Fronts Propagating with Curvature- Dependent Speed: Algorithms ...

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-0.100 -0.060 -0.020 0.020 C.060 0.100 FIG. 5a. Wound spiral collapsing under curvature (Beginning): F(K) = -K, T=O.O, &O@i (@.OQ?). FIG. 5b. Wound spiral collapsing under curvature (Complete history) F(K) = -k (a) T* 0.0, 0.065 (0.01); (b) T=O.O65, 0.130 (0.01); (c) T=0.130, 0.195 (0.01); (d} T=0.195, 0.295 (0.01). - t i- d 35 J i
36 OSHER AND SETHIAN collapse to a circle and eventual disappearance at t = 0.295. (The surface vanishes when !P; < 1 for all ij. ) F. Level Curve, F(K) = 1 - EK, Merging and Breaking Using the wound spiral initial curve in the above example, Fig. 6 shows the results <strong>with</strong> F(K) = 1 - EK, E = 0.01, Npoint = 200, and At = 0.0001. Again, we use the second-order Hamilton-Jacobi scheme. Figure 6a shows the initial curve as the boundary of the shaded region. In Fig. 6b, the spiral expands and pinches off due to the strong convection component, separating into two curves, one propagating outwards and one shrinking in. In Fig. 6c, the front at t = 0.04 is the boundary of the shaded region. The outer front expands and the inner front collapses and disappears. In Fig. 6d, all that remains is the outer front which asymptotically approaches a circle. G. Level Curve, Passive Advection, and Propagation Finally, we solve the passive advection plus propagation equation (Eq. (3.15)) <strong>with</strong> the initial seven-pointed star in Example D, F(K) = 1, and o= (UI(X, Y, t), %(X, Y, t)) = c-y, x)(wx2 -I- y’)). FIG. 6. Burning spiral: merging and breaking F(K) = 1 - EK, E = 0.01: (a) T= 0.0 (initial curve); (b) T=O.OO, 0.03 (0.01); (c) T=0.04, 0.12 (0.01); (d) T=0.13, 0.22, (0.01).
Page 1 and 2: JOURNAL OF COMPUTATIONAL PHYSICS 79
Page 3 and 4: 14 OSHER AND SETHIAN SLIC [26]. In
Page 5 and 6: 16 OSHER AND SETHIAN We notice that
Page 7 and 8: 18 OSHER AND SETHIAN are not unique
Page 9 and 10: 20 OSHER AND SETHIAN (x(si, sz), y(
Page 11 and 12: 22 OSHER AND SETHIAN function of al
Page 13 and 14: 24 OSHER AND SETHIAN (2) Higher Ord
Page 15 and 16: 26 OSHER AND SETHIAN struct a new c
Page 17 and 18: 28 OSHER AND SETHIAN which has a sl
Page 19 and 20: 30 OSHER AND SETHIAN second-order H
Page 21 and 22: 32 FIG. 2. Propagating initial sine
Page 23: I I I I I I I I I -0.250 -*.1m -0.0
Page 27 and 28: 38 OSHER AND SETHIAN according to E
Page 29 and 30: 40 OSJXER AND SETHIAN and upper bac
Page 31 and 32: 42 OSHER AND SETHIAN a EXPANDING TO
Page 33 and 34: a COLLAPSING T0R”S:T:O.O OSHER AN
Page 35 and 36: 46 OSHER AND SETHIAN Following our
Page 37 and 38: 48 OSHER AND SETHIAN To summarize,

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