A Linear Iteration Algorithm for a Second-Order Energy Stable Scheme for a Thin Film Model Without Slope Selection

Abstract

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 $$L^2$$ L 2 gradient flow of the energy $$ \int _\Omega \left( - \frac{1}{2} \ln \left( 1 + | \nabla \phi |^2 \right) + \frac{\epsilon ^2}{2}|\Delta \phi (\mathbf{x})|^2 \right) \mathrm{d}\mathbf{x}$$ ? Ω - 1 2 ln 1 + | ? ? | 2 + ∈ 2 2 | Δ ? ( x ) | 2 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 $$O(s^2)$$ O ( s 2 ) (with $$s$$ 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 $$L^\infty (0, T; H^1) \cap L^2 (0,T; H^3)$$ 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.