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Error Estimate of a Second Order Accurate Scalar Auxiliary Variable (SAV) Numerical Method for the Epitaxial Thin Film Equation

Error Estimate of a Second Order Accurate Scalar Auxiliary Variable (SAV) Numerical Method for the Epitaxial Thin Film Equation
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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 $H^2$ bound for the numerical solution. In addition, this $H^2$ bound is not sufficient for the optimal rate convergence analysis, and we establish a uniform-in-time $H^3$ 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 $H^3$ 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.

Author Details

Qing Cheng

Cheng Wang

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