A Third Order Accurate in Time,BDF-Type Energy Stable Scheme for the Cahn-Hilliard Equation  

作　　者：Kelong Cheng Cheng Wang Steven M.Wise Yanmei Wu 

机构地区：[1]School of Science,Southwest University of Science and Technology,Mianyang,Sichuan 621010,P.R.China [2]Department of Mathematics,The University of Massachusetts,North Dartmouth,MA 02747,USA [3]Department of Mathematics,The University of Tennessee,Knoxville,TN 37996,USA

出　　处：《Numerical Mathematics(Theory,Methods and Applications)》2022年第2期279-303,共25页高等学校计算数学学报（英文版）

基　　金：supported in part by the Computational Physics Key Laboratory of IAPCAM(P.R.China)under Grant 6142A05200103(K.Cheng);the National Science Foundation(USA)under Grant NSF DMS-2012669(C.Wang);Grants NSF DMS-1719854,DMS-2012634(S.Wise).

摘　　要：In this paper we propose and analyze a backward differentiation formula(BDF)type numerical scheme for the Cahn-Hilliard equation with third order temporal accuracy.The Fourier pseudo-spectral method is used to discretize space.The surface diffusion and the nonlinear chemical potential terms are treated implicitly,while the expansive term is approximated by a third order explicit extrapolation formula for the sake of solvability.In addition,a third order accurate Douglas-Dupont regularization term,in the form of−A_(0)△t^(2)△_( N)(φ^(n+1)−φ^(n)),is added in the numerical scheme.In particular,the energy stability is carefully derived in a modified version,so that a uniform bound for the original energy functional is available,and a theoretical justification of the coefficient A becomes available.As a result of this energy stability analysis,a uniform-in-time L_(N)^(6)bound of the numerical solution is obtained.And also,the optimal rate convergence analysis and error estimate are provided,in the L_(△t)^(∞)(0,T;L_(N)^(2))∩L^(2)_(△ t)(0,T;H_(h)^(2))norm,with the help of the L_(N)^(6)bound for the numerical solution.A few numerical simulation results are presented to demonstrate the efficiency of the numerical scheme and the third order convergence.

关 键 词：Cahn-Hilliard equation third order backward differentiation formula unique solvability energy stability discrete L6 N estimate optimal rate convergence analysis 

分 类 号：O17[理学—数学]