Second-Order Unconditionally Stable Direct Methods for Allen–Cahn and Conservative Allen–Cahn Equations on Surfaces

Second-Order Unconditionally Stable Direct Methods for Allen–Cahn and Conservative Allen–Cahn Equations on Surfaces

This paper describes temporally second-order unconditionally stable direct methods for Allen–Cahn and conservative Allen–Cahn equations on surfaces. The discretization is performed via a surface mesh consisting of piecewise triangles and its dual-surface polygonal tessellation. We prove that the pro...

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Main Authors: Binhu Xia, Yibao Li, Zhong Li
Format: Article
Language:English
Published: MDPI AG 2020-09-01
Series:Mathematics
Subjects:
Allen–Cahn equation
conservative Allen–Cahn equation
Laplace–Beltrami operator
triangular surface mesh
unconditionally energy-stable
Online Access:https://www.mdpi.com/2227-7390/8/9/1486
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spelling doaj-2897077d65344c749debd0e80d4890522020-11-25T03:27:11ZengMDPI AGMathematics2227-73902020-09-0181486148610.3390/math8091486Second-Order Unconditionally Stable Direct Methods for Allen–Cahn and Conservative Allen–Cahn Equations on SurfacesBinhu Xia0Yibao Li1Zhong Li2School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, ChinaSchool of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, ChinaSchool of Humanities and Social Science, Xi’an Jiaotong University, Xi’an 710049, ChinaThis paper describes temporally second-order unconditionally stable direct methods for Allen–Cahn and conservative Allen–Cahn equations on surfaces. The discretization is performed via a surface mesh consisting of piecewise triangles and its dual-surface polygonal tessellation. We prove that the proposed schemes, which combine a linearly stabilized splitting scheme, are unconditionally energy-stable. The resulting system of discrete equations is linear and is simple to implement. Several numerical experiments are performed to demonstrate the performance of our proposed algorithm.https://www.mdpi.com/2227-7390/8/9/1486Allen–Cahn equationconservative Allen–Cahn equationLaplace–Beltrami operatortriangular surface meshunconditionally energy-stable
collection DOAJ
language English
format Article
sources DOAJ
author Binhu Xia
Yibao Li
Zhong Li
spellingShingle Binhu Xia
Yibao Li
Zhong Li
Second-Order Unconditionally Stable Direct Methods for Allen–Cahn and Conservative Allen–Cahn Equations on Surfaces
Mathematics
Allen–Cahn equation
conservative Allen–Cahn equation
Laplace–Beltrami operator
triangular surface mesh
unconditionally energy-stable
author_facet Binhu Xia
Yibao Li
Zhong Li
author_sort Binhu Xia
title Second-Order Unconditionally Stable Direct Methods for Allen–Cahn and Conservative Allen–Cahn Equations on Surfaces
title_short Second-Order Unconditionally Stable Direct Methods for Allen–Cahn and Conservative Allen–Cahn Equations on Surfaces
title_full Second-Order Unconditionally Stable Direct Methods for Allen–Cahn and Conservative Allen–Cahn Equations on Surfaces
title_fullStr Second-Order Unconditionally Stable Direct Methods for Allen–Cahn and Conservative Allen–Cahn Equations on Surfaces
title_full_unstemmed Second-Order Unconditionally Stable Direct Methods for Allen–Cahn and Conservative Allen–Cahn Equations on Surfaces
title_sort second-order unconditionally stable direct methods for allen–cahn and conservative allen–cahn equations on surfaces
publisher MDPI AG
series Mathematics
issn 2227-7390
publishDate 2020-09-01
description This paper describes temporally second-order unconditionally stable direct methods for Allen–Cahn and conservative Allen–Cahn equations on surfaces. The discretization is performed via a surface mesh consisting of piecewise triangles and its dual-surface polygonal tessellation. We prove that the proposed schemes, which combine a linearly stabilized splitting scheme, are unconditionally energy-stable. The resulting system of discrete equations is linear and is simple to implement. Several numerical experiments are performed to demonstrate the performance of our proposed algorithm.
topic Allen–Cahn equation
conservative Allen–Cahn equation
Laplace–Beltrami operator
triangular surface mesh
unconditionally energy-stable
url https://www.mdpi.com/2227-7390/8/9/1486
work_keys_str_mv AT binhuxia secondorderunconditionallystabledirectmethodsforallencahnandconservativeallencahnequationsonsurfaces
AT yibaoli secondorderunconditionallystabledirectmethodsforallencahnandconservativeallencahnequationsonsurfaces
AT zhongli secondorderunconditionallystabledirectmethodsforallencahnandconservativeallencahnequationsonsurfaces
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