We present a linear iteration algorithm to implement a second-order energy stable numerical scheme for a model of epitaxial thin film growth without slope selection. the PDE, which is a nonlinear, fourth-order parabolic equation, is the L 2 gradient flow of the energy d x. the energy stability is preserved by a careful choice of the second-order temporal approximation for the nonlinear term, as reported in recent work (Shen et al. in SIAM J Numer Anal 50:105-125, 2012). the resulting scheme is highly nonlinear, and its implementation is non-trivial. in this paper, we propose a linear iteration algorithm to solve the resulting nonlinear system. to accomplish this we introduce an O (s 2) (with s the time step size) artificial diffusion term, a Douglas-Dupont-type regularization, that leads to a contraction mapping property. as a result, the highly nonlinear system can be decomposed as an iteration of purely linear solvers, which can be very efficiently implemented with the help of FFT in a collocation Fourier spectral setting. We present a careful analysis showing convergence for the numerical scheme in a discrete (0, T;) (0,T;) L ∞ (0, T; H 1) L 2 (0, T; H 3) norm. Some numerical simulation results are presented to demonstrate the efficiency of the linear iteration solver and the convergence of the scheme as a whole. © 2013 Springer Science+Business Media New York.


Mathematics and Statistics


National Science Foundation, Grant 91130004

Keywords and Phrases

Contraction Mapping; Energy Stability; Epitaxial Thin Film Growth; Fourier Collocation Spectral; Linear Iteration; Slope Selection

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Document Type

Article - Journal

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Publication Date

01 Jan 2014