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

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a ’ ’ ‘) ALGORITHMSFORSURFACEMOTIONPROBLEMS 37 - ’ b : FIG. 7. Passive advection and propagation: F(K) = 1 + Rotation: (a) T=O.O, 0.03 (0.01); (b) T=O.O3, 0.06 (0.01); (c) I-=0.06, 0.09 (0.01); (d) I-=0.09, 0.12 (0.01). This corresponds to solid body counterclockwise rotation around the origin wi tangential velocity 1 along the circle with radius 0.1 centered at the origin. The si of the numerical parameters and scheme order are the same as in Example Fig. 7, we show the expanding and spinning star at various times. V. MOVING SURFACES In this section, we use PSC algorithms to compute the evolution of several two- dimensional surfaces in three space dimensions. We use the initializing function given in Section 1II.D and the first-order Hamilton-Jacobi scheme given in Eq. (3.11). A. Propagating Function Surfaces, F(K) = 1, F(K) = I- EK, F(K) = K We evolve the initial surface q/(x, y, 0) = -0.25[cos(2nx)- 1 J[cos(27q) - l] -f- 1
38 OSHER AND SETHIAN according to Eqs. (3.1), (3.2), with F(K) = 1, At = 0.01, Npoint = 50 (in each direc- tion), and periodic boundary conditions. This surface is flat in the boundary of the unit square centered at the origin and has a global minimum at (0,O). In Fig. 8a, we plot the surface at various times, showing the focusing of the minimum into a deep dent which then opens up. The surface moves upward with unit speed, asymptotically approaching a flat sheet. Next, we add curvature effects to the speed function and let F(K) = 1 - EK, E = 0.1. The time step is reduced to At =O.OOOl because of stability requirements from the addition of a parabolic term. In Fig. 8b, we plot the surface at various times. Here, the dent is greatly smoothed due to the curvature effects, and the surface becomes flat much faster, similar to the one- dimensional case (Fig. 1). We then consider a saddle surface, described by $(x, y, 0) = cos(2Tcx) - cos(27cy). Again, we start with F(K) = 1, At = 0.01, Npoint = 50, and periodic boundary conditions. In Fig. 9a, we plot the surface at various times. Here, the rising surface develops a discontinuity passing through the saddle point in the y coordinate direc- tion, corresponding to a shock in the tangent vector. Adding curvature (Fig. 9b, F(K) = 1 - EK, E = O.l), the shock is smoothed out, and the surface smoothly approaches a flat sheet. Finally, we move the saddle surface purely under its own curvature (F(K) = -K, where K is the mean curvature). In Fig. 10, we show the front at various times and show the rapid motion toward the steady state flat sheet. 2.2 2.0 1.6 1.6 1.4 1.2 I.0 0.6 0.6 0.4 0.2 0.0 FIG. 8. Propagating surface: F(K) = 1 - EK, mean curvature: (a) F(K) = 1, surface at T= 0.0, 0.3, 0.6; (b) F(K)= 1 --EK, s=O.l, surface at T=O.O, 0.3, 0.6. 2.2 2.0 1.6 1.6 I.4 1.2 1 .o 0.6 0.6 0.4 6.2 0.0
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 and 24: I I I I I I I I I -0.250 -*.1m -0.0
Page 25: 36 OSHER AND SETHIAN collapse to a
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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