Double stabilizations and convergence analysis of a second-order linear numerical scheme for the nonlocal Cahn-Hilliard equation  

作　　者：Xiao Li Zhonghua Qiao Cheng Wang 

机构地区：[1]Department of Applied Mathematics,The Hong Kong Polytechnic University,Hong Kong,China [2]Department of Mathematics,The University of Massachusetts,North Dartmouth,MA 02747,USA

出　　处：《Science China Mathematics》2024年第1期187-210,共24页中国科学（数学）（英文版）

基　　金：supported by the Chinese Academy of Sciences(CAS)Academy of Mathematics and Systems Science(AMSS);the Hong Kong Polytechnic University(PolyU)Joint Laboratory of Applied Mathematics;supported by the Hong Kong Research Council General Research Fund(Grant No.15300821);the Hong Kong Polytechnic University Grants(Grant Nos.1-BD8N,4-ZZMK and 1-ZVWW);supported by the Hong Kong Research Council Research Fellow Scheme(Grant No.RFS2021-5S03);General Research Fund(Grant No.15302919);supported by US National Science Foundation(Grant No.DMS-2012269)。

摘　　要：In this paper,we study a second-order accurate and linear numerical scheme for the nonlocal CahnHilliard equation.The scheme is established by combining a modified Crank-Nicolson approximation and the Adams-Bashforth extrapolation for the temporal discretization,and by applying the Fourier spectral collocation to the spatial discretization.In addition,two stabilization terms in different forms are added for the sake of the numerical stability.We conduct a complete convergence analysis by using the higher-order consistency estimate for the numerical scheme,combined with the rough error estimate and the refined estimate.By regarding the numerical solution as a small perturbation of the exact solution,we are able to justify the discrete?^(∞)bound of the numerical solution,as a result of the rough error estimate.Subsequently,the refined error estimate is derived to obtain the optimal rate of convergence,following the established?∞bound of the numerical solution.Moreover,the energy stability is also rigorously proved with respect to a modified energy.The proposed scheme can be viewed as the generalization of the second-order scheme presented in an earlier work,and the energy stability estimate has greatly improved the corresponding result therein.

关 键 词：nonlocal Cahn-Hilliard equation second-order stabilized scheme higher-order consistency analysis rough and refined error estimate 

分 类 号：O241.8[理学—计算数学]