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An Adaptive Moving Mesh Method for Forced Curve Shortening Flow.

J A Mackenzie1,2, M Nolan1, C F Rowlatt1

  • 1Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, UK.

SIAM Journal on Scientific Computing : a Publication of the Society for Industrial and Applied Mathematics
|December 5, 2019
PubMed
Summary
This summary is machine-generated.

This study introduces an adaptive moving mesh method for geometric evolution equations. The novel approach improves numerical solution accuracy by adapting the mesh to curve features, outperforming uniform meshes.

Keywords:
35K6553C4453C8065M0665M50forced curve shortening flowgeometric partial differential equationsmonitor functionsmoving mesh methodstangential redistribution

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Area of Science:

  • Computational Mathematics
  • Geometric Evolution Equations
  • Numerical Analysis

Background:

  • Geometric evolution equations model evolving curves.
  • Adaptive mesh methods are crucial for resolving complex geometries.
  • Existing methods may struggle with high curvature regions.

Purpose of the Study:

  • To develop a novel adaptive moving mesh method for forced curve shortening geometric evolution equations.
  • To enhance numerical solution accuracy and mesh quality.
  • To provide a robust method for simulating evolving curves.

Main Methods:

  • A tangential mesh velocity derived from a mesh equidistribution principle.
  • A positive adaptivity measure (monitor function) is equidistributed along the curve.
  • Central finite differences for spatial discretization and a second-order implicit scheme for temporal integration.

Main Results:

  • Generated meshes effectively resolve areas of high curvature.
  • The method achieves second-order accuracy in simulations.
  • Demonstrated improved solution accuracy compared to uniform arc-length meshes.

Conclusions:

  • The proposed adaptive moving mesh method offers superior accuracy for geometric evolution problems.
  • Effective mesh control is achieved through the equidistribution principle.
  • This method provides a valuable tool for simulating complex curve evolutions.