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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
Summary
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.
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.
Keywords:
35K6553C4453C8065M0665M50forced curve shortening flowgeometric partial differential equationsmonitor functionsmoving mesh methodstangential redistributionMore Related Videos
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