Convergence of a Crystalline Algorithm for the Motion of a Closed Convex Curve by a Power of Curvature $V=K^\alpha$

Ushijima, Takeo; Yazaki, Shigetoshi

doi:10.1137/s0036142997330135

Cited by 27 publications

(16 citation statements)

References 18 publications

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“…Ishiwata and Tsutsumi [26] investigated numerical treatment for equation (3.2) under Dirichlet boundary condition. Using crystalline approximation, we constructed a numerical scheme for this problem and proved the convergence of the scheme [40]. Our scheme enjoys the discrete versions of the properties of GCF.…”

Section: Approximate Problemmentioning

confidence: 97%

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On the Approximation of Blow-up Time for Solutions of Nonlinear Parabolic Equations

Ushijima¹

2000

Publ. Res. Inst. Math. Sci.

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There are many nonlinear parabolic equations whose solutions develop singularity in finite time, say T. In many cases, a certain norm of the solution tends to infinity as time t approaches T. Such a phenomenon is called blow-up, and T is called the blow-up time. This paper is concerned with approximation of blow-up phenomena in nonlinear parabolic equations. For numerical computations or for other reasons, we often have to deal with approximate equations. But it is usually not at all clear if such wild phenomena as blow-up can be well reflected in the approximate equations. In this paper we present rather simple but general sufficient conditions which guarantee that the blow-up time for the original equation is well approximated by that for approximate equations. We will then apply our result to various examples. §1. Introduction and Main ResultsThere is a large number of nonlinear partial differential equations of parabolic type whose solution for a given initial data cannot be extended globally in time and develop a singularity in finite time, say T. Such a phenomenon is called blow-up and T is called the blow-up time. In many cases, some norms of blow-up solutions tend to infinity as t approaches T.Blow-up is known to occur in various equations including those in combustion theory, chemotaxis models and equations describing crystalline formation involving curvature-driven motion (see [9], [39]). The study of blow-up phenomena is not only interesting from the mathematical point of view but also Communicated by H. Okamoto, March 9, 2000. Revised April 20, 2000. 1991 Generally speaking, it is very difficult to numerically simulate blow-up phenomena accurately. For one thing, one has to deal with numerical data that grow indefinitely as the blow-up time approaches. This is obviously not an easy task. Secondly -and more importantly -it is not at all clear if features of such a wild phenomenon as blow-up can be well reflected in the discretized equation which approximate the original equation. Most of the standard error estimates become useless as t approaches the blow-up time.There have been some attempts to establish numerical methods to capture blow-up phenomena. For example, the rescaling algorithm by Berger and Kohn [10] and the method of MMPDE [12] by Budd et al can observe the shape of blow-up solutions near the blow-up time. There are also some works concerning approximation of blow-up time. Some authors studied numerical blow-up time and its convergence. As for the semilinear parabolic equation ut = u xx + u p , Nakagawa [35] (p = 2) and Chen [13] (p > 1) studied a finite difference scheme for this equation and proved the convergence of the blow-up time of approximate solutions to that of the real solution. They assumed that L q (q = I or 2) norm of the solution diverges at blow-up time. This assumption, however, does not always hold [17]. Recently, Abia, Lopez-Marcos and Martinez [1] considered a one dimensional semi-discrete problem for this equation. Here a semi-discrete problem means the syste...

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Section: Approximate Problemmentioning

confidence: 97%

“…where A$ = 2rj7r/n and This approximate problem is derived by the so-called crystalline approximation (see [40], [41]). Several authors studied numerical methods for problem (3.1) and related problems.…”

Section: Approximate Problemmentioning

confidence: 99%

On the Approximation of Blow-up Time for Solutions of Nonlinear Parabolic Equations

Ushijima¹

2000

Publ. Res. Inst. Math. Sci.

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“…Several papers, e.g. [10], [11], [13], [14], [16], and [24], have shown the convergence of two-dimensional crystalline motions to curve shortening flows in the plane as the number of the edges goes to infinity. We particularly note that the results in [12] and [18] have given the convergence for general curves which are not necessarily convex.…”

Section: P V) ^ P E R(t)}mentioning

confidence: 99%

Convergence of a three-dimensional crystalline motion to Gauss curvature flow

Ushijima

Yagisita

2005

Japan J. Indust. Appl. Math.

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We introduce a three-dimensional crystalline motion whose Wulff shape is a convex polyhedron (W k ). We prove that this crystalline motion converges to the motion by Gauss curvature in Ilk 3 under the assumptions that the polyhedra Wk converge to the unit ball B 3 and are symmetric with respect to the origin.K. Ishii and H. M. Soner showed the convergence of the two-dimensional crystalline motion to the curve shortening flow by a kind of perturbed test function methods. We employ their method to prove our result under aid from the theory of Minkowski problem.

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“…This proof is extendable to other powers provided a classic solution exists for the curvature shortening equation, which is granted for α > 1/3. Numerical schemes for arbitrary powers of curvature are also considered in [14,25], including the case α = 1 3 . Let f be an initial data function.…”

Section: Introductionmentioning

confidence: 99%

Affine Invariant Distance Using Multiscale Analysis

et al. 2015

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In this paper we introduce an affine invariant distance definition from a 2D point to the boundary of a bounded shape using morphological multiscale analysis. We study the mathematical behavior of this distance by examining separately the cases of convex and non-convex shapes. We prove that the proposed distance is bounded in the convex hull of the shape and infinite otherwise. A numerical scheme is given as well as experiments illustrating the behavior of the affine invariant distance.

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Convergence of a Crystalline Algorithm for the Motion of a Closed Convex Curve by a Power of Curvature $V=K^\alpha$

Cited by 27 publications

References 18 publications

On the Approximation of Blow-up Time for Solutions of Nonlinear Parabolic Equations

On the Approximation of Blow-up Time for Solutions of Nonlinear Parabolic Equations

Convergence of a three-dimensional crystalline motion to Gauss curvature flow

Affine Invariant Distance Using Multiscale Analysis

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