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March  2021, 14(3): 893-907. doi: 10.3934/dcdss.2020390

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

 1 Department of Mathematical Sciences, Shibaura Institute of Technology, Fukasaku 309, Minuma-ku, Saitama 337-8570, Japan 2 Division of Pure and Applied Science, Faculty of Science and Technology, Gunma University, Aramaki-machi 4-2, Maebashi, 371-8510 Gunma, Japan

* Corresponding author: Takeshi Ohtsuka

Received  January 2019 Revised  February 2020 Published  March 2021 Early access  June 2020

In this paper, the evolution of a polygonal spiral curve by the crystalline curvature flow with a pinned center is considered from two viewpoints; a discrete model consisting of an ODE system describing facet lengths and another using level set method. We investigate the difference of these models numerically by calculating the area of an interposed region by their spiral curves. The area difference is calculated by the normalized $L^1$ norm of the difference of step-like functions which are branches of $\arg (x)$ whose discontinuities are on the spirals. We find that the differences in the numerical results are small, even though the model equations around the center and the farthest facet are slightly different.

Citation: Tetsuya Ishiwata, Takeshi Ohtsuka. Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 893-907. doi: 10.3934/dcdss.2020390
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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
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
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
Profiles of the square spiral at $t = 1$. The level set method is calculated using $\rho = 0.02$ and $\Delta x = 0.0050$
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)
Profiles of the diagonal spiral at $t = 1$. The level set method is calculated using $\rho = 0.02$ and $\Delta x = 0.0050$
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)
Profiles of the triangle spiral at $t = 0.8$. The level set method is calculated using $\rho = 0.02$ and $\Delta x = 0.0050$
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)
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