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Error Estimate of Full-Discrete Numerical Scheme for the Nonlocal Allen-Cahn Model

Journal of Mathematical Research with Applications 2024年第3期44卷 344-358页

作者： Jun ZHANG Xiaohu YANG Fulin MEI Zhimei JI Yu ZHANG School of Mathematics and Statistics Guizhou University of Finance and Economics The Meteorological Disaster Prevention Center of Guizhou Province Xi'an Institute of Applied Optics Financial Department Guizhou University of Finance and Economics

In this work, we study the error estimates of the fully discrete Fourier pseudospectral numerical scheme for solving the nonlocal volume-conserved Allen-Cahn(AC) *** time marching method of the numerical scheme is based on the well-known Invariant energy Quadratization(IEQ) method. We demonstrate that the proposed fully discrete numerical method is uniquely solvable, unconditionally energy stable, and obtain the optimal error estimate of the scheme for both space and time. Additionally, we conduct several numerical tests to verify the theoretical results.

关键词： nonlocal Allen-Cahn model uniquely solvable unconditionally energy stable error estimate numerical tests

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An Adaptive Time-Stepping Strategy for the Cahn-Hilliard Equation

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Communications in Computational Physics 2012年第4期11卷 1261-1278页

作者： Zhengru Zhang Zhonghua Qiao Laboratory of Mathematics and Complex Systems Ministry of EducationSchool of Mathematical SciencesBeijing Normal UniversityBeijing 100875China Institute for Computational Mathematics&Department of Mathematics Hong Kong Baptist UniversityKowloon TongHong Kong

This paper studies the numerical simulations for the Cahn-Hilliard equation which describes a phase separation *** numerical simulation of the Cahn-Hilliardmodel needs very long time to reach the steady state,and therefore large time-stepping methods become *** main objective of this work is to construct the unconditionally energy stable finite difference scheme so that the large time steps can be used in the numerical *** equation is discretized by the central difference scheme in space and fully implicit second-order scheme in *** proposed scheme is proved to be unconditionally energy stable and *** error estimate for the numerical solution is also obtained with second order in both space and *** using this energy stable scheme,an adaptive time-stepping strategy is proposed,which selects time steps adaptively based on the variation of the free energy against *** numerical experiments are presented to demonstrate the effectiveness of the adaptive time-stepping approach.

关键词： Adaptive time-stepping unconditionally energy stable Cahn-Hilliard equation mass conservation

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A High Order Adaptive Time-Stepping Strategy and Local Discontinuous Galerkin Method for the Modified Phase Field Crystal Equation

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Communications in Computational Physics 2018年第6期24卷 123-151页

作者： Ruihan Guo Yan Xu School of Mathematics and Statistics Zhengzhou UniversityZhengzhouHenan 450001P.R.China School of Mathematical Sciences University of Science and Technology of ChinaHefeiAnhui 230026P.R.China

In this paper,we will develop a first order and a second order convex splitting,and a first order linear energy stable fully discrete local discontinuous Galerkin(LDG)methods for the modified phase field crystal(MPFC)*** which,the first order linear scheme is based on the invariant energy quadratization *** MPFC equation is a damped wave equation,and to preserve an energy stability,it is necessary to introduce a pseudo energy,which all increase the difficulty of constructing numerical methods comparing with the phase field crystal(PFC)*** to the severe time step restriction of explicit timemarchingmethods,we introduce the first order and second order semi-implicit schemes,which are proved to be unconditionally energy *** order to improve the temporal accuracy,the semi-implicit spectral deferred correction(SDC)method combining with the first order convex splitting scheme is *** simulations of the MPFC equation always need long time to reach steady state,and then adaptive time-stepping method is necessary and of paramount *** schemes at the implicit time level are linear or nonlinear and we solve them by multigrid *** experiments of the accuracy and long time simulations are presented demonstrating the capability and efficiency of the proposed methods,and the effectiveness of the adaptive time-stepping strategy.

关键词： Adaptive time-stepping local discontinuous Galerkin method modified phase field crystal equation convex splitting pseudo energy unconditionally energy stable spectral deferred correction

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