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In this paper, we study a novel second-order energy stable BackwardDifferentiation Formula (BDF) finite difference scheme for the epitaxial thinfilm equation with slope selection (SS). One major challenge for the higheroder in time temporal discretization is how to ensure an unconditional energystability and an efficient numerical implementation. We propose a generalframework for designing the higher order in time numerical scheme withunconditional energy stability by using the BDF method with constantcoefficient stabilized terms. Based on the unconditional energy stabilityproperty, we derive an $L^\infty_h (0,T; H_{h}^2)$ stability for the numericalsolution and provide an optimal the convergence analysis. To deal with the4-Laplacian solver in an $L^{2}$ gradient flow at each time step, we apply anefficient preconditioned steepest descent algorithm and preconditionednonlinear conjugate gradient algorithm to solve the corresponding nonlinearsystem. Various numerical simulations are present to demonstrate the stabilityand efficiency of the proposed schemes and slovers.

A Second-Order Energy Stable Backward Differentiation Formula Method for the Epitaxial Thin Film Equation with Slope Selection