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An Energy Stable and Convergent Finite-Difference Scheme for the Modified Phase Field Crystal Equation

Authors:

S. M. WiseAuthors Info & Claims

SIAM Journal on Numerical Analysis, Volume 49, Issue 3

Pages 945 - 969

https://doi.org/10.1137/090752675

Published: 01 May 2011 Publication History

Abstract

We present an unconditionally energy stable finite difference scheme for the Modified Phase Field Crystal equation, a generalized damped wave equation for which the usual Phase Field Crystal equation is a special degenerate case. The method is based on a convex splitting of a discrete pseudoenergy and is semi-implicit. The equation at the implicit time level is nonlinear but represents the gradient of a strictly convex function and is thus uniquely solvable, regardless of time step-size. We present a local-in-time error estimate that ensures the pointwise convergence of the scheme.

References

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Cited By

Qian YZhang YHuang Y(2025)Error analysis of positivity-preserving energy stable schemes for the modified phase field crystal modelApplied Numerical Mathematics10.1016/j.apnum.2024.09.010207:C(470-498)Online publication date: 1-Jan-2025
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Geng YTeng YWang ZJu L(2024)A deep learning method for the dynamics of classic and conservative Allen-Cahn equations based on fully-discrete operatorsJournal of Computational Physics10.1016/j.jcp.2023.112589496:COnline publication date: 27-Feb-2024
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Qian YZhang YHuang Y(2024)Stability and error estimates of GPAV-based unconditionally energy-stable scheme for phase field crystal equationComputers & Mathematics with Applications10.1016/j.camwa.2023.10.029151:C(461-472)Online publication date: 1-Feb-2024
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An Energy Stable and Convergent Finite-Difference Scheme for the Modified Phase Field Crystal Equation
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Published In

cover image SIAM Journal on Numerical Analysis

SIAM Journal on Numerical Analysis Volume 49, Issue 3

May 2011

403 pages

ISSN:0036-1429

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Publisher

Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 May 2011

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Cited By

Qian YZhang YHuang Y(2025)Error analysis of positivity-preserving energy stable schemes for the modified phase field crystal modelApplied Numerical Mathematics10.1016/j.apnum.2024.09.010207:C(470-498)Online publication date: 1-Jan-2025
https://dl.acm.org/doi/10.1016/j.apnum.2024.09.010
Geng YTeng YWang ZJu L(2024)A deep learning method for the dynamics of classic and conservative Allen-Cahn equations based on fully-discrete operatorsJournal of Computational Physics10.1016/j.jcp.2023.112589496:COnline publication date: 27-Feb-2024
https://dl.acm.org/doi/10.1016/j.jcp.2023.112589
Qian YZhang YHuang Y(2024)Stability and error estimates of GPAV-based unconditionally energy-stable scheme for phase field crystal equationComputers & Mathematics with Applications10.1016/j.camwa.2023.10.029151:C(461-472)Online publication date: 1-Feb-2024
https://dl.acm.org/doi/10.1016/j.camwa.2023.10.029
Liu SYu H(2024)Energetic spectral-element time marching methods for phase-field nonlinear gradient systemsApplied Numerical Mathematics10.1016/j.apnum.2024.06.021205:C(38-59)Online publication date: 1-Nov-2024
https://dl.acm.org/doi/10.1016/j.apnum.2024.06.021
Qi L(2024)Numerical analysis of a second-order energy-stable finite element method for the Swift-Hohenberg equationApplied Numerical Mathematics10.1016/j.apnum.2023.11.014197:C(119-142)Online publication date: 1-Mar-2024
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Liu XHong QLiao HGong Y(2024)A multi-physical structure-preserving method and its analysis for the conservative Allen-Cahn equation with nonlocal constraintNumerical Algorithms10.1007/s11075-024-01757-497:3(1431-1451)Online publication date: 1-Nov-2024
https://dl.acm.org/doi/10.1007/s11075-024-01757-4
Doan CHoang TJu L(2024)Low Regularity Integrators for the Conservative Allen–Cahn Equation with a Nonlocal ConstraintJournal of Scientific Computing10.1007/s10915-024-02703-1101:3Online publication date: 21-Oct-2024
https://dl.acm.org/doi/10.1007/s10915-024-02703-1
Xie YLi QMei L(2024)An Unconditionally Energy Stable Method for the Anisotropic Phase-Field Crystal Model in Two DimensionJournal of Scientific Computing10.1007/s10915-024-02543-z100:1Online publication date: 28-May-2024
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Cui MNiu YXu Z(2024)A Second-Order Exponential Time Differencing Multi-step Energy Stable Scheme for Swift–Hohenberg Equation with Quadratic–Cubic Nonlinear TermJournal of Scientific Computing10.1007/s10915-024-02490-999:1Online publication date: 15-Mar-2024
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Yang JKim J(2024)Consistently and unconditionally energy-stable linear method for the diffuse-interface model of narrow volume reconstructionEngineering with Computers10.1007/s00366-023-01935-340:4(2617-2627)Online publication date: 1-Aug-2024
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