Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow

Figure 1.  A figure of two spirals (the solid and dashed lines) and the interposed region by them. The function $ \mathcal{D} (t) $ indicates the area of the gray regions

Figure 2.  Description of $ \Gamma_D = \bigcup_{j = 0}^k L_j (t) $. Note that, for the simplicity, the variable $ t $ of $ L_j $ and $ y_j $ is omitted in the above figure

Figure 3.  Construction of $ \theta_D (t, x) $; we construct a branch of $ \arg (x) $ whose discontinuities are only on $ \Gamma (t) $(the dashed line in (1)). For this purpose we first construct $ \vartheta (x) = \arg (x) $ whose discontinuities are only on $ \mathcal{L}_k (t) $ (the solid line in (2)). Then, we make go down the height of $ \vartheta (x) $ on $ R_{j} (t) $ (the gray region in (3) or (4)) with the jump-height $ 2 \pi $ from $ j = k-1 $ to $ j = 0 $ inductively to remove illegal discontinuities. The solid line in figure (3) or (4) denotes the discontinuity of $ \Theta_{k,k-1} $ or $ \Theta_{k,k-2} $, respectively

Figure 4.  Profiles of the square spiral at $ t = 1 $. The level set method is calculated using $ \rho = 0.02 $ and $ \Delta x = 0.0050 $

Figure 5.  Graphs of functions $ \mathcal{D} (t) $ for the square spiral with a fixed center radius $ \rho = 0.02 $(left), and with a reduced center radius $ \rho = 2 \Delta x $(right)

Figure 6.  Profiles of the diagonal spiral at $ t = 1 $. The level set method is calculated using $ \rho = 0.02 $ and $ \Delta x = 0.0050 $

Figure 7.  Graphs of $ \mathcal{D}(t) $ for the diagonal spiral with a fixed center radius $ \rho = 0.02 $(left), and with a reduced center radius $ \rho = 4 \Delta x $(right)

Figure 8.  Profiles of the triangle spiral at $ t = 0.8 $. The level set method is calculated using $ \rho = 0.02 $ and $ \Delta x = 0.0050 $

Figure 9.  Graphs of $ \mathcal{D} (t) $ for the triangle spiral with a fixed center radius $ \rho = 0.02 $(left), and with a reduced center radius $ \rho = 4\Delta x $(right)