Unconditionally energy stable high-order BDF schemes for the molecular beam epitaxial model without slope selection

Abstract

In this paper, we consider a class of k-order $(3\le k\le 5)$ backward differentiation formulas (BDF-k) for the molecular beam epitaxial (MBE) model without slope selection. Convex splitting technique along with k-th order Douglas-Dupont regularization term ${\tau }_n^k{(-\Delta )}^k{\underset{\_}{D}}_k{\varphi }^n$ (${\underset{\_}{D}}_k$ represents a truncated BDF-k formula) is added to the numerical schemes to ensure unconditional energy stability. The stabilized convex splitting BDF-k $(3\le k\le 5)$ methods are unique solvable unconditionally. Then the modified discrete energy dissipation laws are established by using the discrete gradient structures of BDF-k $(3\le k\le 5)$ formulas and processing k-th order explicit extrapolations of the concave term. In addition, based on the discrete energy technique, the $L^2$ norm stability and convergence of the stabilized BDF-k $(3\le k\le 5)$ schemes are obtained by means of the discrete orthogonal convolution kernels and the convolution type Young inequalities. Numerical results are carried out to verify our theory and illustrate the validity of the proposed schemes.