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

Abstract

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.

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Acknowledgements

This work is supported in part by the Grants NSFC 11671098, 91630309, a 111 Project B08018 (W. Chen), NSF DMS-1418689 (C. Wang), NSFC11871159, Guangdong Provincial Key Laboratory for Computational Science and Material Design 2019B030301001 (X. Wang). C. Wang also thanks the Key Laboratory of Mathematics for Nonlinear Sciences, Fudan University, for support during his visit.

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Correspondence to Xiaoming Wang.

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Dedicated to Professor Andrew Majda on the occasion of his seventieth birthday.

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Chen, W., Li, W., Wang, C. et al. Energy stable higher-order linear ETD multi-step methods for gradient flows: application to thin film epitaxy. Res Math Sci 7, 13 (2020). https://doi.org/10.1007/s40687-020-00212-9

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Keywords

  • Gradient flow
  • Epitaxial thin film growth
  • Exponential time differencing
  • Long-time energy stability
  • Convergence analysis
  • Third-order scheme

Mathematics Subject Classification

  • 65M12
  • 65M70
  • 65Z05