A Third Order BDF Scheme For No-slope-selection Thin Film Model

The no-slope-selection(NSS)equation of the epitaxial thin film growth model is used to describe the continuously epitaxial process of films without slope selection at the atomic level.In this process,the energy decays and keeps stable.In view of the long-time coarsening process and highly nonlinear property of the no-slope-selection(NSS)equation of the epitaxial thin film growth model,it is particularly important to establish a well-posed and energy-stable linear numerical scheme.In this paper we propose and analyze a third order accurate backward differentiation formula(BDF)numerical scheme for the the no-slope-selection(NSS)equation of the epitaxial thin film growth model,with Fourier pseudo-spectral discretization in space.The surface diffusion term is treated implicitly,while the nonlinear chemical potential 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?t~2?_N~2(uⁿ⁺¹-uⁿ),is added in the numerical scheme.A careful energy stability estimate,combined with Fourier eigenvalue analysis,results in the energy stability in a modified version,and a theoretical justification of the coefficient A becomes available.As a result of this energy stability analysis,a uniform in time bound of the numerical energy is obtained.And also,the optimal rate convergence analysis and error estimate are derived in details,in the l_?(0,T;L_h~2)?l₂(0,T;H_h~2)norm,with the help of a linearized estimate for the nonlinear error terms.Some numerical simulation results are presented to demonstrate the efficiency of the numerical scheme and the third order convergence.The long time simulation results for ?=0.02(up to T=3 × 105)have indicated a logarithm law for the energy decay,as well as the power laws for growth of the surface roughness roughness and the mound width.In particular,the power index for the surface roughness and the mound width growth,created by the third order numerical more accurate than those produced by certain second order energy stable schemes in the existing literature.