Abstract
In this paper we are concerned with the stability and convergence analysis of the second order backward differentiation formula (BDF2) scheme with variable time steps for the no-slope-selection (NSS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectral method in physical domain. Under the adjoint time-step ratio condition \(r_k := \tau _k /\tau _{k-1} < 4.864\), the variable-step BDF2 scheme is uniquely solvable and stable in \(L^2\)-norm, also we establish a rigorous error estimate with a novel discrete orthogonal convolution kernels involved in the analysis. Finally, the adaptive time step strategy is used to accelerate the calculation of the steady-state solution, and the theoretical results are verified by numerical examples. In addition, the long time simulation results have indicated a logarithm law for the energy decay, as well as the power laws for growth of the surface roughness and the mound width.
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Acknowledgements
Z.R. Zhang is supported by National Natural Science Foundation of China (NSFC) 11871105, 12231003.
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Meng, X., Zhang, Z. An adaptive BDF2 implicit time-stepping method for the no-slope-selection epitaxial thin film model. Comp. Appl. Math. 42, 124 (2023). https://doi.org/10.1007/s40314-023-02250-9
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DOI: https://doi.org/10.1007/s40314-023-02250-9
Keywords
- Epitaxial thin film growth
- Variable-step BDF2 scheme
- Discrete orthogonal convolution kernels
- Convergence analysis
- Fourier pseudo-spectral method