Abstract
We propose a novel adaptive moving mesh method for the numerical solution of a forced curve shortening geometric evolution equation. Control of the mesh quality is obtained using a tangential mesh velocity derived from a mesh equidistribution principle, where a positive adaptivity measure or monitor function is approximately equidistributed along the evolving curve. Central finite differences are used to discretize in space the governing evolution equation for the position vector, and a second-order implicit scheme is used for the temporal integration. Simulations are presented indicating the generation of meshes which resolve areas of high curvature and are of second-order accuracy. Furthermore, the new method delivers improved solution accuracy compared to the use of uniform arc-length meshes.
Highlights
Within the past 20 years there has been much interest in the numerical approximation of geometric flows
(Ellipse.) Maximum and minimum number of Picard steps required for each scheme, with
We have presented an adaptive moving mesh method to simulate forced curve shortening flow
Summary
Within the past 20 years there has been much interest in the numerical approximation of geometric flows (see, for example, [12, 14]). The numerical solution of geometric evolution laws poses many challenges, and a number of different techniques have been proposed which fall broadly into two categories: embedded methods and sharp interface methods. Examples of embedded techniques include phase-field methods [12] and the level set method [32, 30]. These methods identify the moving interface as the zero level set of an indicator function which is normally evolved through a fixed uniform background mesh.
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