A linear iteration algorithm for a second-order energy stable scheme for a thin film model without slope selection

Wenbin Chen, Cheng Wang, Xiaoming Wang and Steven Wise

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 $\int_\Omega (-\frac12\ln(1+|\nabla\phi|^2) + \frac{\epsilon^2}{2}|\Delta\phi|^2)$ The energy stability is preserved by a careful choice of the second-order temporal approximation for the nonlinear term, as reported in recent work [18]. 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 $L^\infty(0; T;H^21) \Cap L^2(0; T;H^3)$ 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.