We study evolution of a closed embedded plane curve with the normal velocity depending on the curvature, the orientation and the position of the curve. We propose a new method of tangential redistribution of points by curvature adjusted control in the tangential motion of evolving curves. The tangential velocity may not only distribute grid points uniformly along the curve but also produce a suitable concentration and/or dispersion depending on the curvature. Our study is based on solutions to the governing system of nonlinear parabolic equations for the position vector, tangent angle and curvature of a curve. We furthermore present a semi-implicit numerical discretization scheme based on the flowing finite volume method. Several numerical examples illustrating capability of the new tangential redistribution method are also presented in this paper.Keywords Curvature driven flow of a plane curve · Curvature adjusted tangential velocity · Semi-implicit scheme · Flowing finite volume method · Crystalline curvature flow equation Mathematics Subject Classification (2000)35K65 · 65N40 · 53C80

A finite difference scheme is constructed for the evolution of a plane curve driven by a power of curvature: V = K α , α > 0, where V and K are the normal velocity and the curvature, respectively. Here we assume that the curve is closed, strictly convex, and immersed in the plane R 2. This curve-evolution is discretized using a crystalline approximation which is a kind of finite difference scheme for the nonlinear evolution equation. We prove convergence of this scheme and show the rate of convergence. Moreover, some numerical examples are presented.

In this study, the solutions to the one‐phase interior or the classical Hele‐Shaw problem are discretized in space by employing the method of fundamental solutions combined with the uniform distribution method; furthermore, a system of ordinary differential equations is obtained, which is solved using the usual fourth‐order Runge‐Kutta method. The one‐phase interior Hele‐Shaw problem has curve‐shortening (CS), area‐preserving (AP), and barycenter‐fixed (BF) properties. Under our numerical scheme, a discrete version of CS and BF properties holds in an asymptotic sense and AP property in an exact sense, whereas in general, simple boundary element method does not satisfy these properties. Moreover, the one‐phase exterior Hele‐Shaw problem and one‐phase interior Hele‐Shaw problem containing sink/source points can be treated. Therefore, in each problem, a nontrivial exact solution is constructed and an experimental order of convergence is shown.

In this paper, we investigate a time‐dependent family of plane closed Jordan curves evolving in the normal direction with a velocity that is assumed to be a function of the curvature, tangential angle, and position vector of a curve. We follow the direct approach and analyze the system of governing PDEs for relevant geometric quantities. We focus on a class of the so‐called curvature adjusted tangential velocities for computation of the curvature driven flow of plane closed curves. Such a curvature adjusted tangential velocity depends on the modulus of the curvature and its curve average. Using the theory of abstract parabolic equations, we prove local existence, uniqueness, and continuation of classical solutions to the system of governing equations. We furthermore analyze geometric flows for which normal velocity may depend on global curve quantities such as the length, enclosed area, or total elastic energy of a curve. We also propose a stable numerical approximation scheme on the basis of the flowing finite volume method. Several computational examples of various nonlocal geometric flows are also presented in this paper. Copyright © 2012 John Wiley & Sons, Ltd.

We propose a simple and fast numerical method for solving an evolution equation for closed flame/smoldering fronts, equivalent to the Kuramoto-Sivashinsky equation in a scale. Comparison of numerical results and an experiment suggests that our model equation is valid for not only propagating gas-phase flame fronts but also expanding smoldering fronts over thin solids.

We study polygonal analogues of several moving boundary problems and their time discretization which preserves the constant area speed property. We establish various polygonal analogues of geometric formulas for moving boundaries and make use of the geometric formulas for our numerical scheme and its analysis of general constant area speed motion of polygons. Accuracy and efficiency of our numerical scheme are checked through numerical simulations for several polygonal motions such as motion by curvature and area-preserving advected flow etc.

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The Kuramoto-Sivashinsky equation for Jordan curves is used to model the smoldering combustion of a sheet of paper. Here, the behavior of a rotating wave bifurcating from an expanding circle solution to this equation is mathematically analyzed. We also present some numerical examples in which the rotating waves are visualized.