Error Estimate of a Second Order Accurate Scalar Auxiliary Variable (SAV) Numerical Method for the Epitaxial Thin Film Equation
Year: 2021
Author: Qing Cheng, Cheng Wang
Advances in Applied Mathematics and Mechanics, Vol. 13 (2021), Iss. 6 : pp. 1318–1354
Abstract
A second order accurate (in time) numerical scheme is analyzed for the slope-selection (SS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectral discretization in space. To make the numerical scheme linear while preserving the nonlinear energy stability, we make use of the scalar auxiliary variable (SAV) approach, in which a modified Crank-Nicolson is applied for the surface diffusion part. The energy stability could be derived a modified form, in comparison with the standard Crank-Nicolson approximation to the surface diffusion term. Such an energy stability leads to an H2 bound for the numerical solution. In addition, this H2 bound is not sufficient for the optimal rate convergence analysis, and we establish a uniform-in-time H3 bound for the numerical solution, based on the higher order Sobolev norm estimate, combined with repeated applications of discrete Hölder inequality and nonlinear embeddings in the Fourier pseudo-spectral space. This discrete H3 bound for the numerical solution enables us to derive the optimal rate error estimate for this alternate SAV method. A few numerical experiments are also presented, which confirm the efficiency and accuracy of the proposed scheme.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/aamm.OA-2020-0297
Advances in Applied Mathematics and Mechanics, Vol. 13 (2021), Iss. 6 : pp. 1318–1354
Published online: 2021-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 37
Keywords: Epitaxial thin film equation Fourier pseudo-spectral approximation the scalar auxiliary variable (SAV) method Crank-Nicolson temporal discretization energy stability optimal rate convergence analysis.
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