Numerical analysis comparing ODE approach and level set method for evolving spirals by crystalline eikonal-curvature flow

TL;DR

The paper addresses evolving spirals under crystalline curvature flow by comparing a discrete ODE facet-length model with a level-set formulation. It implements both methods for polygonal spirals and quantifies their agreement via an area-based difference using branches of the angle function, finding that the two approaches produce nearly identical evolutions at sufficiently high resolution. The main contribution is a systematic, quantitative cross-validation of the two numerical schemes across square, diagonal, and triangle anisotropies, showing differences stay below about 5% of the domain area under appropriate discretization and center-radius choices, thereby validating the interchangeability of the models for practical computations in crystalline curvature problems.

Abstract

In this paper, the evolution of a polygonal spiral curve by the crystalline curvature flow with a pinned center is considered with two view points, discrete model consist of an ODE system of facet lengths and a level set method. We investigate the difference of these models numerically by calculating the area of the region enclosed by these spiral curves. The area difference is calculated by the normalized L1 norm of the difference of step-like functions which are branches of arg(x) whose discontinuities are only on the spirals. We find the differences of the numerical results considered in this paper are very small even though the evolution laws of these models around the center and the farthest facet are slightly different.

Figures (8)

• Figure 1: Description of $\Gamma_D = \bigcup_{j=0}^k L_j (t)$. Note that the variable $t$ of $L_j$ and $y_j$ is omitted in the above figure for the simplicity.
• Figure 2: Construction of $\theta_D (t, x)$; we construct a branch of $\arg x$ whose discontinuities are only on $\Gamma (t)$(dashed line in (1)). For this purpose we first construct $\vartheta (x) = \arg x$ whose discontinuities are only on $\mathcal{L}_k (t)$ (solid line in (2)). Then, we make go down the height of $\vartheta (x)$ on $R_{j} (t)$ (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 3: Profiles of the square spiral at $t=1$. The level set method is calculated with $\rho = 0.02 - 10^{-8}$ and $\Delta x = 0.0050$.
• Figure 4: Graphs of functions $t \mapsto \mathcal{D} (t)$ for the square spiral with a fixed center radius $\rho = 0.02 - 10^{-8}$(left), and with a reduced center radius $\rho = (2 - 10^{-8}) \Delta x$(right).
• Figure 5: Profiles of the diagonal spiral at $t=1$. The level set method is calculated with $\rho = 0.02 - 10^{-8}$ and $\Delta x = 0.0050$.

Theorems & Definitions (1)

• Remark 1