An adaptive moving mesh method for forced curve shortening flow

J.A. Mackenzie, M. Nolan, C.F. Rowlatt, R.H. Insall

Research output: Contribution to journal › Article

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 discretise 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.
LanguageEnglish
Number of pages27
JournalSIAM Journal on Scientific Computing
StateAccepted/In press - 3 Jan 2019

Fingerprint

Moving Mesh Method
Adaptive Mesh
Adaptive Method
Mesh
Curve
Evolution Equation
Position vector
Mesh Quality
Equidistribution
Second-order Accuracy
Arc length
Implicit Scheme
Adaptivity
Governing equation
Resolve
Finite Difference
Monitor
Curvature
Numerical Solution
Simulation

Keywords

  • geometric partial differential equations
  • monitor functions
  • tangential redistribution
  • moving mesh methods
  • forced curve shortening flow

Cite this

Mackenzie, J.A. ; Nolan, M. ; Rowlatt, C.F. ; Insall, R.H./ An adaptive moving mesh method for forced curve shortening flow. In: SIAM Journal on Scientific Computing. 2019
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Mackenzie, JA, Nolan, M, Rowlatt, CF & Insall, RH 2019, 'An adaptive moving mesh method for forced curve shortening flow' SIAM Journal on Scientific Computing.

An adaptive moving mesh method for forced curve shortening flow. / Mackenzie, J.A.; Nolan, M.; Rowlatt, C.F.; Insall, R.H.

In: SIAM Journal on Scientific Computing, 03.01.2019.

Research output: Contribution to journal › Article

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AU - Mackenzie,J.A.

AU - Nolan,M.

AU - Rowlatt,C.F.

AU - Insall,R.H.

PY - 2019/1/3

Y1 - 2019/1/3

N2 - 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 discretise 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.

AB - 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 discretise 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.

KW - geometric partial differential equations

KW - monitor functions

KW - tangential redistribution

KW - moving mesh methods

KW - forced curve shortening flow

UR - https://epubs.siam.org/journal/sjoce3

M3 - Article

JO - SIAM Journal on Scientific Computing

T2 - SIAM Journal on Scientific Computing

JF - SIAM Journal on Scientific Computing

SN - 1064-8275

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Mackenzie JA, Nolan M, Rowlatt CF, Insall RH. An adaptive moving mesh method for forced curve shortening flow. SIAM Journal on Scientific Computing. 2019 Jan 3.

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