Volume 13, Issue 6
Error Estimate of a Second Order Accurate Scalar Auxiliary Variable (SAV) Numerical Method for the Epitaxial Thin Film Equation

Qing Cheng & Cheng Wang

Adv. Appl. Math. Mech., 13 (2021), pp. 1318-1354.

Published online: 2021-08

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  • 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 H\"older 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.

  • 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.

  • AMS Subject Headings

35K30, 35K55, 65K10, 65M12, 65M70

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COPYRIGHT: © Global Science Press

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@Article{AAMM-13-1318, author = {Cheng , Qing and Wang , Cheng}, title = {Error Estimate of a Second Order Accurate Scalar Auxiliary Variable (SAV) Numerical Method for the Epitaxial Thin Film Equation}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2021}, volume = {13}, number = {6}, pages = {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 H\"older 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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2020-0297}, url = {http://global-sci.org/intro/article_detail/aamm/19425.html} }
TY - JOUR T1 - Error Estimate of a Second Order Accurate Scalar Auxiliary Variable (SAV) Numerical Method for the Epitaxial Thin Film Equation AU - Cheng , Qing AU - Wang , Cheng JO - Advances in Applied Mathematics and Mechanics VL - 6 SP - 1318 EP - 1354 PY - 2021 DA - 2021/08 SN - 13 DO - http://doi.org/10.4208/aamm.OA-2020-0297 UR - https://global-sci.org/intro/article_detail/aamm/19425.html KW - Epitaxial thin film equation, Fourier pseudo-spectral approximation, the scalar auxiliary variable (SAV) method, Crank-Nicolson temporal discretization, energy stability, optimal rate convergence analysis. AB -

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 H\"older 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.

Qing Cheng & ChengWang. (1970). Error Estimate of a Second Order Accurate Scalar Auxiliary Variable (SAV) Numerical Method for the Epitaxial Thin Film Equation. Advances in Applied Mathematics and Mechanics. 13 (6). 1318-1354. doi:10.4208/aamm.OA-2020-0297
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