Abstract
An implicit–explicit linear BDF2 scheme with variable steps is proposed for the molecular beam epitaxial model without slope selection. The fourth-order surface diffusion term is treated implicitly and the nonlinear term is approximated by a second order explicit extrapolation. A new type regularization term Aτ2Δh2D2ϕn (D2ϕn represents the variable-step BDF2 formula) is designed to ensure unconditional energy stability in long-time simulations. Under the time-step ratio constraint 0<τk/τk−1<4.864, the linear BDF2 method preserves the modified discrete energy dissipation law unconditionally if a proper large constant A is chosen. Furthermore, with the help of discrete orthogonal convolution kernels and the corresponding convolution Young inequalities, the L2 norm stability and convergence analysis of the linear BDF2 scheme for the MBE model are established. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed scheme.
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