Energy stable higher-order linear ETD multi-step methods for gradient flows: application to thin film epitaxy

We discuss how to combine exponential time differencing technique with multi-step method to develop higher order in time linear numerical scheme that are energy stable for certain gradient flows with the aid of a generalized viscous damping term. As an example, a stabilized third order in time accurate linear exponential time differencing (ETD) scheme for the epitaxial thin film growth model without slope selection is proposed and analyzed. An artificial stabilizing term $$A\tau ^3\frac{\partial \Delta ^3 u}{\partial t}$$ is added to ensure energy stability, with ETD-based multi-step approximations and Fourier pseudo-spectral method applied in the time integral and spatial discretization of the evolution equation, respectively. Long-time energy stability and an $$\ell ^{\infty }(0,T; \ell ^2)$$ error analysis are provided, based on the energy method. In addition, a few numerical experiments are presented to demonstrate the energy decay and convergence rate.