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.
W. Chen et al., "Energy Stable Higher-Order Linear ETD Multi-Step Methods for Gradient Flows: Application to Thin Film Epitaxy," Research in Mathematical Sciences, vol. 7, no. 3, article no. 13, Springer, Sep 2020.
The definitive version is available at https://doi.org/10.1007/s40687-020-00212-9
Mathematics and Statistics
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)
Article - Journal
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01 Sep 2020