Fast High Order and Energy Dissipative Schemes with Variable Time Steps for Time-Fractional Molecular Beam Epitaxial Growth Model  

作　　者：Dianming Hou Zhonghua Qiao Tao Tang 

机构地区：[1]School of Mathematics and Statistics,Jiangsu Normal University,Xuzhou,Jiangsu 221116,China [2]Department of Applied Mathematics,The Hong Kong Polytechnic University,Hung Hom,Kowloon,Hong Kong [3]Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science,BNU-HKBU United International College,Zhuhai,Guangdong 519087,China

出　　处：《Annals of Applied Mathematics》2023年第4期429-461,共33页应用数学年刊（英文版）

基　　金：supported by NSFC grant 12001248,the NSF of Jiangsu Province grant BK20201020;the NSF of Universities in Jiangsu Province of China grant 20KJB110013;the Hong Kong Polytechnic University grant 1-W00D;supported by Hong Kong Research Grants Council RFS grant RFS2021-5S03 and GRF grant 15302122,the Hong Kong Polytechnic University grant 1-9BCT;CAS AMSS-PolyU Joint Laboratory of Applied Mathematics;supported by the Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science under UIC 2022B1212010006.

摘　　要：In this paper,we propose and analyze high order energy dissipative time-stepping schemes for time-fractional molecular beam epitaxial(MBE)growth model on the nonuniform mesh.More precisely,(2−α)-order,secondorder and(3−α)-order time-stepping schemes are developed for the timefractional MBE model based on the well known L1,L2-1σ,and L2 formulations in discretization of the time-fractional derivative,which are all proved to be unconditional energy dissipation in the sense of a modified discrete nonlocalenergy on the nonuniform mesh.In order to reduce the computational storage,we apply the sum of exponential technique to approximate the history part of the time-fractional derivative.Moreover,the scalar auxiliary variable(SAV)approach is introduced to deal with the nonlinear potential function and the history part of the fractional derivative.Furthermore,only first order method is used to discretize the introduced SAV equation,which will not affect high order accuracy of the unknown thin film height function by using some proper auxiliary variable functions V(ξ).To our knowledge,it is the first time to unconditionally establish the discrete nonlocal-energy dissipation law for the modified L1-,L2-1σ-,and L2-based high-order schemes on the nonuniform mesh,which is essentially important for such time-fractional MBE models with low regular solutions at initial time.Finally,a series of numerical experiments are carried out to verify the accuracy and efficiency of the proposed schemes.

关 键 词：Time-fractional molecular beam epitaxial growth variable time-stepping scheme SAV approach energy stability 

分 类 号：O241.8[理学—计算数学]