A stabilized second order exponential time differencing multistep method for thin film growth model without slope selection

¹ Shanghai Key Laboratory of Mathematics for Nonlinear Sciences, School of Mathematical Sciences, Fudan University, 200433 Shanghai, P.R. China
² School of Mathematical Sciences, Fudan University, 200433 Shanghai, P.R. China
³ Mathematics Department, University of Massachusetts, North Dartmouth, MA 02747, USA
⁴ Department of Mathematics and SUSTech International Center for Mathematics, Southern University of Science and Technology, 518055 Shenzhen, P.R. China

^* Corresponding author: cwang1@umassd.edu

Received: 26 March 2019
Accepted: 22 August 2019

Abstract

In this paper, a stabilized second order in time accurate linear exponential time differencing (ETD) scheme for the no-slope-selection thin film growth model is presented. An artificial stabilizing term $A{\tau }^2\frac{\mathrm{\partial }{\Delta }^{2}u}{\mathrm{\partial }t}$ is added to the physical model to achieve energy stability, with ETD-based multi-step approximations and Fourier collocation spectral method applied in the time integral and spatial discretization of the evolution equation, respectively. Long time energy stability and detailed 𝓁∞(0,T;𝓁²) error analysis are provided based on the energy method, with a careful estimate of the aliasing error. In addition, numerical experiments are presented to demonstrate the energy decay and convergence rate.