experiences surface tension driven oscillations. Figure 5 shows (essentially) conservation of mass for the merging drop problem. A new idea of E. Fatemi and Sussman (private communication) is used to preserve mass in the distance reinitialization step. Figure 6 demonstrates the rupture of a gas bubble at an air water interface. The surface tension eeects cause a high speed jet of water to form. Finally, in gure 7 we see an air bubble rising through a water/oil interface. Surface tension eeects are ignored. Ideas developed in 17] for the motion of multiple junctions, were used here. Figures 8a,b,c,d,e from 13] show volume preserving motion by mean curvature velocity. The topological change presents no diiculty. See 13] for merging and physically realistic acceleration cases. Figure 9 from 3] shows a Stefan problem with super-cooled liquid. The initial solutions are 2 square seeds (temperature T = 0) in a surrounding bath of under-cooled liquid (T = ?:8). No surface temperature or kinetic undercooling terms are incorporated (i.e. T = 0 on the boundary). Time levels shown are increasing in intervals of t = 1:2 up to a time of t = 24:6. Figures 10(a-f) from 9] show the results of our simple vortex sheet evolution algorithm. This is done using a third order ENO scheme on the advection step on a 128 2 grid. Figures 11(a-e) show the roll-up using 1024 2 points. (Figures 10 and 11 were computed by Professor Chi-Wang Shu). Figures 12 and 13 from 28] contrast spectral methods versus third order ENO for compressible isotropic turbulence. Figure 12 shows …