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

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 }^2u}{\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.