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

Abstract

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 \begin{document}$ L^1 $\end{document} norm of the difference of step-like functions which are branches of \begin{document}$ \arg (x) $\end{document} 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.