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

作者：Chen, Wenbin; Wang, Cheng^*; Wang, Xiaoming; Wise, Steven M.

来源：Journal of Scientific Computing, 2014, 59(3): 574-601.

DOI：10.1007/s10915-013-9774-0

• Epitaxial thin film growth
• Slope selection
• Energy stability
• Linear iteration
• Contraction mapping
• Fourier collocation spectral

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 gradient flow of the energy . 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 (with 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 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.

• 出版日期2014-6
• 单位复旦大学