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 Aτ3∂Δ3u∂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 ℓ∞(0 , T; ℓ2) 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.


Mathematics and Statistics


Fudan University, Grant None

Keywords and Phrases

Convergence Analysis; Epitaxial Thin Film Growth; Exponential Time Differencing; Gradient Flow; Long-Time Energy Stability; Third-Order Scheme

International Standard Serial Number (ISSN)

2197-9847; 2522-0144

Document Type

Article - Journal

Document Version


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Publication Date

01 Sep 2020