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 discret...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.展开更多
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).
文摘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.
